Topological Twisting

Yi-hong Gao

Institute of Theoretical Physics, Beijing 100080, China

Lectures presented at Morningside Center of Mathematics, July 2006

Topological twisting is a mathematical proceedure of constructing topological quantum

field theories (TQFT) from (extended) supersymmetric ones. Sketchilly, we have

N ≥ 2 SUSY

N = 1 SUSY, Kahler

twisting−−− −→ Topological Field Theories (0.1)

These lecture notes will be devoted to a pedagogical introduction to topological twisting,

paying particular attentions to the problem of how to topologically twist supersymmetric

Yang-Mills theories. We will divide our discussions into the following three parts:

• What is TQFT?

• Supersymmetries

• Twisting Supersymmetric Theories

1 What is TQFT?

Recall that the basic data in ordinary quantum field theories (QFT) consists of

• A spacetime manifold, (M, gµν)

• A set of fields, Φ = φ1(x), φ2(x), . . ., with x ∈ M

• An action functional, S = S[Φ(x)], which may depend on the spacetime metric gµν(x)

• A “path-integral measure”, [dΦ(x)]

The (Euclidean) partition function as well as correlation functions are defined by

Z =

∫[dΦ] e−S[Φ(x)]

〈φi1(x1)φi2(x2) · · ·〉 =

∫[dΦ] e−S[Φ(x)] φi1(x1)φi2(x2) · · ·

(1.1)

1

In general, both Z and 〈· · ·〉 depend on some geometric data (such as gµν) on M , thus not

giving rise to a topological field theory. Now, by definition, a TQFT is a special kind of QFT

such that (1.1) are topological invariants on the spacetime manifold M .

Thus, our main problem is to construct concrete models in which (1.1) are independent

of the metric gµν as well as other geometric data (if any) on M . To achieve this, one may

have the following three alternatives.

• Quantum Gravity

One may regard the metric as a dynamical field, φ0(x) = gµν(x), and add it to the field set

Φ, so that Φ(x) = φ0(x), φ1(x), . . .. This exactly means that we are dealing with a theory

of quantum gravity, where the so-called general covariance manefests. The path integrals

(1.1) are now to be performed over all metrics on M and hence the resultant should, as long

as renormalizable, not depend on any specific choice of the geometric data on M . In this case

both Z and 〈· · ·〉 are topological invariants. As one knows, however, a theory so constructed

is renormalizable provided M is of dimension two. In that connection one has the famous

equivalence [2] “2D Quantum Gravity ⇐⇒ 2D Topological Gravity”. Unfortunately, this

construction does not directly apply to the higher dimensional situations.

• Theories of the Chern-Simons Type

It is possible to construct topologically invariant actions S[Φ] independent of the metric

gµν . A prototype of this construction is the Chern-Simons theory [3] (in 2 + 1 dimensions),

whose action looks like

S[A] =k

16π2

∫

M

Tr

(A ∧ dA +

2

3A ∧ A ∧ A

), (1.2)

where A is a connection 1-form defined on some G-bundle over M . This action is independent

of the metric, hence (1.1) being topological invariants. A similar construction also applies to

the “gauge theory of gravity” in 2 + 1 dimensions. Again, higher dimensional extensions of

such theories are still eluded.

• Theories of the Cohomological Type

One may also construct field theories of the “cohomological type” [4], which is the main

topic to be discussed below. In such a theory, the action itself is not necessarily independent

of the spacetime metric, but it should have the “BRST-exact” form εS = δW [Φ], where

δ = δε acts on the fields Φ like a differential operator, obeying the nilpotent condition δ2 = 0.

The infinitesimal parameter ε is a Grassmannian (or anticommuting) number. Now δ defines

a BRST complex, whose cohomology ring is described by H = Ker(δ)/Im(δ). Observables

are then defined by the BRST-closed condition δO = 0. Thus, up to the physical equivalence

2

O ∼ O+ δ(· · ·), these observables are in one to one correspondence to the elements of H. As

we will see shortly, this kind of theories actually have a topologically invariant meaning.

1.1 BRST Symmetry and TQFT

The nilpotent condition of the BRST operator δ together with the exactness of the action

S implies δS[Φ] = 0, so δΦ may be thought of as the infinitesimal version of some symmetry

transformation Φ → Φ′ (under which S is invariant). Suppose now the path-integral measure

[dΦ] is also invariant under this transformation. Then, for an arbitrary functional F [Φ] and

an observable O, one has the following ‘obvious’ (though rather formal) identities:

〈F [Φ]O〉 =

∫[dΦ] e−S[Φ] F [Φ]O[Φ] =

∫[dΦ′] e−S[Φ′] F [Φ′]O[Φ′]

=

∫[dΦ] e−S[Φ] F [Φ′]O[Φ] =

∫[dΦ] e−S[Φ] F [Φ + δΦ + O(ε2)]O[Φ]

Although these identities seem to be quite trivial, one can use them to establish some useful

results. For example, it is be possible to expand the functional F [Φ + δΦ + O(ε2)] further in

powers of ε, F [Φ + δΦ + O(ε2)] = F [Φ] + δF [Φ] + O(ε2), to derive

〈δF [Φ]O〉 = 0. (1.3)

Eq.(1.3) shows that in a field theory with BRST symmetry, correlators between a BRST-exact

operator and a BRST-closed operator have to vanish. As a consequence, if two observables

O′, O′′ are physically equivalent, and O, . . . are some other observables, then we have

〈O′O · · ·〉 = 〈O′′O · · ·〉, for O′ = O′′ + δ(something). (1.4)

It follows that correlators between observables depend only on the cohomology classes of the

inserted operators.

Note that in deriving (1.3)-(1.4) we have only used BRST invariance of S[Φ], but not its

BRST-exactness. Now, in a (cohomological) TQFT, the action is actually BRST-exact, and

there exists a fermionic functional W [Φ] such that εS[Φ] = δW [Φ]. For such a theory we can

establish topological invariance by the following considerations. In general, the action may

depend on (in addition to the dynamical variables Φ) certain coupling constants, external

sources, the metric and other possible geometric data on the spacetime manifold M , and let

us collect all of these non-dynamical quantities together to form a “parameter space” Ω. We

can then verify the claim:

For a cohomological TQFT whose action S = S[ω, Φ] depending on ω ∈ Ω,

both the partition function

Z =

∫[dΦ] e−S[ω,Φ] (1.5)

3

const.1 const.2const.3

const.4

const.5

Figure 1: Connected components of Ω, on each of which (1.5) and (1.6) are constants.

and the correlators

〈O1O2 · · ·〉 =

∫[dΦ] e−S[ω,Φ]O1O2 · · · (1.6)

between observables will be locally constant functions on the parameter space. In

other words, they will keep constant on each connected component of Ω (see Figure

1). In particular, if Ω = Ω0 × Ω1 contains a connected factor Ω0, then (1.5) and

(1.6) are both completely independent of the parameter ω0, with ω = (ω0, ω1) ∈Ω0 × Ω1.

Actually, one may write εS[ω, Φ] = δW [ω, Φ] and differentiate Eq.(1.5) to derive:

ε∂Z(ω)

∂ω= −

∫[dΦ] e−S[ω,Φ] ∂

∂ωδW [ω, Φ] = −

⟨δ(

∂

∂ωW [ω, Φ])

⟩

where the last step follows from [δ, ∂/∂ω] = 0 (as δ acts only on dynamical fields, but not on

the parameter ω). From (1.3) one immediately deduces Z ′(ω) = 0 ∀ω ∈ Ω, so that Z(ω) is

indeed a locally constant function. Moreover, since the space Ω0 assumes to be connected,

∂Z(ω0, ω1)/∂ω0 = 0 implies Z does not depend on ω0. A similar argument also applies to

the correlation functions (1.6).

