7/27/2019 TQFTNotes

1/34

NOTES ON TOPOLOGICAL QUANTUM FIELD THEORY

DYLAN G.L. ALLEGRETTI

Abstract. These notes are meant to provide an introduction to topological quantum fieldtheory for mathematicians. Topics include applications of Chern-Simons theory to the studyof knots, Atiyahs definition of a topological quantum field theory, and the invariants ofReshetikhin and Turaev. We also include a self-contained introduction to Yang-Mills theoryfor physical motivation.

1. Introduction

1.1. Feynman integrals. Quantum field theory is the formalism developed by physicists fordescribing quantum mechanical systems that look classically like fields. It provides a unifiedframework for describing many different phenomena. For example, light is understood to bethe result of excitations in a medium called the electromagnetic field, and there is a quantumfield theory called quantum electrodynamics that describes the dynamics of this field and itsinteraction with matter. Other quantum field theories are used to describe the strong andweak nuclear forces and certain condensed matter systems.

Mathematically, a quantum field theory consists of very simple data. To specify a quan-tum field theory, we first of all need a smooth manifold M called spacetime(usually four-dimensional with a Lorentzian metric g). We must also specify a vector space

F(M) whose

elements are called the fields on M(these could be smooth functions or vector fields on Mfor example). Finally, we must specify a function S :F(M) R called the action anda measure D onF(M). To perform computations in quantum field theory, we have tocompute certain integrals called Feynman integrals. In particular, we have to compute theintegral

Z=

F(M)

eiS()D

which is known as the partition function. Once we know the value of this integral, we canuse it to compute the expected values of various physical quantities. By physical quantity,we mean any attribute of a physical system to which we can assign a number by makingthe system interact with a measuring device. Such a quantity is represented by a functionf :F(M) Rwhose valuef() on a field represents the value of the quantity if the fieldconfiguration happens to be. Iffis the function representing some physical quantity, thenthe average measured value of this quantity in the vacuum is

f= 1Z

F(M)

f()eiS()D.

Last update: January 8, 20131

7/27/2019 TQFTNotes

2/34

Performing computations in quantum field theory is thus a simple matter in principle as it allboils down to computing integrals. The problem that makes computations extremely difficultin practice is that the measure D usually cannot be defined rigorously. We are thereforeforced to resort to nonrigorous and heuristic methods to extract physical predictions fromquantum field theory.

Despite this shortcoming, it turns out that we can evaluate Feynman integrals in certainspecial cases. Interestingly, there are quantum field theories in which the values of theseintegrals are topological invariants ofM, and in the case of Chern-Simons theory, knots em-bedded inM. Theories that compute topological invariants in this way are calledtopologicalquantum field theories (TQFTs). The invariants turn out to be easily computable once wehave made sense of the integration, and for this reason, quantum field theory has recentlybecome an extremely important topic in pure mathematics.

1.2. Prerequisites. The purpose of these notes is to give a physically motivated introduc-tion to TQFTs and their applications. They are primarily aimed at mathematicians who wish

to understand recent developments in mathematical physics. The mathematical prerequisitestherefore consist of standard graduate-level mathematics, including some understanding ofbasic differential geometry and category theory. We assume, for example, that the reader isfamiliar with the notion of a smooth manifold, tangent space, and vector bundle. We do notassume any familiarity with more advanced topics in differential geometry such as the the-ory of principal bundles or cobordisms. As far as category theory is concerned, these notesassume the reader is comfortable with the basic ideas of categories, functors, and naturaltransformations, but the prerequisites stop short of any advanced topics like monoidal orhigher categories.

Since these notes are intended primarily for mathematicians, they do not presuppose any

knowledge of physics and they do not discuss the physical ideas related to TQFTs in muchdetail. Instead, the physics is used to motivate the mathematical definition of a TQFT. Thepurpose of these notes is not to teach physics but to explain how the physicists idea of aFeynman integral can serve as a source of inspiration for mathematicians.

1.3. Organization. We begin with a discussion of the physical aspects of quantum fieldtheory in Section 2. In this section, we explain how physical fields are represented mathe-matically and explain the notion of gauge symmetry. We then introduce principal bundlesand principal bundle connections and use these mathematical tools to discuss classical Yang-Mills theory. To illustrate how Yang-Mills theory arises in physics, we briefly discuss its rolein quantum electrodynamics. This discussion of Yang-Mills theory is necessary to motivateour later discussion of Chern-Simons theory and Dijkgraaf-Witten theory. The treatment inthis section is based on the excellent notes [1].

In Section 3, we discuss Chern-Simons theory, a topological theory that gives rise to manytopological invariants of knots. We begin by reviewing the basic invariants of knots and links,namely the writhe, Kauffman bracket, and Jones polynomial. Then we define the Chern-Simons action and introduce the notions of holonomy and Wilson loops. Finally, we sketchthe remarkable connection between Chern-Simons theory and link invariants introducedpreviously. The treatment in this section borrows heavily from [2].

2

7/27/2019 TQFTNotes

3/34

In Section 4, we give a physically motivated introduction to Atiyahs axiomatization of aTQFT, and we explain how a theory satisfying the axioms can be used to compute topologicalinvariants. In the process, we have to introduce the theory of cobordisms and symmetricmonoidal categories. Once we have defined TQFTs, we discuss Dijkgraaf-Witten theory asan example. The material in this section comes from [3], [4], and [5].

Finally, in Section 5, we discuss the theory of Reshetikhin-Turaev invariants, a mathe-matically rigorous counterpart to Chern-Simons theory. To define these invariants of linksand 3-manifolds, we first introduce the notions of ribbon categories and modular tensorcategories. We then explain how such categories give rise to invariants coming from three-dimensional TQFTs. The treatment in this section is based on [6].

1.4. Convention. In these notes, we will follow the Einstein summation convention andimplicitly sum over all indices which appear once as a superscript and once as a subscript.For example, when we write an expression y=cx

, we really mean the sum y=

cx

.

2. Yang-Mills Theory

2.1. The mathematical description of fields. Before we begin our discussion of topo-logical quantum field theories, we will spend some time looking at how quantum field theoryis used in physics. The main example that will motivate our discussion in this section isthe theory of an n-component vector field. In other words, we will model physical fields assmooth functions : M Rn where for simplicity we take M = R4. We usually write afield in terms of its components as = (1, . . . , n). To specify an action for the theory,we first form the expression

L=n

i=1

1

2i

i 12

m22i

where m > 0 is a constant. Here we have defined = gwhere (g) is the inverse of

the matrix (g). An expression of this sort is called a Lagrangian density. Once we have aLagrangian density, the action is obtained by integrating over all of spacetime:

S() =

R4

d4xL.

Observe that in this theory, the fields are represented by sections of a certain fiber bundle,

namelyMRn M. In general, physical fields are represented by smooth sections of somebundle over the spacetime manifold. This point of view will be useful later on when we beginour study of gauge fields.

2.2. Gauge invariance. An important feature of our n-component vector field theory isits invariance under the action of certain transformations. In this theory, the physical fieldsare functions M Rn. Now there is an action of the group O(n) on the n-dimensionalvector space Rn given by matrix multiplication, so there is an action ofO(n) on functions

3

7/27/2019 TQFTNotes

4/34

: M Rn given by (g)(p) =g(p). Let g = (aij) be an orthogonal matrix. Then

(g)i=

nj=1

aijj

=

nj=1

aijj

=g()i

and since orthogonal transformations preserve the inner product on Rn, it follows thati (g)i

(g)i =

i ii. We also have

i(g)

2i =

i

2i , so the action ofO(n)

preserves the full expression for the Lagrangian density. Since the Lagrangian density deter-mines the partition function, it follows that all the physics is invariant under the action ofO(n). In general, whenever we have a groupGacting on the space of fields and preserving theLagrangian density, we say that the theory is invariant under global gauge transformations.

