Non-Abelian Localization and U(1) Chern-Simons Theory · Brendan Donald Kenneth McLellan Doctor of...

Non-Abelian Localization and U(1) Chern-Simons Theory

by

Brendan Donald Kenneth McLellan

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

Copyright c© 2010 by Brendan Donald Kenneth McLellan

Abstract

Non-Abelian Localization and U(1) Chern-Simons Theory

Brendan Donald Kenneth McLellan

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2010

This thesis studies U(1) Chern-Simons theory and its relation to the results of Chris

Beasley and Edward Witten, [BW05]. Using the partition function formalism, we are

led to compare U(1) Chern-Simons theory as constructed in [Man98] to the results of

[BW05]. This leads to an explicit calculation of the U(1) Chern Simons partition function

on a closed Sasakian three-manifold and opens the door to studying rigorous extensions

of this theory to more general gauge groups and three-manifold geometries.

The first main part of this thesis studies an analogue of the work of Beasley and Witten

[BW05] for the Chern-Simons partition function on a Sasakian three-manifold for U(1)

gauge group. A key point is that our gauge group is not simply connected, whereas this

is an essential assumption in Beasley and Witten’s work. We are still able to use Beasley

and Witten’s results, however, to derive a definition of a U(1) Chern-Simons partition

function. We then compare this result to a definition of the U(1) Chern-Simons partition

function given by Mihaela Manoliu [Man98], and find that the two definitions agree up

to some undetermined multiplicative constant. These results lead to a natural interpre-

tation of the Reidemeister-Ray-Singer torsion as a symplectic volume form on the moduli

space of flat U(1) connections over a Sasakian three-manifold.

The second main part of this thesis studies U(1) Chern-Simons theory and its relation

ii

to a construction of Chris Beasley and Edward Witten, [BW05]. The natural geometric

setup here is that of a three-manifold with a Sasakian structure. We are led to study the

stationary phase approximation of the path integral for U(1) Chern-Simons theory after

one of the three components of the gauge field is decoupled. This gives an alternative

formulation of the partition function for U(1) Chern-Simons theory that is conjecturally

equivalent to the usual U(1) Chern-Simons theory, [Man98]. We establish this conjectural

equivalence rigorously using appropriate regularization techniques.

iii

Acknowledgements

There are several people whom I would like to thank here. First, I would like to thank my

thesis advisor Lisa Jeffrey for her patience, understanding, and her insights throughout

the years. This work would not have been possible without her. I would also like to thank

Yael Karshon for several useful discussions. Her guidance and support are greatly ap-

preciated. I would also like to thank Dror Bar-Natan for his unique perspective, insights

and for several helpful discussions. I would especially like to thank Frederic Rochon for

taking the time to explain elliptic PDE theory, the Atiyah-Patodi-Singer theorem, and

the method of heat kernels and eta-invariants in the elliptic case. His help regarding the

study of hypoelliptic operators was also invaluable. I would particularly like to thank

Raphael Ponge and Michel Rumin for their correspondence relating to hypoelliptic opera-

tors and the contact Laplacian. I am indebted to Paul Selick for explaining several useful

concepts related to (co)homology and homotopy theory. Thanks also to Eckhard Mein-

renken for several useful discussions regarding Chern-Weil theory and generally helping

with any questions that I had. I would like to thank Ben Burrington for taking the

time to review the computation of the gravitational Chern-Simons term in [GIJP03]. His

expert understanding of these types of computations proved invaluable for my own com-

putation. My sincerest thanks to John Bland for meeting with me on more than several

occasions to discuss contact and CR-geometry. His keen interest in my work also forced

me to look at some deeper questions more carefully and has certainly deepened my own

understanding. Lastly, I would like to thank Edward Witten for originally suggesting the

topic of my thesis. I appreciate him taking the time to correspond on questions related

to my thesis and also for an invaluable meeting that helped to answer some of my more

general questions.

iv

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Geometry 9

2.1 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Contact and CR Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Seifert Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Vielbein Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 The Space of Connections . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.1 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.2 The U(1)-Bundle Case . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Heisenberg Calculus on Contact Manifolds 43

3.1 Heisenberg Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Heisenberg Calculus and the Rockland Condition . . . . . . . . . . . . . 48

3.3 Some Results in the Heisenberg Calculus . . . . . . . . . . . . . . . . . . 51

4 U(1) Chern-Simons Theory 54

4.1 Induced Principal Bundles and Induced Connections . . . . . . . . . . . 54

v

4.2 The U(1) Chern-Simons Action . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 The U(1) Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Shift Symmetry and the U(1) Partition Function . . . . . . . . . . . . . . 68

5 Non-Abelian Localization for U(1) Chern-Simons Theory 75

5.1 Symplectic Formulation of the U(1) Partition Function . . . . . . . . . . 78

5.2 Orbifolds and Symplectic Volume . . . . . . . . . . . . . . . . . . . . . . 84

5.3 Reidemeister Torsion and Symplectic Volume . . . . . . . . . . . . . . . 86

5.4 The Adiabatic Eta-Invariant . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 Eta-Invariants and Anomalies 98

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2 Structure Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3 The Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4 The Contact Operator D . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.5 Gauge Group and the Isotropy Subgroup . . . . . . . . . . . . . . . . . . 111

6.6 The Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.7 Zeta Function Determinants . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.8 The Eta-Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.9 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.10 Regularizing the Eta-Invariants . . . . . . . . . . . . . . . . . . . . . . . 122

7 Gravitational Chern-Simons and the Adiabatic Limit 130

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.2 Local Formulation of Gravitational Chern-Simons Term . . . . . . . . . . 134

7.3 Computation of Gravitational Chern-Simons Term . . . . . . . . . . . . . 136

A Localization 143

A.1 Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

vi

B Constructions and Computations 148

B.1 Finite Dimensional Analogue of the Shift Symmetry . . . . . . . . . . . . 148

B.2 Gravitational Chern-Simons Calculations . . . . . . . . . . . . . . . . . . 156

B.2.1 Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . 156

B.2.2 Spin Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

B.2.3 Reduced Spin Connection . . . . . . . . . . . . . . . . . . . . . . 162

B.2.4 Reduced Gravitational Chern-Simons . . . . . . . . . . . . . . . . 166

C Miscellaneous Results 169

C.1 Horizontal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

C.2 T dRS = T dC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

C.3 A Standard Result in Cohomology . . . . . . . . . . . . . . . . . . . . . 180

Bibliography 182

vii

Chapter 1

Introduction

1.1 Background

This thesis is generally concerned with U(1) Chern-Simons theory on a closed Sasakian

three-manifold X. We will in fact be interested in a particular choice of quasi-regular K-

contact structure on a given Sasakian three-manifold, and most of our results are stated

assuming that such a choice has been made.

With the exception of §4.3 and §4.4, chapters 2, 3 and 4 are completely review and

serve only to establish notation and some standard definitions and results. There is a

slight novelty in §4.3, which is the extension of the definition of the U(1) Chern-Simons

partition function contained in [Man98] to take into account a dependence on a choice

of framing. §4.4 provides a new heuristic definition of what we call the shifted U(1)

Chern-Simons partition function.

For an excellent overview of the history and background of Chern-Simons theory in

general, we refer the interested reader to [Fre95] and [Fre09]. The background that is

most relevant for this thesis is contained in the papers [Man98] and [BW05]. In par-

1

Chapter 1. Introduction 2

ticular, [Man98] provides a rigorous foundation to study U(1) Chern-Simons theory and

[BW05] provides a heuristic motivation for our work. One of the main aspects of [Man98]

that is crucial for us is the derivation of a rigorous definition of the U(1) Chern-Simons

partition function starting from a heuristically defined path integral. Since the work of

[BW05] uses the path integral approach, this allows us to connect the work of [BW05]

with that of [Man98] very naturally.

1.2 Thesis Results

The first part of this thesis is concerned with the direct comparison of the partition

function of Manoliu [Man98] to the non-abelian localization results of Beasley and Witten

[BW05]. We first derive a definition for the U(1) Chern-Simons partition function that

is closely related to the definition of [Man98]. We obtain the following (see Def. 4.3.22

and §4.3 for all the relevant background)

1.2.1 Definition. Let k ∈ Z be an (even) integer, and X a closed, oriented three-

manifold. The U(1) Chern-Simons partition function, ZU(1)(X, k), is the quantity

ZU(1)(X, k) =∑

p∈TorsH2(X;Z)

ZU(1)(X, p, k) (1.2.2)

where,

ZU(1)(X, p, k) := kmXeπikSX,P (AP )eπi(η(?d)

4+ 1

12CS(Ag)

2π

) ∫MP

(T dRS)1/2 (1.2.3)

where mX = 12(dimH1(X; R)− dimH0(X; R)).

In §5.1 we derive a new definition of the partition function for U(1) Chern-Simons

theory using the methods of [BW05]. We make the following (see Def. 5.1.13 and §5.1

for all the relevant background)


1.2.4 Definition. Let k ∈ Z be an (even) integer, and X a closed, oriented Seifert

three-manifold,

U(1) // X

Σ

,

where Σ = |Σ|,U is an orbifold with underlying space |Σ| a Riemann surface of genus

g. The symplectic U(1) Chern-Simons partition function, ZSU(1)(X, k), is the quantity

ZSU(1)(X, k) =

∑p∈TorsH2(X;Z)

ZSU(1)(X, p, k) (1.2.5)

where,

ZSU(1)(X, p, k) = kmXeπikSX,P (AP )eiπ(

14− 1

2η0)∫MP

ωP . (1.2.6)

One of our main objectives in this thesis is to establish the equivalence of Def. 1.2.1

and Def. 1.2.4. As a first step in this direction, we compare the square root of the Ray-

Singer torsion (T dRS)1/2 (see Appendix C.2) to the symplectic volume form ωP in §5.3.

We establish the following (see Prop. 5.3.19)

1.2.7 Proposition. Let X be a closed, oriented Siefert three-manifold,√τ(X) the square

root of the R-torsion of X, and ωP the symplectic volume form on the moduli space of

flat connections MP ' U(1)2g for some flat U(1)-bundle P over X. Then, there exists

C ∈ R∗, such that √τ(X) = C · ωP .

We then make the following (see Conj. 5.3.20)

1.2.8 Conjecture. C = 1 in Prop. 1.2.7 above.

The next step that we take to establish the equivalence of Def. 1.2.1 and Def. 1.2.4

is to compare the eta-invariants that arise therein. In §5.4 we study the adiabatic eta-

invariant, η0. This invariant is studied in [BW05], [Nic00] and [Bea07], for example. We

find that η0 agrees with the results that we obtain in §6.10, after some observations. On


the one hand, we have (see §5.4)

− η0

2=

[−c1(X)

12+

N∑j=1

s(αj, βj)

], (1.2.9)

and on the other we have (see §6.10 and Prop. 6.10.27)

η(?d)

4+

1

12

CS(Ag)

2π=η0(X, κ)

4=

1

4− c1(X)

12+

N∑j=1

s(αj, βj), (1.2.10)

where,

s(α, β) :=1

4α

α−1∑k=1

cot

(πk

α

)cot

(πkβ

α

)(1.2.11)

is the classical Rademacher-Dedekind sum and [n; (α1, β1), . . . , (αN , βN)] (for (αi, βi) = 1

relatively prime) are the Seifert invariants of X. We observe in §5.4 that the eta-invariant

dependent parts of Def.’s 1.2.1 and 1.2.4 are exactly the same. That is, we have shown

(See Prop. 5.4.11 and §5.4)

1.2.12 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold.

Then,1 we have

η(?d)

4+

1

12

CS(Ag)

2π=

1

4− 1

2η0,

=1

4− d

12+

N∑j=1

s(αj, βj).

The second part of this thesis is contained in our paper [JM10]. Our main objective

here is the rigorous confirmation of a heuristic result utilized in [BW05]. Our results

involve some fairly deep facts about the “contact operator” studied by Michel Rumin,

[Rum94]. Recall that this is the second order operator “D” that fits into the complex

(see §6.4),

C∞(X)dH−→ Ω1(H)

D−→ Ω2(V )dH−→ Ω3(X), (1.2.13)

and in our geometric situation2 can be written as follows:

Dα = κ ∧ [Lξ + dH ?H dH ]α, α ∈ Ω1(H). (1.2.14)

1This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).

2When X is a closed, oriented, quasi-regular K-contact three-manifold.


A somewhat surprising observation is that this operator shows up quite naturally in

U(1) Chern-Simons theory, and this leads us to make several conjectures motivated by

the heuristic constructions of [BW05]. Our main result is the following (see Chapter 6

and Prop. 6.10.24):

1.2.15 Theorem. [JM10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three

manifold. If,

ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4 (η(?D)+ 1

512

∫X R2 κ∧dκ)

∫MP

(T dC)1/2 (1.2.16)

and,

ZU(1)(X, p, k) = kmXeπikSX,P (AP )eπi(η(?d)

4+ 1

12CS(Ag)

2π

) ∫MP

(T dRS)1/2 (1.2.17)

then,3 we have

ZU(1)(X, k) = ZU(1)(X, k)

as topological invariants.

The first main component of this result is the fact that the Ray-Singer analytic

torsion of X, T dRS, is identically equal to the contact analytic torsion T dC . This result is

in fact already known and follows directly from [RS08, Theorem 4.2]. The second main

component of this result is the existence, and the explicit identification, of regularizations

of the eta-invariants of the de-Rham and contact operators, d and D, respectively. This

result is contained in the following (see Chapter 6 and Prop. 6.10.23):

1.2.18 Theorem. [JM10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-

manifold. Then,4 there exists a counterterm, CT , such that eπi4

[η(?D)+CT ] is a topological

invariant that is identically equal to the topological invariant eπi[η(?d)

4+ 1

12CS(Ag)

2π

]. In fact,




we have

CT =1

512

∫X

R2 κ ∧ dκ,

where R ∈ C∞(X) is the Tanaka-Webster scalar curvature of X.

We prove this theorem by appealing to a result that we establish using a “Kaluza-

Klein” dimensional reduction technique, modeled after the paper [GIJP03], for the grav-

itational Chern-Simons term (see Chapter 7 and Prop. 7.3.40).

1.2.19 Theorem. [McL10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-

manifold,

U(1) // X

Σ

.

Let gε := ε−1 κ⊗ κ+ π∗h. After a particular choice of Vielbein,5 then,

CS(Agε) =

(ε−1

2

)∫Σ

r ω +

(ε−2

2

)∫Σ

f 2 ω (1.2.20)

where r ∈ C∞orb(Σ) is the (orbifold) scalar curvature of (Σ, h), ω ∈ Ω2orb(Σ) is the (orbifold)

Hodge form of (Σ, h), and f ∈ Ω0orb(Σ) is the invariant field strength on (Σ, h).6 In

particular, the adiabatic limit of CS(Agε) vanishes:

limε→∞

CS(Agε) = 0. (1.2.21)

Finally, as a consequence of these investigations, we are able to compute the U(1)

Chern-Simons partition function fairly explicitly (see Chapter 6 and Prop. 6.10.27).

5See Equations (7.3.20) and (7.3.21).6In a “special coordinate system,” x0, x1, x2 on U ⊂ X, κ = ϕ0dx

0 + ϕ1dx1 + dx2, and dκ =

(∂0ϕ1 − ∂1ϕ0)dx0 ∧ dx1 = f01dx0 ∧ dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor,

and fαβ =√h εαβ f, where f ∈ Ω0

orb(Σ) is called the invariant field strength on (X,h). See Chapter 7and §2.4 for more background.


1.2.22 Theorem. [JM10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-

manifold. Then,7 we have

η(?d) +1

3

CS(Ag)

2π= η(?D) +

1

512

∫X

R2 κ ∧ dκ

= 1− d

3+ 4

N∑j=1

s(αj, βj),

where 0 < d = c1(X) = n+∑N

j=1βjαj∈ Q and

s(α, β) :=1

4α

α−1∑k=1

cot

(πk

α

)cot

(πkβ

α

)∈ Q

is the classical Rademacher-Dedekind sum, where [n; (α1, β1), . . . , (αN , βN)] (for gcd(αj, βj) =

1) are the Seifert invariants of X. In particular, we have computed the U(1) Chern-

Simons partition function as:

ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4 (1− d

3+4∑Nj=1 s(αj ,βj))

∫MP

(T dC)1/2,

= kmXeπikSX,P (AP )eπi4 (1− d

3+4∑Nj=1 s(αj ,βj))

∫MP

(T dRS)1/2.

These results provide a rigorous confirmation of Beasley and Witten’s heuristic shift

symmetry construction in the case that G = U(1). This work also puts some of the work

of Biquard, Herzlich, Rumin, Seshadri, and Ponge into a physics context. For example,

our work shows why it is natural to expect that the contact Laplacian should be “nice.”8

Our work implies that it is natural to expect that the contact analytic torsion equals the

Ray-Singer analytic torsion in our geometric situation, and that the topological regular-

izations of the contact and de-Rham eta-invariants are in a precise sense equivalent. This

work also establishes a rigorous and explicit computation of the U(1) Chern-Simons par-

tition function for the general class of closed quasi-regular K-contact three-manifolds.9


8In fact, it is maximally hypoelliptic and invertible in the Heisenberg symbolic calculus. See Chapter3.

9This work applies to all closed K-contact three-manifolds, and hence by [Bla76, Corollary 6.5] to allclosed Sasakian three manifolds.


1.3 Future Work

One of my main interests for future work is the extension of these results beyond the

case of quasi-regular K-contact manifolds. Ultimately, I am interested in obtaining an

analogue of these results for any three-manifold with a chosen contact form. It has re-

cently been suggested by Witten that a possible approach to this problem is to consider

the one-loop computation for the shift invariant partition function. He suspects that this

invariant would depend only on the contact structure and not necessarily on the Seifert

structure. It would be interesting to study general contact metric manifolds that do not

necessarily possess a Seifert structure, for example. Additionally, it would be interesting

to look at extending these results to three-manifolds with boundary.

Another interesting direction for future research is the generalization of these results

from a U(1) gauge group to SU(2). As observed in [Wit89], the stationary phase approx-

imation for SU(2) Chern-Simons theory is a straightforward generalization of the U(1)

case. The work of Braverman and Kappeler [BK08] on refined analytic torsion seems to

contain exactly the right tools for such an extension. We note that there is still much work

to be done for other gauge groups (e.g. SU(2)) and future work will be concerned with

finding appropriate generalizations of the results of this thesis to these other cases. As is

noted in [BW05], Chern-Simons theory is intimately related to the theory of knot invari-

ants and the topology of three-manifolds [Wit89], [Ati90a], to two-dimensional rational

conformal field theory [MS90], to three-dimensional quantum gravity [DJT82], [Wit88],

to open string field theory on the topological A-model [Wit95] and to the Gromov-Witten

theory of non-compact Calabi-Yau threefolds [GV98], [GV99].

Chapter 2

Geometry

2.1 Orbifolds

In this section we review orbifolds. We follow primarily [BG08]. Other good references

are [Nic00], and [Sat57].

We begin with the following

2.1.1 Definition. [BG08, Def. 4.1.1] Let |Σ| be a paracompact Hausdorff space. An

orbifold chart or local uniformizing system on |Σ| is a triple (U ,Γ, φ), where U ⊂ Rn is a

connected open subset of Rn, Γ is a finite group acting effectively on U , and φ : U → U

is a continuous map onto an open set U ⊂ |Σ| such that φ γ = φ for all γ ∈ Γ and the

induced natural map of U/Γ onto U is a homeomorphism. An injection or embedding

between two such charts (U ,Γ, φ) and (U ′,Γ′, φ′) is a smooth (or holomorphic) embedding

λ : U → U ′ such that φ′ λ = φ. An orbifold atlas on |Σ| is a family U = Ui,Γi, φi of

orbifold charts such that

i. |Σ| = ∪iφi(Ui),

ii. given two charts (Ui,Γi, φi) and (Uj,Γj, φj) in U with Ui = φi(Ui) and Uj = φj(Uj)

and a point p ∈ Ui ∩ Uj, there exists an open neighbourhood Uk of p, and a

9

Chapter 2. Geometry 10

chart (Uk,Γk, φk) such that there are injections λik : (Uk,Γk, φk)→ (Ui,Γi, φi) and

λjk : (Uk,Γk, φk)→ (Uj,Γj, φj).

An atlas U is a refinement of an atlas V if there exists an injection of every chart of U into

some chart of V . Two orbifold atlases are equivalent if they have a common refinement.

A smooth (complex) orbifold (or V-manifold) is a paracompact Hausdorff space |Σ| with

an equivalence class of smooth (holomorphic) orbifold atlases and we write Σ = (|Σ|,U).

If every finite group Γ consists of orientation preserving diffeomorphisms and there is an

atlas such that all the injections are orientation preserving then the orbifold is orientable.

The finite groups Γi are called local uniformizing groups.

2.1.2 Example. [BH99] Let us define an orbifold structure on S2. We take |Σ| = S2 as

our underlying paracompact Hausdorff space as in Def. 2.1.1 and we define an orbifold

Σ = (S2,U). We view S2 = C ∪ ∞ = U0 ∪ ∞ as the Riemann sphere with two

marked points 0,∞, the south and north pole respectively. Define two orbifold charts,

(U0,Γ0, φ0) = (C,Zm, φ0 : C→ S2\∞),

where φ0(z) = zm, and,

(U∞,Γ∞, φ∞) = (C,Zn, φ∞ : C→ S2\0),

where φ∞(w) = w−n. e2πirm ∈ Zm acts on α ∈ C by rotations e

2πirm : α 7→ e

2πirm ·α in the first

coordinate chart, and similarly for the second. Given any point p ∈ U0∩U∞ = S2\0,∞,

let Up ⊂ U0 ∩ U∞ be any open subset Up ⊂ U0 = C, viewed as a subset of U0 with

coordinate z ∈ C, that is homeomorphic to a disk D ⊂ C. For such sets, let the

corresponding charts be (Up ' Up ⊂ C,Γp := e, φp(z) := z). We claim that the

collection V consisting of the collection of such charts (Up,Γp, φp) along with the charts

(U0,Γ0, φ0), and (U∞,Γ∞, φ∞) forms an orbifold atlas on |Σ| = S2. Condition (ii) in Def.

2.1.1 is clearly satisfied for any two charts of the form (Up,Γp, φp). Let us check this

condition for the pair (U0,Γ0, φ0), and (U∞,Γ∞, φ∞), and leave the other cases to the


reader. Consider, p ∈ U0 ∩ U∞ = S2\0,∞. Let U∩ ⊂ U0 ∩ U∞ be any subset of the

form Up ⊂ U0 = C as described above, with chart:

(U∩ ' U∩ ⊂ C,Γ∩ := e, φ∩(z) := z).

Then,

λ0∩ : (U∩,Γ∩, φ∩)→ (U0,Γ0, φ0),

defined by λ0∩(z) = z1m is our first injection, and

λ∞∩ : (U∩,Γ∩, φ∩)→ (U∞,Γ∞, φ∞),

defined by λ∞∩(z) := (1/z)1n = w

1n is our second injection. Clearly φ∩(z) = φ∞

λ∞∩(z) = z holds, for example. Finally, let U to be the unique maximal atlas containing

V and obtain the orbifold Σ = (S2,U).

2.1.3 Remark. We note that the points 0,∞ in Ex. 2.1.2 are the only points fixed

under the actions of the groups Γ, when |Γ| > 1, in any of the orbifold coordinate charts

contained in the maximal atlas U . The set of points x ∈ |Σ| whose isotropy subgroup

Γx 6= e are called singular points. Those points with Γx = e are called regular points.

See [BH99] for examples of orbifold structures on Riemann surfaces of genus g ≥ 1.

Next we define a natural notion of a “bundle” over an orbifold, which will be of

particular importance in our study of Seifert manifolds. We make the following

2.1.4 Definition. [BG08, Def. 4.2.7] A V-bundle or orbibundle over an orbifold Σ =

(|Σ|,U) consists of a fibre bundle BU over U for each orbifold chart (Ui,Γi, φi) ∈ U with

Lie group G and fiber F a smooth G-manifold which is independent of Ui together with

a homomorphism hUi : Γi → G satisfying:

i. if b lies in the fiber over xi ∈ Ui then for each γ ∈ Γi, bhUi(γ) lies in the fiber over

γ−1xi,


ii. if the map λji : Ui → Uj is an injection, then there is a fixed bundle map λ∗ji :

BUj|λji(Ui) → BUi

satisfying the condition that if γ ∈ Γi, and γ′ ∈ Γj is the unique

element such that λji γ = γ′ λji, then hUi(γ) λ∗ji = λ∗ji hUj(γ′), and if

λkj : Uj → Uk is another such injection then (λkj λji)∗ = λ∗ji λ∗kj.

If the fiber F is a vector space of dimension r and G acts on F as linear transformations

of F , then the V -bundle is called a vector V-bundle of rank r. Similarly, if F is the Lie

group G with its right action, then the V -bundle is called a principal V -bundle.

2.1.5 Remark. Note that the total space E of a V -bundle over an orbifold Σ inherits an

orbifold structure from Σ. We do not describe this here and instead refer the reader to

[BG08, Remark 4.2.1]. We will denote this orbifold structure as (E,U∗).

The notion of sections of orbibundles also makes sense and is important for us since

these are necessary to define vector fields, differential forms, and other such notions. We

have the following

2.1.6 Definition. [BG08, Def. 4.2.9] Let E = (E,U∗) be a V -bundle over an orbifold

Σ. Then a section σ of E over an open set W ⊂ |Σ| is a section σU of the bundle BU for

each orbifold chart (U ,Γ, φ) such that U ⊂ W and for each x ∈ U we have

i. For each γ ∈ Γ, σU(γ−1x) = σU(x)hU(γ).

ii. If λ : (U ,Γ, φ)→ (U ′,Γ′, φ′) is an injection, then λ∗σU ′(λ(x)) = σU(x).

One of the most important examples of an orbibundle for us will the tangent bundle.

We describe this in the following

2.1.7 Example. [BG08, Example 4.2.10] Given an orbifold Σ = (|Σ|,U), we briefly de-

scribe the tangent V -bundle of Σ. For an orbifold chart (Ui,Γi, φi) ∈ U take BUi= TUi

the usual tangent bundle of Ui ⊂ Rn. The fiber is F = Rn with G = GL(n,R) acting via

the Jacobian Jλji for each injection λji : (Ui,Γi, φi) → (Uj,Γj, φj). Since each element

γ ∈ Γi defines an injection (via x 7→ γ · x), the homomorphisms hUi are injective into


G = GL(n,R) and therefore satisfy condition (i) of Def. 2.1.4. The total space of TΣ is

T |Σ| with orbifold atlas U∗ consisting of orbifold charts of the form (Ui×Rn,Γ∗i , φ∗i ), where

Γ∗i = Γi acts on Ui × Rn via (x, v) 7→ (γ−1x, hUi(γ)v), and φ∗i : Ui × Rn → (Ui × Rn)/Γ∗i

is the natural quotient projection. A smooth1 invariant2 section of TΣ is called a vector

field on Σ.

2.1.8 Remark. The notions of cotangent V -bundle, and tensor bundles are constructed

similarly to the construction of Ex. 2.1.7. One can then talk about Riemannian metrics,

symplectic forms, connections, etc., without concern. For example, an orbifold real n-

form ω ∈ Ωnorb(Σ,R) is a collection of Γi invariant n-forms ωi ∈ Ωn(Ui,R) such that

λ∗jiωj = ωi, (2.1.9)

for all injections λji : (Ui,Γi, φi)→ (Uj,Γj, φj).

An important notion for us is contained in the following

2.1.10 Definition. [BG08, Def. 4.2.11] A Riemannian metric g on an orbifold Σ =

(|Σ|,U) is a Riemannian metric gi on each orbifold chart (Ui,Γi, φi) ∈ U that is invariant

under the local uniformizing group Γi and such that the injections λji : (Ui,Γi, φi) →

(Uj,Γj, φj) are isometries, i.e. λ∗ji(gj|λji(Ui)) = gi. Similarly, if Σ is a complex orbifold,

then a Hermitian metric h is a Γi-invariant Hermitian metric on each neighbourhood Ui

such that the injection maps are Hermitian isometries. An orbifold with a Riemannian

(Hermitian) metric is called a Riemannian (Hermitian) orbifold.

The fact that every smooth manifold admits a Riemannian metric also holds in the

orbifold case. Using a modification of the partition of unity argument, we have the

following

1A section σ of a V -bundle is called smooth, continuous, holomorphic, etc. if each of the local sectionsσUi

are smooth, continuous, holomorphic, etc., respectively.2An invariant section σ of a V -bundle is a section such that the local sections can be taken as well-

defined maps from the underlying space |Σ|, σU : U = φ(U)→ BU . Such invariant sections can always beconstructed from local sections σU by “averaging over the group,” Γ, and extended to a global invariantsection.


2.1.11 Proposition. [MM03] Every orbifold admits a Riemannian metric, and every

complex orbifold admits a Hermitian metric.

One of the most important notions that goes through in the orbifold case is that of

integration. Choose a partition of unity (αi) subordinate to the open cover Ui of |Σ|

and define integration of an orbifold n-form ω ∈ Ωnorb(Σ,R) (See Remark 2.1.8) on an

n-dimensional orbifold Σ by ∫Σ

ω :=∑i

1

|Γi|

∫Ui

φ∗i (αi)ωi. (2.1.12)

We have the following nice classification result for orbifolds.

2.1.13 Proposition. [Sat57] Every orbifold Σ = (|Σ|,U) can be presented as a quotient

space of a locally free action of a compact Lie group G on a manifold X.

In light of Prop. 2.1.13, we will in fact be interested in studying manifolds X that

possess a natural U(1) action and may therefore be viewed as principal U(1)-bundles

over an orbifold surface Σ. We note that our main interest in this thesis are complex

and Kahler orbifolds since these structures are essential for understanding quasi-regular

K-contact manifolds (See §2.3). On a complex orbifold there is a Γ-invariant tensor field

J of type (1, 1) which describes the complex structure on the tangent V -bundle TΣ.

Recall the following

2.1.14 Definition. Let Σ = (|Σ|,U) be a complex orbifold with a Hermitian metric

g (See Def. 2.1.10) and corresponding two-form ωg defined by ωg(Y1, Y2) = g(Y1, JY2),

for Y1, Y2 ∈ Γ(TΣ). Then Σ is called a Kahler orbifold if ωg is closed. We call a

Kahler orbifold a Hodge orbifold if [ωg] lies in the image of the coefficient homomorphism

C : H2(|Σ|,Q)→ H2(|Σ|,R).

Finally, we recall that Chern-Weil theory works equally well on orbifolds. Note that

we primarily follow [BG08, §4.3] here. For orbifolds it is also true that every principal

V -bundle P has a connection, κ say, which can be described by an invariant connection


one-form on P with values in the Lie algebra g of G and is a smooth section of Λ1P ⊗ g.

Let I(G) =∑

k Ik(G) denote the algebra of polynomials on g that are invariant under

the adjoint action of G on g. To each polynomial f ∈ Ik(G) one defines a 2k-form on

the orbifold P by

f(Ω)(Y1, . . . , Y2k) =1

2k!

∑σ∈S2k

(−1)|σ|f(Ω(Yσ(1), Yσ(2)), . . . ,Ω(Yσ(2k−1), Yσ(2k))),

for Y1, . . . , Y2k vector fields on P , Ω the curvature of κ, and |σ| denotes the sign of σ as a

permutation. As in the manifold case, this 2k-form is basic and projects to a closed 2-form

on the underlying space |Σ| of the orbifold. Its de Rham cohomology class is independent

of the choice of connection κ as well. This describes the Weil homomorphism

w : I(G)→ H∗DR(|Σ|,R) ' H∗(|Σ|,R).

It turns out that for orbifolds w(I(G)) lies in the image of the coefficient homomorphism

C : H2(|Σ|,Q)→ H2(|Σ|,R).

2.2 Contact and CR Manifolds

In this section we review contact and CR-geometry. We will also review the Tanaka-

Webster torsion and curvature in this section. Our main references are [DT06], [Bla76],

[Gei08], [BG08], and [Bog91].

We begin this section with the following

2.2.1 Definition. [BG08, Def. 6.1.7 ] A (2n + 1)-dimensional manifold X is a contact

manifold if it is equipped with a one-form κ ∈ Ω1(X,R), called a contact one-form, on

X, such that

κ ∧ (dκ)n 6= 0,


everywhere on X. A contact structure on X is an equivalence class of such one-forms,

where κ′ ∼ κ ⇐⇒ ∃ 0 6= f ∈ C∞(X) such that κ′ = fκ. The subbundle ker(κ) =: H ⊂

TX will be called the contact subbundle of X.

2.2.2 Remark. Note that the condition κ ∧ (dκ)n 6= 0 is equivalent to dκ being non-

degenerate on H. There are several different perspectives and more general approaches

to defining a contact structure on an odd dimensional manifold X that we will not pursue

here, as in [Bla76], and [Gei08]. Let the line bundle LH be defined as the annihilator

bundle of the contact subbundle ker(κ) =: H ⊂ TX, i.e. LH := H0. We only note

that a more general definition involves allowing the line bundle LH ⊂ T ∗X over X to be

non-trivial. In Def. 2.2.1 we have assumed that LH is trivial and κ ∈ Γ(LH) represents

a choice of trivializing section. In order to distinguish the two cases, we will refer to

the case where LH is trivial as a strict (or co-orientable) contact structure, and the case

where LH is non-trivial as a non-strict contact structure. Note that the two-fold cover of

a non-strict contact manifold is strict, and in particular, every simply connected contact

manifold is strict.

2.2.3 Example. [BG08, Ex. 6.1.15] One of the most important examples of a contact

manifold is R2n+1 with contact form given by κ = dt−∑

i yidxi. The contact subbundle

H is spanned by ∂∂xi

+ yi∂∂t, ∂∂yi. This clearly defines a contact structure on R2n+1

according to Def. 2.2.1. For example dκ = dxi ∧ dyi is easily seen to be non-degenerate

on H (See Remark 2.2.2). This is called the standard contact structure on R2n+1. By the

contact version of Darboux’s theorem every contact manifold is locally contactomorphic3

to R2n+1 with the standard contact structure, [Bla76].

2.2.4 Example. [BG08, Ex. 6.1.16] Let X = S2n+1, the unit (2n + 1)-sphere. Let

α :=∑n

i=0(xidyi − yidxi) ∈ Ω1(R2n+2,R), where R2n+2 is given the standard Cartesian

3A map Ψ : (X,κ) → (X ′, κ′) between contact manifolds is called a contactomorphism if it is adiffeomorphism that preserves the contact structure, Ψ∗κ′ = f · κ, for some 0 6= f ∈ C∞(X). If thereexists an contactomorphism Ψ : (X,κ)→ (X ′, κ′), then (X,κ) ' (X ′, κ′) are said to be contactomorphic.


coordinates (x0, . . . , xn, y0, . . . , yn). Define

κ := α|S2n+1 .

It is straightforward to see that κ ∧ (dκ)n 6= 0 everywhere on S2n+1. This defines the

standard contact structure on S2n+1.

There is a very interesting generalization of Ex. 2.2.4 above due to Gray, [Gra59].

2.2.5 Proposition. [BG08, Prop. 6.1.17] Let X be an immersed hypersurface in R2n+2

such that no tangent space of X contains the origin of R2n+2. Then X has a contact

structure.

It is interesting to note that the following provides an example of a contact manifold

for which the results of this thesis do not apply. One goal of future work is to find an

analogue of Prop. 6.10.24 in this case.

2.2.6 Example. [BG08, Ex. 6.1.23] Let X = T3 := R3/Z3. Let (x, y, z) denote the

standard Cartesian coordinates in R3 and let

κ := sin(y)dx+ cos(y)dz.

Then κ ∧ dκ = −dx ∧ dy ∧ dz, and the contact subbundle is spanned by ∂∂y, cos(y) ∂

∂x−

sin(y) ∂∂z.

Recall that every oriented surface possesses a symplectic structure. The analogue of

this fairly trivial fact in the three-manifold case is one of the most important facts for

this thesis and is the reason why studying contact structures on three-manifolds is so

natural. We have the following

2.2.7 Theorem. [Mar71] Every orientable three-manifold admits a contact structure.

We recall the following standard fact:


2.2.8 Lemma. [BG08, Lemma 6.1.24] On a contact manifold (X, κ) there is a unique

vector field ξ ∈ Γ(TX) called the Reeb vector field, satisfying the two conditions

ιξκ = 1, ιξdκ = 0.

We note that the Reeb vector field ξ depends strongly on the choice of contact form

κ and one can obtain very different Reeb vector fields for equivalent choices of contact

forms within a contact structure.

2.2.9 Example. [BG08, Ex. 7.1.12] Consider (S2n+1, κ) with the standard contact struc-

ture as in Ex. 2.2.4. Let Hi := xi∂∂yi− yi ∂

∂xi. It is straightforward to see that the Reeb

vector field of κ is given by

ξ =∑i

Hi.

Let w = (w0, . . . , wn) ∈ Rn+1 be some positive vector so that wi > 0 ∀ 0 ≤ i ≤ n. Let

fw(x) :=1∑n

i=0 wi(x2i + y2

i ), for x ∈ S2n+1.