Now we setMET(M) = all metrics gµν(x) on M,Λ = all coupling constants = R+ ×R+ × · · ·

and decompose Ω into the product space Ω = MET(M)×Λ× · · ·. Since the first two factors

of this product are connected spaces, according to the claim, the partition function (1.5) and

correlators (1.6) are both independent of the metric gµν as well as the couplings λ ∈ Λ, thus

being topological invariants.

4

A remarkable feature of cohomological TQFT is that it can be constructed in all d =

0, 1, 2, . . . dimensions. One such model is the (topologically twisted) N = 2 supersymmetric

Yang-Mills theory on a d = 4 manifold [4], whose partition/correlation functions give rise to

Donaldson invariants.

There is a natural conserved scalar chargeQ associated with the symmetry transformation

δ, determined by

δεΦ = [εQ, Φ] =

ε[Q, Φ], for bosonic Φ,

εQ, Φ, for fermionic Φ.(1.7)

This charge is called “BRST operator”. The nilpotent condition δ2 = 0 is fulfilled by requiring

that Q be a fermionic charge, obeying

Q2 = 0. (1.8)

In fact, if Q is fermionic and (1.8) holds, then applying two BRST transformations succeed-

ingly to Φ yields

δεδε′Φ = [εQ, [ε′Q, Φ]] = εQε′QΦ− εQΦε′Q− ε′QΦεQ+ Φε′QεQ

Ones sees that the first and the last terms in the above expression vanish due to εQ = −Qε

and Q2 = 0, while the second and the third terms give rise to

(−1)ηΦ(εε′ + ε′ε)QΦQ, with ηΦ =

0, if Φ is bosonic

1, if Φ is fermionic

which also vanishes because ε, ε′ are Grassmannian numbers. Thus, (1.8) implies the nilpotent

condition δεδε′ = 0.

Since Q is both scalar and fermionic, the usual spin-statistics theorem will be violated

and thus the underlying theory contains ghosts. Note that the symmetry algebra generated

by Q with the relation (1.8) is invariant under rotations Q → eiαQ. Accordingly, the BRST

algebra has a U(1) group as its automorphism group. In many cases, such an automorphism

group is also a symmetry of the action, and if it is, we can label fields by their U(1) charges

Ui = 0,±1,±2, . . . arising in the symmetry transformation laws φi(x) → eiUiαφi(x). These

U(1) charges are called ghost numbers; in particular, Q carries ghost number U = 1.

The ghost number of a field is closely related to the statistics of this field. As an illustra-

tion, suppose we have a bosonic field A with ghost number U = 0. Then, any field ψ with

the same quantum numbers as [Q, A] will carry ghost number U = 1, which is fermionic; A

field φ will carry U = 2 if it transforms as Q, ψ, which is bosonic again. Similarly, χ carries

U = −1 provided the anticommutator Q, χ has the same quantum numbers as A, thus χ

being fermionic; ξ has U = −2 if [Q, ξ] transforms as χ, and so on. These matters may be

summarized in the following table:

5

Fields Quantum Behavior Ghost Numbers Statistics...

......

...φ φ ∼ [Q, ψ] U = 2 Bosonicψ ψ ∼ [Q, A] U = 1 FermionicA U = 0 Bosonicχ [Q, χ] ∼ A U = −1 Fermionicξ [Q, ξ] ∼ χ U = −2 Bosonic...

......

...

One sees from the above that the ghost number U = even (odd) if and only if the underlying

field is bosonic (fermionic). In this connection, the algebra of quantum fields is graded, and

the Q–cohomology ring H∗Q can be decomposed into

H∗Q =

⊕

k

HkQ,

where the k th cohomology group HkQ consists of those observables whose ghost number is

U = k.

1.2 General Covariance

As mentioned, TQFT (of the cohomological type) is more than just a field theory with

BRST symmetry; the action S = Q,W is not only Q-closed but also Q-exact. (The ghost

number of S should be zero, so W has U = −1.) The latter condition leads to theQ-exactness

of the momentum-energy tensor:

Tµν(x) ≡ 2√g

δS

δgµν(x)= Q, Gµν(x), Gµν(x) ≡ 2√

g

δW

δgµν(x). (1.9)

To investigate the physical consequence of (1.9), let us consider a 1-parameter family of

diffeomorphisms M → M close to the idendity

xµ → x′µ = xµ + τvµ(x) + O(τ 2), |τ | ¿ 1. (1.10)

The change in the metric is described by δτgµν = τLvgµν with the Lie derivative

Lvgµν = gµλ∂νvλ + gνλ∂µv

λ + vλ∂λgµν

This together with the definition of Tµν gives

0 = δτS =τ

2

∫

M

d4x√

g T µν(x)Lvgµν = −τ

2

∫

M

d4x√

g vν∇µTµν(x),

which results in the conservation law ∇µTµν(x) = 0. The corresponding charge is then given

by:

T (v) =

∫

Σ

vν(x)T µν(x) dσµ ⇒ T (v) = Q, G(v) (1.11)

6

where Σ ⊂ M is a hyperplane at some t = const and G(v) =∫

Σvν(x)Gµν(x) dσµ. According

to Noether’s theorem, (1.11) generates diffeomorphisms of M acting on the underlying system,

under which an observable O will be transformed into

O′ = O + τδvO, δvO ≡ [T (v),O] = [Q, G(v),O]. (1.12)

Now since O is a BRST invariant, i.e. [Q,O] = 0 for bosonic O and Q,O = 0 for fermionic

O, (1.12) together with the graded Jacobian identities1 shows that δvO takes a BRST-exact

form:

δvO =

Q, [G(v),O], if O is bosonic,

[Q, G(v),O], if O is fermionic

In other words, the infinitesimal transformation δvO of an observable under the spacetime

diffeomorphism (1.10) is proportional to the BRST transformation δεF of another operator

F = [G(v),O ≡ G(v)O − (−1)ηOOG(v). Thus, from (1.3) one sees that

〈δvO ·∏

i

Oi〉 = 0, (1.13)

so that the correlator 〈O · · ·〉 is invariant under the action of Diff(M) : O → O′.

Eq.(1.13) means that even at the quantum level, the spacetime symmetry in a (cohomo-

logical) TQFT is described by the diffeomorphism group Diff(M). This is a huge symmetry

group, much larger than those in ordinary QFT. Recall that Diff(M) invariance is one of the

main features of Einstein’s gravitational theory, known as general covariance. However, in

usual general relativity Diff(M) is only a Lagrangian symmetry, which will be spontaneously

broken to a much smaller subgroup as long as we choose a vacuum, described classically by a

solution gµν of the Einstein field equations. The subgroup of Diff(M) that leaves this vacuum

solution |gµν〉 invariant is generated by all Killing vectors on M with respect to the metric

gµν (namely, those vector fields v obeying Lvgµν = 0). This is the isometry group of (M, gµν),

which we will denote by Isom(M) ⊂ Diff(M). So in ordinary QFT with a background metric

gµν , spacetime symmetry is described by the isometry group Isom(M) (rather than the much

larger group Diff(M)), examples including the Poincare symmetry SO(3, 1) n R3,1 for M =

Minkowski space and the de Sitter symmetry SO(4, 1) for M = dS4. Moreover, for conformal

field theories (CFT) in d > 2 dimensions, spacetime symmetry is a group Conf(M) generated

by conformal Killing vectors, which is, again, much smaller than Diff(M), though it contains

Isom(M) as a subgroup. Even for CFT in d = 2 where the symmetry group is infinite dimen-

sional (e.g. Diff(S1)), spacetime symmetry is still a small subgroup of Diff(M). In all such

“conventional” theories, general covariance is spontaneously broken, but in TQFT discussed

above, Diff(M) invariance manifests (at the quantum mechanical level).