In this case, the group G is called the gauge group of the theory.The idea of Yang-Mills theory is to require the physics to be invariant under not only

global gauge transformations but arbitrary local gauge transformations as well. In otherwords, we wish to consider a theory with gauge group G, and for any function g :M G,we wish to write down a Lagrangian density that is invariant under the transformation(x) g(x)(x). Unfortunately, the Lagrangian density that we have been using is notinvariant under these local gauge transformations because in general

g(x)(x)=g(x)(x).We will therefore have to modify our Lagrangian density, replacing the partial derivatives

by the more sophisticated notion of covariant derivative, which transforms in a nontrivialway when we apply a local gauge transformation.

2.3. Principal bundles. In order to write down a Lagrangian density that is invariant underlocal gauge transformations, we need to develop a theory of bundles with group actions.

Definition 2.3.1. Let Fbe a smooth manifold. We say that a bundle : P M is locallytrivial with standard fiber F if there exists an open covering{U} ofMand for each ahomeomorphism called a local trivializationsuch that

1

(U)

U Fproj

U

commutes where proj is the projection onto the first factor. LetG be a Lie group and let : P Mbe a bundle with a continuous right action P GP such that G preservesthe fibers of and acts freely and transitively on them. Such a bundle is called a principalG-bundle if it is locally trivial and the local trivializations are G-equivariant.

4

7/27/2019 TQFTNotes

5/34

2.4. Associated bundles. Next we are going to study a method of constructing bundlesfrom a principal bundle. Let :P Mbe a principal G-bundle andFa smooth manifoldthat admits a left G-action. Then there is a right action (P F) GP F ofG on theproductP Fgiven by

(u, f)

g= (u

g, g1

f).

Define PGF = (P F)/G with the quotient topology. Since the composition P F projP

M is constant on G-orbits, there is an induced map : PGFM. This inducedmap has a special name.

Definition 2.4.1. The map : PGF M is called the bundle associated to P withfiber F. We use the notation [p, f] for the class of (p, f)P F in the total space of thisbundle.

Proposition 2.4.2. The associated bundle :PG F Mis a locally trivial bundle withthe same trivializing open cover as : P Mand fibers diffeomorphic to F.

Proof. Let U Mbe a trivializing open set for the principal bundle : P M, and let : 1(U) U G be the associated local trivialization. For any point [p, f] 1(U),we have(p) = (x, g) for some xU andgG, and we can define : 1(U)UF by

([p, f]) = (x, g f).It is straightforward to check that this map is well defined. On the other hand, there is amapping :U F 1(U) given by

(x, f) = [1(x, 1), f].

and one can show that is a local trivialization with inverse . This local trivialization

takes the fiber over x to{x} Fand therefore gives rise to a diffeomorphism between fibersof and the space F. This completes the proof.

A special case of this construction is when Fis a vector space on which G has a smoothrepresentation : G GL(F). In this case the bundle associated to P is called a vectorbundle. The following examples and notations will appear over and over in our discussion.

Examples 2.4.3.

(1) IfG is a group ofn n matrices and V is an n-dimensional vector space, then wecan take to be the fundamental representation G GL(V).

(2) Let gbe the Lie algebra ofG and take to be the adjoint representation Ad : G

GL(g). Then the associated bundle PGgM is called the adjoint bundle ofPand denoted Ad(P).

2.5. Transition functions. IfU and U are two sets in our open cover having nonemptyintersectionU =U U, then there are two trivializations , :1(U)U G.These can be written(p) = (m, g(p)) and (p) = (m, g(p)) where p1(m), and wedefine g(p) to be the element G for which g(p) = g(p)g(p).

Lemma 2.5.1. There exist functions g :UG such that g(p) =g((p)).5

7/27/2019 TQFTNotes

6/34

Proof. For any gG, we haveg(pg) =g(pg)g(pg)

1

=g(p)gg1g(p)

1

= g(p).

SinceG acts transitively on the fibers ofP, it follows that g is constant on each fiber. Theresult follows immediately.

Definition 2.5.2. The functionsg :UG in the lemma are calledtransition functions.2.6. g-valued forms. Normally, when we talk about a differential p-form, we mean anoperation that takes pvector fields as input and produces a number. Equivalently, one candefine a p-form to be a smooth section of pTM. Sometimes it is useful to generalize thisnotion and consider an operation that takes p vector fields and produces an element of somevector space X.

Definition 2.6.1. IfXis a vector space and Mis a manifold, then an X

-valued differential

p-form is a smooth section ofXpTM. The space of all p-forms with values in X isdenoted p(M, X).

Many operations and notations from the theory of ordinary real-valued differential formscan also be applied to vector-valued differential forms. For example, if and are vector-valued differential forms, then we define their wedge productby the usual formula with realmultiplication replaced by tensor product:

( )(v1, . . . , vp+q) = 1p!q!

Sp+q

sgn()(v(1), . . . , v(p)) (v(p+1), . . . , v(p+q)).

Thus the wedge product of p-form with values in X and a q-form with values in Y is a(p+q)-form with values in the tensor product X Y. In the special case where and take values in a Lie algebra g, we can compose the wedge product operation with the Liebracket [, ] :g gg to get a new element ofgcalled the bracket:

[, ] = [ ].We can also extend the notion of exterior derivative to X-valued differential forms. Once wechoose a basis{ei}for X, such a form can be written = i ei where i are real-valueddifferential forms, and we define the exterior derivativeof by the formula d = d i ei.Example 2.6.2. For any Lie group G, the Maurer-Cartan form1(G, g) is defined by

g = (Lg1) : TgGT1G= g

where Lg1 :GG, xg1x. One can show that in the special case where Gis a matrixgroup, the Maurer-Cartan form is given by g = g

1dg. Other important properties of theMaurer-Cartan form include the transformation law

R1g = Ad(g1)

where R: GG is the function defined by Rg(x) =xg and the equationd+

1

2[, ] = 0

6

7/27/2019 TQFTNotes

7/34

which is known as the Maurer-Cartan equation.

2.7. Connections. The concept of a connection is central to differential geometry becauseit allows us to differentiate sections of bundles and transport vectors from one tangent spaceto another. To state the definition, we let : P

Mbe a principal G-bundle on M with

trivializing open cover{U}.Definition 2.7.1. LetRg :P Pdenote right multiplication bygG, and letp :GPbe the mapping gpg for pP. For any X g let Xbe the vector field on P given by(X)p = (p)X. A connection 1-formon P is an element of

1(P, g) such that

Rg= Ad(g1)

((X)) =X.

It is a fact, whose proof goes beyond the scope of these notes, that a connection 1-formexists on any principal bundle. Suppose is a connection 1-form on P. For each index,there is a canonical local section s : U1(U) defined so that for each point mUwe have (s(m)) = (m, 1). Define A1(U, g) by

A= s.

This 1-form is known in physics as avector potentialorgauge field. We will use the followingresult to see how the vector potential changes when we pass from one trivializing open setto another.