Define a deformed contact form

κw := fw κ.

It is easy to see that

ξw =∑i

wiHi,

is the corresponding Reeb field for κw. Clearly, the Reeb field changes drastically de-

pending on the choice of vector w. If the components of w are rational numbers, for

example, the orbits of the Reeb field turn out to be all circles. If we choose one of the

components to be irrational, however, we may obtain Reeb orbits that do not close. Yet,

since κw := fw κ differ by a non-zero function fw, these different choices amount to the

same underlying contact structure.

The Reeb vector field is sometimes called the characteristic vector field and the one-

dimensional foliation Fξ uniquely determined by ξ is called the characteristic foliation of


(X, κ).

Let us move on and make the following

2.2.10 Definition. [BG08, Def. 6.2.5] An almost contact structure on a differentiable

manifold X is a triple (ξ, κ, φ), where φ : TX → TX is a tensor field of type (1, 1), ξ is

a vector field, and κ ∈ Ω1(X,R) is a one-form which satisfy

κ(ξ) = 1, φ2 = −I + ξ ⊗ κ,

where I is the identity endomorphism of TX. A smooth manifold with such a structure

is called an almost contact manifold. An almost contact structure is said to be normal if

[φ, φ] + 2dκ⊗ ξ = 0,

where,

[φ, φ](Y1, Y2) := φ2[Y1, Y2]− [φY1, φY2]− φ[φY1, Y2]− φ[Y1, φY2],

is the Nijenhuis torsion of φ.

We make the following

2.2.11 Definition. [BG08, Def. 6.3.1] Let X be an almost contact manifold. A Rie-

mannian metric g on X is said to be compatible with the almost contact structure if for

any vector fields Y1, Y2 ∈ Γ(TX), we have

g(φY1, φY2) = g(Y1, Y2)− κ(Y1)κ(Y2).

An almost contact structure with a compatible metric is called an almost contact metric

structure.

We have the following

2.2.12 Proposition. [BG08] Every almost contact manifold admits a compatible metric.


The following theorem is important for this thesis as it allows us to conclude a par-

ticularly nice form for the metric structures on our contact manifolds.

2.2.13 Theorem. [BG08, Theorem 6.3.6] Let (Σ, J) be an almost complex orbifold (See

§2.1) and let π : X → Σ be a principal U(1) V -bundle over Σ. Suppose that ξ is a

generator of the U(1)-action that corresponds to 1 in the Lie algebra g ' R, and that κ

is a connection one-form on X. Then (ξ, κ, φ), with φ defined as in Eq. 6.3.3 of [BG08],

defines an almost contact structure on X. Furthermore, if h is a Hermitian metric on Σ

compatible with J then

g = κ⊗ κ+ π∗h

is a (bundle-like) Riemannian metric on X compatible with the almost contact structure,

and ξ is a Killing field for g.

We will need the following

2.2.14 Definition. [BG08, Def. 6.4.1] Let (X, κ) be a contact manifold with contact

distribution H. Then an almost contact structure (ξ, κ′, φ) is said to be compatible with

the contact structure if κ = κ′, ξ is the Reeb vector field, and the endomorphism φ

satisfies

dκ(φY1, φY2) = dκ(Y1, Y2), for all Y1, Y2 ∈ Γ(TX),

and,

dκ(φY0, Y0) > 0, for all Y0 ∈ Γ(H).

Denote by AC(κ) the set of compatible almost contact structures on (X, κ).

2.2.15 Proposition. [BG08, Prop. 6.4.3] Let (X, κ) be a contact manifold. The set of

associated Riemannian metrics are in one-to-one correspondence with the set of compat-

ible almost contact structures, AC(κ), on (X, κ).

Finally, the following is the basic definition that we need for this thesis.


2.2.16 Definition. [BG08, Def. 6.4.4] A contact manifold (X, κ) with a compatible

almost contact metric structure (ξ, κ, φ, g) such that

g(Y1, φY2) = dκ(Y1, Y2), for all Y1, Y2 ∈ Γ(TX),

is called a contact metric structure, and (X, ξ, κ, φ, g) is called a contact metric manifold.

We will also need the following

2.2.17 Definition. A normal4 contact metric manifold (X, ξ, κ, φ, g) is called a Sasakian

manifold.

We now move on to study CR structures. We start with the following

2.2.18 Definition. [DT06, Def. 1.1 and 1.2] An almost CR structure on a manifold

X (dim(X) = m) is a subbundle T(1,0) = T(1,0)(X) ⊂ TCX of complex rank n of the

complexified tangent bundle such that

T(1,0)(X) ∩ T(0,1)(X) = 0,

where T(0,1)(X) := T(1,0)(X) the complex conjugate. An almost CR structure is called a

CR structure if

[T(1,0)(X), T(1,0)(X)] ⊂ T(1,0)(X),

so that T(1,0)(X) is an integrable subbundle of TCX. The integers n and k = m − 2n

are called the CR dimension and CR codimension of the almost CR structure and (n, k)

denotes its type. The pair (X,T(1,0)) is called an (almost) CR manifold of type (n, k).

We are mainly interested in almost CR structures of type (1, 1) in this thesis. Consider

the following

2.2.19 Example. Let (X, κ, ξ, φ) be an almost contact manifold with distribution H =

kerκ. The restriction of φ to H determines the decomposition

HC = H(1,0) ⊕H(0,1) ⊂ TCX,

4See Def. 2.2.10.


where H(1,0) and H(0,1) are the +i and −i eigenbundles of φ|H , respectively. Taking

T(1,0)X = H(1,0) determines an almost CR structure on X. This construction clearly

also applies to the case where (X, κ) is a contact manifold and (X, κ, ξ, φ) is choice of

compatible almost complex structure. Whether or not this construction yields a CR

structure (i.e. integrable distribution) has been determined by S. Tanno in [Tan89]. In

this thesis we will always be concerned with the case where this construction yields a CR

structure. This follows because our contact structures will be assumed normal (See Def.

2.2.10).

Notice in the above example that we may recover H from HC by taking the real part

of HC:

H = R(H(1,0) ⊕H(0,1)).

The following is a generalization of this idea.

2.2.20 Definition. Let (X,T(1,0)) be an (almost) CR manifold of type (n, k). Its maximal

complex, or Levi distribution is the real rank 2n subbundle defined as

L(X) = R(T(1,0) ⊕ T(1,0)).

L(X) carries the complex structure JL : L(X)→ L(X) defined by

JL(Y + Y ) = i(Y + Y ),

for any Y ∈ T(1,0).

As noted in Remark 2.2.2 above, given a contact manifold (X, κ) with contact dis-

tribution H, a contact form is naturally viewed a section of the annihilator bundle H0.

Generalizing this to CR manifolds of type (n, 1),5 we let H0 denote the annihilator bun-

dle of the Levi distribution H = L(X). It is easy to see that H0 is a subbundle of T ∗X

that is isomorphic to TX/H. Assume X is orientable. Then since H is oriented by the

5Also called CR manifolds of hypersurface type.


complex structure JL, it follows that H is orientable. Any orientable real line bundle

over a connected manifold is trivial, so there exist globally defined nowhere vanishing

sections θ ∈ Γ(H).

2.2.21 Definition. [DT06, Def. 1.6] Let (X,T(0,1)) be an oriented CR manifold of

type (n, 1) with H = L(X). Then any a choice of θ ∈ Γ(H) is referred to as a pseudo-

Hermitian structure on X. Given a pseudo-Hermitian structure θ on X the Levi form

Lθ is defined by

Lθ(Z,W ) = −idθ(Z,W ),

for any Z,W ∈ T(1,0).

We now make the following

2.2.22 Definition. [DT06, Def. 1.7] Let (X,T(0,1)) be an oriented CR manifold of type

(n, 1) with H = L(X). We say that (X,T(0,1)) is nondegenerate if the Levi form Lθ is

non-degenerate for some (and hence any) choice of pseudo-Hermitian structure θ on X.

If Lθ is positive definite (i.e. Lθ(Z,Z) > 0, ∀ 0 6= Z ∈ T(0,1)) for some θ,6 then (X,T(0,1))

is said to be strictly pseudoconvex.

The main point that we would like to make is that on a contact manifold (X, κ), the

choice of contact form κ defines a pseudo-Hermitian structure on X relative to the almost

CR structure defined in Ex. 2.2.19 above. Going in the other direction, one may obtain

a natural contact metric structure (X, ξ, κ, φ, g) from a CR structure (X,T(0,1)) of type

(n, 1) with pseudo-Hermitian structure κ. First we need the following

2.2.23 Proposition. [DT06, Prop. 1.2] Given (X,T(0,1)) a type (n, 1) CR manifold with

pseudo-Hermitian structure κ, there exists a unique globally defined nowhere zero tangent

vector field ξ on X such that

ιξκ = 1, ιξdκ = 0,

6This does not apply to all choices of θ since Lθ positive definite implies that L−θ is negative definite.


and ξ is transverse to the Levi distribution H = L(X).

We also have

2.2.24 Proposition. [DT06, Prop. 1.4] Given (X,T(0,1)) a type (n, 1) CR manifold

with pseudo-Hermitian structure κ, Levi distribution H = L(X) and ξ as in Prop. 2.2.23

above, then

TX ' H ⊕ Rξ.

By setting φ(Y ) = JLY for all Y ∈ H, and φξ = 0, one can show that (X, κ, ξ, φ)

defines an almost contact manifold. If the Levi form Lκ is non-degenerate then (X, κ) is

a contact manifold. Let

g(Y1, Y2) = dκ(Y1, JLY2).

Then g(JLY1, JLY2) = g(Y1, Y2) since the Nijenhuis tensor of JL vanishes when X is a

CR manifold. We may now extend g to all of TX by using the splitting TX ' H ⊕ Rξ

and defining g(Y, ξ) = 0 and g(ξ, ξ) = 1. The resulting form g is called the Webster

metric of (X, κ). If X is strictly pseudoconvex, then g defines a Riemannian metric and

(X, ξ, κ, φ, g) defines a contact metric structure on X. We may now state the following

2.2.25 Theorem. [DT06, Theorem 1.3] Given (X,T(0,1)) a type (n, 1) CR manifold with

pseudo-Hermitian structure κ, Levi distribution H = L(X), ξ as in Prop. 2.2.23 above,

and JL the complex structure on H = L(X) (extended to φ ∈ End(TX) by requiring that

φ(ξ) = 0). Let g be the Webster metric on (X, κ). There is a unique linear connection

∇ on X satisfying the following axioms:

i. H is parallel with respect to ∇, that is,

∇Y Γ(H) ⊂ Γ(H),

for any Y ∈ Γ(TX).

ii. ∇J = 0, and ∇g = 0.


iii. The torsion T∇7 of ∇ is pure, i.e.

T∇(Y1, Y2) = dκ(Y1, Y2)ξ, and, T∇(ξ, JLY ) + JLT∇(ξ, Y ) = 0,

for all Y1, Y2 ∈ Γ(H).

We then make the following

2.2.26 Definition. [DT06, Def. 1.25] Given (X,T(0,1)) as in Theorem 2.2.25 above, the

connection ∇ in Theorem 2.2.25 is called the Tanaka-Webster connection. The Tanaka-

Webster scalar curvature R ∈ C∞(X) is the scalar curvature8 associated to the Tanaka-

Webster connection.

2.3 Seifert Manifolds

We review the relevant definitions and results in Seifert geometry. We will primarily

study the class of Seifert manifolds that come from the orbifold version of the Boothby-

Wang theorem; that is, we study the class of quasi-regular K-contact manifolds. We will

also study Seifert manifolds in general, and recall some basic facts about their funda-

mental groups and related invariants. Our main references are [BG08], [BW05], [FS92]

and [Orl72].

In this section we briefly review our geometric situation. In particular, we recall the

7Recall that the torsion of a linear connection is defined as

T∇(Y1, Y2) = ∇Y1Y2 −∇Y2Y1 − [Y1, Y2].

8Recall that the scalar curvature is computed via contraction by the Webster metric g of the Riccicurvature

Rij = Rkikj ,

where R is the curvature tensor of the connection ∇,

R(Y1, Y2)Y3 = ∇Y1∇Y2Y3 −∇Y2∇Y1Y3 −∇[Y1,Y1]Y3.

That is, R = Rii = gijRij .


definition of a quasi-regular K-contact manifold and review some standard facts about

these structures in the case of dimension three.

2.3.1 Remark. Our three manifolds X are assumed to be closed throughout this thesis.

2.3.2 Definition. A K-contact manifold is a manifold X with a contact metric structure

(φ, ξ, κ, g) such that the Reeb field ξ is Killing for the associated metric g, Lξg = 0.

where,

• κ ∈ Ω1(X) contact form, ξ = Reeb vector field.

• H := kerκ ⊂ TX denotes the horizontal or contact distribution on (X, κ).

• φ ∈ End(TX), φ(Y ) = JY for Y ∈ Γ(H), φ(ξ) = 0 where J ∈ End(H) complex

structure on the contact distribution H ⊂ TX.

• g = κ⊗ κ+ dκ(·, φ·)

2.3.3 Remark. Note that we will assume that our contact structure is “co-oriented,”

meaning that the contact form κ ∈ Ω1(X) is a global form. Generally, one can take the

contact structure to be defined only locally by the condition H := kerκ, where κ ∈ Ω1(U)

for open subsets U ∈ X contained in an open cover of X.

2.3.4 Definition. The characteristic foliation Fξ of a contact manifold (X, κ) is said

to be quasi-regular if there is a positive integer j such that each point has a foliated

coordinate chart (U, x) such that each leaf of Fξ passes through U at most j times. If

j = 1 then the foliation is said to be regular.

Definitions 6.9.10 and 6.9.11 together define a quasi-regular K-contact manifold, (X,φ, ξ, κ, g).

The following result provides several different perspectives on K-contact structures:

2.3.5 Proposition. [BG08, Prop. 6.4.8] On a contact metric manifold (X,φ, ξ, κ, g),

the following conditions are equivalent:


i. The characteristic foliation Fξ is a Riemannian foliation.

ii. g is bundle-like.

iii. The Reeb flow is an isometry.

iv. The Reeb flow is a CR-transformation.

v. The contact metric structure (φ, ξ, κ, g) is K-contact.

Quasi-regular K-contact three-manifolds are necessarily “Seifert” manifolds that fiber

over a two dimensional orbifold Σ (See §2.1) with some additional structure. Recall:

2.3.6 Definition. A Seifert manifold is a three manifold X equipped with a locally free

U(1)-action.

Thus, a Seifert manifold is simply a U(1)-bundle over an orbifold Σ,

U(1) // X

Σ

.

with smooth total space X. The orbifold base Σ of X is taken to be a Riemann surface

of genus g with N marked points pjNj=1, the exceptional points, and the local model is

C/Zαj centered about the point pj. Here Zαj acts on the local coordinate z at pj as

z 7→ ξ · z, ξ = e2πi/αj

We follow [FS92], and describe a general Seifert manifold X as the total space associated

to the S1 fibration of a line V-bundle over Σ. A line V-bundle over Σ (See Def. 2.1.4) can

be described as a complex line bundle, where the local trivialization over each orbifold

point pj of Σ is now modeled on C × C/Zαj , where Zαj acts on the local coordinates

(z, s) as

z 7→ ξ · z, s 7→ ξβj · s, ξ = e2πi/αj


for some integers 0 ≤ βj < αj. We require that X be smooth, which implies that each

pair of integers (αj, βj) are relatively prime, so that the local action of Zαj on C× S1 is

free (in particular, βj 6= 0). The U(1) action on X is again rotations of the fibres over Σ,

but the points in the S1 fiber over each point pj are fixed by the cyclic subgroup Zαj of

U(1).

It is well known that9 the topological isomorphism class of a Seifert manifold X is given

by the Seifert invariants

[g, n; (α1, β1), . . . , (αN , βN)], gcd(αj, βj) = 1

where g is the genus of Σ, and n is the degree of the line V-bundle.

For later use, we record now the following assumption, that

c1(L) = n+N∑j=1

βjαj

> 0

where L denotes the line V-bundle over Σ which describes X, and c1(L) is the orbifold

first Chern number of L.

2.3.7 Remark. We take c1(L) to be positive by convention, and non-zero so that L is

non-trivial (See Example 2.3.9 for the reason behind this).

Recall also the following description of the fundamental group of X, [Orl72]. π1(X)

is generated by the following elements

ap, bp, p = 1, . . . , g

cj, j = 1, . . . , N

h

9See [Orl72] for example.


which satisfy the relations,

[ap, h] = [bp, h] = [cj, h] = 1 (2.3.8)

cαjj h

βj = 1g∏p=1

[ap, bp]N∏j=1

cj = hn

The generator h is associated to the generic S1 fiber over Σ, the generators ap, bp come

from the 2g non-contractible cycles on Σ, and the generators cj come from the small one

cycles in Σ around each of the orbifold points pj.

We have the following classification result: X is a quasi-regular K-contact three manifold

⇐⇒

• [BG08, Theorem 7.5.1, (i)] X is a U(1)-Seifert manifold over a Hodge orbifold

surface, Σ.

• [BG08, Theorem 7.5.1, (iii)] X is a U(1)-Seifert manifold over a normal projective

algebraic variety of real dimension two.

2.3.9 Example. All 3-dimensional lens spaces, L(p, q) and the Hopf fibration S1 → S3 →

CP1 admit quasi-regular K-contact structures. Note that any trivial U(1)-bundle over a

Riemann surface Σg, X = U(1)×Σg, admits no K-contact structure,10 however, and our

results do not apply in this case.

2.3.10 Remark. Note that in fact our results apply to the class of all closed Sasakian

three-manifolds. This follows from the observation that every Sasakian three manifold is

K-contact, [Bla76, Corollary 6.5], and every K-contact manifold possesses a quasi-regular

K-contact structure, [BG08, Theorem 7.1.10].

A useful observation for us is that for a quasi-regular K-contact three-manifold, the metric

tensor gε must take the following form, [BG08, Theorem 6.3.6]:

gε = ε κ⊗ κ+ π∗h (2.3.11)

10See [Ito97] for example.


where π : X → Σ is our quotient map, and h represents any (orbifold) Kahler metric

(See Def. 2.1.10) on Σ which is normalized so that the corresponding (orbifold) Kahler

form (See Def. 2.1.14), ω ∈ Ω2orb(Σ,R), pulls back to dκ.

2.4 Vielbein Formalism

In this section we outline the basic construction of a Vielbein on an orientable Riemannian

three-manifold (X,G). Our primary references for this are [FF03] and [Car04]. We note

that the contents of this chapter are essential for our computation in Chapter 7 and are

well established in the physics literature.11

2.4.1 Definition. LetM be an orientable manifold of dimension n, Θ an SO(n)-principal

bundle over M , and L(M) the GL(n)-frame bundle of M . A Vielbein is a principal

morphism E : Θ→ L(M) (if one exists).

2.4.2 Remark. Note that if we fix Θ arbitrarily, then it could happen that no principal

morphism E : Θ → L(M) exists. For example, if M is non-parallelizable12 and Θ =

Θtriv := M × SO(n), then there is no global principal morphism E : Θ → L(M). If

such a morphism did exist then E σ : M → L(M) would be a global section for some

trivialization σ of Θtriv, which is impossible since M is non-parallelizable. Of course, if

M is parallelizable, so that σ : M → L(M) is a trivialization, then a Vielbein always

exists by choosing Eσ : Θtriv → L(M) the standard inclusion.

Let E : Θ → L(M) be a Vielbein. Since Θ is an SO(n)-principal bundle over M ,

then by construction E induces a reduction of the structure group of the tangent bundle

TM from GL(n) to SO(n), and hence induces an associated Riemannian metric on M .

Conversely, if a Riemannian metric G is given on M , then the corresponding reduction

of the structure group induces a natural Vielbein.

11See [GIJP03] for example.12A manifold M is called parallelizable if the frame bundle L(M) is trivializable.


Let us henceforth assume that M = (X,G) is an oriented Riemannian three-manifold.

Given a local chart U ⊂ X, a Vielbein may therefore be expressed as a triple

E0, E1, E2 (2.4.3)

where EA ∈ Γ|U(TX) such that

G(EA, EB) = ηAB (2.4.4)

whereA,B ∈ 0, 1, 2. Note that in a Lorentzian spacetime, ηAB represents the Minkowski

metric, of signature (−,+,+) say, and in a Euclidean spacetime it represents the positive-

definite Euclidean metric (so ηAB = δAB, is the Kronecker pairing in this case). We

implicitly assumed a Euclidean signature in our definition of a Vielbein by choosing to

work with SO(n)-bundles. More generally we could have allowed SO(p, q)-bundles of

arbitrary signature. We will work in a Euclidean signature in this thesis. Given a choice

of local coordinate system x0, x1, x2 on X, we define

Gµν := G(∂µ, ∂ν) (2.4.5)

and we define the notation EµA to represent the coordinates of EA in the coordinate

system basis ∂0, ∂1, ∂2:

EA =2∑

µ=0

EµA∂µ. (2.4.6)

Note that we also adopt the Einstein summation convention and write

EA = EµA∂µ, (2.4.7)

for example, where it is understood that a sum is taken over the repeated raised and

lowered indices. We may then express Eq. (2.4.4) in local coordinates:

Gµν EµA E

νB = ηAB (2.4.8)


where µ, ν ∈ 0, 1, 2 are thought of as indexing the spacetime coordinates related to

the manifold coordinates x0, x1, x2, and A,B ∈ 0, 1, 2 are thought of as indexing the

tangent space coordinates that label the Vielbein. By dualizing to the cotangent bundle,

we also consider

EA := EAµ dx

µ ∈ Γ|U(T ∗X). (2.4.9)

which are defined by requiring

EA(EB) = δAB. (2.4.10)

Let Gµν denote the inverse of Gµν , so that GµλGλν = GνλGλµ = δµν . Our relevant

relations for the Vielbein are then:

EµAE

Aν = δµν , EA

µ EµB = δAB (2.4.11)

Gµν EµA E

νB = ηAB , Gµν = EA

µEBν ηAB (2.4.12)

EAµ = Gµνη

ABEνB. (2.4.13)

Note that every closed, oriented three-manifold is parallelizable.13 Thus, if σ : X → L(X)

is a trivialization, one can take Eσ : Θtriv → L(M) the standard inclusion. An important

point for us to note is that there are many possible choices of trivialization σ : X → L(X),

and it is a standard fact that the gravitational Chern-Simons term

CS(A) :=1

4π

∫X

Tr(A ∧ dA+2

3A ∧ A ∧ A), (2.4.14)

where A the Levi-Civita connection on the spin bundle of X, is sensitive to the choice of

trivialization σ up to homotopy equivalence. This dependence is explicitly computable

as the winding number W (E) of the Vielbein, [GIJP03, Eq. 2.21]:

W (E) =1

24π2

∫X

d3x εµνλ tr(VµVνVλ) ∈ Z, (2.4.15)

where (Vµ)σρ := EσA∂µE

Aρ .

13See [LM89].


2.5 The Space of Connections

2.5.1 The General Case

Let X be a compact smooth manifold and G a compact Lie group. Since it will be useful

to deal in both the general case and the special G = U(1) case, we will assume no further

restrictions on X and G, and specialize to the G = U(1) case in the next section. Most

of the material in this section can be found in [AB83].

Let P be a principal G-bundle over X. P is a manifold equipped with a proper and free

action of G for which X = P/G. Define the group of gauge transformations, GP , of P to

be the set of G-equivariant diffeomorphisms of P that preserve fibres:

GP := ψ ∈ (Diff(P, P ))G | π ψ = π (2.5.1)

where,

π : P → X = P/G (2.5.2)

is the standard projection map. We will sometimes identify GP with the infinite dimen-

sional Lie group of G-equivariant smooth functions from P to G,

(C∞(P,G))G := u ∈ C∞(P,G) | u(g · p) = gu(p)g−1 (2.5.3)

The identification is given by,

(C∞(P,G))G → GP (2.5.4)

u 7→ ψ (2.5.5)

where ψ(p) = p · u is a Lie group isomorphism. For a fixed g ∈ G, let ψg ∈ Diff(P, P ) be

the action map,

ψg(p) := p · g (2.5.6)

This G action on P induces an action on the space Ω(P ) of differential forms

g · ω := ψ∗g−1(ω) (2.5.7)


Define the set of horizontal m-forms on P to be

Ωmhor.(P ) := ω ∈ Ωm(P ) | ιξ](ω) = 0, ∀ ξ ∈ g (2.5.8)

We combine the action of G on Ω(P ) with the adjoint action of G of g to obtain an action

of G on the space of g-valued m-forms on P , Ωm(P )⊗ g,

g ·

(∑j

ωj ⊗ ξj

)=∑j

ψ∗g−1(ωj)⊗ Adg(ξj) (2.5.9)

Denote by (Ωm(P ) ⊗ g)G the space of G equivariant g-valued m-forms on P , and by

(Ωmhor.(P ) ⊗ g)G the space of horizontal G-equivariant g-valued m-forms on P . Given a

principal G-bundle P over X, we define the space of connections to be:

AP := A ∈ (Ω1(P )⊗ g)G | A(ξ]) = ξ, ∀ ξ ∈ g (2.5.10)

We have the following,

2.5.11 Proposition. If P is an arbitrary principal G-bundle, then the space of connec-

tions AP is non-empty, and if AP ∈ AP , then AP = AP + (Ω1hor.(P )⊗ g)G

That is, AP is an affine space modeled on the linear space

aP := (Ω1hor.(P )⊗ g)G (2.5.12)

Let g(P ) denote the associated bundle for the adjoint action of G on g,

g(P ) = P ×G g = (g× P )/ ∼ (2.5.13)

where (ξ, p) ∼ (Adg(ξ), g · p). Observe that g(P ) is a vector bundle over X with fibre g.

It is sometimes useful to make the identification

(Ωmhor.(P )⊗ g)G = Ωm(X, g(P )) (2.5.14)

where Ωm(X, g(P )) are vector bundle valued forms onX. Recall that for A =∑

j αj⊗ξj ∈

Ωp(P )⊗ g and B =∑

j βj ⊗ ζj ∈ Ωq(P )⊗ g, that

dA = (d⊗ Idg)(A) =∑j

dαj ⊗ ξj (2.5.15)


and

[A,B] = (∧ ⊗ [·, ·])(A,B) =∑j

∑k

αj ∧ βk ⊗ [ξj, ζk] (2.5.16)

2.5.17 Definition. A connection A ∈ AP will be called flat if its curvature FA =

dA+ 12[A,A] ∈ Ω2(P, g) vanishes, i.e. FA = 0.

Recall that the group of gauge transformations GP acts on the space of connections AP

in the following way;

2.5.18 Proposition. Let u = uψ for the associated gauge transformation ψ ∈ GP . Then

GP acts on AP as follows: ψ ·A = uψ ·A = (ψ)∗A = Adu−1ψA+(uψ)∗ϑ, where ϑ ∈ Ω1(G)⊗g

is the Maurer-Cartan form.14

2.5.20 Remark. Since,

(u∗ϑ)γ(0)

(d

dt|t=0γ(t)

)=

d

dt|t=0((u(γ(0)))−1(u(γ(t)))) (2.5.21)

one often writes u∗ϑ = u−1 · du. Using this notation, the action of the gauge group on a

connection is given by

u · A = Adu−1A+ u−1 · du. (2.5.22)

2.5.23 Proposition. GP acts on (Ω1hor.(P )⊗ g)G as follows: if B ∈ (Ω1

hor.(P )⊗ g)G and

ψ(p) = p · uψ, then ψ ·B = ψ∗B = Adu−1ψB.

The Lie algebra of the gauge group GP is

Lie(GP ) = (Xvert.(P ))G = Z ∈ X(P ) | dπp(Zp) = 0, ψ∗gZ = Z (2.5.24)

and the Lie algebra of (C∞(P,G))G is:

Lie((C∞(P,G))G) = (C∞(P, g))G ' (Ω0hor.(P )⊗ g)G (2.5.25)

We have the following:

14Recall that the Maurer-Cartan form ϑ ∈ Ω1(G)⊗ g is defined by:

ϑg(ξ) := (Lg−1)∗ξ (2.5.19)

for ξ ∈ TgG.


2.5.26 Proposition. The map given by

(C∞(P, g))G → (Xvert.(P ))G (2.5.27)

f 7→ Z (2.5.28)

where Zp = ddt|t=0 p · (exp(−tf(p))) = −(f(p))]p, is a Lie algebra isomorphism.

2.5.29 Proposition. If Z ∈ (Xvert.(P ))G and Zp = −(f(p))]p, define φt ∈ GP by setting

φt(p) = p · exp(−t(f(p))). Then φtt∈R is the flow of Z.

2.5.30 Proposition. The adjoint action of ψ ∈ GP on Z ∈ Lie(GP ) is

ψ · Z = ψ∗Z (2.5.31)

Also, if ψ(p) = p · uψ and Zp = −(f(p))]p, then

(ψ · Z)p = −(Adu(p)f(p))]p. (2.5.32)

2.5.2 The U(1)-Bundle Case

In this section we specialize to G = U(1). We primarily follow sections §2 and §3 of

[Man98].

First, observe that

C∞(X,U(1)) = (C∞(P,G))G (2.5.33)

h ↔ u (2.5.34)

where an element h of C∞(X,U(1)) defines a map uh : P → U(1) by uh(p) = h(π(p)),

and conversely a map u : P → U(1) defines a map hu : X → U(1) by hu(x) = u(p) for

any p ∈ π−1(x). Thus, GP = C∞(X,U(1)). Since GP is independent of the bundle P in

the U(1) case, we define:

GX := C∞(X,U(1)) (2.5.35)


We also have that aP = (Ω1hor.(P ) ⊗ R)U(1) = (Ω1

hor.(P ))U(1) = Ω1(X), where the last

equality comes from the fact that π∗ : Ω1(X)→ Ω1(P ) is an isomorphism onto its image

(Ω1hor.(P ))U(1). In summary, when G = U(1) we identify:

aP = Ω1(X) (2.5.36)

Let AX denote the space of all U(1)-connections on X. An element A ∈ AX is a

connection on a principal U(1)-bundle P . We may write

AX =⊔P

AP (2.5.37)

as the union over all principal U(1)-bundles over X. Recall the notion of equivalence

among principal U(1)-bundles. Two principal U(1)-bundles over X, say P1 and P2, are

equivalent if there exists a U(1)-equivariant map ψ : P1 → P2 that covers the identity

map on X. Two elements A1, A2 ∈ AX are called gauge equivalent if there exists and

isomorphism ψ : P1 → P2 such that A1 = ψ∗A2. This defines an equivalence relation on

AX , where

A1 ∼ A2 ⇐⇒ A1 = ψ∗A2 (2.5.38)

The action of GX = C∞(X,U(1)) on AP can be written out for any U(1)-bundle P

over X. If h : X → U(1) is an element of GX with associated bundle automorphism

ψh : P → P defined by ψh(p) = p · uh(p), the action of h on A ∈ AP is described by:

h · A = (ψh)∗A = A+ u∗hϑ, (2.5.39)

since the adjoint action is trivial is in the U(1) case. Note that ϑ is the Maurer-Cartan

form of U(1), so that ϑ generates H1(U(1),Z) ⊂ H1(U(1),R).

We will now recall some basic facts about curvature and connections for principal U(1)-

bundles. Fix a principal U(1)-bundle π : P → X and let A ∈ AP be a connection on P .

We will be slightly ambiguous in our terminology regarding the curvature of a connection

and we aim to describe this here. First we will need the following


2.5.40 Lemma. Given a connection A ∈ AP on a principal U(1)-bundle π : P → X,

there exists a two form F on X such that

π∗F = dA (2.5.41)

Proof. We shall prove this by showing that the form dA is U(1)-invariant and horizontal,

i.e. that dA is basic. Clearly dA is invariant because A is invariant. dA is horizontal since

for any Y ∈ u(1), ι(Y ]P )dA = dι(Y ]

P )A = 0 since by definition of a connection ι(Y ]P )A = Y

is constant (where Y ]P denotes the vector field on P generated by the infinitesimal action

of Y on P ).

2.5.42 Remark. The ambiguity in our terminology will be the following. The curvature

of a connection A ∈ AP in the U(1)-bundle case will refer to both

FA := dA (2.5.43)

and the form F that satisfies the above lemma 2.5.40

π∗F = dA. (2.5.44)

We will usually use the same notation for both. When there is a chance of confusion, we

will write FA for the form dA and reserve the notation F for the two form on the base

manifold X in lemma 2.5.40.

The notation for the curvature F ∈ Ω2(X,R), being written as independent of the original

connection A used to define it, is partially justified by the following

2.5.45 Lemma. The cohomology class of the curvature F is independent of the connec-

tion form A used to define it.

Proof. Recall from Prop. 2.5.11 that the space of connections is an affine space modeled

on the linear space aP = Ω1(X). Thus, for any two connection forms A,A′ ∈ AP , there

exists a form β ∈ Ω1(X) such that

A− A′ = π∗β.


Taking F, F ′ to be the associated curvature forms of A,A′ respectively, we compute

π∗(F − F ′) = dA− dA′

= π∗(β)

and so F = F ′ + dβ since π∗ : Ω2(X,R)→ Ω2basic(P, u(1)) is an isomorphism, and hence

[F ] = [F ′] ∈ H2DR(X,R).

The following is a standard result in Chern-Weil theory.

2.5.46 Proposition. [MS74] The curvature class [F ] ∈ H2(X,R) is the image of the

first chern class c1(P ) ∈ H2(X,Z) under the natural homomorphism

ψ : H2(X,Z)→ H2(X,R)

so that ψ(c1(P )) = [F ].

Recall that an element in a Z-module is said to be torsion if some integral multiple of it

is zero. We have

2.5.47 Proposition. Let X be a compact 3-manifold. A principal U(1)-bundle π : P →

X has flat connections if and only if the first Chern class, c1(P ) ∈ H2(X,Z), is torsion

c1(P ) ∈ TorsH2(X,Z). (2.5.48)

Proof. Let us first prove that the existence of a flat connection A ∈ AP implies that

c1(P ) ∈ TorsH2(X,Z). By Prop. 2.5.46, we know that ψ(c1(P )) = [F ] and since A is flat

we have F = 0. Thus, c1(P ) ∈ ker(ψ), and by the Universal Coefficient Theorem’s and

the definition of ψ is it elementary to see that ker(ψ) = TorsH2(X,Z). This establishes

the first direction.

To see the other direction, assume that c1(P ) ∈ TorsH2(X,Z) and observe that since

ker(ψ) = TorsH2(X,Z), we must have that F ∈ Ω1(X,R) is exact, i.e. F = dβ for some


β ∈ Ω1(X,R). Since π∗F = dA for some connection A ∈ AP we see that we can find a

flat connection A′ ∈ AP by setting

A′ = A− π∗β

since dA′ = d(A− π∗β) = dA− π∗dβ = π∗(F − F ) = 0. This completes the proof.

Recall that for A =∑

j αj ⊗ ξj ∈ Ωp(P )⊗ g and B =∑

j βj ⊗ ζj ∈ Ωq(P )⊗ g, that

dA = (d⊗ Idg)(A) =∑j

dαj ⊗ ξj (2.5.49)

and

[A ∧B] = (∧ ⊗ [·, ·])(A,B) =∑j

∑k

αj ∧ βk ⊗ [ξj, ζk] (2.5.50)

2.5.51 Definition. We decompose the subspace AfX ⊂ AX of flat connections as:

AfX =⊔P

c1(P )∈TorsH2(X,Z)

AfP (2.5.52)

where AfP ⊂ AP is the space of flat connections A on P with curvature

FA = dA+1

2[A ∧ A] = dA = 0 (2.5.53)

In the case that G = U(1), we have g = R, so that

[ξ, ζ] = 0, ∀ ξ, ζ ∈ g = R (2.5.54)

and hence,

FA = dA+1

2[A ∧ A] = dA = 0 (2.5.55)

as above.

2.5.56 Definition. Let P be a principal U(1)-bundle with c1(P ) ∈ TorsH2(X,Z). The

moduli space of flat connections on P is the quotient space,

MP := AfP/GP (2.5.57)


Note that the action of the gauge group GP on AfP is well defined since if h ∈ GP = GX =

C∞(X,U(1)), and AP ∈ AfP , then

Fh·AP = d(h · AP ) +1

2[h · AP , h · AP ] (2.5.58)

= d(h · AP ), since [h · AP , h · AP ] = 0 for G = U(1) (2.5.59)

= d(AP + u∗hϑ), by definition of h · AP (2.5.60)

= dAP + u∗hdϑ (2.5.61)

= 0, since AP is flat and dϑ = 0 for G = U(1) (2.5.62)

If P1 and P2 are bundles over X with equal first Chern classes, c1(P1) = c1(P2) ∈

TorsHm(X,Z), then MP1 ' MP2 are canonically isomorphic. This can be seen by

choosing a bundle isomorphism, φ : P1 → P2. This induces an isomorphism φ∗ : AP2 →

AP1 , which can be pushed down to an isomorphism of the of the quotients, φ∗ : AP2/GP →

AP2/GP . It can be shown that the isomorphism AP2/GP ' AP2/GP is independent of the

choice of bundle isomorphism φ : P1 → P2.