1The explicit expression for the graded Jacobian identities reads: [Q, G,O] = Q, [G,O]+[Q,O], G =[Q, G,O] + [G, Q,O].

7

A related fact is that the Hamiltonian H =∫

d3xT00 in a TQFT takes the BRST-exact

form Q,∫

d3xG00. Consequently, inserting H into correlators of observables always yields

a vanishing result, 〈HO1O2 · · ·〉 = 0. It follows that only ground states will contribute to the

path-integrals (1.5)-(1.6). Thus, TQFT could be regarded as a simplified version of ordinary

QFT, where all the excited states are truncated.

1.3 Integration over Fermionic Variables

Now, to analyze path-integrals in TQFT, we need to know how to perform integration over

fermionic variables. Here we shall give a discussion of this problem. Let ϑ be a Grassmannian

variable; since the norm of ϑ has no meanings, one can not define∫

dϑ(· · ·) as a Riemann

sum. Nevertheless, it is still possible to consider this integral as a linear functional L acting

on the space of functions in ϑ. Such a functional is well-defined provided the moments L(ϑp)

are consistently specified for all p = 0, 1, . . . . In our present case, we need only to specify the

values of the first two moments L(ϑ0) and L(ϑ1), since ϑp vanishes for each p ≥ 2. As usual,

we may define L(1) = 0, L(ϑ) = 1; namely:∫

dϑ = 0,

∫dϑϑ = 1. (1.14)

To compute the integral of F (ϑ), notice that F (ϑ) can always be expanded as F0 + F1ϑ; so

we have ∫dϑF (ϑ) = F1 =

∂F (ϑ)

∂ϑ.

This definition can be easily extended to multi-variable cases. Let ψ = ψ1, ψ2, . . . , ψdbe a set of d independent fermionic variables, each having ghost number U = 1. Integration

over these variables may be performed with the following specification of the moments:

∫dψ ψi1ψi2 · · ·ψim =

0 if m 6= d

εi1i2···id if m = d(1.15)

here εi1i2···id is the “Levi-Civita tensor”, defined by

εi1i2···id =

1 if (i1, i2, . . . , id) is an even permutation of (1, 2, . . . , d)

−1 if (i1, i2, . . . , id) is an old permutation of (1, 2, . . . , d)

0 otherwise

Note that in order to obtain a non-zero value of (1.15), the total ghost number of the measure

dψ = dψ1 · · · dψd should, as a rule, be equal to the total ghost number of the integrand.

Similarly, if χ = χ1, · · · , χd is another set of independent fermionic variables, with ghost

number possibly different from that of ψ (say, Uχ = −1), we can define

∫dψdχ ψi1ψi2 · · ·ψim χj1χj2 · · ·χjn =

0 if m or n 6= d

εi1i2···idεj1j2···jd if m = n = d(1.16)

8

One necessary condition for (1.16) nonvanishing is that the net ghost numbers of the measure

and the integrand are equal to each other.

Two comments about integration over fermionic ghosts are in order:

(1) If we change the integration variables ψi → ψ′i = J ijψ

j, we will get a Jacobian in the

integral measure, namely dψ = det[J ij] ·dψ′. This Jacobian is the inverse of what appears in

the bosonic case. The reason for introducing such a Jacobian is that the fermionic ghosts ψi

in (1.15) are to be interpreted as dummy variables, so we must have∫

dψ′F (ψ′) =∫

dψF (ψ)

after changing integration variables. Taking the integrand F (ψ) = ψi1ψi2 · · ·ψid as in (1.15)

gives F (ψ′) = det[J ij]F (ψ), and one should specify dψ′ to be det[J i

j]−1dψ, in order to get

the desired relation dψ′F (ψ′) = dψF (ψ).

(2) One can, just as in the bosonic case, introduce a Dirac function δ(ψi) for the fermionic

ghosts, defined by∫

dψ δ(ψi) = 1. But here we get a formula δ(J ijψ

j) = det[J ]δ(ψi) instead

of the usual one δ(J ijφ

j) = | det[J ]|−1δ(φi), since the fermionic Dirac function can be written

as a product δ(ψi) = ψ1ψ2 · · ·ψd.

Now let us consider some concrete examples. We first compute the integral∫

dψ eAijψiψj/2

where A = (Aij) is a non-singular d×d anti-symmetric matrix. To this end, one may expand

the exponential as a Taylor series:

e12Aijψiψj

=∞∑

k=0

1

2kk!Ai1j1Ai2j2 · · ·Aikjk

ψi1ψj1ψi2ψj2 · · ·ψikψjk

Each term in this series is a product of even number of ψ and according to (1.15), only the

term with 2k = d can have a non-zero contribution to the integral. So this integral does not

vanish only if d is even and, if d = 2m, we need only to consider the term with k = m. Thus,

applying (1.15) yields∫

dψ e12Aijψiψj

=1

2mm!Ai1j1Ai2j2 · · ·Aimjmεi1j1i2j2···imjm ≡ Pf [A]. (1.17)

Here Pf [A] is known as the Pfaffian of the anti-symmetric matrix (Aij), which is related to

the determinant of (Aij) via (Pf [A])2 = det[A].

Our second example is the integral∫

dψdχ eBijψiχj

,

with B = (Bij) being a non-singular d×d matrix. In this example the exponential is expanded

as

eBijψiχj

=∞∑

k=0

1

k!Bi1j1Bi2j2 · · ·Bikjk

ψi1χj1ψi2χj2 · · ·ψikχjk

9

Thus, up to an overall factor (−1)d(d−1)/2 arising from exchange of the positions between ψi’s

and χj’s, the integral is computed, with the aid of (1.16), by2

1

d!Bi1j1Bi2j2 · · ·Bidjd

εi1i2···id εj1j2···jd

So using the identities εi1i2···id Bi1j1Bi2j2 · · ·Bidjd= det[B] ·εj1j2···jd

and εj1j2···jd ·εj1j2···jd= d!,

we find ∫dψdχ eBijψiχj

= det[B]. (1.18)

As an application, let us give a simple proof of the algebraic relation (Pf [A])2 = det[A]

mentioned before. We first write, according to (1.17), the Paffian square as an integral

(Pf [A])2 =

∫dψ1 e

12Aijψi

1ψj1

∫dψ2 e

12Aijψi

2ψj2 =

∫dψ1 · dψ2 e

12Aij(ψ

i1+iψi

2)(ψj1−iψj

2).

Next we introduce two sets of fermionic variables ψi = (ψi1+iψi

2)/√

2 and χi = (ψi1−iψi

2)/√

2,

with the Jacobian

dψ1 · dψ2 = (−1)d(d−1)

2 dψ1dψ2 = (−1)d(d−1)

2

[det

( 1√2

i√2

1√2

− i√2

)]d

dψdχ = dψdχ

Now, after changing integration variables, the Paffian square becomes∫

dψdχ eAijψiχj, which

is nothing but just det[A], as one can see from (1.18). This then finishes our proof.

The final example we wish to consider is the integral∫

dudψdχ f(u) e14Cijkl(u)ψiψjχkχl

where u = u1, u2, . . . , ud are usual bosonic variables, having ghost number zero. Again, the

integrand can be expanded into a series

dudψdχ∞∑

n=0

f(u)

22nn!Ci1j1k1l1Ci2j2k2l2 · · ·Cinjnknlnψi1ψj1χk1χl1 · · ·ψinψjnχknχln

We may therefore perform integration over (ψ, χ) using (1.16). Clearly, this integral vanishes

if d = odd, and for d = 2m, only those terms with n = m will contribute, so the result simply

reads

duf(u)

22mm!εi1j1i2j2···imjmCi1j1k1l1Ci2j2k2l2 · · ·Cimjmkmlmεk1l1k2l2···kmlm

Now since duεi1j1i2j2···imjm = dui1∧duj1∧· · ·∧duim∧dujm , we may define differential two-forms

Ckl = 12Cijkldui∧duj and represent the above result as 1

2mm!f(u)Ck1l1∧· · ·∧Ckmlmεk1l1k2l2···kmlm .