Proposition 2.7.2. Let be a connection 1-form, and let be its restriction to 1(U).

Then can be written

= Ad(g1 )

A+ g

where is the Maurer-Cartan form defined above.

Proof. We begin by proving that the two forms agree on the image ofs. Suppose we aregivenmU and p= s(m). Then we have a direct sum decomposition

TpP= im(s ) Vp,so we can write any tangent vectorvTpPuniquely in the formv = (s)(v) + vfor somev Vp. Sinceg s is constant, we have (g s) = 0 and therefore (g)v = (g)v. Itfollows that

Ad(g(p)1)A+ g

(v) = (s)(v) + (g)(v)

=((s)v) +((g)v)

=((s)v) +((g)v)

=((s)v) +(v)

=(v),

7

7/27/2019 TQFTNotes

8/34

so the two forms are agree on the image of s. Next we will show that these two formstransform in the same way under the action ofG. Indeed,

Rg(Ad(g(pg)1)A+ g

) = Ad((g(p)g)

1)RgA+ g

R

g

= Ad(g1g(p)1)s+g

(Ad(g

1))

= Ad(g1)

Ad(g(p)1)s+ g

= Ad(g1).

It follows that equality holds everywhere on 1(U).

Corollary 2.7.3. IfU and U overlap, then on the intersection U, we have

A= Ad(g)A+ g.

Proof. Since and are restrictions of a globally defined 1-form, we know that =on 1(U). Therefore on the intersection U, we have

s=s

=s(Ad(g(s)1)A+ g)

= Ad(g)A+g

In the last step, we used the fact that g s=g. 2.8. Gauge transformations. Like all geometric objects, principal bundles have a naturalnotion of automorphism. Since we ultimately would like to describe physical systems usingthe principal bundle formalism we have developed, we should make sure that all the physicsis invariant under these automorphisms.

Definition 2.8.1. Let : P Mbe a principal G-bundle. A gauge transformationofP isG-equivariant diffeomorphism :P Psuch that the following diagram commutes.

P

P

M

If is a gauge transformation of P, then maps fibers to themselves and thereforerestricts to a gauge transformation of the trivial bundle1(U). Let:

1(U)UGbe the associated local trivialization. Then we can write ((p)) = ((p), g((p))) anduse this to define

(p) =g((p))g(p)1.

The fact that g and are G-equivariant implies (pg) = (p) for all g G. Since Gacts transitively on fibers ofP, it follows that is constant on fibers and hence there exists:UG such that (p) =((p)). Let m be a point ofU and p a point in the fiberoverm. Then we have

(m) =g((p))g(p)1

=g((p))g((p))1g((p))g(p)

1g(p)g(p)1

=g(m)(m)g(m)1

8

7/27/2019 TQFTNotes

9/34

since ((p)) =m. This prove that (m) = Ad(g(m))(m) and therefore the maps define a section of the adjoint bundle PAdG. These maps correspond to the local gaugetransformations that we considered earlier while discussing the physical motivation.

Proposition 2.8.2. Suppose we are given a connection 1-form , and let = (1)be the corresponding gauge transformed 1-form. LetA

and A

be the gauge fields on U

corresponding to and . Then

A= Ad()(A ).

Proof. We can write

p = Ad(g(p)1)A+ g

and

p = Ad(g(p)1)A+ g

.

Letting q= 1(p) and applying (1) to the first of these equations, we find

p = Ad(g(q)

1

)(

1

)

A+ (

1

)

g

= Ad(g(q)

1)( 1)A+ (g 1)= Ad(g(q)

1)A+ (g 1)= Ad(g(p)

1(p))A+ (g 1).

The fact that (g 1)(p) =g(q) = (p)1g(p) implies(g 1)= g Ad(g(p)1(p))

and therefore

p = Ad(g(p)1(m))

(A ) +g.Combining this with our other formula for p yields the desired result.

2.9. Covariant differentiation. Fix a trivializing open set U M, and let Abe a vectorpotential on this open set. If : PG V M is the bundle defined in Example 2.4.3,then there is a local trivialization 1(U)= U V, so we can think of any local sections: U 1(U) of as being essentially the same thing as a smooth function s: UV.

If we choose a basis{ei} for V, then we can write s: U V in terms of coordinates ass= siei withs

i C(M). Since the Lie algebra ofGL(V) is gl(V), we can view the 1-formAas a matrix whose entries Aji are ordinary real-valued 1-forms. One then has the followingderivative operation.

Definition 2.9.1. The covariant derivativeofsis the sectionvsofV defined byvs= dsi(v)ei+ siAji (v)ej.

We think of this new sectionvs as the derivative of s in the direction v. In thespecial case when v = , we writev =. Notice that gauge transformations act oncovariant derivatives, takingv to a new operationv in which the vector potential A isreplaced by the gauge transformed vector potential A. The following proposition says thatcovariant derivatives behave in a particularly nice way under gauge transformations.

9

7/27/2019 TQFTNotes

10/34

Proposition 2.9.2. SupposeG is a matrix group and g:UG are functions defining agauge transformation. Under this gauge transformation,

gs= gs.

Proof. Let g be the g defined on the open set U. Since G is a matrix group, we can writeg in terms of its matrix coefficients, g= (aij). Then

(gs)i =(a

ijs

j)

= (aij)s

j +aij(sj)

= ((g)s)i + (gs)

i,

and the gauge transformed vector potential can be written

A=gAg1 (g)g1

where we denote A=A() andA= A().Therefore

gs= (gs)iei+ (gs)i(A)jiej= ((g)s)

iei+ (gs)iei+ (gs)

i(gAg1)jiej (gg1)ji (gs)iej

= (gs)iei+ (gs)

i(gAg1)jiej

=gsas desired.

2.10. An invariant Lagrangian density. We have now developed the mathematics nec-essary to modify our n-component theory so that the Lagrangian density is invariant underlocal gauge transformations. Let be the vector bundle with fiber Rn associated to a prin-cipal O(n)-bundle :P M. Fix a trivializing open set U M and let{ei} be the basisof local sections of over Ugiven by

ei(p) = (0, . . . , 0, 1, 0, . . . , 0)

with a 1 in the ith position and 0s elsewhere. A physical field is represented mathemat-ically as a section of this vector bundle, so locally we can write =

i iei for smooth

functions i on U. Then we can rewrite our Lagrangian density using covariant instead of

partial derivatives, and we obtain a new expression with the desired invariance property:

L=n

i=1

1

2()i()i+1

2m22i

.

Notice that in order to write down a Lagrangian density which behaves nicely under localgauge transformations, we had to choose a g-valued 1-formA. In quantum field theory, thisobject is interpreted as a new physical field, and if we wish to describe this field quantummechanically, we will need to specify an action.

10

7/27/2019 TQFTNotes

11/34

2.11. Curvature and field strength. The first step in writing down an action for thegauge field A is to construct an auxiliary field called the gauge field strength.

Definition 2.11.1. Let be a connection 1-form. Then thecurvature of is the 2-form = d+ 1

2[, ]. Ifs : U P is one of the canonical sections defined above, then the

gauge field strengthis the pullback F= s.

Explicitly, the field strength F is given by the formula

F=dA+1

2[A, A].

Since the field strength F is defined locally, it is natural to ask how it changes when wepass from one trivializing open set to another. The answer is provided by the followingproposition, which implies that theFpatch together to form a globally defined 2-form withvalues in the adjoint bundle Ad(P).