For each torsion class p ∈ TorsH2(X,Z), define

AfX,p =⊔P

c1(P )=p

AfP (2.5.63)

Once we mod out by the equivalence relation on AX , “∼” as defined in Eq. (2.5.38), the

space

MX,p := AfX,p/ ∼ (2.5.64)

is naturally isomorphic toMP for any principal U(1)-bundle P with c1(P ) = p. We then

have that the moduli space of gauge equivalence classes of flat U(1)-connections on X,

MX = AfX/ ∼, is equal to

MX =⊔

p∈TorsH2(X,Z)

MX,p (2.5.65)

We have the following characterization of MX ,

2.5.66 Proposition. [Man98] Let X be a smooth manifold.


1. There is a natural identification

MX = H1(X,U(1)) (2.5.67)

2. π0(MX) ' TorsH2(X,Z) and each connected component of MX is diffeomorphic

to the torus

H1(X,R)/H1(X,Z). (2.5.68)

Chapter 3

Heisenberg Calculus on Contact

Manifolds

In this chapter we provide the background, notation and terminology that is requisite

for working in the Heisenberg calculus. Since the operators that we will be interested

in studying are not elliptic, we cannot apply the usual pseudodifferential calculus of

elliptic theory to our case, and we use instead the Heisenberg calculus. First we review

general Heisenberg manifolds and some standard operators in this case, and then we

briefly develop the Heisenberg calculus and state some results from this theory that are

needed for this thesis.

3.1 Heisenberg Manifolds

In this section we review Heisenberg manifolds and some standard operators in this set-

ting. We mainly follow [BG88] and [Pon08]. We begin with the following

3.1.1 Definition. 1. A Heisenberg manifold (X,H) consists of a manifold X together

with a distinguished hyperplane bundle H ⊂ TX.

43

Chapter 3. Heisenberg Calculus on Contact Manifolds 44

2. A Heisenberg diffeomorphism φ from a Heisenberg manifold (X,H) to another

Heisenberg manifold (X ′, H ′) is a diffeomorphism φ : X → X ′ such that φ∗H = H ′.

3.1.2 Definition. Let (Xd+1, H) be a Heisenberg manifold. Then:

1. A (local) H-frame for TX is a (local) frame Y0, Y1, . . . , Yd so that Y1, . . . , Yd span

H.

2. A local Heisenberg chart is a local chart with a local H-frame of TX over its

domain.

The main examples of Heisenberg manifolds are the following.

3.1.3 Example. • Heisenberg Group: The (2n+ 1)-dimensional Heisenberg group

H(2n+1) is R(2n+1) = R× R2n equipped with the group law,

x.y := (x0 + y0 +n∑j=1

(xn+jyj − xjyn+j), x1 + y1, . . . , x2n + y2n) (3.1.4)

A left-invariant basis for its Lie algebra h2n+1 is then given by the vector fields

Y0 =∂

∂x0

, (3.1.5)

Yj =∂

∂xj+ xn+j

∂

∂x0

, (3.1.6)

Yn+j =∂

∂xn+j

+ xj∂

∂x0

(3.1.7)

for 1 ≤ j ≤ n. This left-invariant basis for h2n+1 satisfies the relations,

[Yj, Yn+k] = −2δjkY0, [Y0, Yj] = [Yj, Yk] = [Yn+j, Yn+k] = 0 (3.1.8)

for 1 ≤ j, k ≤ n and j 6= k. In particular, the subbundle spanned by the vector

fields Y1, Y2, . . . , Y2n, yields a left-invariant Heisenberg structure on H(2n+1).

• Foliations: Recall that a (smooth) foliation is a manifold X together with a sub-

bundle F ⊂ TX which is integrable in the sense of Frobenius, so that [F ,F ] ⊂ F .

Thus, any codimension one foliation is a Heisenberg manifold.


• Contact Manifolds: Opposite to foliations are contact manifolds, (X(2n+1), H)

where H = kerκ for some one-form κ ∈ Ω1(X). In fact, by Darboux’s theorem

any contact manifold is locally contactomorphic to the Heisenberg group H(2n+1)

equipped with its standard contact form κ0 = dx0 +∑n

j=1(xjdxn+j − xn+jdxj).

• CR Manifolds: Recall that a CR-structure on an orientable manifold X2n+1 is

given by a rank n complex subbundle T1,0 ⊂ TCX which is integrable in Frobenius’

sense and such that T1,0 ∩ T0,1 = 0, where T0,1 = T 1,0. Equivalently, the subbundle

R(T1,0⊗T0,1) (where R denotes the real part) has the structure of a complex bundle

of (real) dimension 2n. In particular, (X,H) is a Heisenberg manifold.

An important object of study for us will be the tangent Lie group bundle of a Heisenberg

manifold (X,H). Let

L : H ×H → TX/H,

be defined by

Lx(Yx, Yx) = [Y, Y ](x) modHx,

for sections Y, Y ∈ Γ(H) near a point x ∈ X. L is well defined by [Pon08, Lemma 2.1.3]

and is called the Levi form of (X,H). The Levi form allows us to define the bundle gX

of graded Lie algebras by endowing (TX/H) ⊕ H with a Lie bracket and grading. Let

Y0, Y0 ∈ Γ(TX/H), Y1, Y1 ∈ Γ(H) and t ∈ R. Then

[Y0 + Y1, Y0 + Y1]x = Lx(Y1, Y1), t · (Y0 + Y1) = t2Y0 + tY1.

One can check that gX is a bundle of two-step nilpotent Lie algebras1 which contains

the normal bundle TX/H in its center. The associated Lie group bundle GX is also

(TX/H)⊕H with the identity as exponential map. Since gX is two-step nilpotent, the

Campbell-Baker-Hausdorff formula implies that the group law is given by

exp(Y )exp(Y ) = exp(Y + Y +1

2[Y, Y ]).

1Recall that a Lie algebra L is said to be two-step nilpotent if its commutator ideal [L,L] is nontrivialand contained in the center of L.


For sections Y0, Y0 ∈ Γ(TX/H), Y1, Y1 ∈ Γ(H), we then have explicitly the group law

(Y0 + Y1) · (Y0 + Y1) = Y0 + Y1 + Y0 + Y1 +1

2L(Y1, Y1).

3.1.9 Definition. The bundles gX and GX are called the tangent Lie group bundle and

the tangent Lie group of X, respectively. We let gx := gX|x and Gx := GX|x denote the

fibres of these bundles with their respective structures.

3.1.10 Remark. It is a basic fact that on a contact manifold the point-wise groups Gx

are isomorphic to the Heisenberg group.2

Our main interest is the study of the contact Laplacian of [RS08]. It is interesting,

however, to note some examples of other differential operators on Heisenberg manifolds.

3.1.11 Example. • Hormander’s sum of squares on a Heisenberg manifold (M,H) of

the form

∆ := ∇∗Y1∇Y1 + · · ·+∇∗Ym∇Ym , (3.1.12)

where the (real) vectors fields Y1, . . . , Ym span H and ∇ is a connection on a vector

bundle E over X and the adjoint is taken with respect to a smooth positive measure

on X and a Hermitian metric on E .

• The Kohn Laplacian 2b;p,q acting on (p, q)-forms on a CR manifold X2n+1 endowed

with a CR compatible Hermitian metric (not necessarily a Levi metric).

• The horizontal sublaplacian ∆b;k acting on horizontal differential forms of degree

k on a Heisenberg manifold (X,H). When X2n+1 is a CR manifold the horizontal

sublaplacian preserves the bidegree and we can consider its restriction ∆b;p,q to

forms of bidegree (p, q).

• The contact Laplacian on a contact manifold X2n+1 associated to the contact com-

plex of [RS08]. There are actually two different working definitions of this Lapla-

cian. One definition can be found in [Rum94], where the contact Laplacian is

2See [Erp10].


defined as a differential operator of order two in degree k 6= n, n + 1 and of order

four in degree n, n+ 1. Recall, [Rum94, pg. 290 Theorem],

∆q =

2d∗HdH if q = 0,

−dHd∗H if q = 3,

D∗D + (dHd∗H)2 if q = 1.

DD∗ + (d∗HdH)2 if q = 2.

(3.1.13)

The definition that we will use is as defined in [RS08] where the contact Laplacian

is a uniformly fourth order operator in all degrees. Recall, [RS08, Eq. 10],

∆q =

(d∗HdH + dHd

∗H)2 if q = 0, 3,

D∗D + (dHd∗H)2 if q = 1.

DD∗ + (d∗HdH)2 if q = 2.

(3.1.14)


3.1.15 Definition. [HN86] Let X be a compact manifold. A differential operator P =∑|α|≤d aα(x)Y α of order d on X is called maximal hypoelliptic at x ∈ X if there exists a

neighbourhood U of x and a constant C ∈ R>0 such that for every distribution u on U ,

∑|α|≤d

||Y αu||L2 ≤ C(||u||L2 + ||Pu||L2).

Recall that maximal hypoellipticity of P implies that P is hypoelliptic, i.e.

Pu smooth⇒ u smooth.

It turns out that the operators above are known to be hypoelliptic in certain cases. For

example, Hormanders sum of squares is hypoelliptic provided that the following bracket

condition is satisfied3:

[Yj1 , [Yj2 , . . . , Yj1 ] . . .],

3See [Hor67].


spans the tangent bundle TX at every point. Kohn, [Koh65], proved that under the

“Y (q)” condition that the Kohn Laplacian 2b;p,q is hypoelliptic. The most important

example for us is the contact Laplacian of Eq. (3.1.14), which Rumin showed in [Rum94]

is maximal hypoelliptic.

3.2 Heisenberg Calculus and the Rockland Condi-

tion

In this section we provide a general overview of the Heisenberg calculus. This theory

was independently introduced by Beals-Greiner [BG88] and Taylor [Tay84]. Our aim is

to develop the basic definitions, terminology, results and ideas that we will need for this

thesis. We primarily follow [BG88], [Pon07], [Pon08], [Erp10] and we refer the reader to

[Pon08] for a much more in depth discussion of the details in the more general case of

Heisenberg manifolds. We will work with differential operators P : C∞(X) → C∞(X)

acting on functions, and defer to [Pon08] the general case of psedodifferential operators,

ΨHDO’s. Also note that we are ultimately interested in contact manifolds (X, κ) in this

thesis and throughout this section the notation gx and Gx should be thought of as the

Heisenberg Lie algebra and Heisenberg group, respectively (See Remark 3.1.10).

The main idea of subelliptic theory on contact manifolds is that vector fields transversal

to the contact hyperplane bundle H are treated as second order operators. The Heisen-

berg calculus is derived quite naturally from this basic idea. This ordering of vector

fields generates a filtration on the space of differential operators P on X. One then as-

sociates a graded algebra D to the filtered algebra P and defines the principal symbol of

an operator P ∈ P as the image of its highest order part in D. The first main deviation

from elliptic theory is the identification of the principal part of the operator P at a point

x ∈ X as an element in a noncommutative algebra Ux. U := Ux |x ∈ X is a bundle


of graded algebras over X, and Ux is realized as the universal enveloping algebra of the

graded nilpotent Lie algebra gx (See Def. 3.1.9). This leads to the idea of replacing the

notion of constant coefficient operators in elliptic theory with Rockland operators and is

why the invertibility of an operator in the Heisenberg calculus involves the representation

theoretic Rockland condition (See Def. 3.2.1). This, for us, summarizes the basic ideas

of the Heisenberg calculus.

First, let us consider the space of differential operators

P = span∏j

Yj |Yj ∈ Γ(TX).

It is natural to define a filtration on P by defining ord(Y ) = 1 for any Y ∈ Γ(H) ⊂ Γ(TX)

(where H ⊂ TX denotes the contact distribution), and ord(Y ) = 2 for any Y /∈ Γ(H).

Then we define,

Pd := span∏j

Yj |∑j

ord(Yj) ≤ d ⊂ P ,

where P0 := C∞(X). We define the (Volterra-Heisenberg) order of an operator P ∈ P

in the Heisenberg calculus to be the integer d such that P ∈ Pd\Pd−1. Recall that the

algebra of symbols for the Heisenberg calculus is the graded algebra D associated to the

filtered algebra P defined in the usual way

D :=⊕Dd, where Dd := Pd/Pd−1.

The principal part of a degree d operator P : C∞(X)→ C∞(X) is the image of P under

the degree d quotient mapping

σdH : Pd → Dd.

As noted above, the principal symbol of an operator P at x ∈ X, σdH(P )(x), is viewed

as an element of an algebra Ux. The algebra Ux is generated by operators in P and

consists of the linear span of their principal symbols at x ∈ X, σdH(P )(x). Thus, Ux =

spanσdH(P )(x) ∈ Ddx | d ∈ N, P ∈ P with the following graded algebra structure. Let


Ax, Bx ∈ Ux be monomials, so that Ax = σdH(P )(x) and Bx = σqH(Q)(x) for some

P,Q ∈ P . Then we define Ax ·Bx ∈ D(d+q)x ⊂ Ux by:

Ax ·Bx := σ(d+q)H (PQ)(x).

It is easy to verify that this is a well defined graded algebra structure on Ux.4 Now ob-

serve that we may naturally view gx ⊂ Ux (See Def. 3.1.9). The identification comes from

observing that the vector fields Γ(TX) ⊂ P are a subset of P and gx = (TX/H)x ⊕Hx

with the Lie algebra structure defined in §3.1. Since the vector fields Γ(TX) generate P ,

gx generates Ux as an algebra. By the Poincare-Birkoff-Witt theorem one can also see

that Ux is naturally isomorphic to the universal enveloping algebra of gx.

The last thing we review in this section is the Rockland condition for differential op-

erators P ∈ P . Our main reason for reviewing this here is that the Rockland condition is

the crucial condition that the contact Laplacian ∆q of Eq. (3.1.14) must satisfy in order

to carry out our analysis in Chapter 6. First we recall that a unitary representation π of

a Lie group G on a Hilbert space Hπ induces a representation dπ of the Lie algebra g on

the space of smooth vectors C∞π ,5 defined by:

dπ(Y )ν =d

dt

∣∣∣t=0π(exp(tY ))ν,

for Y ∈ g and ν ∈ C∞π . We may now make the following, originally due to Rockland

[Roc78]:

3.2.1 Definition. [Pon08, Def. 3.3.8] We say that an operator P ∈ P satisfies the

Rockland condition at x ∈ X if for a nontrivial unitary irreducible representation π of

GxX the operator dπ(P ) is injective on C∞π .

4See [Erp10].5Recall that the space of smooth vectors C∞π ⊂ Hπ is the set of vectors ν ∈ Hπ such that g 7→ π(g)ν

is smooth from G to Hπ.


3.3 Some Results in the Heisenberg Calculus

In this section we list several key results that are needed throughout this thesis. These

results will be applied to the hypoelliptic contact Laplacian, ∆q, defined in Eq. (3.1.14).

In order to be as self contained as possible we have developed most of the requisite

notation and terminology for this section in §3.1 and §3.2. Our first result is the following

3.3.1 Proposition. [Pon08, Chapter 5] Let V be a vector bundle over a compact contact

manifold (X, κ) of dimension 2n + 1. Let P : C∞(X,V) → C∞(X,V) be a differential

operator of even Volterra-Heisenberg order v that is self-adjoint and bounded from below.

If P satisfies the Rockland condition at every point then the principal symbol of P + ∂t is

an invertible Heisenberg symbol and as t 0 the heat kernel kt(x, x) of P on the diagonal

has the following asymptotics in C∞(X, (EndV)⊗ |Λ|(X)):

kt(x, x) ∼∞∑j=0

t2(j−n−1)

v aj(P )(x).

The next proposition is crucial to our construction in Chapter 6 and allows us to

define regularized determinants rigorously using zeta functions.

3.3.2 Proposition. [Pon07, §4] Let P be as in Prop. 3.3.1. Then the zeta function

ζ(P )(s) = dim kerP + Tr∗(P−s), s ∈ C,

is a well defined holomorphic function for Re(s) 1 and admits a meromorphic extension

to C with at worst simple poles occurring at s ∈ 2(n+1−j)v

| j ∈ N\(−N). Moreover

ζ(P )(0) =

∫X

tr(an+1(P ))κ ∧ (dκ)n

is the constant term in the development of Tr(e−tP ) as t 0.

Our main observation is that Prop.’s 3.3.1 and 3.3.2 apply to the contact Laplacian

∆q. That is, ∆q satifies the Rockland condition. As is noted in [RS08], this is observed


in [Rum94, pg. 300] and also shown in [JK95, §5]. Let

ζ0(s) := dim Ker∆0 + ζ0(s)

ζ1(s) := dim Ker∆1 + ζ1(s)

denote the zeta functions as defined in [RS08, §3]. For the next proposition we will need

to make the following

3.3.3 Definition. Let (X, κ) be a contact three-manifold and ∆q the contact Laplacian

(See Eq. (3.1.14)).6 Define the contact torsion function KX : C→ C as follows:

KX(s) :=1∑q=0

(−1)q(2− q)ζq(s)

= 2ζ0(s)− ζ1(s).

We then have the following

3.3.4 Proposition. [RS08, Cor. 3.8 (1)] Let (X, κ) be a three-dimensional contact

manifold and let KX denote the contact torsion function as in Def. 3.3.3. Then KX(0) =

0.

3.3.5 Remark. It is still not known whether or not KX(0) vanishes in all dimensions,

[RS08, pg. 17].

The next theorem is important in our definition of a regularized determinant of the

operator −k ? D : Ω1(H)→ Ω1(H) as in Eq. (6.7.3).

3.3.6 Theorem. [BHR07, Theorem 8.8] Let X be a CR-Seifert manifold,7 ∆H :=

dHd∗H + d∗HdH the horizontal Laplacian, R ∈ C∞(X) the Tanaka-Webster scalar cur-

vature of X and η0(X, κ)8 the renormalized eta invariant. Then,

η0(X, κ) = η(?D)(0) + ζ(∆H)(0)

6Note that this definition makes sense for all contact manifolds, but we will not need this definitionhere.

7See Def. 6.2.1. Equivalently, see §2.3 for a definition of a quasi-regular K-contact manifold; suchstructures are equivalent to CR-Seifert structures.

8See Def. 6.10.7.


with

ζ(∆H)(0) =1

512

∫X

R2 κ ∧ dκ.

Finally, the last result that we need is an explicit identification of the spaces of

harmonic and contact harmonic forms.

3.3.7 Proposition. [Rum94, Prop. 12] Let (E , dH) be the complex defined in Def. 6.4.4,

∆q, 0 ≤ q ≤ 3 denote the contact Laplacian and ∆DRq = d∗d + dd∗ denote the de Rham

Laplacian. If

Hq(E , dH) := α ∈ Eq |∆qα = 0,

and,

Hq(X, d) := α ∈ Ωq(X) |∆DRq α = 0,

denote the contact and de Rham harmonic forms, respectively, then for q ≤ 1,

Hq(E , dH) = Hq(X, d).

Chapter 4

U(1) Chern-Simons Theory

4.1 Induced Principal Bundles and Induced Connec-

tions

In this section we will show how to construct a canonical SU(2)-bundle P over a manifold

X given a U(1)-bundle P over X. This will allow us to define the Chern-Simons action in

the U(1) case, even when our principal U(1)-bundle P is non-trivial. This construction

is crucial to our analysis and is due to [Man98], which we follow here.

We start by viewing U(1) as an embedded maximal torus in SU(2) via a specific inclusion

homomorphism,

ρ : U(1) → SU(2) (4.1.1)

e2πiϕ 7→

e2πiϕ 0

0 e−2πiϕ

(4.1.2)

The induced map on the Lie algebra is,

ρ∗ : u(1) → su(2) (4.1.3)

α 7→

α 0

0 −α

(4.1.4)

54

Chapter 4. U(1) Chern-Simons Theory 55

where u(1) := Lie(U(1)) ' R, and su(2) := Lie(SU(2)). Let ϑ denote the Maurer-Cartan

form on SU(2). Choose a non-degenerate, Ad-invariant bilinear form on su(2),1

Tr : su(2)× su(2) → R (4.1.5)

(a, b) 7→ 1

8πtr(ab) (4.1.6)

The bilinear form Tr is normalized so that the closed three-form 16Tr(ϑ∧[ϑ, ϑ]) represents

an integral cohomology class in H3(SU(2),R). The form Tr on su(2) then restricts to a

symmetric bilinear form on u(1),

〈·, ·〉 : u(1)× u(1) → R (4.1.7)

(α, β) 7→ 〈α, β〉 = Tr(ρ∗α, ρ∗β) =1

4παβ (4.1.8)

As before, let ϑ denote the Maurer-Cartan form on U(1) and observe that ρ∗(ϑ) = ρ∗(ϑ).

Given a principal U(1)-bundle over X we may now use the inclusion map ρ : U(1) →

SU(2) to construct an SU(2)-bundle P on X. We define the induced SU(2)-bundle P

as follows, [Man98, pg. 14]:

Define a right action of U(1) on P × SU(2) by

(p, g) · λ = (p · λ, (ρ(λ))−1g) (4.1.9)

where λ ∈ U(1) and (p, g) ∈ P × SU(2). The natural right SU(2) action on P × SU(2)

given by (p, g) · h = (p, gh), commutes with the U(1) action and therefore passes to a

free action on the quotient P = P ×U(1) SU(2). Thus,

π : P → X (4.1.10)

with projection map π([p, g]) = π(p) is a principal SU(2)-bundle. Define

ιP : P → P (4.1.11)

p 7→ [p, e] (4.1.12)

1This is standard, and follows from the fact that su(2) is a simple Lie algebra.


which is a natural morphism of bundles covering the identity map on X.

Given a connection A ∈ AP on the U(1)-bundle P , there is an induced connection A ∈ AP

on the SU(2)-bundle P . Define, pr1 : P × SU(2) → P and pr2 : P × SU(2) → SU(2)

as the natural projections. A is obtained from A′, which is the following su(2)-valued

one-form on P × SU(2), [Man98, Eq. 3.1],

A′(p,g) := Adg−1(ρ∗pr∗1Ap) + pr∗2(ϑg), (4.1.13)

by pushing A′ down to an su(2)-valued one-form on P = P ×U(1) SU(2). It is not hard

to see that A′ is invariant under the U(1)-action on P × SU(2). We have the following

relationship between A and A [Man98, pg. 14],

ι∗P A = ρ∗A. (4.1.14)

We also have the following relationship between the curvature forms FA = dA and FA =

dA+ 12[A, A], [Man98, pg. 14],

ι∗PFA = ρ∗FA. (4.1.15)

In the next section we will see how to define the Chern-Simons action naturally using

this construction.

4.2 The U(1) Chern-Simons Action

Let X be a closed oriented 3-manifold. Given a principal U(1)-bundle P over X, we

have previously constructed an induced SU(2)-bundle P = P ×U(1)SU(2)→ X as in Eq.

(4.1.10). Also, for any U(1)-connection A ∈ AP we defined in Eq. (4.1.13) an induced

SU(2)-connection A on P . Since for any 3-manifold X, P is trivializable, let s : X → P

be a global section. In this section we define the U(1) Chern-Simons action and study

some of its properties. We primarily follow [Man98] here.


4.2.1 Definition. [Man98, Eq. 3.2] The Chern-Simons action functional of a U(1)-

connection A ∈ AP is defined by:

SX,P (A) =

∫X

s∗α(A) (mod Z) (4.2.2)

where α(A) ∈ Ω3(P ,R) is the Chern-Simons form of the induced SU(2)-connection

A ∈ AP ,

α(A) = Tr(A ∧ FA)− 1

6Tr(A ∧ [A, A]) (4.2.3)

We claim that the definition of the action SX,P does not depend on the choice of section

s : X → P . To see this, we first establish the following property of the SU(2) Chern-

Simons form α(A):

4.2.4 Proposition. [Man98, Prop. 3.3] If s1, s2 : X → P are two global sections of the

SU(2)-bundle P over X, then∫X

s1∗α(A) =

∫X

s2∗α(A) +

∫X

dTr(Ada−1 s2∗A ∧ a∗ϑ)− 1

6

∫X

a∗Tr(ϑ ∧ [ϑ, ϑ]), (4.2.5)

where a : X → SU(2) is the map defined by s2(x) = s1 · a(x), for any x ∈ X.

One can prove this after straightforward manipulations using the standard relation

s∗2A = Ada−1(s1A) + a∗ϑ. Using this result we can prove:

4.2.6 Proposition. [Man98] The definition of the action SX,P does not depend on the

choice of section s : X → P .

Proof. Looking at Eq. (4.2.5), one can see in the case ∂X = ∅ that the second integral

vanishes by Stokes’ theorem because the form is exact. The last integral in Eq. (4.2.5) is

an integer due to the normalization of the bilinear form Tr on su(2). Thus, for any two

sections s1, s2 : X → P , ∫X

s∗1α(A) =

∫X

s∗2α(A) (mod Z) (4.2.7)

and the Chern-Simons action SX,P is well defined.


4.2.8 Remark. [Man98, Remark 3.4] In the case that the U(1)-bundle P over X is trivi-

alizable, we have the expected result:

SX,P (A) =1

4π2

∫X

s∗(A ∧ FA) (mod Z) (4.2.9)

=1

4π2

∫X

s∗(A ∧ dA) (mod Z) (4.2.10)

where s : X → P is a section of the U(1)-bundle P and s : X → P is taken to be

s = ιP s. ιP was defined in Eq. (4.1.11). One can see this with a straightforward

calculation,

s∗α(A) = s∗ι∗Pα(A)

= s∗Tr(ι∗P A ∧ ι∗PFA)− 1

6s∗Tr(ι∗P A ∧ [ι∗P A, ι

∗P A])

= s∗Tr(ρ∗A ∧ ρ∗FA), by eq.’s (4.1.14) and (4.1.15)

=1

4π2s∗(A ∧ FA), by Eq. (4.1.8).

We now collect several properties of the Chern-Simons action SX,P .

4.2.11 Theorem. [Man98, Theorem 3.6] The Chern-Simons functional SX,P : AP →

R/Z defined for a closed oriented 3-manifold X and a principal U(1)-bundle P over X

has the following properties:

1. Functorality

If φ : P1 → P2 is a morphism of principal U(1)-bundles covering an orientation

preserving diffeomorphism φ : X1 → X2 and if A ∈ AP is a connection on P , then

SX1,P1(φ∗A) = SX2,P2(A) (4.2.12)

2. Orientation

If −X denotes the manifold X with the opposite orientation, then

S−X,P (A) = −SX,P (A) (4.2.13)


3. Disjoint Union

Let X = X1tX2 be a disjoint union and P = P1tP2 a principal U(1)-bundle over

X. If Ai ∈ APi , i = 1, 2 are connections, then

SX1tX2,P1tP2(A1 t A2) = SX1,P1(A1) + SX2,P2(A2) (4.2.14)

Proof. 1. Let P1 and P2 be the induced principal SU(2)-bundles coming from P1 and

P2 respectively, as in Eq. (4.1.10). The morphism of U(1) bundles φ : P1 → P2

induces a morphism of SU(2)-bundles φ : P1 → P2 covering φ : X1 → X2. It is

defined by φ([p1, a]) = [φ(p1), a]. Given a section s1 : X1 → P1 we have a section

s2 = φ s1 φ−1 : X2 → P2. We then have

SX2,P2(A) =

∫X2

s∗2α(A) (mod Z) (4.2.15)

=

∫X2

(φ−1)∗s∗1φ∗α(A) (mod Z) (4.2.16)

=

∫X1

s∗1α(φ∗A) (mod Z) (4.2.17)

= SX1,P1(φ∗A), by definition 4.2.1 (4.2.18)

The last equality follows from the fact that φ∗A is the induced SU(2)-connection

on P2 coming from the U(1)-connection φ∗A on P2:

φ∗A(p,g) = Adg−1ρ∗(φ∗pr∗1Ap) + φ∗pr∗2ϑg (4.2.19)

= Adg−1ρ∗(pr∗1φ∗Ap) + pr∗2ϑg (4.2.20)

2. This follows from the fact that integration over a manifold with a chosen orientation

is equivalent to minus the integral over the opposite orientation.

3. This is also an easy consequence of a property of integration. That is, an integral

over a disjoint union is the sum of the integrals over each component in the disjoint

union. In our case, A = A1 t A2 is the extension of A = A1 t A2 on P = P1 t P2


to an SU(2)-connection on P = P1 t P2. Let si : Xi → Pi and s = s1 t s2 : X =

X1 tX2 → P . Then,

SX,P (A) =

∫X1tX2

s∗α(A) (mod Z) (4.2.21)

=

∫X1

s∗1α(A1) +

∫X2

s∗2α(A2) (mod Z) (4.2.22)

= SX1,P1(A1) + SX2,P2(A2) (4.2.23)

As a consequence of part 1) of Theorem 4.2.11, we can see that the Chern-Simons action

is invariant under the action of the gauge group GP on AP ; where the action is defined

in Proposition 2.5.18. We reiterate this in the following

4.2.24 Proposition. [Man98, Prop 3.7] The Chern-Simons functional SX,P for a closed

three-manifold is invariant under the action of the group of gauge transformations,

SX,P (A · h) = SX,P (A) (4.2.25)

for all h ∈ GX . Hence SX,P is a well defined functional on the quotient space AP/GP .


4.2.26 Proposition. [Man98, Lemma 3.18] Let X be a closed, oriented three-manifold

and P a U(1)-bundle over X. If A1 and A2 are connections on P , then

SX,P (A2)− SX,P (A1) =1

2π2

∫X

s∗(FA1 ∧ ω) +1

4π2

∫X

s∗(ω ∧ dω) (mod Z) (4.2.27)

where ω = A2 − A1.

Proof. Let A1 and A2 be the induced connections on P as usual. It follows from the

definition of the induced connection that the su(2)-valued 1-form ω = A2 − A1 on P is

related to ω by

ω[p,g] = Adg−1(ρ∗pr∗1ωp) (4.2.28)


and ι∗P ω = ρ∗ω. It is straightforward to derive the following relation:

α(A2)−α(A1) = −dTr(A1∧ ω)+2Tr(FA1∧ ω)+Tr(ω∧ (dω+[A1, ω]))+

1

3Tr(ω∧ [ω, ω])

(4.2.29)

It is not hard to see from relation 4.2.28 that

Tr(ω ∧ [ω, ω]) = 0 (4.2.30)

and,

Tr(ω ∧ [A1, ω]) = Tr([ω, ω] ∧ A1) = 0 (4.2.31)

where Eq. (4.2.31) follows from the associativity of the Killing form. Recall that since

su(2) is simple, the inner product Tr is a non-zero scalar multiple of the Killing form.

We then have,

α(A2)− α(A1) = −dTr(A1 ∧ ω) + 2Tr(FA1∧ ω) + Tr(ω ∧ dω) (4.2.32)

Thus, we may write

SX,P (A2)− SX,P (A1) =

∫X

s∗(α(A2)− α(A1))

=

∫X

s∗[−dTr(A1 ∧ ω) + 2Tr(FA1

∧ ω) + Tr(ω ∧ dω)]

=

∫X

−s∗dTr(A1 ∧ ω) +

∫X

2s∗Tr(FA1∧ ω) (4.2.33)

+

∫X

s∗Tr(ω ∧ dω)

=

∫X

2s∗Tr(FA1∧ ω) +

∫X

s∗Tr(ω ∧ dω)

=1

2π2

∫X

s∗(FA1 ∧ ω) +1

4π2

∫X

s∗(ω ∧ dω) (mod Z)

where the second last equality comes from Stokes’ theorem and the fact that the first

quantity in Eq. (4.2.33) is exact. The last equation follows from the fact that Tr is

Ad-invariant, s := pr1 s : X → P defines a global section of P and by the definition of

the symmetric bilinear form 〈·, ·〉 on u(1) defined in Eq. (4.1.8).


We further have the following

4.2.34 Proposition. [Man98, Prop. 3.8] The stationary points of the Chern-Simons

functional SX,P : AP → R/Z are the flat connections. In other words, δSX,P (A) = 0 if

and only if FA = dA = 0.

Proof. We consider the variation of SX,P along lines At = A + tω, for t ∈ R and ω ∈

Ω1(X,R). This is sufficient because AP is an affine space. Observe, by Proposition 4.2.26,

we have

SX,P (At) = SX,P (A) +t

2π2

∫X

s∗(FA1 ∧ ω) +t2

4π2

∫X

s∗(ω ∧ dω) (mod Z) (4.2.35)

and we may calculate the variation as

δSX,P (A) =d

dt

∣∣∣t=0SX,P (At) =

1

2π2

∫X

s∗(FA ∧ ω), (4.2.36)

which proves the assertion.

4.3 The U(1) Partition Function

In this section we recall the definition of the partition function, ZU(1)(X, k), for U(1)

Chern-Simons theory. This partition function is defined and studied in both [MPR93]

and [Man98], for example. We follow §7 of [Man98] and “derive”2 a rigorous definition

of the partition function from a heuristically defined path integral (see eq.’s (4.3.2) and

(4.3.3) below). We modify the rigorous definition of the U(1) Chern-Simons partition

function [Man98, Eq. 7.28] to take into account the dependence that the partition func-

tion has on a choice of metric g on X. We follow [Wit89] and revise the definition

of [Man98] by adding a “counterterm,” the gravitational Chern-Simons term (see Eq.

2Note that it would be interesting to make an explicit study of the choices that one needs to makein order to rigorously define the partition function using the methods of this thesis. For example, ourregularization of the eta-invariant is seemingly not unique. Of course, a rigorous definition of the pathintegral measure would probably answer this question.


(4.3.14)), to the eta-invariant (see Eq. (4.3.13)) that shows up in this calculation. By the

Atiyah-Patodi-Singer theorem [APS75a], [APS75b], [APS76], this counterterm effectively

restores topological invariance for the partition function. We note that the existence of

such a counterterm is expected from a physics perspective. That is, the U(1) Chern-

Simons partition function should be a topological invariant a priori, since this theory is

“generally covariant.”3 In order to actually compute the partition function one needs to

make a gauge choice, which in our case is tantamount to a choice of metric g on X. In

physics terminology, this choice introduces a quantum anomaly that is made manifest in

the dependence of the eta-invariant (Eq. (4.3.13)) on the choice of metric g. Introducing

the counterterm (Eq. (4.3.14)) effectively cancels this anomaly.

Before we define the partition function, we recall some notation and terminology. For

each p ∈ TorsH2(X; Z),4 we choose a U(1)-bundle P on X with c1(P ) = p. AP 5 is

the affine space of connections on P modeled on the vector space Ω1(X; R). GP is the

group of gauge transformations and acts on AP in the standard way.6 SX,P (A) is the

Chern-Simons functional of a U(1)-connection A on P → X and is defined in Eq. (4.2.1).

We then heuristically define the partition function as follows.

4.3.1 Definition. [Man98, Eq.’s 7.13, 7.14] Let k ∈ Z be an (even) integer, and X a

closed, oriented three-manifold. The U(1) Chern-Simons partition function, ZU(1)(X, k),

3See the introduction to [Wit89] for a brief discussion of this.4Recall that an element in a Z-module is said to be torsion if some integral multiple of it is zero.

TorsH2(X; Z) denotes the torsion subgroup of H2(X; Z). We take the sum over only torsion classes inEq. (4.3.2) because these are the bundle classes that contain flat connections, by Prop. 2.5.47. We areinterested in summing over bundle classes with flat connections because we expect that these are theonly classes that will contribute non-trivially to the partition function. This is because the integrandof Eq. (4.3.3) is oscillatory, and we expect to get contributions only from the critical points (i.e. flatconnections) of the action via stationary phase.

5See Eq. (2.5.10) and §2.5 for more background.6See Prop. 2.5.18 and §2.5 for a general discussion.


is the heuristic quantity

ZU(1)(X, k) =∑

p∈TorsH2(X;Z)

ZU(1)(X, p, k) (4.3.2)

where,

ZU(1)(X, p, k) =1

Vol(GP )

∫APDAeπikSX,P (A). (4.3.3)

Note that Eq. (4.3.3) is a formal expression, where we heuristically assume the

existence of the measure DA. It is precisely the quantity DAVol(GP )

in Eq. (4.3.3) that is

not well defined. One goal of future work is to find a rigorous definition of this quantity.

For now we follow a different approach to make definition 4.3.1 rigorous. What follows

is mainly a summary of §7 of [Man98], and one should refer to this for more details. To

start, we write

ZU(1)(X, p, k) =1

Vol(GP )

∫APDAeπikSX,P (A) (4.3.4)

=1

VolU(1)

∫AP /GP

DAeπikSX,P (A)[det′(τ ∗AτA)]1/2 (4.3.5)

where the measure DA is formally induced from a choice of metric g on X. That is, g

defines the Hodge-? on the tangent space TAAP ' Ω1(X; R), which in turn induces the

(infinite dimensional) GP -invariant Riemannian metric

〈α, β〉 :=

∫X

α ∧ ?β, (4.3.6)

on AP , for α, β ∈ TAAP ' Ω1(X; R). Clearly, this metric is invariant under the action

of the gauge group GP on AP . Thus, this metric descends to a metric on the quotient

AP/GP , and thereby induces a quotient measure that we denote by DA in Eq. (4.3.5).