So according to the definition of Pfaffian, we finally arrive at∫

dudψdχ f(u) e14Cijkl(u)ψiψjχkχl

=

∫f(u) Pf [Ckl]. (1.19)

2We will drop the overall factor (−1)d(d−1)/2, as it can always be absorbed into the definition (1.16).In fact, the integral measure dψdχ is originally defined by dψ1 · · · dψddχ1 · · · dχd, but one may redefine itto be dψdχ = dψ1dχ1 · · · dψddχd, which differs from the original one (now denoted as dψ · dχ) by a factor(−1)d(d−1)/2, namely dψdχ = (−1)d(d−1)/2dψ · dχ. In what follows we will adopt this new definition.

10

Although the above integrals are all defined on finite-dimensional spaces of “fields”, their

infinite-dimensional extension is straightforward. Let us consider, for instance, the extension

of the gaussian integral (1.18). Suppose ψi(x) ∈ F+, χi(x) ∈ F− are two ghost fields depend-

ing on spacetime points x ∈ M , and let D : F+ → F− be a non-degenerate linear operator

acting on these fields. If the infinite-dimensional field space F− has an inner product 〈·, ·〉defined on it, then (1.18) will be generalized naturally to

∫dψdχ e〈χ,Dψ〉 = det[D], (1.20)

where the determinant det[D] is to be defined by some regularization procedures.

Practically D often appears as a differential operator. In such a case, the non-degenerate

assumption of D is in general invalid; usually one can find some zero modes ψ0, χ0 of ψ, χ,

respectively, such that Dψ0 = D†χ0 = 0, here D† : F− → F+ denotes the formal adjoint of the

operator D with respect to the inner product 〈·, ·〉. When zero modes appear, the integration

formula (1.20) has to be modified. To exploit this, let us decompose ψ, χ orthogonally into

ψ = ψ0 + ψ′, χ = χ0 + χ′

where ψ′ (resp. χ′) summarizes all the non-zero modes of ψ (resp. χ). Accordingly, we have

〈χ,Dψ〉 = 〈χ′,Dψ′〉 and dψ = dψ0dψ′, dχ = dχ0dχ′. Integration over non-zero modes can be

performed as in (1.20), and this gives a factor of det′[D], defined by a regularized product of

all non-zero eigenvalues of D. The remaining integral is∫

dψ0dχ0 ·1, which will contribute to

a null factor due to the net ghost number ∆U of the measure dψ0dχ0. If our fields ψ(x), χ(x)

have ghost numbers Uψ and Uχ, respectively, then ∆U can be counted by:

∆U = #zero modes of ψ · Uψ + #zero modes of χ · Uχ.

In particular, for Uψ = 1 and Uχ = −1, we have

∆U = dim KerD − dim KerD† ≡ Ind(D), (1.21)

here Ind(D) denotes the index of the differential operator.

The condition ∆U 6= 0 will sufficiently force the integral in (1.20) vanishing. In order to

balance ∆U = Ind(D) and thus to get a non-zero result, we may insert some observables into

that integral. Suppose Oi has ghost number U = Ui, then∫

dψdχ e〈χ, Dψ〉 ∏i

Oi

can take a non-vanishing value only if the following “ghost number selection rule” is obeyed:

∑i

Ui = Ind(D). (1.22)

11

We end this discussion with a geometric interpretation of Ind(D). Let B be a configuration

space of some bosonic fields of ghost number zero, and let E π−→ B be a vector bundle over

B. Choose a generic section (thus transverse to zero) s : B → E , which may be regarded as a

function locally defined on B with values in the fibre of E ; so each point u ∈ B is associated to a

vector s(u) ∈ Eu. Now, given such a “function” together with a connection on E , one can take

the covariant derivative DsDu|u : TuB → Eu, which naturally defines a linear map D : F+ → F−,

where F+ and F− are two vector spaces isomorphic to TuB and Eu, respectively. If we restrict

u to the set M of zero points of s, then the covariant derivative will become intrinsic (i.e.

independent of the choice of connections on E), since at each u ∈ M the coupling between

s(u) and the connection vanishes. The assumed transversality condition of s will relate the

dimension of M to the index of D in a natural way. Actually, suppose both the spaces F± are

finite-dimensional, we have dim TuM = dimF+−dimF− by transversality. The vector-space

isomorphism F+/KerD ∼= ImD implies dimF+ − dim Ker D = dimF− − dim Coker D,

so that

dimM = Ind(D). (1.23)

When F± are of infinite-dimensions, however, the naive formula dimM = dimF+ − dimF−no longer makes senses. Nevertheless the identity (1.23) still holds thanks to the Sard-Smale

theory.

1.4 Finite-dimensional Models

We now work out some concrete models of TQFT. As we have seen, the basic data involves

a configuration space B of bosonic fields with ghost number zero, and a vector bundle E π−→ B.

For simplicity, we only consider systems with finite degrees of freedom in this subsection, so

both d ≡ dimB and r ≡ rank E are assumed to be finite here.

Thus, points of B (i.e., those bosonic fields with ghost number zero) are described by the

usual commuting coordinates ui (i = 1, 2, . . . , d), and vectors in the fibre of E are indexed

by a = 1, 2, . . . , r. We choose a fibre metric gab(u) on Eu, pointwisely, along with an SO(r)-

connection Ai = (Aiab) satisfying the condition:

∂

∂uigab(u) = Aiab + Aiba. (1.24)

Mathematically, the meaning of (1.24) is explained as follows. The metric gab(u) pointwisely

defines an SO(r)-invariant inner product 〈s, t〉 = gabsatb on the space of sections of E , while

the connection A defines a covariant derivative ∇A acting on such sections. To make these

two structures compatible, we require a kind of the “Leibniz rules” for ∇A,

∂〈s, t〉∂ui

= 〈∇Ai s, t〉+ 〈s,∇A

i t〉.

12

These rules, in terms of components, are precisely the compatible condition (1.24).

Let us introduce two BRST multiplets. The first is (ui, ψi), with ghost number U = (0, 1),

which consists of the coordinate functions ui on B as well as some tangent vectors ψi. The

BRST transformation law takes a very simple form:

δui = iεψi, δψi = 0 (1.25)

where the nilpotent condition δ2 = 0 is trivially satisfied. The space F+ of the fermionic

ghosts ψ = ψi is thus isomorphic to the tangent space TuB.

The second BRST multiplet is (χa, Ha), with ghost number U = (−1, 0). These fields are

introduced in order to describe the fibre structure of E . In particular the space F− spanned

by χ = χa should be isomorphic to Eu. Thus, if we look at these ghosts at the same point

u of B but in different open covering sets, their components will be changed according to

χa → χ′a = G(u)abχ

b, Ha → H ′a = G(u)abH

b (1.26)

where G(u) ∈ SO(r) is a local gauge transformation. In other words, both χ and H behave

as sections of E .

Since the naive BRST transformation law δχa = εHa, δHa = 0 is not covariant under

(1.26), we shall consider the “gauge covariant” BRST operator δA ≡ δ + iεψiAi acting on

(χa, Ha). It is easy to verify that the nilpotent condition δ2 = 0 is equivalent to the condition

δ2A = 1

2ε1ε2ψ

iψjFij, with Fij ≡ ∂iAj − ∂jAi + [Ai, Aj] being the curvature two-forms. Thus,

the correct transformation law for (χa, Ha) should be δAχa = εHa, δAHa = −12εψiψjFij

abχ

b,

namely

δχa = εHa − iεψiAiabχ

b, δHa = −iεψiAiabH

b − 1

2εψiψjFij

abχ

b (1.27)

By construction, the BRST algebra specified by (1.25) and (1.27) is nilpotent.