Proposition 2.11.2. Let U and U be two trivializing open subsets ofM. Then on theirintersection we have

F= Ad(g)F

Proof. We have seen that on the intersection U the gauge field A is given by

A= Ad(g)A+ g.

Bracketing this 1-form with itself and using graded commutativity of the bracket operation,we see that

[A, A] = [Ad(g)A+g, Ad(g)A+ g

]

= [Ad(g)A, Ad(g)A] + [Ad(g)A, g]

+ [g, Ad(g)A] + [g, g

]

= [Ad(g)A, Ad(g)A] + 2[Ad(g)A, g] + [g, g].

Now this last expression contains

[Ad(g)A, g] = Ad(g)[A, Ad(g

1 )g

]

and we have

Ad(g1 )g= g

Ad(g

1 )

=gRg

= (Rg g).The composition Rg1

g equals the constant function 1, so its differential is zero, and

therefore the pullback of the Maurer-Cartan form by this composition vanishes. It followsthat

[A, A] = [Ad(g)A, Ad(g)A] + [g, g

]

= Ad(g)[A, A] +g[, ]

= Ad(g)[A, A] 2gd.On the other hand, we havedA= Ad(g)dA+g

d, so the desired result is a consequence

of the formula F= dA+ 12

[A, A].

11

7/27/2019 TQFTNotes

12/34

2.12. The Yang-Mills action. In order to write down an action for the fieldA, we need toassume that our Lie algebra gadmits an Ad-invariant inner product, : g g R. Ofcourse an inner product of this sort may not exist on an arbitrary Lie algebra, but from nowon we will only consider gauge groups whose Lie algebras do admit such an inner product.

If we assume our manifold Mcomes equipped with a metric tensor g, then there is alsoa natural inner product operation on differential forms. If =dx and =dx are 1-forms onM, then this inner product is the function, = gonM. More generally,for 1-forms e1, . . . , ep and f1, . . . , f p, we define

e1 ep, f1 fp= det(ei, fj)and extend linearly to get an inner product on the space ofp-forms.

Now, ifS and T are g-valued differential forms written in terms of sections SandT ofg and ordinary p-forms and , then we can combine these notions of inner productand write

S , T =S, T, .In this way, we obtain an inner product operation on g-valued differential forms. We willbe particularly interested in the squared norm|F|2 =F, F where F is the gauge fieldstrength introduced above. By Ad-invariance of the inner product on g, we know that thisnorm agrees with|F|2 on the setU and therefore there is a globally well defined function|F|2 onM whereFis the field strength regarded as a section of Ad(P). With this notation,the Yang-Mills actioncan be written

S(A) =

M

|F|2 vol,provided this integral exists.

To rewrite this action in a form familiar to physicists, we need the following important

operation.

Definition 2.12.1. TheHodge star operatoris the unique linear map : p(M)np(M)such that for all , p(M) we have =, vol. We also define : p(M, g)np(M, g) by(T ) = T where T is a g-valued p-form written in terms of asection T ofgand an ordinaryp-form .

If and are g-valued 1-forms, it is easy to see that the expression, vol is the sameas the composition of with the inner product, : g gR. Abusing notationslightly, we write this composition as Tr(FF). Then the Yang-Mills action can be written

S(A) =M Tr(F F).

2.13. Maxwell theory. To conclude this section, we briefly mention how Yang-Mills theoryarises in quantum electrodynamics, the quantum theory describing electrons, positrons, andphotons. According to this theory, electrons and positrons are the particle-like excitations ina field called the Dirac field. Although more complicated than the n-component vector fieldwe have been discussing, the Dirac field has a similar Lagrangian density, which is invariantunder global gauge transformations with gauge group U(1). As in the example above, wemust use covariant derivatives to ensure that the expression is invariant under local gauge

12

7/27/2019 TQFTNotes

13/34

transformations. As before, this requires introducing a gauge field A, and the associatedfield strength F is called the electromagnetic or photon field. Excitations in this field arephotons, the quantum mechanical particles that make up light.

3. Chern-Simons Theory

3.1. Basic knot theory. We begin this section with a review of some of the classical resultsof knot theory. As we will see, quantum field theory provides an elegant framework in whichthese results can be understood.

Definition 3.1.1. A knot in M is a submanifold ofMdiffeomorphic to a circle. A link isa submanifold diffeomorphic to a disjoint union of circles.

We will typically represent knots and links by drawing two dimensional diagrams. For-mally, ifL is a link, we obtain a link diagramby taking a projectionp onto R2 in such a way

that any point ofp(L) has a neighborhood that looks like a single segment, or two segmentscrossing at an angle. Below are diagrams of the trefoil, the figure-8 knot, and the Hopf link.

We will often consider knots and links equipped with a vector field called a framing.Precisely, ifL is a link and p is any point on L, then we view the tangent space TpL as asubspace ofTpM= R3. Let v be a smooth function L R3 and denote the value ofv at apoint pbyvp.

Definition 3.1.2. We say thatv is a framingfor the linkL ifvpTpLfor all pL. A linkequipped with a framing is called a framed link.

In knot theory, one is interested in understanding when two knots or links are equivalent.The appropriate notion of equivalence in this case is the following.

Definition 3.1.3. We say that two links L and L are ambient isotopicor simply isotopicif there exists a smooth map F : M [0, 1] M such that F(, 0) is the identity on M,F(, 1) takes L to L, and F(, t) is a diffeomorphism for each t [0, 1]. If in addition Land L have orientation or framing, then we also insist that F(, 1) take the orientation orframing ofL to the orientation or framing ofL.

When doing computations in knot theory, we can transform a link diagram into onerepresenting an isotopic link by applying certain special operations at crossings.

13

7/27/2019 TQFTNotes

14/34

Definition 3.1.4. The operations illustrated in the following three diagrams are calledReidemester movesI, II, and III.

When dealing with links equipped with framing, we consider the following operation calledReidemeister move I:

Proposition 3.1.5. Two link diagrams of unframed links represent the same isotopy class ifwe can get from one to the other by applying Reidemeister moves I, II, and III at crossings.Two diagrams of framed links represent the same isotopy class if we can get from one to theother by applying moves I, II, and III.

3.2. Invariants. Now that we have defined isotopy of links, we can start to look for isotopyinvariants which will enable us to classify links up to isotopy. We begin by studying a simpleinvariant of links equipped with an orientation. If we look at a diagram of such a link, thenany crossing looks like

or

and we call these two types of crossings right-handed and left-handed, respectively.

Definition 3.2.1. The writhew(L) of a link L is defined to be the number of right-handedcrossings minus that number of left-handed crossings. That is,

w(L) = #

#

It is easy to see that the writhe is invariant under Reidemeister moves I , II, and III, andthus gives an invariant of framed links.

14

7/27/2019 TQFTNotes

15/34

Definition 3.2.2. The Kauffman bracketof a link L is the functionLin the variables A,B, and ddetermined by the following skein relations.

= 1

=d

=A

+ B

Rather than attempt to explain in words the precise meaning of these relations, we illus-trate the computation of the Kauffman bracket with an example.

Example 3.2.3. The Kauffman bracket of the Hopf link is =A

+ B

=A2

+ AB

+ BA

+ B2

= (A2 +B2)d2 + 2ABd.

Theorem 3.2.4. The Kauffman bracket is an isotopy invariant of framed links forB =A1and d =(A2 + A2).Proof. By Proposition 3.1.5, it suffices to check that the Kauffman bracket is invariant underReidemeister moves I, II, and III. Indeed, we have invariance under move I since

=A

+ B

=Ad

+B

= (Ad+ B)

=A

+ B

=

.