The factor of VolU(1) in Eq. (4.3.5) is identified as the volume of the isotropy subgroup

of GP at any A ∈ AP and is given by, [Man98, Eq. 7.16]:

VolU(1) = [VolX]1/2 =

[∫X

?1

]1/2

. (4.3.7)


Eq. (4.3.7) follows from the definition of the invariant metric on the group GP that is

induced by the inner product on LieGP ' Ω0(X; R) that comes from g:

G(θ, φ) = 〈θ, φ〉 :=

∫X

θ ∧ ?φ, (4.3.8)

for θ, φ ∈ LieGP ' Ω0(X; R). The isotropy subgroup of GP at A ∈ AP is the group of

constant maps from X into U(1) since

θ ∈ LieGP : A 7→ A+ dθ; (4.3.9)

I.e. dθ = 0⇒ θ = constant. Observe that G restricted to the space of constant functions

is simply a scalar at each Ψ ∈ GP ' Maps(X,U(1)):

GΨ(θ, φ) =

∫X

θ ∧ ?φ

= θφ

(∫X

?1

),

since θ, φ ∈ R are constant. We may therefore write GΨ =∫X?1. If

√GΨDΨ denotes

the measure on U(1) < GP , then

Vol(U(1)) =

∫U(1)

√GΨDΨ

=√GΨ, setting

∫U(1)

DΨ = 1,

=

[∫X

?1

]1/2

.

This proves Eq. (4.3.7). τA denotes the differential of the map from GP to AP that

defines the GP -action. The map τA sends LieGP ' Ω0(X; R) to TAAP ' Ω1(X; R), and

can be identified with the differential τA = d : Ω0(X; R)→ Ω1(X; R).

Observe that SX,P is basically a quadratic functional, i.e.

ZU(1)(X, p, k) =eπikSX,P (AP )

Vol(GP )

∫AP

DA exp

[ik

4π

(∫X

A ∧ dA)]

, (4.3.10)

=eπikSX,P (AP )

Vol(GP )

∫AP

DA exp

[ik

4π〈A, ?dA〉

], (4.3.11)

(4.3.12)


where we use Prop. 4.2.26 to rewrite this partition function after identifying AP =

AP + Ω1(X) for a flat base point AP in AP (so that FAP = 0). The inner product 〈·, ·〉

is as defined in Eq. (4.3.6) with respect to a choice of Riemannian metric g on X. Thus,

we use the stationary phase method7 to obtain an exact result, [Man98, Eq. 7.17′′]:


VolU(1)

∫MP

eπi4

sgn(?d) [det′(d∗d)]1/2

[det′(k ? d)]1/2ν,

where ν is the measure induced on the moduli space of flat connections on P , MP (see

Eq. (2.5.57)), and AP ∈ AP is a flat connection on P . This last expression has rigorous

mathematical meaning if the determinants and signatures of the operators therein are

regularized. The signature of the operator ?d on Ω1(X; R) is regularized via the eta-

invariant, so that sgn(?d) = η(?d) + 13

CS(Ag)2π

, where

η(?d) = lims→0

∑λj 6=0

signλj|λj|−s, (4.3.13)

and λj are the eigenvalues of ?d, and

CS(Ag) =1

4π

∫X

Tr(Ag ∧ dAg +2

3Ag ∧ Ag ∧ Ag) (4.3.14)

is the gravitational Chern-Simons term, with Ag the Levi-Civita connection on the spin

bundle of X. The notation det′ refers to regularized determinants and are regularized as

in Remark 7.6 of [Man98].

It is shown in [Man98] that the term inside of the integral

1

VolU(1)

[det′(d∗d)]1/2

[det′(k ? d)]1/2

may be identified, up to a factor depending on k, with the Ray-Singer analytic torsion

of the three-manifold X, T dRS [Man98, Eq. 7.25]. Recall that for a closed three-manifold

X with a given Riemannian metric g, the Ray-Singer analytic torsion can be viewed as

a density on X. Recall the Hodge-de Rham Laplacian:

∆q := d∗d+ dd∗, on Ωq(X; R). (4.3.15)

7See Appendix A.1 for a review of the method of stationary phase.


The analytic Ray-Singer torsion, [RS73],

TRS := exp

(1

2

3∑q=0

(−1)qqζ ′(∆q)(0)

), (4.3.16)

in the case where dim(X) = 3, where ζ denotes the zeta function of ∆q:

ζ(∆q)(s) :=∑

λ∈spec∗(∆q)

λ−s. (4.3.17)

We would like to view TRS as a density T dRS. Consider:

| detH•(Ω(X), d)∗| :=3⊗j=0

(detHj(Ω(X), d))(−1)j (4.3.18)

and,

|| · ||RS := TRS|| · ||L2(Ω(X),d), (4.3.19)

where the L2 norm comes from the identification:

H•(Ω(X), d) ' H•(Ω(X), d). (4.3.20)

Then if δ| detH•((Ω(X),d))∗| = density for || · ||L2(Ω(X),d) on | detH•((Ω(X), d))∗|, then

T dRS := TRS · δ| detH•((Ω(X),d))∗|. (4.3.21)

We then obtain a rigorous definition of the partition function, which is to be contrasted

with [Man98, Eq. 7.27].

4.3.22 Definition. Let k ∈ Z be an (even) integer, and X a closed, oriented three-

manifold. The U(1) Chern-Simons partition function, ZU(1)(X, k), is the rigorous quan-

tity

ZU(1)(X, k) =∑

p∈TorsH2(X;Z)

ZU(1)(X, p, k) (4.3.23)

where,


4+ 1

12CS(Ag)

2π

) ∫MP

(T dRS)1/2 (4.3.24)

where mX = 12(dimH1(X; R)− dimH0(X; R)).


The Atiyah-Patodi-Singer theorem [APS75b] says that the combination

η(?d)

4+

1

12

CS(Ag)

2π(4.3.25)

is a topological invariant depending only on a 2-framing of X. Recall,8 that a 2-framing

is choice of a homotopy equivalence class π of trivializations of TX ⊕ TX, twice the

tangent bundle of X viewed as a Spin(6) bundle. The possible 2-framings correspond to

Z. The identification with Z is given by the signature defect defined by

δ(X, π) = sign(M)− 1

6p1(2TM, π)

where M is a 4-manifold with boundary X and p1(2TM, π) is the relative Pontrjagin

number associated to the framing π of the bundle TX ⊕ TX. The canonical 2-framing

πc corresponds to δ(X, πc) = 0. Either we can choose the canonical framing, and work

with this throughout, or we can observe that if the framing of X is twisted by s units,

then CS(Ag) transforms by

CS(Ag)→ CS(Ag) + 2πs.

The partition function ZU(1)(X, k) is then transformed by

ZU(1)(X, k)→ ZU(1)(X, k) · exp

(2πis

24

). (4.3.26)

Then ZU(1)(X, k) is a topological invariant of framed, oriented three-manifolds, with

a transformation law under change of framing. This is tantamount to a topological

invariant of oriented three-manifolds without a choice of framing.

4.4 Shift Symmetry and the U(1) Partition Function

Our goal in this section is to follow [BW05, §3.1] and obtain a heuristic “shift invariant”

expression for the U(1) Chern-Simons partition function (See Def. 4.4.21 below) by de-

coupling one of the three components of the gauge field A ∈ AP using a shift symmetry

8See [Ati90b].


S. We note that the constructions in this section are largely heuristic and should be

viewed as an initial step in obtaining a rigorous definition for a shift invariant expression

of the U(1) Chern-Simons partition function. For a finite dimensional analogue of the

shift symmetry see Appendix B.1.

First we define S by its variation δS on a field A ∈ AP . Define:

δSA = σκ

where σ ∈ (Ω0(P ) ⊗ u(1))U(1) ' Ω0(X) is an arbitrary form and κ ∈ Ω1(X) is a fixed

contact form on X. Clearly, the Chern-Simons action, SX,P (A),9 does not respect the

shift symmetry. That is,

δSSX,P (A) 6= 0. (4.4.1)

In order to study a shift invariant version of U(1) Chern-Simons theory, we follow [BW05,

§3.1] and introduce a new scalar field Φ ∈ Ω0(X) such that

δSΦ = σ.

We postulate the scaling

Φ→ t−1Φ,

for a non-zero function t ∈ C∞(X), whenever

κ→ tκ

9See §4.2 for a definition and general properties.


so that κΦ ∈ Ω1(X) is invariant under the scaling by t and is a well defined form.

Then for any principal U(1)-bundle P we follow [BW05] and define a new action10

SX,P (A,Φ) := SX,P (A− κΦ)

:=

∫X

α(A− κΦ)

=

∫X

α(A− κΦ) (4.4.2)

= SX,P (A)−∫X

[2κ ∧ Tr(Φ ∧ FA)− κ ∧ dκ Tr(Φ2)] (4.4.3)

where Eq. (4.4.2) follows from the definition of A and Φ (where Φ|[p,g] := Adg−1(ρ∗pr∗1Φ|p))

on P = P ×U(1) SU(2) (See §4.1). It is easy to see that the new action, SX,P (A,Φ), is

invariant under the shift symmetry:

δSSX,P (A,Φ) =δSX,PδA

(A− Φκ) · δS(A− Φκ) (4.4.4)

=δSX,PδA

(A− Φκ) · (δS(A)− δS(Φ)κ) (4.4.5)

=δSX,PδA

(A− Φκ) · (σκ− σκ) (4.4.6)

= 0. (4.4.7)

Now define a “new”11 partition function:

ZU(1)(X, p, k) :=1

Vol(S)

1

Vol(GP )

∫AP

DADΦ eπikSX,P (A,Φ), (4.4.8)

where DΦ is defined by the invariant, positive definite quadratic form, [BW05, Eq. 3.8],

(Φ,Φ) = − 1

4π2

∫X

Φ2 κ ∧ dκ. (4.4.9)

As observed in [BW05], the new partition function of Eq. (4.4.8) should be identically

equal to the original partition function defined for U(1) Chern-Simons theory as in Eq.

(4.3.3)

ZU(1)(X, p, k) =1

Vol(GP )


10See §4.1 for a definition of the “hat” notation used here.11The point is that this partition function is really not new, per se, but is expressed in a different form

using an a priori choice of contact structure.


This is seen by fixing Φ = 0 using the shift symmetry, δΦ = σ, which will cancel the

pre-factor Vol(S) from the resulting group integral over S and yield exactly our original

partition function:

ZU(1)(X, p, k) =1

Vol(GP )

∫APDA eπikSX,P (A).

Thus, we obtain the heuristic result,

ZU(1)(X, p, k) = ZU(1)(X, p, k). (4.4.11)

On the other hand, we obtain another description of ZU(1)(X, p, k) by integrating Φ out.

We will briefly review this computation here. Our starting point is the formula for the

shifted partition function (See Eq. (4.4.3))

ZU(1)(X, p, k) =1

Vol(S)

1

Vol(GP )

∫A(P )

DADΦ eπikSX,P (A,Φ) (4.4.12)

where

SX,P (A,Φ) = SX,P (A)−∫X

[2κ ∧ Tr(Φ ∧ FA)− κ ∧ dκ Tr(Φ2)]. (4.4.13)

We formally complete the square with respect to Φ as follows:∫X

[κ ∧ dκ Tr(Φ2) − 2κ ∧ Tr(Φ ∧ FA)]

=

∫X

[Tr(Φ2)− 2κ ∧ Tr(Φ ∧ FA)

κ ∧ dκ

]κ ∧ dκ

=

∫X

Tr

(Φ2 − 2κ ∧ FA

κ ∧ dκΦ

)κ ∧ dκ

=

∫X

Tr

([Φ− κ ∧ FA

κ ∧ dκ

]2

−[κ ∧ FAκ ∧ dκ

]2)κ ∧ dκ

We then only need to compute the Gaussian∫DΦ exp

[πik

∫X

Tr

([Φ− κ ∧ FA

κ ∧ dκ

]2)κ ∧ dκ

]

=

∫DΦ exp

[πik

∫X

Tr(Φ2)κ ∧ dκ]

=

∫DΦ exp

[ik

4π

∫X

Φ2κ ∧ dκ]

=

∫DΦ exp

[−1

2(Φ, TΦ)

]


where we take T = 2πikI acting on the space of fields Φ and the inner product (Φ,Φ) is

defined as in Eq. (4.4.9). We then formally get∫DΦ exp

[−1

2(Φ, TΦ)

]=

√(2π)∆G

detT(4.4.14)

=

(−ik

)∆G/2

(4.4.15)

where the quantity ∆G is formally the dimension of the gauge group G. Note that we

follow [BW05] and abuse notation slightly throughout by writing 1κ∧dκ . We have done

this with the understanding that since κ ∧ dκ is non-vanishing, then κ ∧ FA = φκ ∧ dκ

for some function φ ∈ Ω0(X, su(2)),12 and we identifyκ∧FAκ∧dκ := φ.

Our new description of the partition function is now,

ZU(1)(X, p, k) = C

∫AP

DA exp

[πik

(SX,P (A)−

∫X

Tr[(κ ∧ FA)2]

κ ∧ dκ

)](4.4.16)

where C = 1Vol(S)

1Vol(GP )

(−ik

)∆G/2. Using Prop. 4.2.26, we may rewrite this partition

function after identifying AP = AP + Ω1(X) for a flat base point AP in AP (so that

FAP = 0). We then obtain

ZU(1)(X, p, k) = C1

∫AP

DA exp

[ik

4π

(∫X

A ∧ dA−∫X

(κ ∧ dA)2

κ ∧ dκ

)](4.4.17)

where

C1 =eπikSX,P (AP )

Vol(S) Vol(GP )

(−ik

)∆G/2

.

Note that the critical points of this action, up to the action of the shift symmetry, are

precisely the flat connections, [BW05, Eq. 5.3]. In our notation, A ∈ TAPAP . Let us

define the notation

S(A) :=

∫X

A ∧ dA−∫X

(κ ∧ dA)2

κ ∧ dκ(4.4.18)

12Note that we also abuse notation here again and make the identification of forms on P with formson X via the pullback of some trivializing section of P .


for the new action that appears in the partition function. Also, define

S(A) :=

∫X

(κ ∧ dA)2

κ ∧ dκ(4.4.19)

so that we may write

S(A) = CS(A)− S(A) (4.4.20)

The primary virtue of Eq. (4.4.17) above is that it is heuristically equal to the original

Chern-Simons partition function of Def. 4.3.1 and yet it is expressed in such a way that

the action S(A) is invariant under the shift symmetry. This means that S(A+σκ) = S(A)

for all tangent vectors A ∈ TAP (AP ) ' Ω1(X) and σ ∈ Ω0(X). We may naturally view

A ∈ Ω1(H), the subset of Ω1(X) restricted to the contact distribution H ⊂ TX. If ξ

denotes the Reeb vector field of κ, then Ω1(H) = ω ∈ Ω1(X) | ιξω = 0. The remaining

contributions to the partition function come from the orbits of S in AP , which turn out

to give a contributing factor of Vol(S), [BW05, Eq. 3.32]. We thus reduce our integral

to an integral over AP := AP/S and obtain:


Vol(GP )

∫AP

DA exp

[ik

4π

(∫X

A ∧ dA−∫X

(κ ∧ dA)2

κ ∧ dκ

)]=

eπikSX,P (AP )

Vol(GP )

∫AP

DA exp

[ik

4πS(A)

]

where DA denotes an appropriate quotient measure on AP , and we can now assume that

A ∈ Ω1(H) ' TAP AP .

We therefore make the following heuristic definition:

4.4.21 Definition. Let k ∈ Z be an (even) integer, and (X, κ) a closed, oriented contact

three-manifold. The shifted U(1) Chern-Simons partition function, ZU(1)(X, k), is the

heuristic quantity

ZU(1)(X, k) =∑

p∈TorsH2(X;Z)

ZU(1)(X, p, k) (4.4.22)


where,


Vol(GP )

∫AP

DA exp

[ik

4πS(A)

], (4.4.23)

where S(A) :=∫XA ∧ dA−

∫X

(κ∧dA)2

κ∧dκ is the shifted Chern-Simons action.

Note that we are justified in excluding the factor(−ik

)∆G/2from Eq. (4.4.23) since we

may redefine the heuristic partition function to cancel this factor.

Chapter 5

Non-Abelian Localization for U(1)

Chern-Simons Theory

In [BW05] the authors study the Chern-Simons partition function, [BW05, Eq. 3.1],

Z(k) =1

Vol(G)

(k

4π2

)∆G ∫DA exp

[ik

4π

∫X

Tr

(A ∧ dA+

2

3A ∧ A ∧ A

)], (5.0.1)

where,

• A ∈ AP = A ∈ (Ω1(P )⊗ g)G | A(ξ]) = ξ, ∀ ξ ∈ g is a connection on a principal

G-bundle π : P → X1 over a closed three-manifold X,

• g = LieG and ξ] ∈ Γ(TP ) is the vector field on P generated by the infinitesimal

action of ξ ∈ g on P ,

• k ∈ Z, thought of as an element of H4(BG,Z) that parameterizes the possible

Chern-Simons invariants,

• G := ψ ∈ (Diff(P, P ))G | π ψ = π is the gauge group,

1In fact, [BW05] consider only G compact, connected and simple, and for concreteness one mayassume G = SU(2).

75

Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 76

• ∆(G) is formally defined as the dimension of the gauge group.2

In general, the partition function of Eq. (5.0.1) does not admit a general mathematical

interpretation in terms of the cohomology of some classical moduli space of connections, in

contrast to Yang-Mills theory for example, [Wit92]. The main result of [BW05], however,

is that if X is assumed to carry the additional geometric structure of a Seifert manifold,

then the partition function of Eq. (5.0.1) does admit a more conventional interpretation

in terms of the cohomology of some classical moduli space of connections. Using the

additional Seifert structure on X, [BW05] decouple one of the components of a gauge

field A, and introduce a new partition function, [BW05, Eq. 3.7],

Z(k) = K ·∫DADΦ exp

[ik

4π

(CS(A)−

∫X

2κ ∧ Tr(ΦFA) +

∫X

κ ∧ dκ Tr(Φ2)

)],

(5.0.2)

where

• K := 1Vol(G)

1Vol(S)

(k

4π2

)∆G,

• κ ∈ Ω1(X,R) is a contact form associated to the Seifert fibration of X, [BW05,

§3.2],

• Φ ∈ Ω0(X, g) is a Lie algebra-valued zero form on X,

• DΦ is a measure on the space of fields Φ,3

• S is the space of local shift symmetries4 that “acts” on the space of connections

AP and the space of fields Φ, [BW05, §3.1],

2Note that the definition of the Chern-Simons partition function in Eq. (5.0.1) is completely heuristic.The measure DA has not been defined, but only assumed to “exist heuristically,” and the volume anddimension of the gauge group, Vol(G) and ∆(G), respectively, are at best formally defined.

3The measure DΦ is defined independently of any metric on X and is formally defined by the positivedefinite quadratic form

(Φ,Φ) := −∫X

κ ∧ dκ Tr(Φ2),

which is invariant under the choice of representative for the contact structure (X,H) on X, i.e. underthe scaling κ 7→ fκ, Φ 7→ f−1Φ, for some non-zero function f ∈ Ω0(X,R).

4S may be identified with Ω0(X, g), where the “action” on AP is defined as δσ(A) := σκ, and on thespace of fields Φ is defined as δσ(Φ) := σ, for σ ∈ Ω0(X, g). δσ denotes the action associated to σ.


• FA ∈ Ω2(X, g) is the curvature of A, and

• CS(A) :=∫X

Tr(A ∧ dA+ 2

3A ∧ A ∧ A

)is the Chern-Simons action. 5

[BW05] then give a heuristic argument showing that the partition function computed

using the alternative description of Eq. (5.0.2) should be the same as the Chern-Simons

partition function of Eq. (5.0.1). In essence, they show, [BW05, pg.13]:

Z(k) = Z(k), (5.0.3)

by gauge fixing Φ = 0 using the shift symmetry (See §4.4 for a description of the shift

symmetry in the U(1) gauge group case). [BW05] then observe that the Φ dependence

in the integral can be eliminated by simply performing the Gaussian integral over Φ in

Eq. (5.0.2) directly. They obtain the alternative formulation:

Z(k) = Z(k) = K ′ ·∫DA exp

[ik

4π

(CS(A)−

∫X

1

κ ∧ dκTr[(κ ∧ FA)2

])], (5.0.4)

where K ′ := 1Vol(G)

1Vol(S)

(−ik4π2

)∆G/2. Note that we follow [BW05, Eq. 3.9] here and abuse

notation slightly by writing 1κ∧dκ . We have done this with the understanding that since

κ ∧ dκ is non-vanishing (since κ is a contact form), then κ ∧ FA = φκ ∧ dκ for some

function φ ∈ Ω0(X, g), and we identify κ∧FAκ∧dκ := φ.

The original argument of [BW05] was to decouple one of the components of the gauge

field A ∈ AP 6 by introducing a local shift symmetry,7 and then to translate the Chern-

Simons partition function into a “moment map squared” form using this symmetry. The

general “moment map squared” form for the partition function is a symplectic integral

5Note that the partition functions of Eq.’s 5.0.1 and 5.0.2 are defined implicitly with respect thepullback of some trivializing section of the principal G-bundle P . Of course, every principal G-bundleover a three-manifold for G compact, connected and simple is trivializable. It is basic fact that thepartition functions of Eq.’s 5.0.1 and 5.0.2 are independent of the choice of such trivializations.

6See §2.5 for notation and terminology.7See [BW05, §3.1].


of the canonical form

Z(ε) =1

Vol(H)

(1

2πε

)∆H/2 ∫Y

exp

[Ω− 1

2ε(µ, µ)

]where Y is a symplectic manifold with symplectic form Ω, and H is a Lie group that acts

on Y in a Hamiltonian fashion with moment map µ. ∆H = dim(H) and ε = 2πk

.8 The

technique of non-abelian localization [Wit92] can then be applied to study such integrals.

The goal of this chapter is to develop analogous results for the case of a U(1) gauge group.

5.1 Symplectic Formulation of the U(1) Partition Func-

tion

The goal of this section is to provide a rigorous definition of the U(1) Chern-Simons

partition function using the technique of non-abelian localization following [BW05].

We study the U(1) Chern-Simons partition function using the final results of [BW05],

which a priori are only valid for a simply connected gauge group G (e.g. G = SU(2)). Us-

ing the results of [BW05], we derive an alternative definition for the U(1) Chern-Simons

partition function on a closed, oriented Seifert three-manifold X, called the symplectic

U(1) Chern-Simons partition function, denoted as ZSU(1)(X, k) (See Def. 5.1.13). We note

that the results in this section are intended to serve as motivation for a definition of the

U(1) partition function using the results of [BW05], and do not comprise a rigorous anal-

ysis of the method of non-abelian localization for a U(1) gauge group. It is surprising,

however, that this method yields a definition that is closely9 related with our previous

definitions, although we could not infer this from Beasley and Witten’s calculation be-

cause our situation does not satisfy the hypotheses of [BW05] (i.e. G simply connected).

The results of this section also further support the conclusions of [BW05]. Our starting

8Note that we have chosen to express the quantities in this thesis in terms of the Chern-Simonscoupling constant, k ∈ Z.

9See §5.3 and §5.4 for a detailed study of the precise relations.


point is the following, [BW05, Eq. 5.172]:

ZX := Z(ε)|M0=

1

|Γ|exp

(−iπ

2η0

)∫M0

A(M0)exp

[1

2πεΩ +

1

2c1(TM0) +

in

4π2εrΘ

].

(5.1.1)

We have:

• M0 is a smooth component of the moduli space of irreducible flat connections on

a Seifert manifold X (this formula is derived in [BW05] under the assumption that

X is a principal U(1) bundle of degree n over Σ),

• Γ = Z(G) is the center of G,

• η0, is called the adiabatic eta-invariant, and is recalled in §5.4,10

• A(M0) =∏dimM0

j=1xj/2

sinh(xj/2)where xj(1 ≤ j ≤ n), are the Chern roots of TM0, so

that c(TM0) =∏n

j=1(1 + xj), xj ∈ H2(M0,Z),

• Ω is the symplectic form on M0,

• εr = 2πk+cg

, where cg is the dual Coxeter number of G,

• Θ ∈ H4(M0) is the cohomology class corresponding to the degree 4 element

−(φ, φ)/2 in the equivariant cohomology H4G(pt) (for φ ∈ g using the Cartan model

of equivariant cohomology), Θ can also be described in terms of the universal bundle

U

C // U

Jac(Σ)× Σ

in other words

Θ = −1

2c1(U)2|pt.∈Σ

where Jac(Σ) is the Jacobian of Σ.

10We have chosen the opposite sign convention to [BW05] and [Bea07] in defining η0, and their eta-invariant is the negative of ours.


Our first observation is that the overall constant |Γ| does not really make sense for

G = U(1), if interpreted as the cardinality of U(1). We observe, however, that Eq.

(5.1.1) is derived by localization on a smooth component of the moduli space of flat

connections, and since such a component consists of irreducible connections, the isotropy

group Γ arises solely from the center of G. In U(1) Chern-Simons theory, the isotropy

group is Γ = H ' U(1) and as in [Man98] the correct quantity to replace |Γ| with here is

|Γ| = Vol(H) =

[∫X

?1

]1/2

, (5.1.2)

where ? is the Hodge star relative to some choice of metric g on X (See §4.3). We consider

e(−iπ2η0)∫M0

A(M0)exp

[1

2πεΩ +

1

2c1(TM0) +

in

4π2εrΘ

]. (5.1.3)

By Prop. 2.5.66 and our observations in §5.2, MX ' U(1)2g × HT , where MX is the

moduli space space of flat connections on X and HT ' Tors(H2(X,Z)). We posit that

M0 = MP ' U(1)2g and Eq. (5.1.1) applies to each connected component of MX .

That is, we take each component of the U(1) partition function to be proportional to Eq.

(5.1.3), and weighted by eπikSX,P (AP ) for some flat connection in a bundle P over X, with

p := c1(P ). We do this in complete analogy with the considerations in §4.3. In summary,

we make the initial definition

ZSU(1)(X, k) =

∑p∈TorsH2(X;Z)

ZSU(1)(X, p, k) (5.1.4)

where,

ZSU(1)(X, p, k) =

eπikSX,P (AP )

Vol(H)e(−

iπ2η0)∫MP

A(MP )exp

[1

2πεΩP +

1

2c1(TMP ) +

in

4π2εrΘ

],

(5.1.5)

where MP is the moduli space of flat connections on the principal U(1)-bundle P and

ΩP :=∑

1≤i≤g dθi ∧ dθi, is the standard symplectic form on U(1)2g 'MP .

5.1.6 Remark. We note that ΩP |A is naturally viewed as an element of ∧2H1(X), where

H1(X) is the space of harmonic one-forms and is canonically identified as the tangent


space to the moduli space of flat connections at A, i.e. H1(X) ' TAMP . In particular,

for α, β ∈ H1(X), differential forms,

ΩP |A(α, β) :=

∫X

κ ∧ α ∧ β, (5.1.7)

where κ ∈ Ω1(X) is our chosen contact form as usual.

This is not our final definition of ZSU(1)(X, k), however. The first thing we observe is

that Θ = 0 in the case that G = U(1). This follows since the universal bundle U for U(1)-

bundles is the classical Poincare line bundle, and the Poincare line bundle is normalized

to have degree d = 0 when restricted to the Jacobian of Σ. Since c1(U) = d[Σ] ∈ H2(Σ),

this implies c1(U) = 0, and hence Θ = 0. Also, since MP ' U(1)2g, we know

c(MP ) := c(TMP ),

=

g∏i=1

c(Li) =

g∏i=1

(1 + xi), where Li = TΣi, and xi = c1(Li) ∈ H2(Σi,Z)

where Σi ' (U(1))2. Then the tangent bundles TΣi are trivial, and hence

xi = c1(TΣi) = 0

Thus

A(MP ) =

dimMP∏j=1

xj/2

sinh(xj/2)= 1 (5.1.8)

Clearly, c1(TMP ) = 0 as well. Recalling that ε = 2πk

, we have∫M0

exp

[k

(ΩP

(2π)2

)]=∫M0

kg(

ΩP(2π)2

)g/g!

= kg∫M0

(ΩP

(2π)2

)g/g!

Define

ωP :=ΩgP

g!(2π)2g Vol(H).

Our revised definition of the U(1) Chern-Simons partition function becomes

ZSU(1)(X, p, k) = kgeπikSX,P (AP )e(−

iπ2η0)∫MP

ωP . (5.1.9)


Eq. (5.1.15) is still not a final definition of ZSU(1)(X, p, k). We notice in Def. 4.3.22 that

aside from the term eπikSX,P (AP ), the k-dependence of ZSU(1)(X, p, k) should be of the form

kmX where mX = 12(dimH1(X; R) − dimH0(X; R)). Eq. (5.1.15) has a k-dependence

of kg, however. Since we are assuming that X is connected, we have dimH0(X; R) = 1.

We account for this extra factor of k−1/2 by observing that the dimension of the stabi-

lizer of the gauge group action (for U(1) gauge group) is dim(H0(X,R)) = 1. A similar

phenomenon occurs in Yang-Mills theory at the higher non-flat critical points of the

Yang-Mills action. As observed in [BW05, Eq. 4.45], there is a k1/2 dependence on the

partition function coming from the fact that the gauge group G doesn’t act freely on the

locus of non-flat Yang-Mills solutions. This k1/2 dependence comes from the U(1) < G

subgroup generated by the non-flat Yang-Mills solution, which acts as a stabilizer at the

corresponding connection. U(1) Chern-Simons theory also comes with a U(1) stabilizer

subgroup; the subgroup of constant gauge transformations with values in U(1). Here we

get a factor of k−1/2. In the computation of Eq. 5.172 of [BW05] it is assumed that one

is localizing at the locus of irreducible flat connections, and therefore the isotropy group

of A, ΓA = u ∈ G | u(A) = A, is finite. We do not get a factor of k−1/2 here because

the dimension of the stabilizer is zero. We take this difference of the k-dependence into

account when we make our rigorous definition of the partition function below (See Def.

5.1.13).

Finally, as is noted in §5 of [BW05],11 the computation of their partition function is

done implicitly with respect to a choice of the so called Seifert framing on X. This

choice of framing results in a difference of a factor of eiδΨ · e(−iπ2η0) in the partition

function for the canonical framing, where for general gauge group, [BW05, Eq. 5.101]:

eiδΨ = exp

(iπ∆G

4− iπ∆Gcg

12(k + cg)θ0 +

iπ

2η0

). (5.1.10)

11See [BW05, Pages 89-92].


For the case of a U(1) gauge group, we take ∆G = 1 and we posit that the term iπ∆Gcg

12(k+cg)θ0

vanishes in the exponential. In this thesis, we assume the choice of canonical framing.

Thus, we should replace

e(−iπ2η0), (5.1.11)

in the partition function with,

e(iπ4− iπ

2η0). (5.1.12)

Before we make our rigorous definition, we recall the following:

mX :=1

2(dimH1(X; R)− dimH0(X; R)),

η0 is the adiabatic eta-invariant defined in §5.4, AP ∈ AP is a flat connection on P ,MP ,

is the moduli space of flat connections on P , and ωP :=ΩgP

g!(2π)2g Vol(H)is the symplectic

volume form on MP with

ΩP :=∑

1≤i≤g

dθi ∧ dθi,

the standard symplectic form on U(1)2g 'MP .

We are now ready to make the following

5.1.13 Definition. Let k ∈ Z be an (even) integer, and X a closed, oriented Seifert

three-manifold,

U(1) // X

Σ

,

where Σ = |Σ|,U is an orbifold with underlying space |Σ| a Riemann surface of genus

g. The symplectic U(1) Chern-Simons partition function, ZSU(1)(X, k), is the rigorous

quantity

ZSU(1)(X, k) =

∑p∈TorsH2(X;Z)

ZSU(1)(X, p, k) (5.1.14)


where,


14− 1

2η0)∫MP

ωP . (5.1.15)


and Def. 4.3.22.

5.2 Orbifolds and Symplectic Volume

In order to be more explicit in our computation of the U(1) Chern-Simons partition

function we find an explicit form for the moduli space of flat connections, MX (See Eq.

(2.5.65)), over a Seifert manifold X (See Def. 2.3.6) in this section. The main result of

this section is contained in Prop. 5.2.4 below.

Recall that the fundamental group of X ([Orl72]), π1(X), is generated by the follow-

ing elements

ap, bp, p = 1, . . . , g

cj, j = 1, . . . , N

h

which satisfy the relations,

[ap, h] = [bp, h] = [cj, h] = 1

cαjj h

βj = 1g∏p=1

[ap, bp]N∏j=1

cj = hn.

The generator h is associated to the generic S1 fiber over Σ, the generators ap, bp come

from the 2g non-contractible cycles on Σ, and the generators cj come from the small one

cycles in Σ around each of the orbifold points pj.


The moduli space of flat connections over X can be realized as the space of homomor-

phisms from π1(X) to U(1). Consider,

ρ : π1(X)→ U(1)

and let, Aj = ρ(cj), B = ρ(h). Then since U(1) is abelian, the generating relations for

π1(X) translate into the following restrictions on ρ:

Aαjj B

βj = 1, j = 1, . . . , N

N∏j=1

Aj ·B−n = 1,

where, Aj = e2πiψj , j = 1, . . . , N , and B = e2πiψ0 for some ψj ∈ R, j = 0, . . . , N .

We condense the above restrictions and write,

N∏j=0

e2πiψlKj,l = 1 (5.2.1)

where Kj,l is the following matrix

K =

−n β1 β2 · · · βN

1 α1 0 · · · 0

... 0. . . 0

...

1... 0 αN−1 0

1 0 · · · 0 αN .

(5.2.2)

Let v = (ψ0, ψ1, . . . , ψN) ∈ RN+1. Then v solves the above equation, 5.2.1, if and only if

K · v = w ∈ ZN+1

Observe that detK = (−∏N

j=1 αj) ·(n+

∑Nj=1

βjαj

)= (−

∏Nj=1 αj) · c1(L) 6= 0, since

αj 6= 0, ∀ 1 ≤ j ≤ N , and the orbifold first chern number c1(L) > 0 by assumption (See

Remark 2.3.7). Thus, K is invertible, and

v = K−1w =(CofK)T

detKw


where the cofactor matrix CofK ∈MN+1(Z) and detK ∈ Z. So,

v =1

detKw ∈ 1

detKZN+1

where w = (CofK)T ·w ∈ ZN+1.

In general, once we fix the values of ρ(ap), ρ(bp) ∈ U(1) there are (detK)N+1 possibilities

for ρ. Thus, for each point in U(1)2g there is a factor of order |(detK)N+1| in the moduli

space of flat connections of X. It is interesting to note that it has been independently

shown in [Man98] (See Prop. 2.5.66) that

π0(MX) ' Tors(H2(X,Z)). (5.2.3)

Thus, we have HT ' Tors(H2(X,Z)) and in particular |(detK)N+1| = |Tors(H2(X,Z))|.

We have therefore established the following

5.2.4 Proposition. Let X be a closed, oriented Seifert three-manifold, and MX denote

the moduli space of flat connections on X. Then

MX ' U(1)2g × Tors(H2(X,Z)) (5.2.5)

where |Tors(H2(X,Z))| = |(detK)N+1| for the matrix K defined in Eq. (5.2.2). Note

that the special case where N = 0, we recover det K = −n and MX ' U(1)2g × Zn.

5.3 Reidemeister Torsion and Symplectic Volume


and Def. 4.3.22. This boils down to proving the equivalence of the quantities


14− 1

2η0)∫MP

ωP , (5.3.1)

and,


4+ 1

12CS(Ag)

2π

) ∫MP

(T dRS)1/2, (5.3.2)


where all the requisite notation for Eq. (5.3.1) is defined in §5.1 and similarly the no-

tation for Eq. (5.3.2) is defined in §4.3. In this section we compare the square root of

the Ray-Singer torsion (T dRS)1/2 (see Appendix C.2) to the symplectic volume form ωP .

Superficially, these two quantities are both elements of Ω2g(MP ,R). The goal of this

section is to establish that in fact these two quantities agree up to some undetermined

constant. Ultimately, we would like to show these quantities are identically equal

(T dRS)1/2 = ωP , (5.3.3)

as differential forms on Ω2g(MP ,R).

Our first observation for this section is that it is much more useful for us to work with

the combinatorially defined Reidemeister torsion (R-torsion), τ(X) (see Remark 5.3.12

below), than directly with the Ray-Singer torsion, T dRS. It is a well known fact that

τ(X) = T dRS. This was independently shown by Cheeger, [Che79] and Muller, [Mul78],

and more recently a new proof has been given by Braverman, [Bra02].

We begin by reviewing the Reidemeister torsion, (R-torsion), and provide some relevant

examples. Recall that the R-torsion is an invariant for a CW-complex and a representa-

tion of its fundamental group. Before we define the R-torsion, we recall the definition of

the torsion of a chain complex. We primarily follow [JCW93] and [Yam08].