Now, according to the standard formalism, the action S is described by the BRST trans-

form of a certain fermionic “potential” W , of ghost number U = −1. One may take

W =1

2λ〈χ,H + 2is〉 (1.28)

where s ∈ Γ(E) is a section of the vector bundle. Computing δW = δAW then gives

S =1

2λ〈H, H + 2is〉+

1

λ〈χ, Dψ〉+

1

4λFijabψ

iψjχaχb (1.29)

with

(Dψ)a = (∇Ai sa)ψi. (1.30)

The parameter space Ω of this theory consists of R+×MET(E)×Γ(E)×connections on E .

This space is of course connected, implying that the partition function

Z =

(1

2π

)d ∫du dψ dχ dH e−S (1.31)

13

is independent of the choices of s ∈ Γ(E), λ ∈ R+, and gab ∈ MET(E). In other words (1.31)

depends only on B and E but not on their geometric data, and thus constitutes a topological

invariant.

The net ghost number of the integration measure in (1.31) is ∆U = d−r, so the partition

function can have a non-zero value if d = r. In that case, it is natural to ask which invariant

(1.31) actually describes. Thus, let us first evaluates the gaussian integral over H. This leads

to the following integral representation of Z:

Z =

(λ

2π

)d/2 ∫dudψdχ

1√g(u)

exp

(−〈s, s〉

2λ− 1

λ〈χ,∇A

i s〉ψi − 1

4λFijabψ

iψjχaχb

)(1.32)

Since Z does not depend on s, one may choose s to be identical to zero. With this choice the

integral becomes exactly the type (1.19) we considered before, so for d = even the partition

function is simply given by (up to an irrelevant overall factor (−1)d/2):

Z =1

(2π)d/2

∫

B

1√g(u)

Pf [Fab] =1

(2π)d/2

∫

BPf [Fa

b], (1.33)

where Fab ≡ 1

2Fij

abdui ∧ duj is the curvature two-form on E . The right hand side of (1.33) is

noting but the Euler characteristic χ(E) of the vector bundle, which is indeed a topological

invariant.

On the other hand, one may evaluate (1.32) by taking λ → 0 while keeping s generic. In

this case only the zero-set M of s will contribute to the integral, so we may fix an arbitrary

point uα ∈ M and consider its contribution. As indicated at the end of the last subsection,

the connection A decouples from s at uα. Also, since a generic section is transverse to zero,

we can assume det[∂sa/∂ui] 6= 0 at u = uα. Now expanding

s(u) = s(uα) +∂s

∂ui(uα) · (ui − ui

α) + O((u− uα)2) ≈ ∂s

∂ui(uα) · (ui − ui

α)

we compute the contribution of uα to the integral (1.32) as:

(λ

2π

)d/21√

g(uα)

∫du exp

(− 1

2λgab(uα)

∂sa

∂ui(uα)

∂sb

∂uj(uα)(ui − ui

α)(uj − ujα)

)

×∫

dψ dχ exp

(−1

λgab(uα)χa ∂sb

∂ui(uα)ψi − 1

4λFijabψ

iψjχaχb

). (1.34)

The fermionic modes in (1.34) are all massive, and the curvature term is suppressed in the

weak coupling limit λ → 0, so the integral is essentially gaussian. Performing this explicitly

and summing the resultant over all uα ∈M, we obtain:

Z =∑

s(u)=0

sign det

(∂sa

∂ui(u)

)≡

∑α

εα. (1.35)

14

As a consequence, the Euler characteristic χ(E) can be alternatively computed by the number

of zero points of s, counted with signs εα = ±1. This is known as the Hopf-Poincare theorem.

In deriving (1.35) we have assumed that the section is generic, namely s is transverse to

zero. This means that the zero-set of s consists of isolated points. But for a special choice

of s ∈ Γ(E), the zero-set M may become a disjoint union of some (connected) submanifolds

Mα, with dim(Mα) = dα. Upon trivialization of the bundle, we can choose a local coordinate

system of u ≈ uα ∈ Mα, ui = (um‖ , up

⊥), with um‖ (m = 1, . . . , dα) in the direction tangent to

Mα and up⊥ (p = dα + 1, . . . , d) in the direction normal to Mα, such that

sa(ui) ≈(

0 00 sp

q

) (um‖ − um

α

uq⊥ − uq

α

)(1.36)

where (spq) is a (d− dα)× (d− dα) matrix. We assume further that the behavior of s in the

directions normal to Mα is non-degenerate: det[spq] 6= 0. Accordingly, the fermionic ghosts

ψi, χa are decomposed into the massless components ψm‖ , χm

‖ and the massive components

ψp⊥, χp

⊥. This decomposition depends of course on how we trivialize E ; globally the massless

components χm‖ will behave as a section of some vector bundle Eα over Mα, which is called

“vector bundle of antighost zero modes” [7]. Now, for small λ, one may evaluate the partition

function by integrating out the massless and massive modes respectively. The former gives the

Euler class χ(Eα), while the latter provides a factor εα = ±1, determined by εα = sign det[spq].

One thus finds [7]

Z =∑

α

εαχ(Eα). (1.37)

• Elimination of the Signs

So far we have derived a couple of explicit expressions for the partition function. These

expressions have a common feature: when they count contributions from different connected

componentsMα ofM⊂ B, a sign εα = ±1 will appear. Such a sign originates from the BRST

symmetry between bosonic and fermionic fields in the massive sector, whose appearance is

essential for suppressing geometric dependence.

However, sometimes we wish to eliminate this εα and count the connected components

without signs. Let us consider a vector bundle E π−→ B of rank r = d ≡ dim(B). Suppose s

is a section of E transverse to zero. The zero-set M of s thus consists of isolated points in

B, as discussed above. In terms of local data, this set can be described by the roots of the

equations

sa(ui) = 0 (1 ≤ a, i ≤ d). (1.38)

Now, in order to count the total number of such roots, we introduce a set of variables ya

and defines uI = (ui, ya) (1 ≤ I ≤ 2d) as local coordinates on the extended manifold B ≡ E .

15

We then extend sa(ui) arbitrarily to the functions

sA(uI) ≡ (sa(uj, yb), hi(u

j, yb))

(1 ≤ A ≤ 2d), (1.39)

where sa(u, y), hi(u, y) obey the following conditions:

sa(uj, yb) = sa(uj) + O(y), hi(uj, yb) = ya

∂sa

∂ui+ O(y2). (1.40)

The above conditions are motivated by half of the “Riemann-Cauchy equations”

∂sa

∂ui=

∂hi

∂ya

(mod y), (1.41)

which makes sA(uI) look like a holomorphic map between two complex manifolds. Thus, it’s

likely that the number of solutions to the equation sA = 0 is a topological invariant, although

the number of solutions of (1.38) may not be so.

Globally, the functions sa(u, y) may be regarded as a section of the pull-back bundle π∗(E)

over B, here π : B ≡ E → B is the canonical projection of E onto its base manifold B. In a

similar spirit, since hi(u, y) transforms under ui → u′i as a component of a certain cotangent

vector field on B, the functions hi may be identified with a section of the pull-back bundle

π∗(T ∗B) over B. So sA(u) defines a section of the vector bundle E = π∗(E)⊕

π∗(T ∗B).