15

7/27/2019 TQFTNotes

16/34

We also have AB = 1 and (A2 + B(Ad+ B)) = 0, and therefore =A

+ B

=A

A

+ B

+B(Ad+ B)

=AB

+ (A2 + B(Ad+ B))

=

.

Hence the Kauffman bracket is invariant under move II. Finally, it is invariant under move IIIsince

=A

+ B

=A

+ B

=

.

This completes the proof.

The proof of Theorem 3.2.4 shows in particular that the Kauffman bracket is not invariantunder Reidemeister move I. Indeed, our proof of invariance under the modified Reidemeistermove I shows that

= (Ad+B)

=A3

,

and a nearly identical computation shows =A3

.

Although the Kauffman bracket is not invariant under Reidemeister move I, these computa-tions show that if we multiply it by the factor (A3)w(L), we will have a Laurent polynomialwhich is invariant under this move. This motivates the following definition.

Definition 3.2.5. Let L be an oriented link. Then the Jones polynomialassociated to L is

VL(A) = (A3)w(L)L(A).It is immediate from the above discussion that the Jones polynomial is an invariant of

unframed oriented links.16

7/27/2019 TQFTNotes

17/34

3.3. The Chern-Simons form. In the previous section, we formulated Yang-Mills theoryin terms of the Yang-Mills action and indicated its use in quantum electrodynamics. Weare now going to study a different field theory called Chern-Simons theory. We will see thatthis theory is similar to Yang-Mills theory, but instead of using it to describe a physicalsystem, we will talk about its applications to topology. Specifically, we will discuss how

Chern-Simons theory computes topological invariants of knots embedded in M.To begin, recall that we defined the Yang-Mills action to be the integral over M of the

Lagrangian density Tr(FF). Likewise in Chern-Simons theory, the action is obtained byintegrating a certain 3-form on an orientable 3-manifold M. To define this 3-form, we willassume that the gauge group G is simply connected. In this case every principal G-bundleis trivial, and once we choose a trivialization, we can identify any connection on the bundlewith a g-valued 1-form Aon M. Then the action is obtained by integrating the 3-form

Tr(A dA +23

A A A).which is known as the Chern-Simons form. Let us explain the notation in the above ex-

pression. As in Yang-Mills theory, we are assuming our Lie algebra g admits an invariantinner product, . The first term of the Chern-Simons form is obtained by wedging Awith dA to get a g g-valued 3-form, and then composing with the inner product to get anordinary real-valued 3-form which we can integrate over M. The second term should reallybe thought of as

1

3Tr([A, A] A)

where we first construct the g g-valued 3-form [A, A] Aand then compose with the innerproduct to get a real-valued 3-form.

3.4. Holonomy. Suppose that : [0, T]

M is a smooth path from p to qand for eacht [0, T] let u(t) be a vector in the fiber ofPGV over (t). Recall that the covariantderivative of a section scan be written as partial derivatives plus a vector potential:

s=n

i=1

(si)ei+As.

We would also like to be able to differentiate u(t) in the direction is going, namely (t).In analogy with the above formula, we define

(t)u(t) = ddt

u(t) +A((t))u(t).

Definition 3.4.1. We will say that u(t) is parallel transported along if(t)u(t) = 0 forall t[0, T].

Intuitively, the statement that u(t) is parallel transported along means we can dragthe vectoru(0) along, andu(t) is the resulting vector at(t). Suppose we are given a vectoruin the fiber over p. By the familiar existence result for ordinary differential equations, wecan find a function u(t) which satisfies

d

dtu(t) +A((t))u(t) = 0

17

7/27/2019 TQFTNotes

18/34

with the initial conditionu(0) =u. This means we can parallel translate u along any smoothpath. LetH(, )u = u(T) denote the result of parallel translating u along the path tothe point q. Since the differential equation defining parallel transport is linear, the mapH(, ) :VV is a linear transformation.Definition 3.4.2.

The map H(, ) :VV is called the holonomy

along the path .

Let us investigate the effect of a local gauge transformation on the holonomy. Supposeu(t) satisfies the parallel transport equation

(t)u(t) = 0.In terms of a vector potential A, this equation says

d

dtu(t) =A((t))u(t)

or in components

ddtu(t) =(t)Au(t).Now a local gauge transformation g : M G takes u(t) to a function w(t) = g((t))u(t).Differentiating this new function, we obtain

d

dtw(t) =

d

dtg((t))

u(t) +g((t))

d

dtu(t)

=(t)(g)u(t) g(t)Au(t)=(t)(g)g

1w(t) (t)gAg1w(t)We are assuming that the gauge group G is a group of matrices, so the gauge transformed

vector potential A

can be written A= gAg

1

(g)g1

and therefored

dtw(t) =(t)Aw(t).

This proves that w satisfies the parallel transport equation(t)w(t) = 0 where is theconnection obtained by applying the gauge transformation g to. By definition then, theholonomyH(, ) maps g(0)u(0) to g(T)u(T), and we have

H(, ) =g((T))H(, )g((0))1.

3.5. Wilson loops. Let be a loop inMbased atp. Then the transformation law that we

derived above for holonomy becomes

H(, ) =g(p)H(, )g(p)1

and therefore the trace of the holonomy is invariant under gauge transformations:

Tr H(, ) = Tr(g(p)H(, )g(p)1) = Tr H(, ).Definition 3.5.1. The trace of the holonomy along is called aWilson loop and is denoted

W(, D) = Tr(H(, )).18

7/27/2019 TQFTNotes

19/34

3.6. Synthesis. Finally, we are going to relate Chern-Simons theory to the link invariantsthat we introduced earlier in this section. Recall that in quantum field theory any physicalquantity is represented by a function f :F(M) R on the space of fields, and the vacuumexpectation value of such a quantity is given by

f= 1

ZF(M)

f()eiS()

D.

In a theory with gauge symmetry, two fields are physically equivalent if they are related by agauge transformation, so in this setting we require the function fto be gauge invariant. Wehave already seen that Wilson loops are invariant under gauge transformations, and indeed,they are among the simplest observables in gauge theory. Our main claim in this section isthat the expectation values of Wilson loops in Chern-Simons theory are isotopy invariants oflinks. More precisely, ifL is a framed oriented link in Mwith components 1, . . . , n, thenthe unnormalized expectation value

A

W(1, A) . . . W (n, A)eik4S(A)DA.

is an invariant ofL. Herek is an integer called the level,Sdenotes the Chern-Simons action,andA denotes the space of all connections on M. As a special case, we can consider thisintegral for the empty link, and we find that the partition function

Z=

A

eik4

S(A)DA

is an invariant of the 3-manifold M. In order to turn these claims into completely preciseresults, we would need to rigorously define the measure DA appearing in the integrals andmake a number of technical modifications to the statements. Although we will not attempt

to do this here, we emphasize that these integrals have been made precise in certain specialcases, and they are known to give rise to familiar invariants. For example, let L(L) denote ouralleged knot invariant above, and consider the case where M=S3 and is the fundamentalrepresentation ofU(1). Then we have

L(L) =eiw(L)/k

wherew denotes the writhe ofL. Alternatively, if we take to be the fundamental represen-tation ofS U(2), thenL(L) is simply the Kauffman bracket ofL evaluated atA = q1/4 whereq=e

2ik+2 . By varying the gauge group of the theory, we obtain a wealth of other interesting

invariants.