Let C∗ =(

0→ Cndn−→ Cn−1

dn−1−−−→ · · ·C1d1−→ C0 → 0

)be a chain complex over F (either

R or C). Let Zi denote the cycles of this complex, Bi denote the boundaries, and Hi the

homology. We say that C∗ is acyclic if Hi = 0 for all i.

Let ci be a basis of Ci and c be the collection cii≥0. We call a pair (C∗, c) a based chain

complex, c the preferred basis of C∗ and ci the preferred basis of Ci. Let hi be a basis of

Hi.

We construct another basis as follows. By the definitions of Zi, Bi, and Hi, the following


two split exact sequences exist

0→ Zi → Cidi−→ Bi−1 → 0,

0→ Bi → Zi → Hi → 0.

Let Bi−1 be a lift of Bi−1 to Ci and Hi a lift of Hi to Zi. Then we can decompose Ci as

follows.

Ci = Zi ⊕ Bi−1

= Bi ⊕ Hi ⊕ Bi−1

= di+1Bi ⊕ Hi ⊕ Bi−1

Choose a basis bi for Bi. We write bi+1 = bi+1j

nij=1 for a lift of bi and hi = hij

mij=1 for

a lift of hi. By construction, the set bi ∪ di+1(bi+1) ∪ hi forms another ordered basis of

Ci. Denote this basis as bidi+1(bi+1)hi. The definition of the R-torsion, Tor(C∗, c), is as

follows:

Tor(C∗, c)h = (−1)|C∗| ·n∏i=1

[bidi+1(bi+1)hi/ci](−1)i+1 ∈ F∗. (5.3.4)

where [bidi+1(bi+1)hi/ci] denotes the determinate of the change of basis matrix from

the basis ci to the basis bidi+1(bi+1)hi; |C∗| =∑

i≥0 αi(C∗) · βi(C∗), where αi(C∗) =∑ik=0 dimCk and βi(C∗) =

∑ik=0 dimHk.

An alternative definition12 that we will also use is to equip our complex C∗ with volumes

µi ∈ (∧maxCi)′, one for each i, and then define

Tor(C∗, µ)h =

∧i even µi [b

i ∧ di+1(bi+1) ∧ hi]∧i odd µi [b

i ∧ di+1(bi+1) ∧ hi]. (5.3.5)

where we take bi = ∧nij=1bij, di+1b

i+1 = ∧nij=1di+1bi+1j , and hi = ∧mij=1h

ij. The torsion is an

element

Tor(C∗, µ) ∈ ⊗2i+1| ∧max H2i+1(C∗)| ⊗2i | ∧max H2i(C∗)′ |,

12See [JCW93].


where |·| denotes absolute values in the determinant lines. It is well known that the torsion

is independent of the choices of bi, and lifts bi, hi.13 The latter definition specializes to

the former definition when we choose the canonical volumes associated to a choice of

preferred basis c for C. In the case that our complex C∗ is acyclic, we define

Tor(C∗, µ) =

∧i even µi [b

i ∧ di+1(bi+1)]∧i odd µi [b

i ∧ di+1(bi+1)]∈ R∗. (5.3.6)

We will be interested in a specific chain complex C∗. In particular, let N be a cell

complex, and ρ a representation of π1(N) in G. The Lie algebra g is acted on by π1(N)

under the composition of the adjoint action of G and the representation ρ. Let gρ denote

g with the π1(N)-module structure from ρ. Let N denote the universal cover of N . Since

the fundamental group π1(N) acts on N by covering transformations, the chain complex

C∗(N) also has a natural π1(N)-module structure. The chain complex of interest is then

C∗(N, gρ), defined as the quotient of C∗(N)⊗ g under the equivalence:

σ ⊗ Y ∼ σa⊗ Ad(ρ(a))−1Y, (5.3.7)

where a ∈ π1(N), σ ∈ C∗(N), and Y ∈ g. The usual differential on C∗(N) is compatible

with the equivalence relation, and thus descends to a differential δρ on C∗(N, gρ). By

dualizing one obtains the corresponding cochain complex C∗(N, gρ) with differential dρ =

δ∗ρ. We have the following

5.3.8 Lemma. [JCW93] Suppose h ∈ G. If ρ and hρh−1 are conjugate representations of

π1(N) in G, then the map Ad(h) : g→ g induces an isomorphism of the chain complexes

C∗(N, gρ) and C∗(N, ghρh−1). Hence, one obtains a natural isomorphism between the

cohomology groups H i(C∗(N, gρ)) and H i(C∗(N, ghρh−1)).

We will mainly be interested in the zeroth and first cohomology groups of this complex.

We recall the following

13See [RS73].


5.3.9 Proposition. [JCW93] Let [ρ] ∈ Hom(π1(N), G)/G. The choice of a particular

ρ ∈ Hom(π1(N), G) in the conjugacy class [ρ] identifies the Zariski tangent space to the

space Hom(π1(N), G)/G at ρ with the first cohomolgy group H1(N, gρ).

Furthermore, the Lie alegbra of the isotropy group of ρ(the subgroup of G fixing the

representation ρ under conjugation) is H0(N, gρ).

Using the definition of the R-torsion in equation (5.3.5) above, we may define volumes

on C∗(N, gρ) by using the natural metric on g. We take σij ⊗ Yk to be an orthonormal

basis of C∗(N, gρ), where σij are the i-cells in the universal cover N and the Yk are an

orthonormal basis of g. This volume is well defined since the adjoint representation is

an orthogonal representation of G, and hence compatible with the equivalence relation

in Eq. (5.3.7). We then have

Tor(C∗(N, gρ), µ) ∈ ⊗2i+1| ∧max H2i+1(N, gρ)| ⊗2i | ∧max H2i(N, gρ)′|.

Since H i(N, gρ) ' Hi(N, gρ)′, we have

Tor(C∗(N, gρ), µ) ∈ ⊗2i+1| ∧max H2i+1(N, gρ)′ | ⊗2i | ∧max H2i(N, gρ)|. (5.3.10)

Define the determinant line,

| detH•(N, gρ)∗| :=

⊗j

| detHj(N, gρ)(−1)j |,

where,

detHj(N, gρ) := ∧maxHj(N, gρ),

and H−1 := H ′ denotes the dual space. Then we may also write

Tor(C∗(N, gρ), µ) ∈ | detH•(N, gρ)∗|. (5.3.11)

5.3.12 Remark. The isomorphisms in Lemma 5.3.8 identify the torsion Tor(C∗(N, gρ))

with Tor(C∗(N, ghρh−1), so the torsion descends to an equivalence class τ(N, ρ) depending

only on the conjugacy class [ρ] ∈ Hom(π1(N), G)/G.


It is instructive to consider the case when N is a genus g surface Σg.

5.3.13 Example. From the relation (5.3.10) above, we see that the torsion τ(Σg; ρ) of a

surface Σg takes values in

∧maxH1(Σg, gρ)′ ⊗ ∧maxH2(Σg, gρ)⊗ ∧maxH0(Σg, gρ).

Note that [Wit91] assumes that the flat connection ρ is irreducible, which in particular

implies that H0(Σ, adE) ' H2(Σ, adE) ' 0.14 Thus, we have

τ(Σg; ρ) ∈ ∧maxH1(Σg, gρ)′.

Observe that H1(Σg, gρ) is a symplectic vector space with the symplectic form given

by the cup product H1(Σg, gρ) ⊗ H1(Σg, gρ) → H2(Σg,R) ' R. One can show that in

fact the torsion may be identified with the symplectic volume on a smooth component

of irreducible flat connections in the moduli space Hom(π1(Σ), G)/G. Note that this

identification naturally follows from Prop. 5.3.9 above. A rigorous proof of this is given

in [Wit91].

5.3.14 Remark. The case of interest is when N = X is a closed, oriented Seifert manifold,

and G = U(1). In particular, we observe that H•(N, gρ) ' H•(Ω(X), d) for any flat U(1)

connection ρ, where Hj(Ω(X), d) denotes the usual de Rham cohomology of X. We

henceforth impose these conditions in this section.

As above, we observe that the torsion τ(X; ρ) takes values in

| detH•(Ω(X), d)∗| :=3⊗j=0

| detHj(Ω(X), d)(−1)j |,

where by Poincare duality H3(Ω(X), d)′ is canonically isomorphic to H0(Ω(X), d), and

H1(Ω(X), d)′ is canonically isomorphic to H2(Ω(X), d). Thus,

τ(X; ρ) ∈ | detH1(Ω(X), d)′|⊗2⊗| detH0(Ω(X), d)|⊗2.

14E denotes a principal G bundle over Σ and adE denotes the corresponding adjoint bundle. Notealso that [Wit91] restricts to the case where the gauge group G is compact, semi-simple and connected.


Choosing the canonical trivialization | detH0(Ω(X), d)| ' R+ with respect to an or-

thonormal basis for H0(Ω(X), d), we may then view the square root of the R-torsion as

a density, √τ(X; ρ) ∈ | detH1(Ω(X), d)′|.

Define,

√τ(X) : MP → | detH1(Ω(X), d)′|, by, (5.3.15)√

τ(X)(ρ) :=√τ(X)|ρ :=

√τ(X; ρ). (5.3.16)

Note that MX ' MP × TorsH2(X; Z) by Prop. 2.5.66, and we are implicitly choosing

only one bundle class [P ] to work with here in this decomposition ofMX , so that c1(P ) ∈

TorsH2(X; Z). Then we may identify√τ(X) ∈ Ω2g(MP ,R) since By Prop. 4.1.2 of

[JM09], H1(X) is identified with the Zariski tangent space to the moduli space MP '

U(1)2g.

5.3.17 Remark. Note that we are assuming that√τ(X) defined above is a smooth density

here with respect to the natural smooth structure on MP and the smooth structure on

H1(X) as identified with the Zariski tangent space to the moduli space MP ' U(1)2g.

We could not find an explicit proof of this fact in the literature, and it is of interest to

the author to show this in the future. For now we simply take this for granted.

We would like to see√τ(X) is proportional up to a constant to the symplectic volume

ωP on the moduli space MP , i.e.

√τ(X) = C · ωP , (5.3.18)

where C ∈ R∗ is some non-zero constant. Recall that ωP :=ΩgP

g!(2π)2g Vol(H)with

ΩP :=∑

1≤i≤g

dθi ∧ dθi,

the standard symplectic form on U(1)2g ' MP . It will be sufficient to identify√τ(X)

and ωP at a single point of the moduli space since√τ(X) and ΩP (and hence ωP ) are


invariant under left multiplication, i.e. invariant under the action,

Lδ : U(1)2g → U(1)2g

defined by Lδ : γ 7→ δ · γ for γ, δ ∈ U(1)2g. Let us show this presently. First, we ob-

serve that√τ(X) is invariant under this action. This follows directly from the fact that

in the G = U(1) case√τ(X; ρ) '

√τ(X; ρ′) by definition (see (5.3.5)). In particular,√

τ(X; ρ) '√τ(X; δ · ρ) for all δ, ρ ∈ U(1)2g. It is also clear that ΩP :=

∑1≤i≤g dθi∧dθi

is invariant under this action since this action just represents rotations in each copy of

U(1) on U(1)2g.

Thus, let e denote the identity element of U(1)2g. Then at the point e,√τ(X) and

ωP must agree up to a non-zero constant (recall that√τ(X) 6= 0 is a volume form by

definition, and similarly ωP 6= 0):

√τ(X)|e = C · ωP |e

for some C ∈ R∗. By left invariance, we therefore have:

√τ(X)|ρ = C · ωP |ρ, ∀ρ ∈ U(1)2g.

Thus, √τ(X) = C · ωP , for C ∈ R∗.

We have shown the following

5.3.19 Proposition. Let X be a closed, oriented Siefert three-manifold,√τ(X) the

square root of the R-torsion of X, and ωP the symplectic volume form on the moduli

space of flat connections MP ' U(1)2g for some flat bundle P over X. Then, there

exists C ∈ R∗, such that √τ(X) = C · ωP .

We make the following


5.3.20 Conjecture. C = 1 in Prop. 5.3.19 above.

5.3.21 Remark. We propose to prove Conjecture 5.3.20 by modeling our proof on [Wit91]

for the analogous result in two dimensions (see Ex. 5.3.13). We note that Conjecture

5.3.20 would yield a calculation of the R-torsion for the class of closed, oriented Siefert

three manifolds. Consider,∫MP

√τ(X) =

∫MP

ωP , assuming Conjecture 5.3.20 is true, (5.3.22)

=

∫MP

ΩgP

g!(2π)2g Vol(H), (5.3.23)

=1

Vol(H), (5.3.24)

where (see §6.5),

Vol(H) =

[∫X

κ ∧ dκ]1/2

=

[n+

N∑j=1

βjαj

]1/2

,

for [n; (α1, β1), . . . , (αN , βN)] the Seifert invariants of our Seifert manifold X. Also, recall

that (see §2.3)

c1(L) = n+N∑j=1

βjαj

> 0.

Thus, by our observations in Appendix C.2 and the Cheeger-Muller theorem, [Che79],

[Mul78], we would have

TRS = TC = (c1(L))−1.

5.4 The Adiabatic Eta-Invariant

In this section we continue with our aim of proving the equivalence of Def. 5.1.13 and

Def. 4.3.22, i.e. that


14− 1

2η0)∫MP

ωP , (5.4.1)

and,


4+ 1

12CS(Ag)

2π

) ∫MP

(T dRS)1/2, (5.4.2)


are equal. Our focus in this section is to compare the eta-invariants that show up in

Eq.’s 5.4.1 and Def. 5.4.2. We will see that these invariants are closely related, and in

fact differ by the exact amount required to establish the equivalence of the eta-invariant

dependent parts of Def. 5.1.13 and Def. 4.3.22. The regularized eta-invariant of Eq.

(5.4.2)

η(?d)

4+

1

12

CS(Ag)

2π, (5.4.3)

is studied in §6.10 for example, and is computed in Prop. 6.10.27


Then,15

η(?d) +1

3

CS(Ag)

2π= η(?D) +

1

512

∫X

R2 κ ∧ dκ

= 1− d

3+ 4

N∑j=1

s(αj, βj),

where 0 < d = c1(X) = c1(L) = n+∑N

j=1βjαj∈ Q, and

s(α, β) :=1

4α

α−1∑k=1

cot

(πk

α

)cot

(πkβ

α

)∈ Q





3+4∑Nj=1 s(αj ,βj))

∫MP

(T dC)1/2,


3+4∑Nj=1 s(αj ,βj))

∫MP

(T dRS)1/2.

Thus, we turn our attention to the study of η0, which was named the adiabatic eta-

invariant by Nicolaescu in [Nic00]. This invariant shows up in Def. 5.1.13 for the

symplectic U(1) Chern-Simons partition function, ZSU(1)(X, k), precisely due to the con-

siderations in §5.2 of [BW05]. See [Bea07, App. C] for an explicit computation of η0.



The main result that we will need is [Bea07, Eq. C.31]:

η0 =∆G

6

[c1(L)− 12

N∑j=1

s(αj, βj)

], (5.4.5)

where ∆G = dim(G) is the dimension of the gauge group. Note that the adiabatic

eta-invariant η0 is a topological invariant defined and computed on a general Seifert

three-manifold X.

5.4.6 Remark. We have chosen the opposite sign convention to [BW05] and [Bea07] in

defining η0, and their eta-invariant is the negative of ours.

In our case G = U(1), ∆G = 1, and we have

− η0

2= −c1(L)

12+

N∑j=1

s(αj, βj). (5.4.7)

L denotes the line V-bundle over the orbifold Σ associated to the Siefert manifold X (i.e.

X is the boundary of the disc bundle of L). Setting d = c1(L) in Eq. (5.4.7), we obtain

− η0

2= − d

12+

N∑j=1

s(αj, βj). (5.4.8)

Inserting this into Eq. (5.4.1), we have

ZSU(1)(X, p, k) = kmXeπikSX,P (AP )e

iπ4 (1− d

3+4∑Nj=1 s(αj ,βj)))

∫MP

ωP . (5.4.9)

This precisely matches the eta-invariant dependence in Eq. (5.4.2),

ZU(1)(X, p, k) := kmXeπikSX,P (AP )eπi4

(η(?d)+ 1

3CS(Ag)

2π

) ∫MP

(T dRS)1/2, (5.4.10)

by Prop. 5.4.4, i.e.

η(?d) +1

3

CS(Ag)

2π= 1− d

3+ 4

N∑j=1

s(αj, βj).

Thus, we have shown the following



Then,16

η(?d)

4+

1

12

CS(Ag)

2π=

1

4− 1

2η0,

=1

4− d

12+

N∑j=1

s(αj, βj).


Chapter 6

Eta-Invariants and Anomalies

This chapter studies U(1) Chern-Simons theory and its relation to a construction of

Chris Beasley and Edward Witten, [BW05]. The natural geometric setup here is that

of a three-manifold with a Seifert structure (see Chapter 2). Based on a suggestion of

Edward Witten we are led to study the stationary phase approximation (see A.1) of the

path integral for U(1) Chern-Simons theory after one of the three components of the

gauge field is decoupled. This gives an alternative formulation of the partition function

for U(1) Chern-Simons theory that is conjecturally equivalent to the usual U(1) Chern-

Simons theory (see §4.3). We establish this conjectural equivalence rigorously using

appropriate regularization techniques. This approach leads to some rather surprising

results and opens the door to studying hypoelliptic operators (see Chapter 3) and their

associated eta-invariants in a new light.

6.1 Introduction

The objective in this chapter is to study the partition function for U(1) Chern-Simons

theory using the analogue of Eq. (5.0.4) in this case. Thus, we are also assuming here that

X is a Seifert manifold with a “compatible” contact structure, (X, κ), [BW05, §3.2]. Note

that any compact, oriented three-manifold possesses a contact structure and one aim of

98

Chapter 6. Eta-Invariants and Anomalies 99

future work is to extend our results to all closed, oriented three-manifolds using this fact.

For now, we restrict ourselves to the case of closed three-manifolds that possess contact

compatible Seifert structures (see Definition 6.2.1 for example). We restrict to the gauge

group U(1) so that the action is quadratic and hence the stationary phase approximation

is exact (see Appendix A.1). A salient point is that the group U(1) is not simple, and

therefore may have non-trivial principal bundles associated with it. This makes the U(1)-

theory very different from the SU(2)-theory in that one must now incorporate a sum over

bundle classes in a definition of the U(1)-partition function. Recall (definition 4.3.1) that

the partition function for U(1) Chern-Simons theory is heuristically defined as:

ZU(1)(X, k) =∑

p∈TorsH2(X;Z)

ZU(1)(X, p, k), (6.1.1)

where,

ZU(1)(X, p, k) =1

Vol(GP )


Recall that the torsion subgroup TorsH2(X; Z) < H2(X; Z)1 enumerates the U(1)-bundle

classes that have flat connections. Note that the bundle P → X in Eq. (6.1.2) is taken to

be any representative of a bundle class with first Chern class c1(P ) = p ∈ TorsH2(X; Z).

Also note that some care must be taken to define the Chern-Simons action, SX,P (A), in

the case that G = U(1). More details of this construction can be found in section 4.3,

where a rigorous version of this definition is also constructed (see definition 4.3.22).

The main results of this chapter may be summarized as follows. First, our main ob-

jective is the rigorous confirmation of the heuristic result of Eq. (5.0.3) in the case where

the gauge group is U(1). This statement is certainly non-trivial and involves some fairly

deep facts about the “contact operator” as studied by Michel Rumin, [Rum94]. Recall

1Recall the definition of the torsion of an abelian group is the collection of those elements which havefinite order.


that this is the second order operator “D” that fits into the complex,


D−→ Ω2(V )dH−→ Ω3(X), (6.1.3)

and in our case2 is defined by:

Dα = κ ∧ [Lξ + dH ?H dH ]α, α ∈ Ω1(H). (6.1.4)

This operator is elaborated upon in §6.4 below. A somewhat surprising observation is

that this operator shows up quite naturally in U(1) Chern-Simons theory (see Prop.

6.4.14 below), and this leads us to make several conjectures motivated by the rigorous

confirmation of the heuristic result of Eq. (5.0.3). Our main result is the following (see

§6.10 below for the relevant definitions, and Prop. 6.10.24 in particular):

6.1.5 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three mani-

fold. If,


512

∫X R2 κ∧dκ)

∫MP

(T dC)1/2, (6.1.6)

and (see §4.3),


4+ 1

12CS(Ag)

2π

) ∫MP

(T dRS)1/2 (6.1.7)

then,3

ZU(1)(X, k) = ZU(1)(X, k)


Following [Man98], we rigorously define ZU(1)(X, k) in §6.6 using the fact that the

stationary phase approximation for our path integral should be exact. This necessitates

the introduction of the regularized determinant of D in Eq. (6.7.3), which in turn nat-

urally involves the hypoelliptic Laplacian of Eq. (6.7.5). The rigorous quantity that

2The case where X is taken to be a closed, oriented quasi-regular K-contact three-manifold (see §2.3).3This result follows after a particular choice of Vielbein for the gravitational Chern-Simons term

CS(Ag). See Equations (7.3.20) and (7.3.21).


we obtain for the integrand of Eq. (6.6.2) in §6.6 is derived in Prop. 6.7.15. Using an

observation from §6.5 that identifies the volume of the isotropy subgroup of the gauge

group GP , we identify the integrand of Eq. (6.6.2) with the contact analytic torsion T dC

defined in Def. 6.9.4. After formally identifying the signature of the contact operator D

with the η-invariant of D in §6.8, we obtain our fully rigorous definition of ZU(1)(X, k)

in Eq. (6.9.17) below, which is repeated in Eq. (6.1.6) above.

On the other hand, [Man98] provides a rigorous definition of the partition function

ZU(1)(X, k) that does not involve an a priori choice of a contact structure on X. The

formula for this is recalled in Eq. (6.9.18) below, and is the term ZU(1)(X, p, k) in Eq.

(6.1.7) of Prop. 6.1.5 above.

Our first main step in the proof of Prop. 6.1.5 is confirmation of the fact that the

Ray-Singer analytic torsion (See §4.3, and Eq.’s 4.3.16, 4.3.21) of X, T dRS, is identically

equal to the contact analytic torsion T dC (See Def. 6.9.4).4 We observe that this result

follows directly from [RS08, Theorem 4.2].

We also observe in Remark 6.7.20 that the quantities mX and nX that occur in Prop.

6.1.5 are equal. This leaves us with the main final step in the confirmation of Prop.

6.1.5, which involves a study of the η-invariants, η(?d), η(?D), that naturally show up

in ZU(1)(X, k), ZU(1)(X, k), respectively. This analysis is carried out in §6.10, where we

observe that the work of Biquard, Herzlich, and Rumin, [BHR07] is our most pertinent

reference. Our main observation here is that the quantum anomalies that occur in the

computation of ZU(1)(X, k) and ZU(1)(X, k) should, in an appropriate sense, be com-

pletely equivalent. In our case, these quantum anomalies are made manifest precisely in

4We consider the square roots thereof, viewed as densities on the moduli space of flat connectionsMP (see Eq. (2.5.57) for a definition of MP ).


the failure of the η-invariants to represent topological invariants. As observed by Wit-

ten in [Wit89], this is deeply connected with the fact that in order to actually compute

the partition function, one needs to make a choice that is tantamount to either a valid

gauge choice for representatives of gauge classes of connections, or in some other way by

breaking the symmetry of our problem. Such a choice for us is equivalent to a choice

of metric, which is encoded in the choice of a quasi-regular K-contact structure on our

manifold X. Witten observes in [Wit89] that the quantum anomaly that is introduced by

our choice of metric may be canceled precisely by adding an appropriate “counterterm”

to the η-invariant, η(?d). This recovers topological invariance and effectively cancels the

anomaly.5 This counterterm is found by appealing to the Atiyah-Patodi-Singer theorem

[APS75b], and is in fact identified as the gravitational Chern-Simons term

CS(Ag) :=1

4π

∫X

Tr(Ag ∧ dAg +2

3Ag ∧ Ag ∧ Ag), (6.1.8)

where Ag is the Levi-Civita connection on the spin bundle of X for the metric,

g = κ⊗ κ+ dκ(·, J ·), (6.1.9)

on our quasi-regular K-contact three manifold, (X,φ, ξ, κ, g). In particular, we use the

fact that,

η(?d)

4+

1

12

CS(Ag)

2π, (6.1.10)

is a topological invariant of X, after choosing the canonical framing. As is discussed in

§6.10, this leads us to conjecture that there exists an appropriate counterterm for the η-

invariant associated to the contact operator D that yields the same topological invariant

as in Eq. (6.1.10). More precisely, we conjecture that there exists a counterterm, CT ,

such that

eπi[η(?d)

4+ 1

12CS(Ag)

2π

]= e

πi4

[η(?D)+CT ], (6.1.11)

as topological invariants. We establish the following in Proposition 6.10.23,

5In this case, topological invariance is recovered only up to a choice of two-framing for X. Of course,there is a canonical choice of such framing as observed in [Ati90b].



Then6 there exists a counterterm, CT , such that eπi4

[η(?D)+CT ] is a topological invariant

that is identically equal to the topological invariant eπi[η(?d)

4+ 1

12CS(Ag)

2π

]. In fact, we have

CT =1

512

∫X

R2 κ ∧ dκ,


This proposition is proven in §6.10 by appealing to the following result, which is

established using a “Kaluza-Klein” dimensional reduction technique for the gravitational

Chern-Simons term. This result is modeled after the paper [GIJP03], and is listed as

Proposition 6.10.20 (see Chapter 7 for a detailed study of this result).

6.1.13 Proposition. [McL10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact

three-manifold,

U(1) // X

Σ

.


CS(Agε) =

(ε−1

2

)∫Σ

r ω +

(ε−2

2

)∫Σ

f 2 ω (6.1.14)




limε→∞

CS(Agε) = 0. (6.1.15)




(∂0ϕ1 − ∂1ϕ0)dx0 ∧ dx1 = f01dx0 ∧ dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor,

and fαβ =√h εαβ f, where f ∈ Ω0



Finally, as a consequence of these investigations, we are able to compute in Proposition

6.10.27 the U(1) Chern-Simons partition function fairly explicitly.


Then,9

η(?d) +1

3

CS(Ag)

2π= η(?D) +

1

512

∫X

R2 κ ∧ dκ

= 1− d

3+ 4

N∑j=1

s(αj, βj),

where 0 < d = c1(X) = c1(L) = n+∑N

j=1βjαj∈ Q and

s(α, β) :=1

4α

α−1∑k=1

cot

(πk

α

)cot

(πkβ

α

)∈ Q





3+4∑Nj=1 s(αj ,βj))

∫MP

(T dC)1/2,


3+4∑Nj=1 s(αj ,βj))

∫MP

(T dRS)1/2.

6.2 Structure Operators

At this point, we restrict the structure on our three-manifold and assume that the Seifert

structure is compatible with a contact metric structure (φ, ξ, κ, g) on X. In particular,

we restrict to the case of a quasi-regular K-contact manifold (see §2.3). Note that the as-

sumption that the Seifert structure on X comes from a quasi-regular K-contact structure

(φ, ξ, κ, g) is equivalent to assuming that X is a CR-Seifert manifold (see Prop. 2.3.5).

Recall the following



6.2.1 Definition. A CR-Seifert manifold is a three-dimensional compact manifold en-

dowed with both a strictly pseudoconvex CR structure (H, J) and a Seifert structure,

that are compatible in the sense that the circle action ψ : U(1)→ Diff(X) preserves the

CR structure and is generated by a Reeb field ξ. In particular, given a choice of contact

form κ, the Reeb field is Killing for the associated metric g = κ⊗ κ+ dκ(·, J ·).

The assumption that X is CR-Seifert (hence quasi-regular K-contact) is sufficient to

ensure that the assumption in [BW05, Eq. 3.27], which states that the U(1)-action on

X, ψ : U(1)→ Diff(X), acts by isometries, is satisfied.

We now employ the natural Hodge star operator ?, induced by the metric g on X,

that acts on Ω•(X) taking k forms to 3 − k forms. We have chosen a normalization

convention such that ?1 = κ ∧ dκ and ?κ = dκ.10 Now define

6.2.2 Definition. Define the horizontal Hodge star operator to be the operator:

?H : Ωq(X)→ Ω2−q(H) q = 0, 1, 2,

defined for β ∈ Ωq(X) by

?H β = ?(κ ∧ β), (6.2.3)

where ? is the usual Hodge star operator on forms for the metric g = κ⊗ κ+ π∗h on X.

We have

6.2.4 Proposition. (See Prop. C.1.21) ?Hα = (−1)qιξ(?α) = ?(κ∧α) for all α ∈ Ωq(X),

0 ≤ q ≤ 2.11

We also have the following

10We choose h to be any (orbifold) Kahler metric on Σ which is normalized so that the corresponding(orbifold) Kahler form, ω ∈ Ω2

orb(Σ,R), pulls back to dκ. See §2.3 for more details.11Note that the statement is true for q = 3, but only trivially so.


6.2.5 Proposition. (See C.1.22) The following equalities hold

?H κ = 0 (6.2.6)

?H(κ ∧ dκ) = 0 (6.2.7)

?H1 = dκ. (6.2.8)

Lastly, we have the following

6.2.9 Proposition. (See Prop. C.1.19) ?2H = −1 on Ω1(H).

6.3 The Action

Our starting point in this section is the analogue of Eq. (5.0.4) for the U(1) Chern-Simons

partition function:


Vol(GP )

∫AP

DA exp

[ik

4πS(A)

](6.3.1)

where SX,P (AP ) is the Chern-Simons invariant associated to P for AP a flat connection

on P . The derivation of Eq. (6.3.1) can be found in §4.4. It is obtained by expanding

the U(1) analogue of Eq. (5.0.4) around a critical point AP of the action. Our key

observation is that the action S(A) =∫XA ∧ dA −

∫X

(κ∧dA)2

κ∧dκ may now be expressed in

terms of the horizontal quantities of §6.2. Define the notation

S(A) :=

∫X

(κ ∧ dA)2

κ ∧ dκ(6.3.2)

so that we may write

S(A) = CS(A)− S(A), (6.3.3)

where CS(A) :=∫XA ∧ dA. Let us start with the term S(A). First, the term κ ∧ dA

in S(A) is equivalent to κ ∧ dHA since the vertical part of dA is annihilated by κ in the

wedge product. The term κ∧dAκ∧dκ is equivalent to ?(κ∧ dHA) by the properties of ? above.


By the definition of ?H , ?(κ∧ dHA) = ?HdHA (See Prop. C.1.26 for a proof of this). We

then have,

S(A) =

∫X

(κ ∧ dA)2

κ ∧ dκ

=

∫X

?H(dHA) ∧ κ ∧ dHA

=

∫X

κ ∧ [dHA ∧ ?H(dHA)]

We claim that S(A) is now expressed in terms of an inner product on Ω2H. More

generally, we define an inner product on Ωl(H) for 0 ≤ l ≤ 2:

6.3.4 Definition. Define the pairing 〈·, ·〉lκ : ΩlH × ΩlH → R as

〈α, β〉lκ := (−1)l∫X

κ ∧ [α ∧ ?Hβ] (6.3.5)

for any α, β ∈ ΩlH, 0 ≤ l ≤ 2.

6.3.6 Proposition. The pairing 〈·, ·〉lκ is an inner product on ΩlH.

Proof. It can be easily checked that this pairing is just the restriction of the usual L2-inner

product, 〈·, ·〉 : ΩlX × ΩlX → R,

〈α, β〉 :=

∫X

α ∧ ?β (6.3.7)

restricted to horizontal forms. i.e. for any β ∈ ΩlH, 0 ≤ l ≤ 2, we have ?β = κ ∧ ?Hβ.

We then have α ∧ ?β = (−1)lκ ∧ [α ∧ ?Hβ] for any α, β ∈ ΩlH, 0 ≤ l ≤ 2. Thus,

〈·, ·〉lκ = 〈·, ·〉 on ΩlH and therefore defines an inner product.

By our definition, we may now write S(A) = 〈dHA, dHA〉2κ. We make the following

6.3.8 Definition. Define the formal adjoint of dH , denoted d∗H , via:

〈d∗Hγ, φ〉l−1κ = 〈γ, dHφ〉lκ

for γ ∈ Ωl(H), φ ∈ Ωl−1(H) where l = 1, 2 and d∗Hγ = 0 for γ ∈ Ω0(H).


6.3.9 Proposition. d∗H = (−1)l ?H dH?H : Ωl(H) → Ωl−1(H), 0 ≤ l ≤ 2, where

Ω−1(H) := 0.

Proof. This just follows from the definition of d∗ relative to the ordinary inner product

〈·, ·〉, and the facts that 〈·, ·〉l−1κ is just this ordinary inner product restricted to horizontal

forms and d∗ = (−1)l ? d?.

Thus, we may now write S(A) = 〈A, d∗HdHA〉1κ and identify this piece of the action with

the second order operator d∗HdH on horizontal forms.

Now we turn our attention to the Chern-Simons part of the action CS(A) =∫XA ∧ dA.

We would like to reformulate this in terms of horizontal quantities as well. This is

straightforward to do; simply observe that dA = κ∧LξA+ dHA (See Prop. C.1.27 for a

proof). Thus, we have:

CS(A) =

∫X

A ∧ dA (6.3.10)

=

∫X

A ∧ [κ ∧ LξA+ dHA] (6.3.11)

=

∫X

A ∧ [κ ∧ LξA] +

∫X

A ∧ dHA (6.3.12)

=

∫X

A ∧ [κ ∧ LξA] (6.3.13)

where the last line follows from the fact that A∧dHA = 0 since both forms are horizontal.

Putting this all together, we may now express the total action S(A) in terms of horizontal

quantities as follows:

S(A) = CS(A)− S(A)

=

∫X

A ∧ [κ ∧ LξA] +

∫X

A ∧ [κ ∧ dH ?H dHA]

=

∫X

A ∧ [κ ∧ (Lξ + dH ?H dH)A]


6.4 The Contact Operator D

A surprising observation is that κ ∧ (Lξ + dH ?H dH) turns out to be well known. It is

the second order operator “D” that fits into the complex,


D−→ Ω2(V )dH−→ Ω3(X) (6.4.1)

where,

Ω2(V ) := κ ∧ α | α ∈ Ω1(H) = κ ∧ Ω1(H), (6.4.2)

and for f ∈ C∞(X), dHf ∈ Ω1(H) stands for the restriction of df to H as usual, while

dH : Ω2(V )→ Ω3(X) (6.4.3)

is just de Rham’s differential restricted to Ω2(V ) in Ω2(X).

6.4.4 Definition. Denote the complex in Eq. (6.4.1) as (E , dH). Thus, we denote

D := dH in the middle degree.12

D is defined as follows:13 Since d induces an isomorphism

d0 : Ω1(V )→ Ω2(H), with d0(fκ) = fdκ|Λ2(H) (6.4.5)

then any α ∈ Ω1(H) admits a unique extension l(α) in Ω1(X) such that dl(α) belongs to

Ω2(V ); i.e. given any initial extension α of α, one has

l(α) = α− d−10 (dα)|Λ2(H) (6.4.6)

We then define

Dα := dl(α) (6.4.7)

12Note that we have abused notation in a few places in this thesis regarding the operator dH in middledegree. We will only view D = dH in terms of the complex (E , dH), and otherwise dH : Ω1(X)→ Ω2(H)is meant to denote the operator d : Ω1(X) → Ω2(X) followed by the projection of Ω2(X) onto Ω2(H).That is, dH : Ω1(X) → Ω2(H) is defined as dH := π d where π : Ω2(X) → Ω2(H) is the projectiondefined in Eq. (C.1.13). The notation should also be clear from the context.

13See [BHR07, §6].


We then have, [BHR07, Eq. 39]:

Dα = κ ∧ [Lξ + dH ?H dH ]α (6.4.8)

for any α ∈ Ω1(H). Thus,

S(A) =

∫X

A ∧ [κ ∧ (Lξ + dH ?H dH)A] (6.4.9)

=

∫X

A ∧DA (6.4.10)

= 〈A, ?DA〉 (6.4.11)

where 〈·, ·〉 is the usual L2 inner product on Ω1(X).

Alternatively, we make the following

6.4.12 Definition. Let D1 : Ω1(H)→ Ω1(X) denote the operator

D1 := Lξ + dH ?H dH (6.4.13)

and observe that we can also write S(A) = 〈A, ?HD1A〉1κ, identifying S(A) with the

operator ?HD1 on Ω1(H). Thus, we have proven the following

6.4.14 Proposition. The new action, S(A), as defined in Eq. (4.4.18), for the “shifted”

partition function of Eq. (6.3.1) can be expressed as a quadratic form on the space of

horizontal forms Ω1(H) as follows:

S(A) = 〈A, ?DA〉 (6.4.15)

or equivalently as,

S(A) = 〈A, ?HD1A〉1κ (6.4.16)

where D and D1 are the second order operators defined in Eq.’s 6.4.8 and 6.4.13, re-

spectively. 〈·, ·〉 is the usual L2 inner product on Ω1(X), and 〈·, ·〉1κ is defined in Eq.