We can construct a model whose partition function computes the weighted sum∑

α εα

with the signs

εα = sign det

(∂sA

∂uI(uα)

)

2d×2d

, (1.42)

where uα are determined by sA(uI) = 0 or, equivalently, by

sa(uj, yb) = hi(uj, yb) = 0. (1.43)

With the aid of (1.40), it is easy to see that the solutions of (1.43) at y = 0 are precisely

the solutions of the original system (1.38) and, moreover, at these solutions the determinant

det(

∂esA

∂euI (uα))

in (1.42) is computed by:

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∂s1

∂u1 · · · ∂s1

∂ud∂s1

∂y1· · · ∂s1

∂yd...

. . ....

.... . .

...∂sd

∂u1 · · · ∂sd

∂ud∂sd

∂y1· · · ∂sd

∂yd∂h1

∂u1 · · · ∂h1

∂ud∂h1

∂y1· · · ∂h1

∂yd...

. . ....

.... . .

...∂sd

∂u1 · · · ∂hd

∂ud∂hd

∂y1· · · ∂hd

∂yd

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∂s1

∂u1 · · · ∂s1

∂ud ∗ · · · ∗...

. . ....

.... . .

...∂sd

∂u1 · · · ∂sd

∂ud ∗ · · · ∗0 · · · 0 ∂s1

∂u1 · · · ∂sd

∂u1

.... . .

......

. . ....

0 · · · 0 ∂s1

∂ud · · · ∂sd

∂ud

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

> 0,

which leads to εα = +1 for all α. Thus, if we have a vanishing theorem ensuring that all the

solutions of (1.43) are at y = 0, then the partition function Z =∑

α εα agrees with the

total number of solutions to Eq.(1.38).

16

The above discussion may be generalized as follows. Again, let B be a manifold of di-

mension d, with local coordinates ui. Let E π−→ B be now a vector bundle of rank d′ < d.

For a section sa(ui) of E with nice transversality properties, we can consider the system of

equations:

sa(ui) = 0 (1 ≤ a ≤ d′, 1 ≤ i ≤ d). (1.44)

By the implicit function theorem, solutions of Eq.(1.44) will form a disjoint union of smooth,

compact submanifolds Mα of B, each of which has dimension d − d′. One can, as before,

extend both ui and sa to

uI ≡ (uj, yb), sA(uI) ≡ (sa(uj, yb), hi(u

j, yb)), (1 ≤ I, A ≤ d + d′) (1.45)

here the extension of s obeys a condition similar to that given in (1.40). From the geometric

point of view, uI defines a local coordinate system of the (d + d′)-dimensional manifold

B ≡ E , while sA provides a section of the vector bundle E ≡ π∗(E)⊕

π∗(T ∗B) over B.

The zero-set of s may be decomposed into a disjoint union of manifolds Mα, of dimen-

sions dα. Each of such Mα contains Mα, a connected component of the zero-set of s, as a

submanifold at y = 0. So we must have the inequality dα ≥ d− d′ (the equality dα = d− d′

will hold provided we have the vanishing theorem Mα = Mα). The partition function then

has the form of (1.37), which in the present situation can be represented by

Z =∑

α

εα · χ(Eα) (1.46)

where Eα denotes the vector bundle of antighost zero modes along Mα and εα is the sign of the

determinant det[Qα], with Qα being the maximal non-singular part of the matrix(

∂esA

∂euI (uα)).

Now if there exists a vanishing theorem to guarantee Mα = Mα for each α, namely, if the

solutions of sA(u, y) = 0 are all at y = 0, then due to cancellation between ∂s/∂u and ∂h/∂y,

the sign εα will always take the value +1. So (1.46) becomes

Z =∑

α

χ(Eα). (1.47)

There is an interesting geometrical interpretation [7] of Eq.(1.47). From the standard

BRST construction, we know that the field content must contain a set of extended antighosts

χA = (χa, ρi) associated with the section sA = (sa, hi). The components ρi transform as hi

under diffeomorphisms of B, hence behaving as cotangent vector fields on B. To explore the

geometric meaning of Eα, we have to analyze the zero modes of χ along Mα. The analysis

may be done as follows. According to the assumed vanishing theorem Mα = Mα, the

manifold Mα consists of the zeros of both the sections s and s. The transversality condition

for s implies that there are totally d′ directions in B normal to Mα, each corresponding

17

to a massive mode χa, 1 ≤ a ≤ d′, arising from the antighosts in the original system.

Consequently, all the massless modes of χ must be contained in the extended part of the

antighosts, ρi. They should therefore transform as cotangent vector fields on B and, in turn,

they should be in the directions cotangent to Mα. One may thus ask whether such massless

directions are sufficient to span the whole cotangent space of Mα. Note that we have totally

(d+d′− dα) = 2d′ directions in B normal to Mα, in which only k directions correspond to the

massive modes of χA, here k ≡ rank(Qα) ≤ 2d′3 . Accordingly, the total number of massless

modes of χ is d+d′−k, which is no less than d−d′, and we do have sufficiently many massless

directions to span the whole cotangent space of Mα. The bundle Eα of anighost zero modes

is thus precisely the cotangent bundle T ∗Mα∼= T ∗Mα. So (1.47) gives

Z =∑

α

χ(T ∗Mα) =∑

α

χ(Mα) = χ

(⋃α

Mα

)= χ(M), (1.48)

and the Euler class of the zero-set M of s : B → E allows an integral representation.

1.5 Incorporation of gauge invariance

To contact supersymmetric Yang-Mills theories, we will discuss how to incorporate gauge-

invariance into a system with BRST symmetry. Let us consider a finite-dimensional, compact

Lie group G acting freely on some field configuration space A, with dimA ≡ d. The quotient

manifold B = A/G is thus smooth, having dimension d\ = d − t, here t ≡ dim(G) denotes

the dimension of the gauge transformation group. Clearly, any vector fields on the manifold

A can be decomposed into directions tangent and normal to the G-orbits. If we choose a

basis Txtx=1 of the Lie algebra of G, then the action of Tx on A gives an infinitesimal

diffeomorphism of A, which is generated by a vector field Ux = U ix ∂/∂ui tangent to the

G-orbit. Such vector fields span the whole tangent space to the G-orbit at u and they must

obey the same algebra as Tx’s:

[Tx, Ty] = c zxy Tz, [Ux, Uy] = c z

xy Uz (1.49)

where c zxy denote the structure constants of Lie(G).

In order to describe the degrees of freedom arising from gauge transformations, we intro-

duce a new bosonic ghost field φ = φx Tx valued in the Lie algebra of G, with ghost number

U = 2. Their components φx (1 ≤ x ≤ t) play a role similar to the Lie parameters τx

in G; so two quantities A, B are called gauge equivalent provided A = B + [φ, ·]. Thus,

to establish a G-invariant theory, the underlying BRST algebra should be formulated up to

3That k may be smaller then 2d′ reflects that fact that, unlike the section s of E , the section s in generaldoes not have any nice transversality properties; so its behavior in the normal directions to Mα may bedegenerate.

18

gauge equivalence. In particular, the nilpotent condition δ2 = 0 for BRST transformations

should be replaced by the “G-equivariant” nilpotent condition

δ2 ∝ [φ, ·], δφ = 0. (1.50)

The action S in this theory may be construsted by εS = δW as before. However, since

now δ is no longer nilpotent, the action so construted does not automatically constitute a

BRST-closed form. The BRST invariance comes from the additional requirement that W

should be a G-invariant functional4. If this requirement is obeyed, we can easily extend the

earlier discussions to the present case and establish a formula similar to (1.3). Consequently,

both the partition function Z =∫

e−S and the correlators 〈O1O2 · · ·〉 =∫

e−S O1O2 · · · of

observables are topological invariants, which do not depend on the couplings λ, λ′, . . . as well

as all other non-dynamical data (in particular the spacetime metric). We shall now illustrate

this construction by the following consideration.

Our problem is now to construct the Euler class of a vector bundle E \ π→ B in terms of

the data (A,G, E , s), where E = pr∗(E \), s = pr∗(s\), with pr : A → B being the canonical

projection and s\ a section of E \. As in the last subsection, we may take rank(E \) = d\ = even.