4. Atiyahs Definition

4.1. Cobordisms. One of the major advantages of doing topology with TQFTs is the factthat fields are defined locally. This means that we can compute invariants by breaking ourmanifolds into simpler pieces and computing invariants on the pieces. In order to studyTQFTs axiomatically, we should therefore talk about how to break a manifold into simplepieces, or equivalently, how to build up a closed n-manifold by gluing together simpler n-manifolds along their (n 1)-dimensional boundaries.

19

7/27/2019 TQFTNotes

20/34

The manifolds that we will use to build up more complicated manifolds by gluing are calledcobordisms. Roughly speaking, a cobordism is an n-manifold that connects together two(n 1)-dimensional manifolds.Definition 4.1.1. Let 0 and 1 be closed (n 1)-manifolds. A cobordismbetween 0 and1 is an n-dimensional manifold with boundary M= 0

1.

Example 4.1.2. The illustrations below show two different ways to connect the manifold0 = S

1

S1 to the manifold 1 = S1

S1

S1 by a two-dimensional cobordism.

As illustrated in the picture on the right, neither the i nor the cobordismMis required tobe a connected manifold.

Usually, we think of a cobordism as having a direction so that some of its boundarycomponents can be viewed as the source and the others can be viewed as the target ofthe cobordism. To make this idea precise, consider an oriented manifold M with boundary,and let be a connected component of the boundary of M. Given a point x , let[v1, . . . , vn1] be a positive basis for the tangent space Tx. Then a vector w TxM iscalled a positive normal if [v1, . . . , vn1, w] is a positive basis for TxM. If a positive normalpoints inward, then is called an in-boundary, while if it points outward, then is calledan out-boundary. (It is a nontrivial theorem in differential topology that these notions do

not depend on the choice of the point x.)Definition 4.1.3. Let 0and 1be closed oriented (n1)-manifolds. Anoriented cobordismMfrom 0 to 1 is a compact oriented manifold with a map 0Mtaking 0 diffeomor-phically onto the in-boundary ofMand a map 1 M taking 1 diffeomorphically ontothe out-boundary ofM.

For example, either of the manifolds illustrated in the above example can be regarded asan oriented cobordism once we choose an orientation. When drawing pictures of orientedcobordisms, we will always think of the cobordism as going from top to bottom so that theboundary components near the top of the page form the in-boundary, and the ones near the

bottom of the page form the out-boundary.Definition 4.1.4. We say that two cobordisms 0 M 1 and 0 M 1 areequivalent if there exists an orientation-preserving diffeomorphism : MM making thefollowing diagram commute.

0

M

1

M

20

7/27/2019 TQFTNotes

21/34

4.2. Composition of cobordisms. Since every oriented cobordism has a source and target,it is natural to think of such cobordisms as the morphisms in a category whose objects are(n 1)-manifolds. Given a cobordismM0 from 0 to 1and a cobordism M1from 1 to 2,we would like to define their composite to be the cobordism obtained by gluing M0 to M1along the manifold 1. The illustration below shows an example for n = 2.

g

f

=

gf

While the space M0

1M1 that we obtain by gluing two cobordisms does have a natural

manifold structure, it is not immediately obvious that it can be given smooth structure. Wewill now define the smooth structure in two steps. The first step is to define the smoothstructure when the cobordisms being composed are of a special type.

Definition 4.2.1. Acylinderis a manifold of the form [a, b] where is a closed manifoldof dimension n 1.

We will always regard such a cylinder as an oriented cobordism from to itself. LetM0andM1be cobordisms that are equivalent (in the sense of Definition 4.1.4) to cylinders 0 [0, 1]and 1 [1, 2], the equivalences being given by 0:M0 [0, 1] and1 : M1 [1, 2].Then there is a homeomorphism

1

2 : M0

M1 [0, 2]

The manifold [0, 2] has smooth structure which agrees with that of 0 [0, 1] and1[1, 2], so we can put a smooth structure on M0

M1 by pulling back the atlas of

[0, 2] along this homeomorphism.The next step is to use the idea of the last paragraph to define the smooth structure on

an arbitrary composition of oriented cobordisms. Here we require a technical notion fromthe subject known as Morse theory.

Definition 4.2.2. Let Mbe a compact manifold and f :M [0, 1] a smooth map to theclosed unit interval. A critical pointx of this function f is said to be nondegenerate if thematrix

2f

xixjis nonsingular in any coordinate system, and f is called aMorse functionif all of its criticalpoints are nondegenerate and f1({0, 1}) =M.

A theorem of differential topology states that Morse functions always exist, so if M0andM1are the oriented cobordisms we wish to compose, we can take Morse functionsf0:M0[0, 1]andf1 : M1[1, 2]. By choosing >0 to be small, we can ensure that f0andf1are regularon [1 , 1] and [1, 1 +]. Then the preimages of these two intervals are diffeomorphic tocylinders, and we know how to get a smooth structure on the composition.

21

7/27/2019 TQFTNotes

22/34

4.3. Composition of cobordism classes. Finally, we show that the composition of twocobordisms does not depend on the actual cobordisms chosen, but only on their equivalenceclasses. For this we need the following result from differential topology.

Proposition 4.3.1. Let M0 and M1 be composable cobordisms with common boundarycomponent , and let M

1M

0 = M

0

M

1 be their composition. If and are two

smooth structures on M1 M0 which both induce the original smooth structure on M0 andM1, then there is a diffeomorphism M1 M0M1 M0 taking to .

Suppose we are given composable cobordisms M0 and M1 which are equivalent (rel theboundary) to cobordisms M0 and M

1:

0

M0

0

1

M1

1

2

M0 M1

Then we have the composition M1 M0 and the composition M1 M0, and the maps0 and1glue to give a homeomorphism : M1M0M1M0which restricts to a diffeomorphismon each of the original cobordisms:

0

M1 M0

2

M1 M0We can use this homeomorphism to define a smooth structure on M1 M0. Although thissmooth structure may not coincide with the original, Proposition 4.3.1 implies that the twostructures are diffeomorphic rel the boundary. Thus we see that there is a well defined wayto compose equivalence classes of oriented cobordisms.

4.4. The category of cobordisms. The notions introduced so far allow us to define acategory Cob(n) which is an important ingredient in Atiyahs definition of a TQFT. Theobjects of this category are closed oriented (n 1)-dimensional manifolds. Given two suchobjects 0 and 1, a morphism 0 1 is a diffeomorphism class of oriented cobordismsfrom 0 to 1. It is simple to check that the composition law defined above is associative.IfM is a cobordism from 0 to 1 and C is the cylinder on 0, then one can check thatM C= Mup to equivalence, and similarly C M = M ifC is the cylinder on 1. Thisproves that Cob(n) is in fact a category. This category is called thecategory ofn-dimensional

cobordisms.

4.5. Some terminology. Before we get to the definition of a TQFT, we have to discusssome ideas from category theory. In the discussion that follows, we will encounter definitionsinvolving many commutative diagrams, and we would like to use the notion of a naturaltransformation to simplify these definitions. Recall that a natural transformation betweena functor F :C D and a functor G:C D consists of a morphism X : F(X)G(X)for each object X ofC. These morphisms are called the components of. In the following,ifF, G:C1 Cn D are functors and X1,...,Xn :F(X1, . . . , X n)G(X1, . . . , X n) are

22

7/27/2019 TQFTNotes

23/34

morphisms, then these s are called natural isomorphisms or a natural transformationif they are the components of a natural isomorphism or natural transformation F G.