(6.3.5).


6.5 Gauge Group and the Isotropy Subgroup

In order to extract anything mathematically meaningful out of this construction we will

need to divide out the action of the gauge group GP on AP . At this point we observe

that the gauge group GP ' Maps(X → U(1)) naturally descends to a “horizontal” action

on AP , which infinitesimally can be written as:

θ ∈ Lie(GP ) : A 7→ A+ dHθ (6.5.1)

Following [Sch79b], we let HA denote the isotropy subgroup of GP at a point A ∈ AP .

Note that HA can be canonically identified for every A ∈ AP , and so we simply write

H for the isotropy group. The condition for an element of the gauge group h(x) = eiθ(x)

to be in the isotropy group is that dHθ = 0, given definition 6.5.1 above. By [Rum94,

Prop. 12], we see that Lξθ = 0 since θ is harmonic.14 Therefore we have dθ = 0 since

d = dH + κ ∧ Lξ. Thus, the group H can be identified with the group of constant maps

from X into U(1); hence, is isomorphic to U(1). We let Vol(H) denote the volume of the

isotropy subgroup, computed with respect to the metric induced from GP , so that

Vol(H) =

[∫X

κ ∧ dκ]1/2

=

[n+

N∑j=1

βjαj

]1/2

, (6.5.2)

where [n; (α1, β1), . . . , (αN , βN)] are the Seifert invariants of our Seifert manifold X (See

§2.3). The last equality in Eq. (6.5.2) above follows from Eq. 3.22 of [BW05]. Let us

briefly derive Eq. (6.5.2). Let u, v ∈ Lie(GP ) ' Ω0(X) and recall that the metric on GP

is given by:

G(u, v) :=

∫X

u ∧ ?v. (6.5.3)

To calculate the volume of H ' U(1) < GP we let dΘ denote the standard measure on

U(1) such that∫U(1)

dΘ = 1 and obtain the induced measure√GΘdΘ on U(1) < GP .

14Note that θ is harmonic since dHθ = 0. We leave the details of this fact to the reader, who shouldconsult the actual proof of [Rum94, Prop. 12].


Since u, v ∈ H are viewed as constant functions in Lie(GP ) ' Ω0(X), we have

G(u, v) :=

∫X

u ∧ ?v = uv

∫X

?1 = uv

∫X

κ ∧ dκ. (6.5.4)

So√GΘ is constant on TH and may be identified as

√GΘ =

[∫Xκ ∧ dκ

]1/2. Then,

Vol(H) :=

∫U(1)

√GΘdΘ,

=√GΘ

∫U(1)

dΘ,

=√GΘ, since

∫U(1)

dΘ = 1,

=

[∫X

κ ∧ dκ]1/2

,

and we have justified Eq. (6.5.2).

6.6 The Partition Function

We now have


Vol(GP )

∫AP

DA e[ik4πS(A)]

=Vol(GP )

Vol(H)

eπikSX,P (AP )

Vol(GP )

∫AP /GP

e[ik4πS(A)] [det′(d∗HdH)]

1/2µ

=eπikSX,P (AP )

Vol(H)

∫AP /GP

e[ik4πS(A)] [det′(d∗HdH)]

1/2µ (6.6.1)

where µ is the induced measure on the quotient space AP/GP and det′ denotes a regu-

larized determinant to be defined later. Since S(A) = 〈A, ?DA〉 is quadratic in A, we

may apply the method of stationary phase15 to evaluate the oscillatory integral (6.6.1)

exactly. We obtain,

(6.6.2)


Vol(H)

∫MP

eπi4

sgn(?D) [det′(d∗HdH)]1/2

[det′(k ? D)]1/2

ν

15Our main references for the method of stationary phase are [GS77], [Sch79a] and [Sch79b]. SeeAppendix A.1 for more details.


whereMP denotes the moduli space of flat connections modulo the gauge group16 and ν

denotes the induced measure on this space. Note that we have included a factor of k in

our regularized determinant since this factor occurs in the exponent multiplying S(A).

6.7 Zeta Function Determinants

We will use the following to define the regularized determinant of k ? D

6.7.1 Proposition. [Sch79b] Let H0, H1 be Hilbert spaces, and S : H1 → H1 and

T : H0 → H1 such that S2 and TT ∗ have well defined zeta functions with discrete spectra

and meromorphic extensions to C that are regular at 0 (with at most simple poles on

some discrete subset). If ST = 0, and S2 is self-adjoint, then

det′(S2 + TT ∗) = det′(S2) det′(TT ∗) (6.7.2)

Proof. This equality follows from the facts that S2TT ∗ = 0 and TT ∗S2 = 0 (i.e. these

operators commute), which both follow from ST = 0 and the fact that S2 and TT ∗ are

both self-adjoint.

Following the notation of Eq.’s (3)-(6) in section 2 of [Sch79b], we set the operators

S = k ?D and T = kdHd∗H on Ω1(H) and observe that ST = 0 since (6.4.1) is a complex.

With Prop. 6.7.1 as motivation, we make the formal definition

det′(k ? D) := C(k, J) · [det′(S2 + TT ∗)]1/2

[det′(TT ∗)]1/2(6.7.3)

where S2 + TT ∗ = k2(D∗D + (dHd∗H)2), TT ∗ = k2(dHd

∗H)2 and

C(k, J) := k(− 11024

∫X R2 κ∧dκ) (6.7.4)

is a function of R ∈ C∞(X), the Tanaka-Webster scalar curvature of X, which in turn

depends only on a choice of a compatible complex structure J ∈ End(H). That is, given

16See Def. 2.5.57 for a definition of MP .


a choice of contact form κ ∈ Ω1(X), the choice of complex structure J ∈ End(H) de-

termines uniquely an associated metric (See Prop. 2.2.15). We have defined det′(k ? D)

in this way to eliminate the metric dependence that would otherwise occur in the k-

dependence of this determinant. The motivation for the definition of the factor C(k, J)

comes explicitly from Prop. 6.7.13 below.

The operator

∆ := D∗D + (dHd∗H)2 (6.7.5)

is actually equal to the middle degree Laplacian defined in Eq. (10) of [RS08] and has

some nice analytic properties.17 In particular, it is maximally hypoelliptic and invertible

in the Heisenberg symbolic calculus.18 We define the regularized determinant of ∆ via

its zeta function19

ζ(∆)(s) :=∑

λ∈spec∗(∆)

λ−s. (6.7.6)

Note that our definition agrees with [RS08] up to a finite constant term, dimH1(E , dH).20

This is a consequence of the following

6.7.7 Proposition. [RS08, Prop. 2.2] Let (X, κ) be a contact three-manifold. The con-

tact complex (E , dH), defined in Def. 6.4.4 above, forms a resolution of the constant sheaf

R and its cohomology therefore coincides with the de Rham cohomology of X. Moreover,

the natural projection π : Ωk(X) → Ek,21 for k ≤ 1, and inclusion i : Ek → Ωk(X), for

k ≥ 2, induce an isomorphism between the two cohomologies.

Also, ζ(∆)(s) admits a meromorphic extension to C that is regular at s = 0 (See

17See Eq. (6.9.3) below for the full definition, and also Chapter 3 for some more background.18See Prop. 3.3.1 and also §3.2 for the relevant definitions.19We follow [RS08, Pg. 10] here.20Recall Def. 6.4.4 for a definition of the complex (E , dH). Also, it is noted in [RS08, Pg. 11] that

dimH1(E , dH) is finite by hypoellipticity.21Define π(α) := α− ιξ(α)κ, for α ∈ Ωk(X), for k ≤ 1.


Prop. 3.3.2). Thus, we define the regularized determinant of ∆ as

det′(∆) := e−ζ′(∆)(0) (6.7.8)

Let ∆0 := (d∗HdH)2 on Ω0(X), ∆1 := ∆ on Ω1(H) and define ζi(s) := ζ(∆i)(s). We claim

the following

6.7.9 Proposition. For any real number 0 < c ∈ R,

det′(c∆i) := cζi(0) det′(∆i) (6.7.10)

for i = 0, 1.

Proof. To prove this claim, recall that ζi(s) = ζ(∆i)(s) for i = 0, 1, scale as follows:

ζ(c∆i)(s) = c−sζ(∆i)(s). (6.7.11)

From here we simply calculate the scaling of the regularized determinants using the

definition

det′(∆i) := e−ζ′(∆i)(0) (6.7.12)

and the claim is proven.

The following will be useful.

6.7.13 Proposition. For ∆0 := (d∗HdH)2 on Ω0(X), ∆1 := ∆ on Ω1(H) (See. Eq.

(6.7.5)) and ζi(s) := ζ(∆i)(s), we have

ζ0(0)− ζ1(0) =

(− 1

512

∫X

R2 κ ∧ dκ)

+ dim Ker∆1 − dim Ker∆0 (6.7.14)

=

(− 1

512

∫X

R2 κ ∧ dκ)

+ dimH1(E , dH)− dimH0(E , dH).

where R ∈ C∞(X) is the Tanaka-Webster scalar curvature of X and κ ∈ Ω1(X) is our

chosen contact form as usual.


Proof. Let

ζ0(s) := dim Ker∆0 + ζ0(s)

ζ1(s) := dim Ker∆1 + ζ1(s)

denote the zeta functions as defined in [RS08]. From Prop. 3.3.4, one has that

ζ1(0) = 2ζ0(0)

for all 3-dimensional contact manifolds. By Theorem 3.3.6, one knows that on CR-Seifert

manifolds that

ζ0(0) = ζ(∆0)(0) = ζ(∆H)(0) =1

512

∫X

R2 κ ∧ dκ

Thus,

ζ1(0) =1

256

∫X

R2 κ ∧ dκ

By our definition of the zeta functions, which differ from that of [RS08] by constant

dimensional terms, we therefore have

ζ0(0) =1

512

∫X

R2 κ ∧ dκ− dim Ker∆0

ζ1(0) =1

256

∫X


Hence,

ζ0(0)− ζ1(0) =

[1

512

∫X


]−[

1

256

∫X


]=

(− 1

512

∫X

R2 κ ∧ dκ)

+ dim Ker∆1 − dim Ker∆0

=

(− 1

512

∫X

R2 κ ∧ dκ)

+ dimH1(E , dH)− dimH0(E , dH).

and the result is proven.

We now have the following


6.7.15 Proposition. The term inside of the integral of Eq. (6.6.2) has the following

expression in terms of the hypoelliptic Laplacians, ∆0 and ∆1, as defined in Prop. 6.7.13:

[det′(d∗HdH)]1/2

[det′(k ? D)]1/2

= knX[det′(∆0)]1/2

[det′(∆1)]1/4

(6.7.16)

where

nX :=1

2(dimH1(E , dH)− dimH0(E , dH)). (6.7.17)

Proof.

(6.7.18)

[det′(d∗HdH)]1/2

[det′(−k ?H D1κ)]

1/2= C(k, J)−1 · [det′(d∗HdH)2]

1/4 · [det′ k2(dHd∗H)2]

1/4

[det′(k2∆)]1/4

= C(k, J)−1 · kζ0(0)/2 [det′(∆0)]

1/4 · [det′(∆0)]1/4

kζ1(0)/2 [det′(∆1)]1/4

(6.7.19)

= C(k, J)−1 · k12

(ζ0(0)−ζ1(0)) [det′(∆0)]1/2

[det′(∆1)]1/4

= C(k, J)−1 · C(k, J) · knX [det′(∆0)]1/2

[det′(∆1)]1/4, Prop. 6.7.13,

= knX[det′(∆0)]1/2

[det′(∆1)]1/4

where the second last line comes from Eq. (6.7.14). Also note that d∗HdH and dHd∗H

have the same eigenvalues (by standard arguments), which allows us to proceed to Eq.

(6.7.19) from Eq. (6.7.18).

6.7.20 Remark. Note that by Prop. 6.7.7, the definition of nX (see Eq. (6.7.17)) here

is exactly equal to the quantity mX := 12(dimH1(X, d) − dimH0(X, d)) of [Man98, Eq.

5.18]. This shows that our partition function has the same k-dependence as that in

[Man98].


6.8 The Eta-Invariant

Next we regularize the signature sgn(?D) via the eta-invariant and set sgn(?D) =

η(?D)(0) := η(?D) where

η(?D)(s) :=∑

λ∈spec∗(?D)

(sgnλ)|λ|−s (6.8.1)

Finally, we may now write the result for our partition function

(6.8.2)

ZU(1)(X, p, k) = knXeπikSX,P (A0)eπi4η(?D)

∫MP

1

Vol(H)

[det′(∆0)]1/2

[det′(∆1)]1/4

ν

where nX := 12(dimH1(E , dH) − dimH0(E , dH)). Note that ν is a measure on MP

22

relative to the horizontal structure on the tangent space of MP .

6.9 Torsion

Now we will study the quantity 1Vol(H)

[det′(∆0)]1/2

[det′(∆1)]1/4ν inside of the integral in Eq. (6.8.2),

and in particular how it is related to the analytic contact torsion TC . First, recall that,

[RS08, Eq. 16]:

TC := exp

(1

4

3∑q=0

(−1)qw(q)ζ ′(∆q)(0)

)(6.9.1)

where

w(q) =

q if q ≤ 1,

q + 1 if q > 1.

(6.9.2)

22MP denotes the moduli space of flat connections modulo the gauge group. See Def. 2.5.57.


in the case where dim(X) = 3. Note that we have chosen a sign convention that leads to

the inverse of the definition of TC in [RS08]. Recall, [RS08, Eq. 10]:

∆q =

(d∗HdH + dHd

∗H)2 if q = 0, 3,

D∗D + (dHd∗H)2 if q = 1.

DD∗ + (d∗HdH)2 if q = 2.

(6.9.3)

We would, however, like to work with torsion when viewed as a density on the determinant

line

| detH•(E , dH)∗| := | detH0(E , dH)| ⊗ | detH1(E , dH)∗|

⊗ | detH2(E , dH)| ⊗ | detH3(E , dH)∗|

We follow [RS73] and [Man98] and make the analogous definition.

6.9.4 Definition. Define the analytic torsion as a density as follows

T dC := TC · δ| detH•(E,dH)|

where TC is as defined in Eq. (6.9.1), and

δ|detH•(E,dH)| := ⊗dimXq=0 |νq1 ∧ · · · ∧ ν

qbq|(−1)q

where νq1 , · · · , νqbq is an orthonormal basis for the space of harmonic contact forms

Hq(E , dH) with the inner product defined in Eq. (6.3.5). Note that Hq(E , dH) is canon-

ically identified with the cohomology space Hq(E , dH), and bq := dim(Hq(E , dH)) is the

qth contact Betti number.

Let

ν(q) := νq1 ∧ · · · ∧ νqbq

and write the analytic torsion of a compact connected Seifert 3-manifold X as

T dC = TC × |ν(0)| ⊗ |ν(1)|−1 ⊗ |ν(2)| ⊗ |ν(3)|−1. (6.9.5)


In terms of regularized determinants, we have

TC =[(det′(∆0))0 · (det′(∆1))1 · (det′(∆2))−3 · (det′(∆3))4

]1/4(6.9.6)

where ∆q, 0 ≤ q ≤ 3, denotes the Laplacians on the contact complex as defined in [RS08,

Eq. 10] and recalled in Eq. (6.9.3) above. This notation agrees with our notation for

∆0, ∆1 as in Eq. (6.7.10). The Hodge ?-operator induces the equivalences ∆q ' ∆3−q

and allows us to write

TC =[(det′(∆0))0 · (det′(∆1))1 · (det′(∆2))−3 · (det′(∆3))4

]1/4(6.9.7)

=det′(∆0)

(det′(∆1))1/2(6.9.8)

Also, from the isomorphisms Hq(X,R) ' Hq(E , dH) of Prop. 6.7.7, we have Poincare

duality Hq(E , dH) ' H3−q(E , dH)∗, and therefore

T dC = TC × |ν0|⊗2 ⊗ (|ν1|−1)⊗2 (6.9.9)

Moreover, by Prop. 3.3.7, Hq(E , dH) = Hq(X,R) ⊂ Ωq(X), and thus any orthonormal

basis ν(0) of H0(E , dH) ' R is a constant such that

|ν(0)| =[∫

X

κ ∧ dκ]−1/2

. (6.9.10)

Let us briefly justify Eq. (6.9.10). This is a direct consequence of the definition of the

L2 metric || · ||L2 on H0(E , dH) and the fact that we are taking ||ν(0)||L2 = 1. That is,

consider

1 = ||ν(0)||2L2 ,

:=

∫X

ν(0) ∧ ?ν(0),

= |ν(0)|2∫X

?1,

= |ν(0)|2∫X

κ ∧ dκ.

Thus, |ν(0)| =[∫Xκ ∧ dκ

]−1/2and we have justified Eq. (6.9.10). Also, recall that the

tangent space TAMP ' H1(E , dH) ' H1(X,R), at any point A ∈MP . The measure ν on


MP that occurs in Eq. (6.8.2) is defined relative to the metric on H1(E , dH) ' H1(E , dH),

which can be identified with the usual L2-metric on forms. Thus the measure ν may

be identified with the inverse of the density |ν(1)| by dualizing the orthogonal basis

ν11 , . . . , ν

1b1 for H1(E , dH); i.e.

ν = |ν(1)|−1 = |ν11 ∧ · · · ∧ ν1

b1|−1 (6.9.11)

Putting together equations 6.9.8, 6.9.10, 6.9.11 into Eq. (6.9.9), we have

T dC = TC × |ν0|⊗2 ⊗ (|ν1|−1)⊗2 (6.9.12)

=det′(∆0)

(det′(∆1))1/2·[∫

X

κ ∧ dκ]−1

ν⊗2 (6.9.13)

= Vol(H)−2 det′(∆0)

(det′(∆1))1/2· ν⊗2 (6.9.14)

We have thus proven the following,

6.9.15 Proposition. The contact analytic torsion, when viewed as a density T dC as in

definition 6.9.4, can be identified as follows:

(T dC)1/2 =1

Vol(H)

[det′(∆0)]1/2

[det′(∆1)]1/4

ν (6.9.16)

Our partition function is now

ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4η(?D)

∫MP

(T dC)1/2 (6.9.17)

This partition function should be completely equivalent to the partition function defined

in [Man98, Eq. 7.27]:

ZU(1)(X, p, k) = kmXeπikSX,P (AP )eπi4η(?d)

∫MP

(T dRS)1/2. (6.9.18)

Our goal in the remainder is to show that this is indeed the case. Our first observation is

that (T dC)1/2 is equal to the Ray-Singer torsion (T dRS)1/2 that occurs in [Man98, Eq. 7.27].

This follows directly from [RS08, Theorem 4.2]; note that their sign convention makes

TC the inverse of our definition (See Prop. C.2.21 for a proof and §C.2 in general).


6.10 Regularizing the Eta-Invariants

Since we have seen that our k-dependence matches that in [Man98] (i.e. mX = nX ; cf.

Remark 6.7.20), the only thing left to do is to reconcile the eta invariants, η(?D) and

η(?d). As observed in [Wit89], the correct quantity to consider is

η(?d)

4+

1

12

CS(Ag)

2π. (6.10.1)

where,

CS(Ag) =1

4π

∫X

Tr(Ag ∧ dAg +2

3Ag ∧ Ag ∧ Ag) (6.10.2)

is the gravitational Chern-Simons term, with Ag the Levi-Civita connection on the spin

bundle of X for a given metric g on X. See §4.3 for an exposition on the regularization

of η(?d) in Eq. (6.10.1).23 It was noticed in [Wit89] that in the quasi-classical limit,

quantum anomalies can occur that can break topological invariance. Invariance may be

restored in this case only after adding a counterterm to the eta invariant. Our job then

is to perform a similar analysis for the eta invariant η(?D), which depends on a choice

of metric. Of course, our choice of metric is natural in this setting and is adapted to the

contact structure. One possible approach is to consider variations over the space of such

natural metrics and calculate the corresponding variation of the eta invariant, giving us

a local formula for the counterterm that needs to be added. Such a program has already

been initiated in [BHR07].

Our starting point is the conjectured equivalence that results from the identification

of Eq.’s 6.9.17 and 6.9.18:

eπi[η(?d)

4+ 1

12CS(Ag)

2π

]“=”e

πi4

[η(?D)+CT ] (6.10.3)

where CT is some appropriate counterterm that yields an invariant comparable to the

left hand of this equation. As noted in §4.3, the left hand side of this equation depends

23See Eq. (4.3.25) in particular.


on a choice of 2-framing on X, and since we have a rule (See Eq. (4.3.26)) for how the

partition function transforms when the framing is twisted, we basically have a topological

invariant. Alternatively, as also noted in §4.3, one can use the main result of [Ati90b]

and fix the canonical 2-framing on TX ⊕ TX. We therefore expect the same type of

phenomenon for the right hand side of this equation, having at most a Z-dependence on

the regularization of our eta invariant, along with a rule that tells us how the partition

function changes when our discrete invariants are “twisted,” once again yielding a topo-

logical invariant.

Let us first make the statement of the conjecture of Eq. (6.10.3) more precise. We

should have the following

6.10.4 Conjecture. (X,φ, ξ, κ, g) a closed quasi-regular K-contact three-manifold. Then

there exists a counterterm, CT , such that

eπi4

[η(?D)+CT ]

is a topological invariant that is identically equal to the topological invariant

eπi[η(?d)

4+ 1

12CS(Ag)

2π

],

where CS(Ag) and all relevant operators are defined with respect to the metric g on X

and we use the canonical 2-framing [Ati90b].

Our regularization procedure for η(?D) will be quite different than that used for

η(?d). Since we are restricted to a class of metrics that are compatible with our contact

structure, we are really only concerned with finding appropriate counterterms for η(?D)

that will eliminate our dependence on the choice of contact form κ and complex structure

J ∈ End(H). In the case of interest, we observe that our regularization may be obtained

in one stroke by introducing the renormalized η-invariant, η0(X, κ), of X that is discussed

in [BHR07, §3]. Before giving the definition of η0(X, κ), we require the following


6.10.5 Lemma. [BHR07, Lemma 3.1] Let (X, J, κ) be a strictly pseudoconvex pseudo-

hermitian 3-manifold. The η-invariants of the family of metrics gε := ε−1κ⊗κ+dκ(·, J ·)

have a decomposition in homogeneous terms:

η(gε) =2∑

i=−2

ηi(X, κ)εi. (6.10.6)

The terms ηi for i 6= 0 are integrals of local pseudohermitian invariants of (X, κ), and

the ηi for i > 0 vanish when the Tanaka-Webster torsion, τ , vanishes.

We then make the following

6.10.7 Definition. Let (X, κ) be a compact strictly pseudoconvex pseudohermitian 3-

dimensional manifold. The renormalized η-invariant η0(X, κ) of (X, κ) is the constant

term in the expansion of Eq. (6.10.6) for the η-invariants of the family of metrics gε :=

ε−1κ⊗ κ+ dκ(·, J ·).

Our assumption that X is K-contact ensures that the Reeb flow preserves the metric.

In this situation, it is known that the Tanaka-Webster torsion necessarily vanishes.24 In

the case where the torsion of (X, κ) vanishes, the terms ηi(X, κ) in Eq. (6.10.6) vanish

for i > 0, so that when ε→∞, one has

η0(X, κ) = limε→∞

η(gε) := ηad (6.10.8)

The limit ηad is known as the adiabatic limit and has been studied in [BC89] and [Dai91],

for example. The adiabatic limit is the case where the limit is taken as ε goes to infinity,

ηad := limε→∞

η(gε), (6.10.9)

while the renormalized η-invariant, η0(X, κ), is naturally interpreted as the constant term

in the asymptotic expansion for (η(gε)) in powers of ε, when ε goes to 0. This reverse

process of taking ε to 0 is also known as the diabatic limit. When torsion vanishes (i.e.

24See [BHR07, §3].


when the Reeb flow preserves the metric), Eq. (6.10.8) is the statement that the diabatic

and adiabatic limits agree. One of the main challenges for our future work will be to

extend beyond the case where torsion vanishes. This will naturally involve the study of

the diabatic limit. For now, we are restricted to the case of vanishing torsion. In this

case, the main result that we will use is the following

6.10.10 Theorem. [BHR07, Theorem 1.4] Let X be a compact CR-Seifert 3-manifold,

with U(1)-action generated by the Reeb field of an U(1)-invariant contact form κ. If R

is the Tanaka-Webster curvature of (X, κ) and D is the middle degree operator of the

contact complex (cf. Eq. (6.4.1) and 6.4.13), then

η0(X, κ) = η(?D) +1

512

∫X

R2 κ ∧ dκ. (6.10.11)

Theorem 6.10.10 compels us to conjecture that CT = 1512

∫XR2 κ∧dκ. Our motivation

for this comes from the fact that η0(X, κ) is a topological invariant in our case. We have

the following,

6.10.12 Theorem. [BHR07, Remark 9.6 and Eq. 27] If X is a CR-Seifert manifold,

then η0(X, κ) is a topological invariant and

η0(X, κ) = 1− d

3+ 4

N∑j=1

s(αj, βj), (6.10.13)

where 0 < d ∈ Q is the degree of X as a compact U(1)-orbifold bundle and

s(α, β) :=1

4α

α−1∑k=1

cot

(πk

α

)cot

(πkβ

α

)(6.10.14)

is the classical Rademacher-Dedekind sum, where [n; (α1, β1), . . . , (αN , βN)] (for (αi, βi) =

1 relatively prime) are the Seifert invariants of X.

6.10.15 Remark. Note that we have chosen the opposite sign convention of [BHR07] for

the orbifold first Chern number d = c1(X) in this thesis. Thus, d appears with a plus

sign in Theorem 6.10.12 in [BHR07].


Thus, we are led to consider the natural topological invariant eπi4

[η0(X,κ)] and how it

compares with the topological invariant eπi[η(?d)

4+ 1

12CS(Ag)

2π

]. We consider the limit

limε→∞

eπi[η(?εd)

4+ 1

12CS(Agε )

2π

](6.10.16)

where gε = ε−1κ⊗ κ+ dκ(·, J ·) is the natural metric associated to X. On the one hand,

since this is a topological invariant, and is independent of the metric, we must have

limε→∞

eπi[η(?εd)

4+ 1

12CS(Agε )

2π

]= e

πi[η(?d)

4+ 1

12CS(Ag)

2π

]. (6.10.17)

where we take g1 := g so that ?g1 := ?.

On the other hand, since η(gε) = η(?εd) by definition, and we know that its limit

exists as ε→∞ (in fact η0(X, κ) = limε→∞ η(gε)), we have

limε→∞

eπi[η(?εd)

4+ 1

12CS(Agε )

2π

]= e

πi[η0(X,κ)

4+

limε→∞112

CS(Agε )2π

]. (6.10.18)

Thus, we have

eπi[η(?d)

4+ 1

12CS(Ag)

2π

]= e

πi[η0(X,κ)

4

]eπi

limε→∞112

CS(Agε )2π

. (6.10.19)

We therefore see that if we can understand the limit limε→∞112

CS(Agε )2π

, we will obtain

crucial information for our problem. The following has been established using a “Kaluza-

Klein” dimensional reduction technique modeled after the paper [GIJP03] (see Chapter

7 and Prop. 7.3.40),

6.10.20 Proposition. [McL10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact

three-manifold,

U(1) // X

Σ

.


CS(Agε) =

(ε−1

2

)∫Σ

r ω +

(ε−2

2

)∫Σ

f 2 ω (6.10.21)

25See Equations (7.3.20) and (7.3.21).





limε→∞

CS(Agε) = 0. (6.10.22)

Proposition 6.10.20 combined with Eq. (6.10.19) and Theorem 6.10.10 gives us the

following,


Then27 there exists a counterterm, CT , such that eπi4

[η(?D)+CT ] is a topological invariant

that is identically equal to the topological invariant eπi[η(?d)

4+ 1

12CS(Ag)

2π

]. In fact, we have

CT =1

512

∫X

R2 κ ∧ dκ,


Prop. 6.10.23 is the final equivalence that was needed to establish our main equiva-

lence of the partition functions Z and Z. We thus have the following


If,


512

∫X R2 κ∧dκ)

∫MP

(T dC)1/2 (6.10.25)

and (see §4.3),


4+ 1

12CS(Ag)

2π

) ∫MP

(T dRS)1/2 (6.10.26)

26In a “special coordinate system,” x0, x1, x2 on U ⊂ X, κ = ϕ0dx0 + ϕ1dx

1 + dx2, and dκ =(∂0ϕ1 − ∂1ϕ0)dx0 ∧ dx1 = f01dx

0 ∧ dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor,and fαβ =

√h εαβ f, where f ∈ Ω0




then,28

ZU(1)(X, k) = ZU(1)(X, k)


Given Proposition 6.10.23 and Theorem 6.10.12, we conclude the following as an

immediate consequence,


Then,29

η(?d) +1

3

CS(Ag)

2π= η(?D) +

1

512

∫X

R2 κ ∧ dκ

= 1− d

3+ 4

N∑j=1

s(αj, βj),

where 0 < d = c1(X) = c1(L) = n+∑N

j=1βjαj∈ Q and

s(α, β) :=1

4α

α−1∑k=1

cot

(πk

α

)cot

(πkβ

α

)∈ Q





3+4∑Nj=1 s(αj ,βj))

∫MP

(T dC)1/2,


3+4∑Nj=1 s(αj ,βj))

∫MP

(T dRS)1/2.

Finally, we are justified in making the following

6.10.28 Definition. Let k ∈ Z be an (even) integer, and (X,φ, ξ, κ, g) a closed, quasi-

regular K-contact three-manifold. The shifted U(1) Chern-Simons partition function,




ZU(1)(X, k), is the rigorous quantity

ZU(1)(X, k) =∑

p∈TorsH2(X;Z)

ZU(1)(X, p, k) (6.10.29)

where,

ZU(1)(X, p, k) := knXeπikSX,P (AP )eπi4 (η(?D)+ 1

512

∫X R2 κ∧dκ)

∫MP

(T dC)1/2 (6.10.30)

where nX = 12(dimH1(E , dH)− dimH0(E , dH)).

Of course, Prop. 6.10.24 is the statement that Def.’s 4.3.22 and 6.10.28 agree.

Chapter 7

Gravitational Chern-Simons and the

Adiabatic Limit

In this chapter we compute the gravitational Chern-Simons term explicitly for an adi-

abatic family of metrics using standard methods in general relativity. We use the fact

that our base three-manifold is a quasi-regular K-contact manifold heavily in this com-

putation. Our key observation is that this geometric assumption corresponds exactly to

a Kaluza-Klein Ansatz for the metric tensor on our three manifold, which allows us to

translate our problem into the language of general relativity. Similar computations have

been performed in [GIJP03], although not in the adiabatic context.

7.1 Introduction

The primary goal of this chapter is to explicitly compute the adiabatic limit

limε→0

CS(Agε)

2π, (7.1.1)

where,

CS(Agε) :=1

4π

∫X

Tr(Agε ∧ dAgε +2

3Agε ∧ Agε ∧ Agε), (7.1.2)

130

Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 131

is the gravitational Chern-Simons term with Agε the Levi-Civita connection on the spin

bundle of X for the family of metrics,

gε = ε κ⊗ κ+ dκ(·, J ·), (7.1.3)

on a quasi-regular K-contact three manifold, (X,φ, ξ, κ, g).

The main motivation for the considerations of this chapter come directly from our work

in Chapter 6. In particular, this chapter provides one possible proof of Prop. 6.10.20,

which is the key to establishing Prop.’s 6.10.23, 6.10.24 and 6.10.27. In §6.10, we are

naturally led to compute the adiabatic limit of the regularized eta-invariant,

η(?εd)

4+

1

12

CS(Agε)

2π, (7.1.4)

by considering the limits of each term in Eq. (7.1.4) separately. By the Atiyah-Patodi-

Singer theorem Eq. (7.1.4) is a topological invariant,1 and one of our objectives in

Chapter 6 was to compute this topological invariant explicitly. It turns out that the

computation of adiabatic limit of Eq. (7.1.4) is all that is required to obtain this explicit

identification. This fact led us to study the limit of Eq. (7.1.1).

7.1.5 Remark. In this chapter we consider ε as opposed to ε−1 in our computations (as

in Eq.’s (7.1.1), (7.1.2) and (7.1.3)). This is simply a matter of notational convenience.

The gravitational Chern-Simons term was first introduced in the physics literature

by S. Deser, R. Jackiw, G. ’t Hooft and S. Templeton (cf. [DJT82], for example). The

reduction of the gravitational Chern-Simons term from three to two dimensions was sub-

sequently investigated in [GIJP03] for an abstract three-dimensional space-time. A key

observation in [GIJP03] is that setting a Kaluza-Klein Ansatz for the metric tensor ef-

fects a reduction from three to two dimensions for the gravitational Chern-Simons term.

1Note that Eq. (7.1.4) actually depends on a choice of 2-framing for X, and by a result of Atiyah[Ati90b] there exists a canonical choice of such framing. In our case, we explicitly choose a framing viaa particular choice of Vielbein (see Eq. (7.3.20) below), and work with this throughout this chapter. Inparticular, we may view Eq. (7.1.4) as a topological invariant without ambiguity.


Our first observation is that the natural associated metric for a quasi-regular K-contact

three-manifold satisfies this Ansatz. Although the work in [GIJP03] necessarily computes

the gravitational Chern-Simons term for our class of three-manifolds, we must extend the

results of [GIJP03] to perform this calculation for the adiabatic family of metrics in Eq.

(7.1.3).

We note that the terminology adiabatic limit in this chapter is meant to describe the

limit in Eq. (7.1.1) with respect to the adiabatic family of metrics in Eq. (7.1.3). We

borrow this terminology from [BHR07], where one considers the adiabatic limit for the

family of eta-invariants,

η(?εd), (7.1.6)

where d : Ω1(X)→ Ω2(X) is the standard exterior derivative on forms, ?ε are the Hodge

star operators for the family of metrics in Eq. (7.1.3), and the eta-invariant is defined as

usual,

η(?d)(s) :=∑

λ∈spec∗(?d)

(sgnλ)|λ|−s, (7.1.7)

so that,

η(?d) := η(?d)(0). (7.1.8)

It is shown in [BHR07], for example, that the adiabatic limit,

limε→0

η(?εd), (7.1.9)

exists, and in fact, for the case of quasi-regular K-contact three-manifolds, is a topologi-

cal invariant of X. As is noted in [BHR07], the adiabatic limit has been known for some

time, and has been studied in [BC89] and [Dai91], for example.

We note that our computations are modeled on [GIJP03]. The novelty here is the

introduction of the adiabatic family of metrics in Eq. (7.1.3), for which one must be

careful to keep track of the ε dependence in the explicit computation of the gravitational


Chern-Simons term of Eq. (7.1.2).

The first thing we do is find an explicit formula for the family of metrics in Eq. (7.1.3)

relative to a special coordinate system that is adapted to our geometric situation. We

then observe that the result we obtain is precisely the Kaluza-Klein Ansatz of [GIJP03].

This allows us to carry the rest of our analysis out in parallel with [GIJP03]. We then

compute the Christoffel symbols for the Levi-Civita connection for the family of metrics

in Eq. (7.1.3). Using the formulae for the Christoffel symbols and Vielbein, we compute

the components of the spin connection Agε and directly evaluate CS(Agε). Our compu-

tation yields a formula for CS(Agε) in “reduced” terms as in [GIJP03] and also provides

an explicit identification of the ε-dependence for this quantity. We note that the class of

quasi-regular K-contact three-manifolds are necessarily U(1)-bundles that fiber over an

orbifold surface Σ = Σ (See §2.3). Our main result is the following:

7.1.10 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold,

U(1) // X

Σ

.

Let gε := ε κ⊗ κ+ π∗h. After a particular choice of Vielbein,2 then,

CS(Agε) =( ε

2

)∫Σ

r ω +

(ε2

2

)∫Σ

f 2 ω (7.1.11)


Hodge form of (Σ, h), and f ∈ Ω0orb(X) is the invariant field strength on (Σ, h).3 In


limε→0

CS(Agε) = 0. (7.1.12)



(∂0ϕ1−∂1ϕ0)dx0∧dx1 = f01dx0∧dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor, and

fαβ =√h εαβ f, where f ∈ Ω0

orb(X) is called the invariant field strength on (X,h). See §2.4 for morebackground.


As outlined in chapter 6, this allows us to compute the regularized eta-invariant

η(?εd)

4+

1

12

CS(Agε)

2π, (7.1.13)

explicitly, and in fact is an important step in the computation of the U(1) Chern-Simons

partition function as a topological invariant for the class of quasi-regular K-contact three

manifolds.

7.2 Local Formulation of Gravitational Chern-Simons

Term

In order to perform the computation of the gravitational Chern-Simons term in Eq.