We also pick a system of local coordinates ui (1 ≤ i ≤ d) on A and a G-invariant metric gab(u)

(1 ≤ a, b ≤ d\) on E , together a connection A = pr∗(A\) compatible with this metric, here

A\ denotes some connection one-form on E \. Introduce a field φ in the adjoint representation

of G, with ghost number U = 2, which can be expanded in the form φ =∑t

x=1 φx Tx with

Lie algebra generators Tx of G. The action of Tx on A is described by the vector field Uxi,

and its lifting to act on the bundle E is described by an action on sections, specified as

V a → V ′a = Uxi∇A

i V a + YxabV

b with some matrix Yxab.

The G-equivariant BRST algebra of this system may be set up as follows. On the world

manifold A, we have the multiplet (ui, ψi, φx), of ghost number U = (0, 1, 2), which obey the

BRST transformation laws

δui = iε ψi, δψi = ε φx U ix , δφ = 0. (1.51)

On the vector bundle E → A, we have the mutiplet (χa, Ha), of ghost number U = (−1, 0).

For this multiplet we shall consider the covariant BRST transformation δA = δ + iεψiAi as

in the last subsection. However, since now δ2 ∝ [φ, ·] 6= 0, the relation δ2A ∝ F (A) is no

longer valid; instead we have δ2A = ε1ε2(

12ψiψjFij(A) + iφxYx), where the additional Y -term

comes from δ2 = iε1ε2φx(U i

x Ai + Yx) acting on (χ,H) as well as the cancellation between

iε1ε2φxU i

x Ai and (δ2ui)Ai. Accordingly, the BRST transformation laws for (χa, Ha) are given

by

δχa = εHa − iεψiA ai bχ

b, δHa = −1

2εψiψjF a

ij bχb − iεφxY a

x bχb − iεψiA a

i bHb. (1.52)

4In fact, if Tx(W ) = 0, then according to (1.50) we have εδS = δ2W ∝ φxTx(W ) = 0.

19

Finally, to get a BRST neutral measure, we have to introduce a third multiplet (φ, η) in

the adjoint representation of G, with ghost number U = (−2,−1), which obeys the BRST

transformation laws

δφ = iεη, δη = ε[φ, φ]. (1.53)

Eqs.(1.51)–(1.53) then specify the full G-equivariant BRST algebra.

Given now a G-invariant Riemannian metric gij(u) on A, we define

W =1

2λgab(u) χa

(Hb + 2isb(u)

)+

1

λ′φxgij(u)U i

x ψj + W ′, (1.54)

where W ′ is a possible non-minimal term. The action is constructed by εS = δW ; so with

εS ′ ≡ δW ′, one easily derives:

S =1

2λHaHa +

i

λHasa(u) +

1

λχa∇A

i sa(u)ψi +1

4λFijabψ

iψjχaχb

+i

2λχaχbφxYxab +

i

λ′ηxgij(u)U i

x ψj +1

λ′φxgij(u)U i

x U jy φy

+i

2λ′φx

(∂Uxj

∂ui− ∂Uxi

∂uj

)ψiψj + S ′ (1.55)

Since the W -function (1.54) is invariant under the action of G, namely Tx(W ) = 0, we deduce

that the action S is BRST-closed: εδS = δ2W =∝ φxTx(W ) = 0. Thus, from the reasoning

that leads to Eq.(1.3), we conclude that the partition function

Z =1

(2π)d(−i)t · V ol(G)

∫dφ dφ dη du dψ dχ dH e−S (1.56)

is a numeric invariant, which depends only on A, E and the action of G on them.

Before committing ourselves to the evaluation of (1.56), let us consider the problem of

measuring volumes of the G-orbits in A. For the sake of convenience, we introduce a metric

gxy = U ix U j

y gij(u) on each G-orbit in A, and define the quadratic form dτ 2 = gxydτxdτ y for

the Lie parameters τx ∈ G to measure the induced distances from the Riemannian metric gij

on A. The induced volume of such an orbit (passing through the point u ∈ A) is thus give

by

V ol′(G) =

∫

Gdτ

√det[gxy(ui + τxU i

x + O(τ 2)]. (1.57)

As G acts freely on M , the vector field Ux = U ix ∂/∂ui is non-zero anywhere, thus gxy being

non-degenerate (and in fact positive definite), with the well-defined inverse gxy. A crucial

property of this induced metric is that det[gxy(u)] keeps constant on each G-orbit. To see

this, one considers an infinitesimal move of ui along some G-orbit, ui → ui + τxU ix , with the

Lie parameters τx ≈ 0. The change in the determinant det[gxy(u)], up to the first order of τ ,

is computed by:

det[gxy(ui + τxU i

x )]− det[gxy(ui)]

det[gxy(ui)]= gxy(u)

(gxy(u

i + τ zU iz )− gxy(u

i))

20

= gxy(u)((LτUx)

i U jy gij(u) + U i

x (LτUy)j gij(u) + U i

x U jy (Lτg)ij(u)

)(1.58)

here Lτ is the Lie derivative with respect the vector field τ ≡ τxU ix ∂/∂ui. Clearly, the

last term in (1.58) vanishes due to G-invariance of gij(u). The remaining terms may then

be analyzed as follows: Using the definition of the Lie derivative and Eq.(1.49), we have

(LτUx)i = [τ, Ux]

i = τ y[Uy, Ux]i = τ yc z

yx U iz , so that the first term in (1.58) is determined

by

τx cxyz U iy U j

z gij(u),

which vanishes due to the anti-symmetrical property cxyz = −cxzy of the structure constants.

This result also applies to the second term in (1.58), and we conclude that det[gxy(u)] is

indeed invariant under the move ui → ui + τx U ix along the G-orbit. As a consequence of this

invariance, we may reduce (1.57) to

V ol′(G) =√

det[gxy(u)] · V ol(G). (1.59)

We now return to the partition function (1.56). We will integrate out some field variables

to derive an effective theory without gauge invariance. Our analysis follows closely the

arguments presented in [7], which may be divided into the following four steps.

Step (I). According to (1.55), all terms involving (φ, φ) in the action can be isolated as

S1(φ, φ) =1

λ′φx gxy(u) φy +

i

2λΞx φx +

i

2λ′Ψx φx (1.60)

where the “linear sources” Ξx, Ψx are given by

Ξx = Yxabξaξb, Ψx =

(∂Uxj

∂ui− ∂Uxi

∂uj

)ψiψj. (1.61)

Thus, writing φx ≡ (ϕx1 + iϕx

2), φx ≡ (ϕx1 − iϕx

2) as well as

Y xab

(∂Uxj

∂ui− ∂Uxi

∂uj

)≡ Ωijab, (1.62)

we can evaluate the integral over φ, φ in (1.56) explicitly to simplify our partition function.