4.6. Monoidal categories. The main reason for the present categorical digression is thatwe need to equip our categories with an operation so that for any two objects XandY wecan form the product XY. The example to keep in mind is the disjoint union operationin the category ofn-dimensional cobordisms. We want this operation to be unital andassociative so that our category is in some ways like a monoid. It is unnatural, however, torequire the operation to be strictly unital and associative. Instead, we would like for to beunital and associative up to isomorphism, and the isomorphisms (XY)Z= X(YZ)and 1X= X= X1 should be part of the structure, satisfying certain axioms calledcoherence conditions.

Definition 4.6.1. Amonoidal categoryis a category Ctogether with a functor :CC C,a distinguished object 1 C, and natural isomorphisms X,Y,Z : (XY)Z X(YZ),X : 1XX, andX :X1X. These isomorphisms are required to make the diagram

(WX)(YZ)

W(X(YZ))

W((XY)Z)(W(XY))Z

((WX)Y)Z

W,X,YZ

1WX,Y,Z

W,XY,Z

W,X,Y1Z

WX,Y,Z

and the diagram

(X1)YX,1,Y

X1Y

X(1Y)

1XY

XY

commute for all objects X, Y, and Z.

It is easy to see that the above definition gives a categorical analog of the notion of amonoid. In these notes, what we really need is a categorical analog of the notion of acommutativemonoid. We can get this by requiring our category to include isomorphismsXYYXthat reverse the order of factors. These isomorphisms are part of the structureand are required to satisfy additional coherence conditions.

23

7/27/2019 TQFTNotes

24/34

Definition 4.6.2. A braided monoidal category is a monoidal categoryC together withnatural isomorphisms X,Y :XYYX, such that the diagrams

X(YZ)X,YZ(YZ)X

Y,Z,X

(XY)Z

X,Y,Z

X,Y1Z

Y(ZX)

(YX)ZY,X,ZY(XZ)

1YX,Z

(XY)ZXY,ZZ(XY)

1

Z,X,YX(YZ)

1

X,Y,Z

1XY,Z

(ZX)Y

X(ZY)1X,Z,Y

(XZ)Y

X,Z1Y

commute for all X, Y, and Z. A braided monoidal category is called a symmetric monoidalcategoryif in addition we have Y,X X,Y = 1XY.Examples 4.6.3. The following symmetric monoidal categories arise frequently in categorytheory and the study of TQFTs.

(1) The category Cob(n) or n-dimensional cobordisms with =

.(2) The category Vect(k) of vector spaces over a field k with =k.(3) The category Setof sets with =

.

(4) The category Setof sets with =.

Now that we have defined symmetric monoidal categories and given some examples, wecan define functors that preserve the structure of these categories.

Definition 4.6.4. Let (C, , 1C) and (D, , 1D) be two monoidal categories. A monoidalfunctorbetween these categories is a functor F :

C D, together with a natural trans-

formation X,Y : F XF Y F(XY) and a morphism : 1D F1C such that thediagram

(F XF Y)F Z

X,Y1FZ

F X(F YF Z)

1FXY,Z

F(XY)F Z

XY,Z

F XF(YZ)

X,YZ

F((XY)Z)F

F(X(YZ))

and the diagrams

F X1D1FX

FX

F XF1C

X,1C

F X F(X1C)FX

1DF Y 1FY

FY

F1CF Y

1C ,Y

F Y F(1CY)FY

commute for all X, Y, Z C. A monoidal functor between two symmetricmonoidal cate-gories is called a symmetric monoidal functorif, in addition to satisfying the axioms above,

24

7/27/2019 TQFTNotes

25/34

the following diagram is commutative for all X and Y inC.

F XF YFX,FY

X,Y

F YF X

Y,X

F(XY) FX,YF(YX)

4.7. Topological quantum field theories. We have now introduced the language of sym-metric monoidal categories, and we are ready to talk about TQFTs. Recall that in orderto specify a quantum field theory, we must specify a spacetime manifold M, a vector spaceF(M) of fields onM, an action S:F(M) R, and a measure D. The goal of topologicalfield theory is to construct a topological invariant of a closed manifold M from the actionSand measure D.

In fact, we will associate an invariant not only to closed manifolds, but to any n-dimensionalcobordism as well. Let 1 and 2 be closed (n

1)-dimensional manifolds, and let M be a

cobordism from 1 to 2. Write Fun(F(i)) for the space of functions onF(i). Then theinvariant that we associate to M is the integral operator ZM : Fun(F(1))Fun(F(2))with kernel

KM(1, 2) =

F(M)

|1=1,|2=2

eiS()D.

As a special case, we can take Mto be a closed n-manifold, regarded as a cobordism fromthe empty (n 1)-manifold to itself. By convention, the vector spaceF() is the zero spaceso that Fun(F())= R. Thus ZM is a linear map R R, which is the same thing asan element ofR. In this way, our construction assigns a numerical invariant to any closedoriented n-manifold.

To describe this invariant more explicitly, let M be a closed n-manifold, regarded as acobordism from 1 =to 2=. We write 0 for the unique field inF(1) and 0+ for theunique field inF(2). For anyf Fun(F(1)), we have

ZM(f)(0+) =

F(1)

f()

F(M)

|1=,|2=0+

eiS()D

D

=

F(M)

eiS()D

f(0)

since the total measure ofF(1) equals 1. The invariant that we are assigning to M istherefore just the partition function

F(M) e

iS()

D of the field theory.The reason for making our field theory assign invariants to all n-dimensional cobordisms

and not just closed n-manifolds is that this enables us to compute invariants by breakingour manifolds into simpler pieces and computing invariants on the pieces. For example, let1, 2, and 3 be closed (n 1)-manifolds, and let M1 be a cobordism from 1 to 2, M2a cobordism from 2 to 3. Then

(ZM2 ZM1)(f)(+) =F(2)

F(1)

f()K(, )K(, +)DD

25

7/27/2019 TQFTNotes

26/34

LetMbe the cobordism obtained by gluingM1and M2along 2. We know that ifis a fieldonMand1=|M1,2=|M2, then the action satisfiesS() =S(1)+S(2). Moreover,since fields are defined locally, we know that such a field is completely determined by itsrestriction to the manifolds Mi and the i. Therefore

F(2)

K(, )K(, +)D

=

F(2)

1F(M1)

1|1=,1|2=

eiS(1)D1

2F(M2)

2|2=,2|3=+

eiS(2)D2

D

=

F(M)

|1=,|3=+

ei(S(|M1)+S(|M2))D

=

F(M)

|1=,|3=+

eiS()D

=K(, +).

Plugging this result back into the previous equation gives Z(M2 M1) = ZM2 ZM1. Wecan express this fact more abstractly by saying that a topological quantum field theory is afunctor from Cob(n) to Vect(k) which assigns the vector space Fun(F()) to an object and assigns the linear map ZM to a cobordism M.

In addition, it is possible to convince oneself that a topological quantum field theoryshould preserve the monoidal structure of these categories. Locality of fields implies thatF(1

2) = F(1) F(2), and it follows that Fun(F(1

2)) = Fun(F(1))

Fun(F(2)). One can argue, using the sort of heuristic arguments involving Feynman inte-grals that we have been using, that Z(M1

M2) =Z M1 ZM2. We are thus led to make

the following definition.