(7.1.2) explicitly, we will adopt the conventions of [GIJP03]; indexing everything in sight

and working in local coordinates. To this end, we express the gravitational Chern-Simons

term in a coordinate system x0, x1, x2 for a local chart U ⊂ X as follows:

CS(Agε) :=1

4π

∫X

d3x εµνλ Tr

((Agε)µ∂ν(A

gε)λ +2

3(Agε)µ(Agε)ν(A

gε)λ

), (7.2.1)

where µ, ν, λ ∈ 0, 1, 2, and εµνλ is the three-dimensional Levi-Civita symbol, normalized

so that ε012 = 1, and defined for any σ ∈ S3 = permutations of 0, 1, 2 by:

εσ(012) := (−1)|σ|, (7.2.2)

where |σ| denotes the sign of σ as a permutation, so that

|σ| :=

0 if σ is even,

1 if σ is odd.

(7.2.3)

Note that repeated indices are allowed, and our definition of εµνλ implies that εµνλ = 0

whenever any of µ, ν, λ ∈ 0, 1, 2 are equal.


Let ∇G : Γ(TX) → Γ(T ∗X ⊗ TX) denote the standard Levi-Civita connection asso-

ciated to the metric G on X. Let

Γλµν :=1

2Gλρ (∂νGρµ + ∂µGρν − ∂ρGµν) (7.2.4)

be the Christoffel symbols relative to the coordinate basis x0, x1, x2 for the Levi-Civita

connection, i.e.

(∇G)∂µ∂ν = Γλµν∂λ. (7.2.5)

Our computation is facilitated by the basic relationship between the spin connection AG,4

and the Levi-Civita connection ∇G, [GIJP03, Eq. 2.17]:

[(AG)µ]AB = EAν E

λBΓνµλ − Eλ

B∂µEAλ . (7.2.6)

where EA ∈ Γ|U(TX) and EA ∈ Γ|U(T ∗X) are the Vielbein defined in §2.4. Our goal

then is to compute the Christoffel symbols for the family of metrics G = gε, which will

give us an explicit formula for the spin connections [(AG)µ]AB. We note that in three

dimensions, the spin connection is anti-symmetric in A,B,

[(AG)µ]AB := ηAC [(AG)µ]CB = −[(AG)µ]BA, (7.2.7)

and we may use this fact to write

[(AG)µ]AB := ηAC [(AG)µ]CB = εABCACµ , (7.2.8)

where ACµ is a vector-valued one-form defined by this relation. We then obtain a slightly

simpler expression for CS(AG), [GIJP03, Eq. 2.22]:

CS(Agε) = − 1

4π

∫X

d3x εµνλ(

2ηABAAµ∂νA

Bλ −

2

3εABCA

AµA

Bν A

Cλ

)(7.2.9)

= − 1

2π

∫X

d3x εµνλ(ηABA

Aµ∂νA

Bλ

)+

1

π

∫X

d3x detAAµ (7.2.10)

Note that we have suppressed the Agε notation in our integrals to just A. Eq. (7.2.8)

will be the result that we use to perform our computation directly.

4Note that we will generally write the notation G for our metric interchangeably with gε. We willneed to explicitly make the identification G = gε later, but for now this simplfies notation.


7.3 Computation of Gravitational Chern-Simons Term

The first quantities that we wish to compute are the Christoffel symbols. Before we

can do this, however, we will need to find a useful expression for our family of metrics,

G = gε, in an “arbitrary” coordinate system x0, x1, x2 on X that reflects our geometric

situation. We follow [Ber47, pg. 265], and introduce a “special coordinate system.” Such

a coordinate system is adapted to our geometric situation in the following sense: We

define our coordinates via the local decomposition of X given by a local trivialization

from its bundle structure

π−1(U) ' U × S1, (7.3.1)

where U ⊂ Σ is any open subset of Σ.

7.3.2 Remark. Note that Σ is an orbifold. All of our considerations are completely valid

for the orbifold case by the results of Ichiro Satake [Sat57]. In particular, the notions

of (co-)tangent bundles, (co-)tangent vectors, forms, curvature, integration, Riemannian

metrics, orthogonal frames, etc... all have rigorously defined orbifold counterparts. See

§2.1 for more details on orbifolds. For example, the decomposition π−1(U) ' U × S1

assumes that U ⊂ R2 is an orbifold coordinate chart on Σ.

If ξ denotes the Reeb vector field on our quasi-regular K-contact manifold, (X,φ, ξ, κ, g),

then our coordinate system is chosen such that ξp = [0, 0, 1], for any p ∈ U , and the

first two coordinates x0, x1 coincide with the coordinates on our base orbifold Σ. We

should note that such a choice of coordinates does not necessarily respect the contact

structure, TX ' H⊕Rξ, of our K-contact manifold (X,φ, ξ, κ, g); i.e. for the coordinates

x0, x1, the associated vector fields ∂∂x0 ,

∂∂x1 are not necessarily horizontal vector fields.

The vector fields ∂∂x0 ,

∂∂x1 may have components in the Reeb direction:

∂

∂xα= hα + ϕα

∂

∂x2, α ∈ 0, 1, (7.3.3)

where hα is the horizontal component of ∂∂xα

, and ϕα∂∂x2 is its vertical component in the

direction of the Reeb field. Clearly, we have chosen our coordinates so that the local


vector field ∂∂x2 coincides with the Reeb direction. We now wish to express our family of

metrics in this coordinate system. By definition:

gε = ε κ⊗ κ+ π∗h. (7.3.4)

Evaluating this in our coordinate system yields:

Gµν = gε =

hαβ + εϕαϕβ εϕα

εϕβ ε

. (7.3.5)

Our matrix is indexed with the understanding that α, β ∈ 0, 1 and µ, ν ∈ 0, 1, 2

index the entire matrix.5

7.3.6 Remark. Note that the K-contact condition (i.e. the Reeb field is Killing for the

metric G = gε) is crucial for our analysis and ensures that the quantities, hαβ, ϕα, are

independent of the third local coordinate x2, [Ber47, Eq. 17.60]. It is precisely this

condition that makes our computation of the gravitational Chern-Simons term feasible.

This result is easy to see in our special coordinate system x0, x1, x2 described above.

Recall the definition of the Lie derivative,

Lξg :=d

dt

∣∣∣t=0φ∗tg, (7.3.7)

where φt : X → X is the flow of ξ. In our coordinate system, φt(x0, x1, x2) = (x0, x1, x2 +

t) and the condition ddt

∣∣∣t=0φ∗tg = 0 is equivalent to d

dt

∣∣∣t=0gφt = d

dt

∣∣∣t=0g(x0, x1, x2 +t) = 0.

Using our explicit expression for the matrix g in Eq. (7.3.5) above we have

d

dt

∣∣∣t=0ϕα(x0, x1, x2 + t) = ∂2ϕα(x0, x1, x2) = 0, (7.3.8)

5We follow [GIJP03] in our notation; letters from the middle Greek alphabet (λ, µ, ν, . . .) will de-note spacetime components on our three-manifold, while beginning Greek letters (α, β, γ, . . .) will de-note spacetime components on our reduced two-manifold. Tangent space components are generallydescribed by Latin letters, upper case (A,B,C, . . .) for three-dimensions and lower case (a, b, c, . . .) fortwo-dimensions.


and,

d

dt

∣∣∣t=0

(hαβ + ϕαϕβ)(x0, x1, x2 + t) = ∂2hαβ + ∂2(ϕαϕβ)

= ∂2hαβ, since ∂2ϕα(x0, x1, x2) = 0 by Eq. (7.3.8),

= 0.

Thus, ∂2ϕα = 0 and ∂2hαβ = 0.

Thus, we can now see that the Kaluza-Klein Ansatz of [GIJP03, Eq. 3.28] is implied

by our geometric situation. Note that our sign conventions differ, and we follow [Ber47,

Eq. 17.53]. Also, it is not difficult to show that our inverse metric is given by:

Gµν =

hαβ −hαδϕδ

−hβδϕδ ε−1 + hδζϕδϕζ

, (7.3.9)

where hαβ denotes the inverse of the (orbifold) Kahler metric h on Σ. After some calcu-

lation, we find that the Christoffel symbols

Γλµν =1

2Gλρ (∂νGρµ + ∂µGρν − ∂ρGµν) , (7.3.10)

for the metric G = gε may be computed as (see B.2.1):

Γδαβ = γδαβ −ε

2hδζ(ϕβfζα + ϕαfζβ) (7.3.11)

Γ2αβ =

1

2(Dαϕβ +Dβϕα) +

ε

2ϕζ(ϕβfζα + ϕαfζβ), (7.3.12)

Γδ2β =ε

2hδζfβζ (7.3.13)

Γ22β =

ε

2ϕζfζβ (7.3.14)

Γδ22 = Γ222 = 0. (7.3.15)

Some explanation of notation is in order. First, α, β, δ, ζ ∈ 0, 1 index the coordinates

on Σ. The γδαβ are the Christoffel symbols for the Levi-Civita connection of the metric h

on Σ:

γδαβ :=1

2hδζ (∂βhζα + ∂αhζβ − ∂ζhαβ) . (7.3.16)


D is the covariant derivative for the Levi-Civita connection of the metric h on Σ:

Dαϕβ := ∂αϕβ − γζαβϕζ . (7.3.17)

fαβ is the “abelian field strength” tensor:

fαβ := ∂αϕβ − ∂βϕα, (7.3.18)

and finally, all two-dimensional indices are raised and lowered with the metric h; i.e.

ϕα := hαζϕζ . (7.3.19)

In order to compute our spin connection using Eq. (7.2.6), we need an explicit formula

for the Vielbein. We choose these as follows:

Eaα = eaα, E

22 =√ε, E2

α =√εϕα, E

a2 = 0 (7.3.20)

Eαa = eαa , E

22 =

1√ε, E2

a = −ϕζ eζa, Eα2 = 0, (7.3.21)

where eaα, eαa are the Vielbein (i.e. Zweibein) for the two-dimensional metric tensor h on

Σ. Note that a, α, ζ ∈ 0, 1 are indices in two-dimensions, and a ∈ 0, 1 denotes the

“tangent space coordinates” that index the Zweibein, as usual. We leave the straight-

forward confirmation that these formulae define a local orthogonal trivialization to the

reader.

Thus, using Eq. (7.2.6),

[Aµ]AB = EAν E

λBΓνµλ − Eλ

B∂µEAλ , (7.3.22)

and our formulae for the Vielbein and the connection components, we may compute the


spin connections [(AG)µ]AB := [Aµ]AB (see B.2.2):

[Aα]ab = eζb

[−Dαe

aζ −

ε

2eaδh

δρϕαfρζ

](7.3.23)

[Aα]a 2 = −[Aα]2 a = ηa b

√ε

2ebδh

δζfαζ (7.3.24)

[A2]ab =ε

2eζbe

aδh

δρfζρ (7.3.25)

[A2]2 a = [A2]a 2 = 0 (7.3.26)

[Aα]22 = [A2]22 = 0 (7.3.27)

where we have lowered the two dimensional index on [Aα]a 2 := ηab[Aα]b2 above. ηab is

the two-dimensional Kronecker pairing. Then using Eq. (7.2.8) we may compute the

quantities ACµ as follows (see B.2.3):

A2α = −ωα −

ε

2fϕα , A2

2 = − ε2f (7.3.28)

Aaα =

√ε

2eaαf , Aa2 = 0 (7.3.29)

where ωα is defined by the relation ηac(ωα)cb =: ωα,ab = εabωα, and (ωα)ab is the spin

connection on Σ:

(ωα)ab := eζb∂αeaζ − eaδ e

ζbγ

δαζ (7.3.30)

= eζbDαeaζ (7.3.31)

Also, f ∈ C∞orb(Σ) is the invariant field strength defined by the relation:

fαβ =√h εαβ f. (7.3.32)

Thus, using Eq.’s (7.3.28) and (7.3.29) and the formula for CS(Agε) given by Eq. (7.2.10),

we find that (see B.2.4):

CS(Agε) = − 1

4π

∫S1

dx2

∫Σ

dx0 ∧ dx1√h(εfr + ε2f 3) (7.3.33)

= −1

2

∫Σ

dx0 ∧ dx1√h(εfr + ε2f 3) (7.3.34)


where we take the volume of S1 to be 2π, r ∈ Ω0orb(Σ) is the (orbifold) scalar curvature,

and ω ∈ Ω2orb(Σ) is the (orbifold) Kahler form of Σ. Recall that the Reeb vector field is

dual to the one-form κ under our metric gε when ε = 1:

κ(·) = g1(ξ, ·). (7.3.35)

In our coordinate system, this means:

κ = ϕ0dx0 + ϕ1dx

1 + dx2. (7.3.36)

We then have:

dκ = (∂0ϕ1 − ∂1ϕ0)dx0 ∧ dx1 = f01dx0 ∧ dx1, (7.3.37)

=√hfdx0 ∧ dx1 (7.3.38)

We have implicitly identified the (orbifold) Kahler form ω on Σ with its pullback under

π : X → Σ here in this coordinate system. Strictly speaking we have:

dκ = π∗ω. (7.3.39)

By reversing the orientation of Σ, and substituting ω in for√hfdx0∧dx1 in Eq. (7.3.34),

we obtain:

7.3.40 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold,

U(1) // X

Σ

.

Let gε := ε κ⊗ κ+ π∗h. After a particular choice of Vielbein,6 then,

CS(Agε) =( ε

2

)∫Σ

r ω +

(ε2

2

)∫Σ

f 2 ω (7.3.41)

6See Equations (7.3.20) and (7.3.21).





limε→0

CS(Agε) = 0. (7.3.42)

7In a “special coordinate system,” x0, x1, x2 on U ⊂ X, κ = ϕ0dx0 + ϕ1dx

1 + dx2, and dκ =(∂0ϕ1−∂1ϕ0)dx0∧dx1 = f01dx

0∧dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor, andfαβ =

√h εαβ f, where f ∈ Ω0

orb(Σ) is called the invariant field strength on (X,h). See §2.4 for morebackground.

Appendix A

Localization

A.1 Stationary Phase

In this section we review a finite dimensional model for the method of stationary phase.

First, we recall the classical case where our functional is non-degenerate and then follow-

ing [GS77], [Sch79a], [Sch79b], and [Man98], we briefly review the case of a degenerate

functional.

Take k ∈ R>0, M a manifold of dimension n = 2l, f ∈ C∞(M), and dx a density

on M . The primary reason why the method of stationary phase is crucial for this thesis

is that it gives us a way to study integrals of the form:

F (k) :=

∫M

eikfdx. (A.1.1)

Of course, the study of such integrals is rigorously understood in the case that M is finite

dimensional.1 In this thesis, the main quantity of interest is the U(1) Chern-Simons path

integral (See §4.3 and §4.4), given heuristically as:

ZU(1)(X, k) =∑

p∈TorsH2(X;Z)

ZU(1)(X, p, k), (A.1.2)

1See [BGV92, §7.4].

143

Appendix A. Localization 144

where,

ZU(1)(X, p, k) =1

Vol(GP )

∫APDAeπikSX,P (A). (A.1.3)

The similarities between Eq.’s (A.1.1) and (A.1.3) are clear. In fact, this analogy is so

strong that we formally use the method of stationary phase to obtain a rigorous definition

of the U(1) Chern-Simons partition function for this thesis (See §4.3 and §6.6).

As a warm up, let us first assume that f ∈ C∞(M) is non-degenerate, meaning that

the set M0 where the differential of f vanishes is finite and the Hessian Hp := ∇pdf of

f is non-degenerate for all p ∈ M0. Let sgn(Hp) be the signature of the quadratic form

Hp. Let T+p , T−p denote the subspaces of TpM on which Hp is positive/negative definite,

respectively. Then sgn(Hp) = dimT+p − dimT−p . The following is standard, [BGV92,

§7.4]:

F (k) =

∫M

eikfdx =∑p∈M0

(2π

k

)leikf(p)eπi sgn(Hp)/4 1

| detHp|1/2+O(k−(l+1)). (A.1.4)

A.1.5 Example. The best finite dimensional analogue for us is when M = Rn, n = 2l and

f(x) = 〈x, Tx〉 is quadratic, for some symmetric, non-degenerate matrix T . This is a

nice analogue because AP is an affine space and the U(1) Chern-Simons action SX,P (A)

may be viewed as a quadratic functional (See Eq. (4.3.10)):


Vol(GP )

∫AP

DA exp

[ik

4π

(∫X

A ∧ dA)]

, (A.1.6)

=eπikSX,P (AP )

Vol(GP )

∫AP

DA exp

[ik

4π〈A, ?dA〉

], (A.1.7)

(A.1.8)

so that basically,

SX,P (A) =

(∫X

A ∧ dA)

= 〈A, ?dA〉. (A.1.9)

In this case, the stationary phase approximation of Eq. (A.1.16) is exact (i.e. the error

term O(k−(l+1)) vanishes). This can be seen by evaluating the following Gaussian integral


directly: ∫Rneik〈x,Tx〉dnx. (A.1.10)

Since T is symmetric, we change variables and let x = Py for an orthogonal matrix P

such that D = P TTP is diagonal. Let λj, 1 ≤ j ≤ n denote the eigenvalues of T . We

have ∫Rneik〈x,Tx〉dnx =

∫Rneik〈y,P

TTPy〉dny (A.1.11)

=

∫Rneik

∑nj=1 λjy

2j dny (A.1.12)

=n∏j=1

∫Reikλjy

2j dy. (A.1.13)

Individually, we have ∫Reikλjy

2j dy =

√2π

k

1

|√λj|

eiπ4

sgn(λj). (A.1.14)

Thus, ∫Rneik〈x,Tx〉dnx =

(2π

k

)leπi sgn(T )/4 1

| detT |1/2. (A.1.15)

Clearly, this matches the stationary phase formula Eq. (A.1.16) when f(x) = 〈x, Tx〉,

since the only critical point of f is when x = 0 and the Hessian is H0 = T .

Example A.1.5 demonstrates our main motivation for making the rigorous definitions,

Def.’s 4.3.22 and 6.10.28, and why we make these definitions with the expectation that

the partition functions are exact. In order to be more complete in our finite dimensional

analogue we must consider the case where our functional is allowed to be non-degenerate.

This is the case for the U(1) Chern-Simons action (See Eq. (A.1.9)). This case was first

considered in [Sch79a], [Sch79b]. We will thence consider a slightly more general class of

integrals:

F (k) :=1

VolG

∫M

eikfdµ. (A.1.16)

where:

• (M, g) is a Riemannian manifold with a compact Lie group G acting as a group of

isometries.


• G is assumed to be endowed with an invariant inner product that defines an invari-

ant Riemannian metric on G. Vol(G) denotes the volume of G with respect to this

metric. Let g := LieG denote the Lie algebra of G as usual.

• The action G M generates a map τp : g→ TpM for each p ∈M via:

τp(Y ) :=d

dt

∣∣∣t=0

exp(tY ) · p, for Y ∈ g. (A.1.17)

• Hp := g ∈ G | g · p = p is the isotropy subgroup of G at p. Assume that Hp is

conjugate to a fixed subgroup H < G ∀p ∈M .2 So Vol(Hp) = Vol(H).

• Let gM/G denote the Riemannian metric on the quotient space M/G induced from

the metric g on M . Then µ, µM/G denote the associated measures for the metrics

on M and M/G, respectively.

• f is assumed to be a G-invariant function on M such that the stationary points

of f (i.e. p ∈ M 3: dfp = 0) form a G-invariant submanifold F of M , and

f(p) = Q ∀p ∈ F , say.

• Assume that TpF = Ker(Hessf)p for all p ∈M.

• Assume thatM := F/G is a manifold and we let µM denote the naturally induced

measure on M.

Eq. (A.1.16) is the true analogue of Eq. (A.1.3) that we use to make our rigorous

definitions of the partition functions. All of the ingredients match up as in the following

dictionary:

2This is the case as in §6.5.


Chern-Simons theory, Eq. (A.1.3) Finite dimensional analogue, Eq.

(A.1.16)

AP = A ∈ (Ω1(P ) ⊗ g)G | A(ξ]) =

ξ, ∀ ξ ∈ g, and g = 〈·, ·〉 =∫X· ∧ ?·

(M, g).

GP = ψ ∈ (Diff(P, P ))G | π ψ =

π ' Maps(X,U(1)).

G.

LieGP ' Ω0(X), TAAP ' Ω1(X), d :

Ω0(X)→ Ω1(X).

τp : g→ TpM .

SX,P (A) : AP → R/Z. f : M → R.

MP ⊂ AP/GP . M⊂M/G.

The main result that we use in §4.3 and §6.6 is the following stationary phase approxi-

mation, [Sch79a], [Sch79b]:

1

VolG

∫M

eikfµ = C0 ·∫Meπi4

Sgn(Hessf)p| det′(τ ∗p τp)|1/2

| det′(Hessf)p|1/2µM +O(k−r) (A.1.18)

for some r ∈ R>0, and where,

C0 :=1

k(dimM−dimF )/2

eπiQk

Vol(H), (A.1.19)

and det′ denotes a regularized determinant, which is intended to represent a product

of non-zero eigenvalues. Our basic assumption is that O(k−r) vanishes since SX,P (A) :

AP → R/Z is quadratic.

Appendix B

Constructions and Computations

B.1 Finite Dimensional Analogue of the Shift Sym-

metry

In this section we present a finite dimensional analogue of the shift symmetry construc-

tion of [BW05, §3.1]. We exhibit a specific example that should help to clarify the basic

elements of the heuristic infinite dimensional construction of [BW05]. In order to depict

this analogue more clearly, we will set up a dictionary between our finite dimensional

example and the infinite dimensional Chern-Simons theory.

148

Appendix B. Constructions and Computations 149

Chern-Simons theory Finite dimensional analogue

AP = A ∈ (Ω1(P ) ⊗ g)G | A(ξ]) =

ξ, ∀ ξ ∈ g.

R3.

G = ψ ∈ (Diff(P, P ))G | π ψ = π. e = trivial group.

Contact form κ ∈ Ω1(X,R). Fixed vector 0 6= v0 ∈ R3.

S(A) =∫X

Tr(A ∧ dA+ 2

3A ∧ A ∧ A

),

for A ∈ AP .

S(w) = ‖w‖2 + ‖w‖2 · ‖w × v0‖, for

w ∈ R3 and fixed 0 6= v0 ∈ R3.

Lie algebra-valued zero form on X, Φ ∈

Ω0(X, g).

Φ ∈ R scalar coordinate.

The space of local shift symmetries S

that “acts” on the space of connections

AP and the space of fields Φ.

S = R; For any r ∈ S, r ·w := w+rv0,

w ∈ R3 and r · Φ := Φ + r, Φ ∈ R.

Vol(S) =∫S DΦ :=∫

S exp(−∫Xκ ∧ dκ Tr(σ2)

)DS.

Vol(S) =∫

RDΦ :=∫

R exp (−σ2) dσ =

√π, where σ ∈ S = R.

Our finite dimensional analogue of the partition function of Eq. (5.0.1) is defined as

follows:

Z :=

∫R3

dw exp

[−1

2S(w)

]. (B.1.1)

where S(w) = ‖w‖2 + ‖w‖2 · ‖w× v0‖ is as defined above. Note that we will work with

a “Wick rotated” Euclidean action in our finite dimensional analogue. This just means

that the factor of i =√−1 is replaced by −1 in the exponential. These two viewpoints

are formally equivalent, and it suffices to consider such an action our purposes.

The first thing that we notice is that the action S(w) = ‖w‖2 + ‖w‖2 · ‖w × v0‖ is

not invariant under the action of the shift symmetry S = R on R3; i.e. S(r · w) =

S(w + rv0) 6= S(w) for general r ∈ S = R and w ∈ R3. This is exactly analogous to


the fact that S(A) =∫X

Tr(A ∧ dA+ 2

3A ∧ A ∧ A

)is not invariant under the local shift

symmetry action, δσ(A) := σκ, for σ ∈ S. In order to recast their partition function,

Z(k), into a form that is invariant under the shift symmetry, [BW05] introduce a new

scalar field Φ ∈ Ω0(X, g) which also has a transformation rule under the shift symmetry;

δσ(Φ) := σ, for σ ∈ S. The shifted partition Z(k) with shifted action:

S(A,Φ) := S(A− Φκ) (B.1.2)

is then considered since it is the only shift invariant action that incorporates both Φ

and the shift symmetry. To see that S(A − Φκ) is invariant under the shift symmetry,

observe:

δσS(A− Φκ) =δS

δA(A− Φκ) · δσ(A− Φκ) (B.1.3)

=δS

δA(A− Φκ) · (δσ(A)− δσ(Φ)κ) (B.1.4)

=δS

δA(A− Φκ) · (σκ− σκ) (B.1.5)

= 0, (B.1.6)

After introducing some normalization factors and expanding the action S(A−Φκ) in their

partition function, [BW05] obtain Eq. (5.0.2) and establish the equivalence Z(k) = Z(k)

heuristically. [BW05] then observe that S(A−Φκ) is quadratic in Φ, which allows them

to evaluate the Gaussian integral over Φ easily, and obtain the completely equivalent

formulation for their partition function as in Eq. (5.0.4). The action in Eq. (5.0.4)

CS(A)−∫X

1

κ ∧ dκTr[(κ ∧ FA)2

],

is now manifestly invariant under the shift symmetry, allowing one to quotient out by

the shift symmetry, and reduce the problem from an integral over AP to an integral over

AP/S. It turns out that AP/S is a symplectic space and [BW05] are able to show that

their problem can now be solved using the technique of non-abelian localization as in

SU(2)-Yang-Mills theory (cf. [Wit92]).


We analogously define:

S(Φ,w) := S(w− Φv0), (B.1.7)

and consider,

Z :=1

Vol (S)

∫R×R3

DΦ dw exp

[−1

2S(Φ,w)

]. (B.1.8)

where the action of S = R is extended to R× R3 via:

r · (Φ,w) := (Φ + r,w + rv0).

S(Φ,w) = S(w− Φv0) : R× R3 → R is clearly invariant under this action:

S[r · (Φ,w)] = S(r · Φ, r ·w)

= S[(r ·w)− (r · Φ)v0)]

= S[(w + rv0)− (Φ + r)v0)]

= S(Φ,w).

Also, notice that we can trivially fix Φ = 0 in our action by simply changing the inde-

pendent coordinate w to w′ = w− Φv0. Our partition function is:

Z =1

Vol (S)

∫R×R3

DΦ dw′ exp

[−1

2S(w′)

]=

[1

Vol (S)

∫RDΦ

] ∫R3

dw exp

[−1

2S(w)

]=

∫R3

dw exp

[−1

2S(w)

]= Z

where the third line comes from the fact that Vol(S) =∫

RDΦ :=∫

R exp (−σ2) dσ =√π,

as noted in our table above. Hence, our new description of the partition, Z, is completely

equivalent to our old description, Z; i.e. Z = Z.

B.1.9 Remark. Our primary motivation for choosing an action of the form:

S(w) = ‖w‖2 + ‖w‖2 · ‖w× v0‖,


comes directly from the analogue of our finite dimensional construction with [BW05].

We choose this action so that it is cubic in w, and S(w − Φv0) is quadratic in Φ. The

fact that S(w− Φv0) is quadratic in Φ simply follows from the observation that

(w− Φv0)× v0 = w× v0,

since the cross product of v0 with itself vanishes.

At this point, we will continue with our analogy and evaluate the Gaussian integral

over Φ directly. For simplicity, we assume that ‖v0‖ = 1. We compute:

S(Φ,w) = S(w− Φv0)

= ‖w− Φv0‖2 + ‖w− Φv0‖2 · ‖(w− Φv0)× v0‖

= ‖w− Φv0‖2 · [1 + ‖w× v0‖]

=[Φ2 − (2w · v0)Φ + ‖w‖2

]· [1 + ‖w× v0‖]

=[(Φ− (w · v0))2 + (‖w‖2 − (w · v0)2)

]· [1 + ‖w× v0‖] ,

where we have completed the square in the last line. We then make the change of

variables:

Φ 7→ Φ′ = Φ− (w · v0),

in the partition function, and obtain:

Z = Z =1

Vol (S)

∫R×R3

DΦ dw exp

[−1

2S(Φ,w)

]=

1√π

∫R×R3

DΦ dw exp

[−1

2([(Φ− (w · v0))2 + (‖w‖2 − (w · v0)2)

]· [1 + ‖w× v0‖])

]=

1√π

∫R×R3

dw DΦ′ exp

[−1

2[1 + ‖w× v0‖] Φ′2

]×

× exp

[−1

2([‖w‖2 − (w · v0)2

]· [1 + ‖w× v0‖])

].

There are a couple of natural and equivalent ways to proceed from here. Since it is

our intention to elaborate on the way a physicist performs these kinds of computations,

we will first perform a “gauge fixing” for the shift symmetry to evaluate the partition


function Z. Recall that a gauge choice for the shift symmetry is defined by choosing a

function

c : R3 → R (B.1.10)

such that c is one-to-one when restricted to the gauge orbits and is normalized so that

0 ∈ Im(c|Orbit) on each gauge orbit. Clearly the gauge orbits of the shift symmetry are

just lines of the form x + r · v0 ∈ R3 | r ∈ R, and we can choose a representative x for

each gauge orbit in the plane defined by

Pv0 := x ∈ R3 |x · v0 = 0. (B.1.11)

Our natural choice for the gauge function is

c(w) := w · v0; (B.1.12)

i.e. c(w) is just the distance from w ∈ R3 to the plane Pv0 (recall that we set ‖v0‖ = 1).

We may then evaluate our partition function by inserting the delta function

δ(c(w)) (B.1.13)

and the measure fixing “Faddeev-Popov” determinant

∆c(w) :=

∣∣∣∣ ∂∂rc(r ·w)

∣∣∣∣r=r0

, (B.1.14)

into the integrand of the partition function. Note that r0 ∈ S = R is defined by the

relation c(r0 ·w) = 0. It is then usually assumed in the physics literature that one may

simply write:

Z =

∫R3/S

dw δ(c(w))∆c(w)

[2π

1 + ‖w× v0‖

]1/2

× (B.1.15)

× exp

[−1

2([‖w‖2 − (w · v0)2

]· [1 + ‖w× v0‖])

].

after computing the Guassian integral with respect to Φ′ and the group integral over S,

which produces a factor of Vol(S) =√π that cancels the factor out front of the integral.


∆c(w)dw is interpreted in the integral as a measure on the quotient space R3/S, dx.

After applying the delta function, δ(c(w)), observing that the Jacobian ∆c(w) = 1 is

trivial, and identifying x = r0w (where r0 is defined by the relation c(r0 · w) = 0), we

obtain:

Z =

∫Pv0'R2

dx

[2π

1 + ‖x‖

]1/2

exp

[−1

2(‖x‖2 · [1 + ‖x‖])

]We further observe that this integral may be computed by making a change of variables

to polar coordinates

x = (rcos(θ), rsin(θ)).

We obtain:

Z =√

2π

∫ 2π

0

dθ

∫ ∞0

drr√

1 + rexp

[−1

2(r2 · (1 + r))

]= (2π)3/2

∫ ∞0

drr√

1 + rexp

[−1

2(r2 · (1 + r))

]Although the gauge fixing procedure that we have just demonstrated yields the correct

answer, it is instructive to consider an alternative method of computing Z that will shed

some light on some of the steps that are being skipped by the physicists. Our starting

point is the integral:

Z =1√π

∫R×R3

dw DΦ′ exp

[−1

2[1 + ‖w× v0‖] Φ′2

]×

× exp

[−1

2([‖w‖2 − (w · v0)2

]· [1 + ‖w× v0‖])

].

A point that was missed in our computation is that one must be careful to observe that

formally

DΦ = exp[−σ2]dσ (B.1.16)

where σ is viewed as a variable on S = R, and that the change of variables Φ 7→ Φ′ =

Φ− (w · v0) gives

DΦ′ = exp[−(σ′ + (w · v0))2]dσ′ (B.1.17)


where σ′ is a new variable on S. We cannot simply “perform the Gaussian integral

over Φ′” here, as there are mixed terms in the measure DΦ′. In order to carry out the

computation, we should change coordinates from w = (x, y, z) to (r,x), where r = w ·v0

and x = w− (w · v0)v0 = ProjPv0(w), so that

w = x + rv0. (B.1.18)

Then our measure becomes, dw = |J(w)|drdx = drdx, where the Jacobian |J(w)| = 1

since J(w) ∈ SO(3,R) can be taken to be a rotation in R3. We are then essentially

performing a gauge fixing where x = w−(w·v0)v0 becomes our gauge orbit representative

for each w ∈ R3 and the measure fixing Faddeev-Popov determinant ∆c(w) is identified

as the Jacobian |J(w)|. Making this substitution yields:

Z =1√π

∫R×R2×

dr dx dσ′ exp[−(σ′ + r)2] ×

× exp

[−1

2[1 + ‖x‖]σ′2

]exp

[−1

2(‖x‖2 · [1 + ‖x‖])

].

Now we make the independent substitution, r 7→ r′ = σ′ + r, which gives:

Z =1√π

∫Rdr′exp[−(r′)2]

∫R2×R

dx dσ′ exp

[−1

2[1 + ‖x‖]σ′2

]× exp

[−1

2(‖x‖2 · [1 + ‖x‖])

].

This last step justifies the cancelation of Vol(S) from the group integral over S; i.e.∫Rdr′exp[−(r′)2] =

√π. (B.1.19)

Performing the Gaussian integral over σ′ then yields our result:

Z =

∫Pv0'R2

dx

[2π

1 + ‖x‖

]1/2

exp

[−1

2(‖x‖2 · [1 + ‖x‖])

].

The main reason why this analysis is possible is because we have expressed our partition

function in the shift invariant form∫R3

dw

[2π

1 + ‖w× v0‖

]1/2

exp

[−1

2([‖w‖2 − (w · v0)2

]· [1 + ‖w× v0‖])

]. (B.1.20)


i.e. if,

f(w) :=

[2π

1 + ‖w× v0‖

]1/2

exp

[−1

2([‖w‖2 − (w · v0)2

]· [1 + ‖w× v0‖])

], (B.1.21)

then f(r ·w) = f(w) for all r ∈ S and w ∈ R3. Beasley and Witten obtain the analogue

of this result as in Eq. (5.0.4) by “evaluating the Gaussian integral with respect to Φ” in

Eq. (5.0.2). The shift invariant partition function that they obtain in Eq. (5.0.4) then

allows them to reduce their integral modulo the shift symmetry as in Eq. (B.1.15).

B.2 Gravitational Chern-Simons Calculations

B.2.1 Levi-Civita Connection

In this section we explicitly compute the Christoffel symbols

Γλµν =1

2Gλρ (∂νGρµ + ∂µGρν − ∂ρGµν) , (B.2.1)

for the Levi-Civita connection ∇G for the family of metrics

Gµν := gε =

hαβ + εϕαϕβ εϕα

εϕβ ε

. (B.2.2)

with inverse metric

Gµν =

hαβ −ϕα

−ϕβ ε−1 + ϕζϕζ

. (B.2.3)

B.2.4 Remark. We will use the comma notation to denote partial derivatives; i.e. Gρβ,α :=

∂αGρβ, and the semi-colon notation to denote covariant derivatives, so ωαβ;ρ := ∂ρωαβ −

Γζαρωζβ − Γζρβωαζ for some (0, 2) tensor ωαβ, for example.

We break this computation down into cases.

I. Γδαβ, δ, α, β ∈ 0, 1. Reading off the components of the metric from Eq.’s (B.2.2)

and (B.2.3):


• ρ ∈ 0, 1:

1

2Gδρ (Gρβ,α +Gρα,β −Gαβ,ρ)

=1

2hδρ ([hρβ,α + ε(ϕρϕβ),α] + [hρα,β + ε(ϕρϕα),β]− [hαβ,ρ + ε(ϕαϕβ),ρ])

=1

2hδρ ([hρβ,α + ε(ϕρ,αϕβ + ϕρϕβ,α)] + [hρα,β + ε(ϕρ,βϕα + ϕρϕα,β)]) +

− 1

2hδρ ([hαβ,ρ + ε(ϕα,ρϕβ + ϕαϕβ,ρ)])

= γδαβ +ε

2hδρ ((ϕρ,αϕβ + ϕρϕβ,α) + (ϕρ,βϕα + ϕρϕα,β)− (ϕα,ρϕβ + ϕαϕβ,ρ))

= γδαβ +ε

2

(hδρ[ϕβ(ϕρ,α − ϕα,ρ) + ϕα(ϕρ,β − ϕβ,ρ)] +

ϕδϕα,β + ϕδϕβ,α

)where

γδαβ :=1

2hδρ (hρβ,α + hρα,β − hαβ,ρ) (B.2.5)

are the Christoffel symbols for the two-dimensional metric tensor h.

• ρ = 2:

1

2Gδ2 (G2β,α +G2α,β −Gαβ,2)

=ε

2(−ϕδ)

(ϕα,β + ϕβ,α − [ε−1hαβ,2 + (ϕαϕβ),2]

)=−ε2

ϕδϕα,β + ϕδϕβ,α

where the last line follows from the fact that

ε−1hαβ,2 + (ϕαϕβ),2 = 0, (B.2.6)

since the ∂2 derivatives of hαβ, and ϕα vanish.