A step-by-step computation goes as follows: We first make the change of the integration

variables (φ, φ) → (ϕ1, ϕ2); the resultant is a standard gaussian integral of the form

∫dφ dφ e−S1(φ,φ) =

∫dϕ1 dϕ2 exp

− 1

2λ′(ϕx

1 gxy(u) ϕy1 + ϕx

2 gxy(u) ϕy2)

− i

2√

2

(Ψx

λ′+

Ξx

λ

)ϕx

1 −1

2√

2

(Ψx

λ′− Ξx

λ

)ϕx

2

Then, performing this integral exactly, yields

(2πλ′)t

det[gxy(u)]exp

(− 1

4λgxy(u)ΞxΨy

),

21

which can be rewritten, with the definitions (1.61)–(1.62), in terms of the four-fermion inter-

actions Ω · ψψξξ. So we simply obtain:∫

dφ dφ e−S1(φ,φ) =(2πλ′)t

det[gxy(u)]exp

(− 1

4λΩijab ψi ψj ξa ξb

). (1.63)

Now using this result and Eq.(1.59), we may express the partition function (1.56) as

Z =(λ′)t

(2π)d\(−i)t

∫dη du dψ dχ dH

exp(−S(I)eff )

V ol′(G)√

det[gxy(u)](1.64)

here S(I)eff is the effective action

S(I)eff =

1

2λgab(v)HaHb +

i

λHasa(u) +

1

4λFijab ψi ψj χa χb

+1

λχa∇A

i sa(u)ψi +i

λ′ηxgij(u)U i

x ψj + S ′, (1.65)

with

Fijab = Fijab + Ωijab. (1.66)

Step (II). The effective action (1.66) is linear in the ghost field η. So when performing

the integral over η, we will get a factor of(− i

λ′

)t

δ(gij(u)U ix ψj) (1.67)

To explore the meaning of this factor, let us decompose the Riemannian metric on A into

vierbeins: gij(u) = eki(u)ek

j(u) (here summation on the repeated index k = 1, . . . , d is

implied). Such vierbeins provide a local coframe ek = ekidui on A, which satisfies the

orthogonality relations < ek, el >= δkl, where the inner product < ·, · > is specified by

< dui, duj >= gij(u). With respect to this basis, any vector fields V = V i∂/∂ui on A has the

orthogonal components V k ≡ V · ek = ekiV

i. In particular, the vector fields Ux = U ix ∂/∂ui

tagent to the G-orbit through u ∈ A have the orthogonal components U kx = ek

iUi

x , from

which we can form the induced metric gxy(u) = U kx · U k

y . The delta function in (1.67)

can then be rewritten as δ(U kx · ψk) with ψk = ek

iψi. Notice that the frame ek is uniquely

determined up to a local SO(d) transformation ek → e′k = G(u)kle

l. Thus, by a suitable

SO(d) rotation, one can find an orthogonal frame such that ek1≤k≤t are in the directions

tangent to the G-orbit, and ekt+1≤k≤d are in the normal directions to that orbit. The

orthogonality of ek implies:

U kx =

0 if k = p, t < p ≤ d

U zx if p = z, 1 ≤ z ≤ t

(1.68)

where U zx constitute an invertible t×t matrix, whose determinant is equal to

√det[gxy] (since

Eq.(1.68) implies gxy = U zx · U z

y ). Now we can decompose ψ into the tangent part ψz ≡ eziψ

i

22

and the normal part ψp ≡ epiψ

i. As U kx · ψk = U z

x · ψz and δ(U zx · ψz) = det[U z

x ]δ(ψz), the

factor (1.67) reduces to (− i

λ′

)t √det[gxy(u)] · δ(ψz). (1.69)

So after integratng out the η field, (1.64) becomes

Z =

(1

2π

)d\ ∫du dψ dχ dH δ(ψz)

exp(−S(II)eff )

V ol′(G)(1.70)

here ψz denotes the orthogonal components of ψ tangent to the G-orbit, and S(II)eff is the

effective action

S(II)eff =

1

2λgab(v)HaHb +

i

λHasa(u) +

1

4λFijab ψi ψj χa χb +

1

λχa∇A

i sa(u)ψi + S ′. (1.71)

Step (III). Now, to investigate the ψ-integral in (1.70), one may change the integration

variables ψi → ψk = eki · ψi = (ψz, ψp). This will contribute to a Jacobian in the integration

measure, namely

[dψi] = det[eki] · [dψk] =

√det[gij(u)] · [dψz][dψp]

Clearly, the integral∫

[dψz]δ(ψz)(· · ·) in (1.70) has the effect of taking the tangent components

of ψ to be zero. Let [Eik] be the inverse matrix of [ek

i]. We may write ψi = Eikψ

k, and

perform the integral over ψ in (1.70) to derive:

Z =

(1

2π

)d\ ∫du [dψp] dχ dH

√det[gij(u)]

V ol′(G)exp(−S

(III)eff ) (1.72)

here S(III)eff is the effective action

S(III)eff =

1

2λ< H, H + 2is > +

1

λ< χ,∇A

i s > Eipψ

p

+1

4λFijabE

ipE

jqψ

pψqχaχb. +1

4λΩijabE

ipE

jqψ

pψqχaχb. (1.73)

We will show that the last term in (1.73) vanishes. In fact, using the definition (1.62), we

have:

Ωijab Eip(u) Ej

q(u) = Y xab (∇iUxj −∇jUxi) Ei

p(u) Ejq(u)

= Y xab (∇i( Ei

p(u) Ejq(u)Uxj)−∇j( Ei

p(u) Ejq(u)Uxi)) = 0. (1.74)

Here ∇i is the covariant derivative specified by the Levi-Civita connection on A, which

obeys the metricity condition ∇iEjk = 0. That (1.74) vanishes comes from the identities

Ei · Uxi = EipgijU

jx = (Ei

peki)(e

kjU

jx ) = δk

p · U kx = U p

x = 0 (the last identity is a part of

Eq.(1.68)). Note that the geometric data Y xab plays no roles in the above effective action.

This could be expexted, as the underlying partition function is a topological invariant.

23

Step (IV ). Finally, we consider the u-integral in (1.72). The measure is given by [dui] =

du1∧· · ·∧dud with the basis dui of one-forms on A dual to ∂/∂ui. Of course, it is equivalently

good to use a different basis duk, where uk are the “orthogonal” coordinates on M , locally

determined by solving the differential equations duk = ekidui. The change of basis then raises

a Jacobian, so that [dui] = [duk]/√

det[gij(u)]. This Jacobian will be canceled exactly by the

factor√

det[gij(u)] in (1.72). Moreover, one can decompose the measure [duk] into a product

of the normal part [dup] and the tangent part [duz]. Since both the effective action S(III)eff

and the volume V ol′(G) in (1.72) take the constant values along G-orbits, the integral over

the tangent variables uz gives a factor of V ol′(G), which will be cancelled in (1.72). On the

other hand, the normal part of the orthogonal coordinates on A, up, will provide a natural

system of local cordinates on B = A/G, and we have the pull-back relations s = pr∗(s\),

A = pr∗(A\) and F = pr∗(F \) (with pr : A → B being the canonical projection) have the

explicit representations sa(u) = s\a(up), Ai(u) = A\q(u

p) · eqi, and Fij(u) = F \

qr(up) · eq

ierj.

Substituting these into (1.73), the effective action becomes:

S(IV )eff =

1

2λ< H, H + 2is\ > +

1

λ< χ,∇A\

p s\ > ψp +1

4λF \

pqabψpψqχaχb. (1.75)

This form is clearly independent of the coordinates ux tangent to the G-orbit. Thus, after

the change of integration variables5 ui → uk, we find:

Z =

(1

2π

)d\ ∫[dup] [dψp] dχ dH exp(−S

(IV )eff ), (1.76)

which is precisely the Euler characteristic χ(E \) of the vector bundle E \.

5 This will raise a Jacobian cancelling the factor√

det[gij(u)] in (1.72).

24

References

[1] A. Kapustin and E. Witten, “Electric-Magnetic Duality And The Geometric Langlands

Program”, hep-th/0604151.

[2] E. Witten, Proceedings, Surveys in Differential Geometry (1990) 243–310.

[3] E. Witten, Commun. Math. Phys. 121 (1989) 351.

[4] E. Witten, Commum. Math. Phys. 117 (1988) 353.

[5] N. Seiberg and E. Witten, Nucl. Phys. B 426 (1994) 19.

[6] E. Witten, “Monopoles and Four-Manifolds”, IASSNS-HEP-94-96, hep-th/9411102.

[7] C. Vafa and E. Witten, hep-th/9408074.

25