Clearly, the terms in curly brackets cancel in the sum of the ρ ∈ 0, 1 and

ρ = 2 cases above, and we have for δ, α, β ∈ 0, 1:

Γδαβ = γδαβ −ε

2hδζ (ϕβfζα + ϕαfζβ) , (B.2.7)

where,

fαβ := ϕβ,α − ϕα,β. (B.2.8)


II. Γ2αβ, α, β ∈ 0, 1.

• ρ ∈ 0, 1:

1

2G2ρ (Gρβ,α +Gρα,β −Gαβ,ρ)

=1

2(−ϕρ) ([hρβ,α + ε(ϕρϕβ),α] + [hρα,β + ε(ϕρϕα),β]− [hαβ,ρ + ε(ϕαϕβ),ρ])

=1

2(−ϕρ) ([hρβ,α + ε(ϕρ,αϕβ + ϕρϕβ,α)] + [hρα,β + ε(ϕρ,βϕα + ϕρϕα,β)]) +

− 1

2(−ϕρ) ([hαβ,ρ + ε(ϕα,ρϕβ + ϕαϕβ,ρ)])

= −ϕργραβ −ε

2ϕρ ((ϕρ,αϕβ + ϕρϕβ,α) + (ϕρ,βϕα + ϕρϕα,β)− (ϕα,ρϕβ + ϕαϕβ,ρ))

= −ϕργραβ +ε

2(ϕρ[ϕβfρα + ϕαfρβ)]− ϕρϕρϕα,β + ϕρϕρϕβ,α)

• ρ = 2:

1

2G22 (G2β,α +G2α,β −Gαβ,2)

=ε

2(ε−1 + ϕζϕζ) (ϕα,β + ϕβ,α) , since Gαβ,2 = 0,

=1

2(ϕα,β + ϕβ,α) +

ε

2

ϕζϕζϕα,β + ϕζϕζϕβ,α

Clearly, the terms in curly brackets cancel in the sum of the ρ ∈ 0, 1 and

ρ = 2 cases, and we have:

Γ2αβ =

1

2(ϕα,β + ϕβ,α)− ϕζγζαβ +

ε

2ϕζ [ϕβfζα + ϕαfζβ]

=1

2(Dαϕβ +Dβϕα) +

ε

2ϕζ(ϕβfζα + ϕαfζβ),

where the last line follows from the fact that the Levi-Civita connection is

symmetric (i.e. γζαβ = γζβα), and

Dαϕβ := ϕβ,α − ϕζγζαβ. (B.2.9)

III. Γδ2β, δ, β ∈ 0, 1.


• ρ ∈ 0, 1:

1

2Gδρ (Gρβ,2 +Gρ2,β −G2β,ρ)

=ε

2hδρ (ϕρ,β − ϕβ,ρ) , since Gρβ,2 = 0,

=ε

2hδζfβζ .

• ρ = 2: It is easy to see that

1

2Gδ2 (G2β,2 +G22,β −G2β,2) = 0. (B.2.10)

This follows from observing that G22,β = ∂βε = 0, and the ∂2 derivative G2β,2

vanishes. Thus,

Γδ2β =ε

2hδζfβζ . (B.2.11)

IV. Γ22β, β ∈ 0, 1.

• ρ ∈ 0, 1:

1

2Gδρ (Gρβ,2 +Gρ2,β −G2β,ρ)

=ε

2(−ϕρ) (ϕρ,β − ϕβ,ρ) , since Gρβ,2 = 0,

=ε

2ϕζfζβ.

• ρ = 2: As above, it is not difficult to see that

1

2Gδ2 (G2β,2 +G22,β −G2β,2) = 0. (B.2.12)

Thus,

Γ22β =

ε

2ϕζfζβ. (B.2.13)

V. Γλ22, λ ∈ 0, 1, 2. It is also easy to see that

1

2Gλρ (Gρ2,2 +Gρ2,2 −G22,ρ) = 0. (B.2.14)


B.2.2 Spin Connection

In this section we use our formulae for the Vielbeins,

Eaα = eaα, E

22 =√ε, E2

α =√εϕα, E

a2 = 0 (B.2.15)

Eαa = eαa , E

22 =

1√ε, E2

a = −ϕζ eζa, Eα2 = 0, (B.2.16)

and our formulae for the Christoffel symbols for the Levi-Civita connection associated to

our family of metrics G := gε to compute the spin connections [(AG)µ]AB := [Aµ]AB using

Eq. (7.2.6),

[Aµ]AB = EAν E

λBΓνµλ − Eλ

B∂µEAλ . (B.2.17)

We break this down into cases.

I. [Aα]ab , a, b, α ∈ 0, 1. Plug in the appropriate quantities from Eq.’s (B.2.15) and

(B.2.16) and for the Christoffel symbols.

• λ, ν ∈ 0, 1:

Eaν E

λb Γναλ − Eλ

b ∂αEaλ = eζbe

aδ

(γδαζ −

ε

2hδρ (ϕζfρα + ϕαfρζ)

)− eζb∂αe

aζ

= eζb

[(eaδγ

δαζ − ∂αeaζ)−

ε

2eaδh

δρϕαfρζ

]− ε

2eζbe

aδh

δρϕζfρα

= eζb

[−Dαe

aζ −

ε

2eaδh

δρϕαfρζ

]− ε

2eζbe

aδh

δρϕζfρα

• λ = 2, ν ∈ 0, 1:

Eaν E

λb Γναλ − Eλ

b ∂αEaλ = eaδ(−ϕζ e

ζb)( ε

2

)hδρfαρ

= ε

2eζbe

aδh

δρϕζfρα

These two cases, λ, ν ∈ 0, 1 and λ = 2, ν ∈ 0, 1, are the only cases for which we

get a non-zero contribution to [Aα]ab , since the term Eaν E

λb Γνµλ always vanishes for

ν = 2 when a, b ∈ 0, 1 by our formulae for the Vielbeins, and the term Eλb ∂µE

aλ

vanishes for λ = 2 for the same reason. After observing that the terms in the curly


brackets from the cases, λ, ν ∈ 0, 1 and λ = 2, ν ∈ 0, 1, cancel in our sum, we

obtain:

[Aα]ab = eζb

[−Dαe

aζ −

ε

2eaδh

δρϕαfρζ

]. (B.2.18)

II. [Aα]a2, a, α ∈ 0, 1. The only terms to contribute to the sum over λ, ν ∈ 0, 1, 2 in

the sum of Eq. (B.2.17) are when λ = 2 and ν ∈ 0, 1, since Eλ2 = 0 for λ ∈ 0, 1.

Even when λ = 2, the sum Eλ2 ∂αE

aλ = 0 in Eq. (B.2.17) since Ea

2 = 0 for a ∈ 0, 1.

Thus,

[Aα]a2 =1√εeaδΓ

δα2

=1√εeaδε

2hδζfαζ

=

√ε

2eaδh

δζfαζ

By lowering the two dimensional index [Aα]a 2 := ηab[Aα]b2, with ηab the two-

dimensional Kronecker pairing, we obtain

[Aα]a 2 = −[Aα]2 a = ηa b

√ε

2ebδh

δζfαζ . (B.2.19)

III. [A2]ab , a, b ∈ 0, 1. The only terms to contribute to the sum over λ, ν ∈ 0, 1, 2

in the sum of Eq. (B.2.17) are when λ, ν ∈ 0, 1. All other terms vanish by

our formulae for the Vielbeins and the Christoffel symbols. Note that the sum

Eλb ∂2E

aλ = 0 in Eq. (B.2.17) since all ∂2 derivatives vanish for the Vielbeins. Thus,

[A2]ab = eζbeaδΓ

δζ2

= eζbeaδ

ε

2hδρfζρ

=ε

2eζbe

aδh

δρfζρ

IV. [A2]a2, a, α ∈ 0, 1. First, the sum Eλ2 ∂2E

aλ = 0 in Eq. (B.2.17) since all ∂2

derivatives vanish for the Vielbeins. The only term to contribute to the sum over


λ, ν ∈ 0, 1, 2 in the sum of Eq. (B.2.17) is ν ∈ 0, 1 and λ = 2. Thus,

[A2]a2 = E22eaδΓ

δ22

= 0

since Γδ22 = 0. By lowering the two dimensional index [A2]a 2 := ηab[Aα]b2, and using

our anti-symmetry properties, we see

[A2]2 a = [A2]a 2 = 0. (B.2.20)

V. [Aµ]22, µ ∈ 0, 1, 2. Lastly, since [Aµ]a b is anti-symmetric in a, b, we see trivially

that

[Aµ]22 = 0. (B.2.21)

for all µ ∈ 0, 1, 2.

B.2.3 Reduced Spin Connection

In this section we compute the corresponding vector valued one-forms ACµ for our spin

connections [(A)µ]AB defined by the relation

[(A)µ]AB := ηAC [Aµ]CB = εABCACµ . (B.2.22)

Contracting with the Levi-Civita symbol εABC , we have

ACµ =1

2εABCηAD[Aµ]DB . (B.2.23)

Eq. (B.2.23) combined with our formulae for the spin connections [Aµ]DB are the main

relations that we use to compute ACµ throughout this section. As usual, we do this

computation in cases.


I. A2α, α ∈ 0, 1. Let ηab denote the two-dimensional Kronecker pairing as usual.

Eq. (B.2.23) gives us

A2α =

1

2εAB2ηAD[Aα]DB

=1

2[η0a[Aα]a1 − η1a[Aα]a0]

=1

2

[η0ae

ζ1

[−Dαe

aζ −

ε

2eaδh

δρϕαfρζ

]− η1ae

ζ0

[−Dαe

aζ −

ε

2eaδh

δρϕαfρζ

]]=

−1

2(η0ae

ζ1 − η1ae

ζ0)Dαe

aζ

+− ε

4eaδh

δρϕαfρζ(η0aeζ1 − η1ae

ζ0)

(B.2.24)

We compute the quantities in the curly brackets of Eq. (B.2.24) separately. First

we recall that the spin connection (ωα)ab on Σ is defined by:

(ωα)ab := eζb∂αeaζ − eaδ e

ζbγ

δαζ (B.2.25)

= eζbDαeaζ (B.2.26)

Then ωα is defined by the relation

ωα,ab = εabωα = ηac(ωα)cb (B.2.27)

Thus, we compute the first term in Eq. (B.2.24):−1

2(η0ae

ζ1 − η1ae

ζ0)Dαe

aζ

= −1

2(ωα,01 − ωα,10), by Eq.’s (B.2.26) and (B.2.27),

= −1

2(2ωα,01), by anti-symmetry of ωα,ab,

= −ωα, by Eq. (B.2.27).

Before computing the second term in Eq. (B.2.24), we recall the following relations:

eζq = hζληqbebλ, (B.2.28)

fρζ =√hfερζ , (B.2.29)

εab = η0aη1b − η1aη0b, (B.2.30)

εδλεabeaδebλ = 2

√h. (B.2.31)


where

εδλ = |h|hρδhζλερζ (B.2.32)

in Eq. (B.2.31). Using Eq.’s (B.2.28) and (B.2.29), we compute the second term

in curly brackets from Eq. (B.2.24):

− ε

4eaδh

δρϕαfρζ(η0aeζ1 − η1ae

ζ0)

= − ε4

√hfϕα(eaδe

bλ)(hρδhζλερζ

)(η0aη1b − η1aη0b)

= − ε4

(√h)−1fϕα

(eaδe

bλεδλεab

), by (B.2.30), (B.2.32),

= − ε2

(√h)−1fϕα

√h, by Eq. (B.2.31),

= − ε2fϕα

Thus, we have:

A2α = −ωα −

ε

2fϕα. (B.2.33)

II. A22. As before, we compute:

A22 =

1

2εAB2ηAD[A2]DB

=1

2[η0a[A2]a1 − η1a[A2]a0]

=1

2

[η0a

[ ε2eζ1e

aδh

δρfζρ

]− η1a

[ ε2eζ0e

aδh

δρfζρ

]]= − ε

4

√hfebλe

aδ(h

δρhζλερζ) (η0aη1b − η1aη0b) , by Eq.’s (B.2.28) and (B.2.29),

= − ε4

(√h)−1f(eaδe

bλεδλεab), by Eq.’s (B.2.30) and (B.2.32),

= − ε2

(√h)−1f

√h, by Eq. (B.2.31),

= − ε2f

Thus, we have:

A22 = − ε

2f. (B.2.34)

III. Aaα, a, α ∈ 0, 1. Before we perform this computation we recall:

eδa = (√h)−1ελδεabe

bλ (B.2.35)


Let a ∈ 0, 1\a be the element of 0, 1 that represents the compliment of

a ∈ 0, 1. We then compute:

Aaα =1

2εABaηAD[Aα]DB

=1

2εaa[ηaD[Aα]D2 − η2D[Aα]Da

]=

1

2εaa [[Aα]a2 − [Aα]2a]

= εaa[Aα]a2, by anti-symmetry of [Aα]ab,

= εaaηa b

√ε

2ebδh

δζfαζ

= εaa

√ε

2

√h f eζa εαζ , by Eq.’s (B.2.28) and (B.2.29),

= εaa

√ε

2

√h f [(

√h)−1ελζεabe

bλ] εαζ , by Eq. (B.2.35),

=

√ε

2f [ελζ εαζεaaεabe

bλ]

=

√ε

2f [δλαe

aλ]

=

√ε

2f eaα.

Thus, we have:

Aaα =

√ε

2f eaα. (B.2.36)

IV. Aa2, a ∈ 0, 1.

Aa2 =1

2εABaηAD[A2]DB

=1

2εaa[ηaD[A2]D2 − η2D[A2]Da

]=

1

2εaa [[A2]a2 − [A2]2a]

= εaa[A2]a2, by anti-symmetry of [Aα]ab,

= 0, since [A2]a2 = 0 in general.

Thus,

Aa2 = 0. (B.2.37)


B.2.4 Reduced Gravitational Chern-Simons

In this section we compute the gravitational Chern-Simons term CS(Agε) in terms of

reduced quantities using Eq. (7.2.10):

CS(AG) = − 1

2π

∫X

d3x εµνλ(ηABA

Aµ∂νA

Bλ

)+

1

π

∫X

d3x detACµ . (B.2.38)

I. We first compute εµνληABAAµ∂νA

Bλ from the first integral term. First observe that

Aaµ∂νAaλ = 0 for any a ∈ 0, 1 and any permutation σ(012) = µνλ, since if ν = 2,

then all ∂2 derivatives vanish, and if ν 6= 2 then Aa2 = 0 by our previous results.

Thus, we only need compute the term εµνλA2µ∂νA

2λ, where ν 6= 2. We do this in

cases.

• (µ, ν, λ) = (2, 0, 1):

ε201A22∂0A

21 = [− ε

2f ] · ∂0[−ω1 −

ε

2fϕ1]

= [ε

2f ] · ∂0[ω1 +

ε

2fϕ1]

• (µ, ν, λ) = (2, 1, 0):

ε210A22∂1A

20 = −[− ε

2f ] · ∂1[−ω0 −

ε

2fϕ0]

= −[ε

2f ] · ∂1[ω0 +

ε

2fϕ0]

• (µ, ν, λ) = (0, 1, 2):

ε012A20∂1A

22 = [−ω0 −

ε

2fϕ0] · ∂1[− ε

2f ]

= [ω0 +ε

2fϕ0] · ∂1[

ε

2f ]

• (µ, ν, λ) = (1, 0, 2):

ε102A21∂0A

22 = −[−ω1 −

ε

2fϕ1] · ∂0[− ε

2f ]

= −[ω1 +ε

2fϕ1] · ∂0[

ε

2f ]


Adding these four cases together and grouping terms by powers of ε we obtain:

εµνλA2µ∂νA

2λ =

( ε2

)[f(∂0ω1 − ∂0ω1) + (ω0∂1f − ω1∂0f)] +

+( ε

2

)2

[(f∂0(fϕ1)− f∂1(fϕ0)) + (fϕ0∂1f − fϕ1∂0f)]

=( ε

2

)[2f(∂0ω1 − ∂1ω0) + ∂1(ω0f)− ∂0(ω1f)] +

+( ε

2

)2

[f 2(∂0ϕ1 − ∂1ϕ0)]

The term in curly brackets in the second last line above yields a global exact form

on Σ, and since ∂Σ = ∅, Stokes’ theorem implies that this term vanishes in the

integral of Eq. (B.2.38). It is also well known that the term in the second last line

above, ∂0ω1− ∂1ω0 = 12

√hr, where r ∈ Ω0

orb(Σ) is the (orbifold) scalar curvature of

(Σ, h). Also, the term ∂0ϕ1 − ∂1ϕ0 = f01 =√hf . Thus, we may write:∫

X

d3x εµνλ(ηABA

Aµ∂νA

Bλ

)=

∫S1

dx2

∫Σ

dx0 ∧ dx1√h

[( ε2

)fr +

( ε2

)2

f 3

](B.2.39)

This completes our computation of the first integral term in Eq. (B.2.38).

II. We now compute the second integral term detACµ from Eq. (B.2.38). For this we

note that detACµ may be computed directly, since:

ACµ =

√ε

2eaαf 0

−ωα − ε2fϕα − ε

2f

. (B.2.40)

Thus,

detACµ = det

(√ε

2eaαf

)·(− ε

2f)

=( ε

4f 2 det(eaα)

)·(− ε

2f)

= −1

2

( ε2

)2√hf 3, since det(eaα) =

√h.

Thus, ∫X

d3x detACµ =

∫S1

dx2

∫Σ

dx0 ∧ dx1√h

[−1

2

( ε2

)2

f 3

]. (B.2.41)


Adding our main results from Eq.’s (B.2.39) and (B.2.41), we obtain:

CS(Agε) = − 1

4π

∫S1

dx2

∫Σ

dx0 ∧ dx1√h(εfr + ε2f 3). (B.2.42)

Appendix C

Miscellaneous Results

C.1 Horizontal Operators

In this section we collect some results used in this thesis. First we recall some notation

and terminology. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold

(See §2.3) and

Ωq(H) := α ∈ Ωq(X) | ιξα = 0,

for q = 1, 2, be the space of horizontal forms on X (for q = 0 we take Ω0(H) = C∞(X)).

Recall that ξ ∈ Γ(TX) is the Reeb vector field for κ. Let H ⊂ TX denote the contact

distribution on X and H∗ ⊂ T ∗X denote the dual bundle of H. Also let H0 ⊂ T ∗ denote

the annihilator subbundle of H in TX. The cotangent bundle has the natural splitting

induced by the choice of κ:

T ∗X ' H0 ⊕H∗. (C.1.1)

Let Ω1(V ) := Γ(H0). In this way, sections of T ∗X are identified as

Γ(T ∗X) = Ω1(X) ' Ω1(V )⊕ Ω1(H). (C.1.2)

Clearly,

Ω1(V ) := fκ ∈ Ω1(X) | f ∈ C∞(X). (C.1.3)

169

Appendix C. Miscellaneous Results 170

Let

Ω2(V ) := κ ∧ α | α ∈ Ω1(H) = κ ∧ Ω1(H). (C.1.4)

From Eq. (C.1.1) we also have the decomposition

Γ(∧2T ∗X) = Ω2(X) ' Ω2(V )⊕ Ω2(H). (C.1.5)

Let ? : Ωq(X) → Ω3−q(X) be the Hodge star operator associated to the natural metric

g = κ ⊗ κ + π∗h (See Eq. (2.3.11)) on X. We recall the definition and some properties

of the Hodge star operator presently. Let (U, φ) be some local coordinate chart on X

such that x0, x1, x2 are the coordinates about a given point x ∈ X. We choose this

coordinate system so that ξx = ∂∂x0 |x and

∂

∂x0,∂

∂x1,∂

∂x2

,

forms a positively oriented orthonormal basis for our metric g at the point x. Let

dx0, dx1, dx2 denote the dual basis. Let

µx := dx0 ∧ dx1 ∧ dx2,

denote the volume form associated to g at x. Recall that the Hodge star operator may

be defined pointwise on Ωq(X). That is, given β ∈ Ωq(X), we may define ?β ∈ Ω3−q(X)

uniquely by specifying (?β)|x pointwise at x via:

α|x ∧ (?β)|x = µx,

for all α ∈ Ωq(X). Note that κ|x = dx0|x and dκ|x = dx1 ∧ dx2 follows easily from the

fact that our coordinate system can be taken such that κ = dx0 + x1dx2 locally about x

(and x is centered about the origin in our coordinate system). Thus, µx = κ ∧ dκ|x and

µ = ?(1) = κ ∧ dκ. (C.1.6)

Recall that on a three-manifold X

? ?β = β, ∀ β ∈ Ωq(X). (C.1.7)

We will find it convenient to define the Hodge star at x via:


i. ?dx0 = dx1 ∧ dx2,

ii. ?dx1 = −dx0 ∧ dx2,

iii. ?dx2 = dx0 ∧ dx1.

Note that Eq. (i) is the global statement

? κ = dκ. (C.1.8)

Note that Eq. (C.1.7) combined with Eq.’s (C.1.6), (i), (ii), and (iii) together define the

Hodge star operator for any β ∈ Ω3(X). Let us collect some further useful observations.

First, since Ω1(H) = Γ(H∗), we have by definition

β ∈ Ω1(H) ⇐⇒ β|x ∈ spandx1, dx2 ∀ x ∈ X, (C.1.9)

in our choice of coordinate system about a given point x. Similarly,

β ∈ Ω2(H) ⇐⇒ β|x ∈ spandx1 ∧ dx2 ∀ x ∈ X. (C.1.10)

β ∈ Ω1(V ) ⇐⇒ β|x ∈ spandx0 ∀ x ∈ X. (C.1.11)

β ∈ Ω2(V ) ⇐⇒ β|x ∈ spandx0 ∧ dx1, dx0 ∧ dx2 ∀ x ∈ X. (C.1.12)

We will find it useful to give a different characterization of horizontal vectors β ∈ Ωq(H),

for q = 1, 2, using the projection operator

π : Ωq(X) → Ωq(H) (C.1.13)

β 7→ β − κ ∧ ιξ(β). (C.1.14)

It is not difficult to see, for q = 1, 2, that

β ∈ Ωq(H) ⇐⇒ π(β) = β. (C.1.15)

Finally, we recall the following


C.1.16 Definition. Define the horizontal Hodge star operator to be the operator:

?H : Ωq(X)→ Ω2−q(H) q = 0, 1, 2,

defined for β ∈ Ωq(X) by

?H β = ?(κ ∧ β), (C.1.17)

where ? is the usual Hodge star operator on forms for the metric g = κ⊗ κ+ π∗h on X.

It is easy to verify by the properties of ? that ?H maps strictly to Ω2−q(H) q = 0, 1, 2.

We are now ready to prove the following

C.1.18 Proposition. ? : Ω1(H)→ κ ∧ Ω1(H) =: Ω2(V ) and ? : Ω2(H)→ Ω1(V ).

Proof. To prove the first part, let β ∈ Ω1(H). By C.1.9, for any x ∈ X ∃ ax, bx ∈ R 3:

β|x = axdx1 + bxdx

2.

By Eq.’s (ii) and (iii) we have

?β|x = dx0 ∧ (−axdx2 + bxdx1),

which by C.1.9, C.1.11 and C.1.3 says that ?β = κ∧β′ for some β′ ∈ Ω1(H). This proves

? : Ω1(H)→ κ ∧ Ω1(H).

To see the second part, let α ∈ Ω2(H). Then by C.1.10 ∃ cx ∈ R 3:

α|x = cxdx1 ∧ dx2.

By C.1.7 and i,

?α|x = cxdx0,

and by C.1.11 this says that ?α ∈ Ω1(V ). This proves ? : Ω2(H) → Ω1(V ), and we are

done.

C.1.19 Proposition. ?2H = −1 on Ω1(H).


Proof. To see this, let β ∈ Ω1(H). Consider

?2H β = ?(κ ∧ (?(κ ∧ β)), (C.1.20)

by definition C.1.16. By C.1.9 and C.1.11, at a given x ∈ X,

κ ∧ β|x = dx0 ∧ (axdx1 + bxdx

2),

for some ax, bx ∈ R. By ii, iii and C.1.7 we have

?(κ ∧ β)|x = axdx2 − bxdx1.

Similarly,

κ ∧ ?(κ ∧ β)|x = dx0 ∧ (axdx2 − bxdx1),

and so,

?(κ ∧ ?(κ ∧ β))|x = −axdx1 − bxdx2 = −β|x.

Thus,

?(κ ∧ (?(κ ∧ β)) = ?2Hβ = −β,

and ?2H = −1 as claimed.

C.1.21 Proposition. ?Hα = (−1)qιξ(?α) = ?(κ ∧ α) for all α ∈ Ωq(X), 0 ≤ q ≤ 2.1

Proof. We prove this in cases.

i. Let us first prove this for α ∈ C∞(X) = Ω0(X).

On the one hand,

?(κ ∧ α) = αdκ,

since α ∈ C∞(X) and by C.1.8. On the other hand,

ιξ(?α) = αιξ(κ ∧ dκ), by C.1.6,

= αdκ, by property of the contraction ιξ.

1Note that the statement is true for q = 3, but only trivially so.


Thus, ιξ(?α) = ?(κ ∧ α) for α ∈ C∞(X) = Ω0(X).

ii. Assume α ∈ Ω1(X) ' Ω1(V )⊕ Ω1(H).

Write

α = αV + αH ,

uniquely for αV ∈ Ω1(V ) and αH ∈ Ω1(H). Let

α := ?(κ ∧ α) = ?(κ ∧ αH) ∈ Ω1(H).

We want to show that

α = −ιξ(?α).

To see this, observe that by Prop. C.1.19,

?(κ ∧ α) = ?(κ ∧ (?(κ ∧ αH)), by def. of α.

= ?2HαH , by def. of ?2

H ,

= −αH , by Prop. C.1.19.

Thus, if we operate with −ιξ? on both sides of the equation

− ? (κ ∧ α) = αH ,

we obtain

−ιξ(?α) = −ιξ(?αH), since −ιξ(?αV ) = 0 by Prop. C.1.18 and Eq. (C.1.7),

= ιξ(?(?(κ ∧ α)),

= ιξ(κ ∧ α)), by C.1.7,

= α, by property of the contraction ιξ and α ∈ Ω1(H).

Thus, we have shown

α = −ιξ(?α),


and therefore −ιξ(?α) = ?(κ ∧ α) for α ∈ Ω1(X).

iii. Assume α ∈ Ω2(X) ' Ω2(V )⊕ Ω2(H).

Write

α = αV + αH ,

uniquely for αV ∈ Ω2(V ) and αH ∈ Ω2(H). First observe that we can write

αH = φdκ,

for some φ ∈ C∞(X). This follows from observing that κ∧ αH = φκ∧ dκ for some

φ ∈ C∞(X) since κ ∧ dκ is non-vanishing. Then contract the equation κ ∧ αH =

φκ ∧ dκ by ιξ to obtain ιξ(κ ∧ αH) = αH on the one hand, and ιξ(φκ ∧ dκ) = φdκ

on the other hand. Thus, αH = φdκ as claimed. Now,

ιξ(?α) = ιξ(?(αV + αH))

= ιξ(αH + ?(φdκ)), αH := ?αV ∈ Ω1(H), by Prop. C.1.18,

= ιξ(φκ), since ιξαH = 0 and ?dκ = κ by C.1.7 and C.1.8,

= φ, since ιξκ = 1.

Also,

?(κ ∧ α) = ?(κ ∧ (αV + αH)

= ?(κ ∧ αH), since αV ∈ Ω2(V ) = κ ∧ Ω1(H),

= ?(φκ ∧ dκ),

= φ, since ?(κ ∧ dκ) = 1 by C.1.6 and C.1.7.

Thus, ιξ(?α) = ?(κ ∧ α) for α ∈ Ω2(X), as claimed.

This completes the proof.


C.1.22 Proposition. The following equalities hold

?H κ = 0 (C.1.23)

?H(κ ∧ dκ) = 0 (C.1.24)

?H1 = dκ. (C.1.25)

Proof. These are just an easy consequence of C.1.6 and C.1.8 and from Prop. C.1.21

?Hα = (−1)qιξ(?α) = ?(κ∧α). First, ?Hκ = −ιξ(?κ) = −ιξ(dκ), by C.1.8, and ιξ(dκ) = 0

since ξ is the Reeb field for κ. Thus, ?Hκ = 0. Second, ?H(κ∧dκ) = ?(κ∧κ∧dκ) = ?(0) =

0. Thus, ?H(κ ∧ dκ) = 0. Last, ?H1 = ιξ(?1) = ιξ(κ ∧ dκ) by C.1.6, and ιξ(κ ∧ dκ) = dκ

by property of ιξ. Thus, ?H1 = dκ.

C.1.26 Proposition. κ∧dAκ∧dκ = ?HdHA for A ∈ Ω1(X).

Proof. For this proof we first recall what is meant by the notation κ∧dAκ∧dκ . By definition,

κ ∧ dAκ ∧ dκ

:= φ ∈ C∞(X),

where κ∧dA = φκ∧dκ. Since dA = dVA+dHA for some dVA ∈ Ω2(V ) and dHA ∈ Ω2(H)

(note that we define dH : Ω1(X) → Ω2(H) as the restriction of d to the horizontal

distribution H), we see that κ ∧ dA = κ ∧ dHA. We have

κ ∧ dAκ ∧ dκ

= φ,

= ?(φκ ∧ dκ), by C.1.6 and C.1.7,

= ?(κ ∧ dA),

= ?(κ ∧ dHA),

= ?HdHA, by Def. C.1.16 of ?H .

Thus, κ∧dAκ∧dκ = ?HdHA as claimed.

C.1.27 Proposition. dA = κ ∧ LξA+ dHA for A ∈ Ω1(H).


Proof. To see this we recall that the definition of dH : Ω1(X)→ Ω2(H) is the restriction

of d to the horizontal distribution H. In other words,

dHA = π(dA),

where π : Ω2(X)→ Ω2(H) is defined in Eq. (C.1.13) above. We have

dHA = π(dA),

= dA− κ ∧ ιξ(dA).

Observe that LξA = ιξ(dA) + d(ιξA) = ιξ(dA) since ιξA = 0 for A ∈ Ω1(H). Thus,

dHA = dA− κ ∧ ιξ(dA),

= dA− κ ∧ LξA.

Rearranging this last equation we obtain our result, dA = κ ∧ LξA+ dHA.

C.2 T dRS = T dC

Next we would like to prove T dRS = T dC . As we will see, this follows directly from [RS08,

Theorem 4.2]. First we recall the definition of the contact Laplacian,:

∆q =

(d∗HdH + dHd

∗H)2 if q = 0, 3,

D∗D + (dHd∗H)2 if q = 1,

DD∗ + (d∗HdH)2 if q = 2.

(C.2.1)

on (E , dH). The analytic contact torsion is given by,

TC := exp

(1

4

3∑q=0

(−1)qw(q)ζ ′(∆q)(0)

)(C.2.2)

where,

w(q) =

q if q ≤ 1,

q + 1 if q > 1.

(C.2.3)


in the case where dim(X) = 3. Also recall the Hodge-de Rham Laplacian:

∆DRq := d∗d+ dd∗, on Ωq(X; R). (C.2.4)

The analytic Ray-Singer torsion is given by,

TRS := exp

(1

2

3∑q=0

(−1)qqζ ′(∆DRq )(0)

)(C.2.5)

in the case where dim(X) = 3. Want to see T dRS = T dC as densities. Consider:

detH•(Ω(X), d)∗ :=3⊗j=0

(detHj(Ω(X), d))(−1)j (C.2.6)

detH•((E , dH))∗ :=3⊗j=0

(detHj(E , dH))(−1)j (C.2.7)

and,

|| · ||RS := TRS|| · ||L2(Ω(X),d), (C.2.8)

|| · ||C := TC || · ||L2(E,dH), (C.2.9)

where the L2 norms come from the identifications:

H•(Ω(X), d) ' H•(Ω(X), d), (C.2.10)

H•(E , dH) ' H•(E , dH). (C.2.11)

We have the following

C.2.12 Proposition. [RS08, Prop. 2.3]

detH•(Ω(X), d)∗ ' detH•((E , dH))∗. (C.2.13)

We also have the following

C.2.14 Theorem. [RS08, Theorem 4.2] On any quasi-regular K-contact three manifold

(X,φ,R, κ, g):

|| · ||RS ' || · ||C . (C.2.15)


Let us make the following

C.2.16 Definition. Let

δ| detH•(E,dH)|∗ := ⊗dimXq=0 |νq1 ∧ · · · ∧ ν

qbq|(−1)q

be the density for ||·||L2(E,dH) on | detH•((E , dH))∗| where νq1 , · · · , νqbq is an orthonormal

basis for the space of harmonic contact forms Hq(E , dH). Similarly, let

δ| detH•(Ω(X),d)|∗ := ⊗dimXq=0 |νq1 ∧ · · · ∧ ν

qbq|(−1)q

be the density for || · ||L2(Ω(X),d) on | detH•(Ω(X), d)∗| where νq1 , · · · , νqbq is an orthonor-

mal basis for the space of harmonic forms Hq(Ω(X), d). Then we define

T dC := TC · δ| detH•((E,dH))∗|, (C.2.17)

and,

T dRS := TRS · δ| detH•(Ω(X),d)|∗ . (C.2.18)

Recall that the absolute value of the determinant lines are defined as

| detH•(Ω(X), d)∗| :=3⊗j=0

| detHj(Ω(X), d)(−1)j | (C.2.19)

| detH•((E , dH))∗| :=3⊗j=0

| detHj(E , dH)(−1)j |, (C.2.20)

so that T dC ∈ | detH•((E , dH))∗| and T dRS ∈ | detH•(Ω(X), d)∗|. Our desired result is the

following

C.2.21 Proposition. T dRS ' T dC.

Proof. This follows precisely from the fact that the isomorphism in C.2.12 comes directly

from an isomorphism Ψ : (Ω(X), d) → (E , dH), which we can take to preserve the bases

νq1 , · · · , νqbq, νq1 , · · · , ν

qbq. That is, we take Ψ(νql ) = νql . Thus, Ψ preserves the densities

δ|detH•(Ω(X),d)|∗ , and δ| detH•(E,dH)|∗ and also induces the equivalence Ψ : || · ||RS ' || ·

||C by Theorem C.2.14. Therefore, by these observations, we see that T dRS = TRS ·

δ|detH•(Ω(X),d)|∗ ' TC · δ| detH•((E,dH))∗| = T dC .


C.3 A Standard Result in Cohomology

The result that we would like to prove in this section is the following

C.3.1 Proposition. [FS92], [Nic00, Theorem 1.3] Let X be a Seifert manifold over an

orbifold Σ = (|Σ|,U) (See §2.1). Then,

dimH1(X,R) =

2g n ≥ 1

2g + 1 n = 0

where n is the degree of X over Σ,

U(1) // X

Σ

,

and g is the genus of the base space |Σ|.

Proof. We provide a proof of this fact in the case that X is a principal U(1) bundle

and leave the general case to the references [FS92], and [Nic00, Theorem 1.3]. By the

Universal Coefficient Theorem (UCT),

H1(X,R) ' Hom(H1(X,Z),R) (C.3.2)

i.e. the UCT implies that:

0→ Ext(H0(X,Z),R)→ H1(X,R)→ Hom(H1(X,Z),R)→ 0

is exact. Also

Ext(H0(X,Z),R) ' Ext(Z,R) ' 0

since Z is free. Thus we compute Hom(H1(X,Z),R). By Hurewicz,

H1(X,Z) ' π1(X)

[π1(X), π1(X)].

Hence we have the following presentation of π1(X), [Orl72]:

π1(X) ' 〈ap, bp, h|[ap, h] = [bp, h] = 1,

g∏p=1

[ap, bp] = hn〉


where g is the genus of the base space Σ of our X,

U(1) // X

Σ

and n = c1(X) is the Chern number of the U(1)-bundle X. The generator h arises

from the generic fibre over Σ. Observe that in the abelianization of π1(X) the following

relation is satisfied:g∏p=1

[ap, bp] = hn (C.3.3)

We have

[π1(X), π1(X)] = 〈[ap, bp]|g∏p=1

[ap, bp] = hn〉, (C.3.4)

and therefore,

π1(X)

[π1(X), π1(X)]= 〈ap, bp, h | [ap, bp] = hn = 1〉 =

g⊕p=1

〈ap〉g⊕p=1

〈bp〉⊕(

〈h〉〈hn〉

), (C.3.5)

where ap, bp, h now represent equivalence classes in the abelianization and 〈ap〉 ' Z,

〈bp〉 ' Z, 〈h〉〈hn〉 ' Z/nZ ' Zn. Thus,

π1(X)

[π1(X), π1(X)]'

Z2g × Zn n ≥ 1

Z2g+1 n = 0.

(C.3.6)

Finally we have,

H1(X,R) ' Hom(H1(X,Z),R) ' Hom

(π1(X)

[π1(X), π1(X)],R)'

Hom(Z2g × Zn,R) n ≥ 1

Hom(Z2g+1,R) n = 0

'

R2g n ≥ 1

R2g+1 n = 0

In conclusion,

dimH1(X,R) =

2g n ≥ 1

2g + 1 n = 0

(C.3.7)

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