Definition 4.7.1. An n-dimensional topological quantum field theoryis a symmetric monoidalfunctor Cob(n)Vect(k).

We should emphasize at this point that the above computations involving Feynmanintegrals are not really well defined because we do not know how to define the measure D.The only reason for including them in the discussion is that they provide some physicalmotivation for the definition of a TQFT. On the other hand, Definition 4.7.1 is completelyprecise, and it provides a rigorous framework for constructing quantum invariants of spaces.

4.8. Dijkgraaf-Witten theory. A simple example of a TQFT in the sense of Atiyah is

the theory introduced by Dijkgraaf and Witten. This example is similar in some ways to agauge theory but has a well defined partition function. Precisely, we let G be a finite groupandManyn-dimensional cobordism. We take F(M) to be the set of all isomorphism classesof principal G-bundles on M. IfM has boundary M = 0

1, then the kernel defining

ZM: Fun(F(1))Fun(F(2)) is

KM(0, 1) =

F(M)

|0=0,|1=1

1

| Aut()|26

7/27/2019 TQFTNotes

27/34

where Aut() is the finite group of automorphisms of. Then

ZM(f) =

0F(0)

KM(0, 1)f(0).

Since the group G is finite, it follows that the sums in these expressions are finite and thereare no problems of convergence. One can easily check that these data define a TQFT in thesense of Definition 4.7.1.

5. Reshetikhin-Turaev Invariants

5.1. Semisimple categories. In this section, we will see how to construct interesting quan-tum invariants starting from certain categories. These categories, known as modular tensorcategories, come equipped with a lot of of extra structure, and we will spend much of thissection defining the extra structures. We begin by recalling the following definition (for

details see [4]).

Definition 5.1.1. Let k be a field. A categoryC is additiveover k if(1) Each set hom-set inC is a k-vector space and the compositions are k-bilinear.(2) There exists an object 0 C such that HomC(0, V) = HomC(V, 0) = 0 for every

object V C.(3)C has finite direct sums.

An additive categoryC is said to be abelianif it has the following additional property:

(4) Every morphism inC has a kernel and cokernel and is the composition of anepimorphism followed by a monomorphism. If ker = 0 then = ker(coker ), andif coker = 0 then = coker(ker ).

In an abelian category C, it makes sense to say to that an object U issimpleif any injectionV U is either 0 or an isomorphism. We say that an abelian category issemisimpleif everyobject is isomorphic to a direct sum of simple ones. If an additive category is also monoidal,we will require that the monoidal structure be bilinear and that the unit object 1 besimple and End(1)= k. In this section, we will assume that all categories are semisimplewith the property End(V)=k for simple V.

5.2. Braids and braided monoidal categories. Recall that in the previous section, wedefined a braided monoidal category to be a category with the extra structures, 1, , ,, and satisfying certain coherence conditions. We will now discuss how these categoriesare related to three-dimensional topology.

Definition 5.2.1. A braid in n strands is the isotopy class of a union ofn nonintersectingsegments of smooth curves in R3 called strandswith endpoints in{1, . . . , n} {0} {0, 1}such that for each segment the third coordinate is strictly decreasing from 1 to 0.

27

7/27/2019 TQFTNotes

28/34

Example 5.2.2. The following diagrams represent equivalent braids in four strands.

If we are given two braids in n strands, we can compose them by putting one on top of theother. This operation turns the set of all braids in n strands into a group called the braidgroup and denoted Bn.

LetC be a braided monoidal category, and let V1, . . . , V n be objects ofC . Consider theexpressions obtained by forming the tensor product Vi1

Vin where (i1, . . . , in) is a

permutation of (1, . . . , n) and then inserting parentheses and factors of 1. If X1, X2 areobjects constructed in this way and : X1X2is any isomorphism obtained by composing, , , , and their inverses, then we can associate to an element ofBn as follows: Toeach factor of, , and , we assign the identity braid

and to each factor ofViVi+1, we assign the braid which interchanges the ith and (i+ 1)ststrands

One then has the following result.

Proposition 5.2.3. The morphism depends only on its image in the braid group Bn.

For a proof of this theorem, see [4].

5.3. Ribbon categories. Just as the notion of braided monoidal category is related to thetopology of braids, there is a notion of ribbon category, which is related to the topology of

braids equipped with a framing.Definition 5.3.1. LetC be a monoidal category, and let Vbe an object ofC. A right dualto V is an object V ofC together with morphisms eV : V V1 and iV : 1V Vsuch that the compositions

V iV1VV V V 1VeVV

and

V1ViVV V VeV1VV

28

7/27/2019 TQFTNotes

29/34

are identities. Here and throughout this section we skip the canonical associativity and unitisomorphisms so that, for example, the morphism V VV V above is really thecomposition

V1V 1 V iV1V(V V) V V (V V).

Aleft dualtoV is defined to be an objectV ofC together with morphismseV :V V1and i V : 1V Vsatisfying similar axioms. Finally, a monoidal categoryC is said to berigidif every object inC has right and left duals.Definition 5.3.2. A ribbon category is a rigid braided monoidal category with naturalisomorphisms V :VV satisfying

VW =V W1 = 1

V = (V)

1

for all objects V and W.

In any rigid braided monoidal categoryC, there is a natural isomorphism V : V Vgiven by the composition

V i1V V V 1

1V V V 1eV

and thus ifCis a ribbon category, we can define a natural isomorphism V =VV :VV.This operation is called a balancing isomorphismor twist, and can be shown to satisfy thefollowing axioms:

VW =WVVW(V W)1= 1

V = (V).

For any object V in a ribbon categoryC and any endomorphism f ofV, we define thetraceTr f Endk(1)=k to be the composition

1 iV V V f1V V V1V V eV1.

In particular, for f= 1V, this lets us define the dimension ofV by dim V= Tr1V.

5.4. The graphical calculus. Let Cbe a ribbon category. A useful way of proving theoremsabout aboutC is to assign a diagram to each morphism and then argue pictorially usingthe diagrams. In this scheme, if f : V W is any morphism inC , then the diagramcorresponding to f looks like

f

V

W29

7/27/2019 TQFTNotes

30/34

The identity morphism can also be written as

V

V

1V.

= V

where the symbol .= means that the two diagrams represent the same morphism in C. Also,

if f : U V and g : V W are two morphisms, the diagram corresponding to theircomposition can be written

U

W

gf .

=

U

V

f

W

g

If V and V are objects inC then the diagram corresponding to their tensor product isobtained by drawing the corresponding arrows parallel to one another. Iff : V W andf : V W are morphisms, then the diagram corresponding to their tensor product canbe written

VV

WW

ff .

= f

V

W

V

W

f

In our diagrams, arrow corresponding to the dual V of an object Vis obtained by reversing

the arrow corresponding to V. We will also omit the arrows corresponding to 1 C andidentifyV with V via the isomorphism V. Then the diagrams foreV : V V 1 andiV : 1V V can be written as

eV

V

V

1

.=

and

iV

1

V

V

.

=

Finally, the diagrams corresponding to V V and 1V V can be written as

V V

V

V

V

V

.=

30

7/27/2019 TQFTNotes

31/34

and

1V V

V

V

V

V

.=

5.5. Ribbon tangles. In addition to providing an intuitive graphical notation, the graphicalcalculus introduced above actually helps us study the topology of certain braid-like objects.Let us now define precisely what these objects are.

Definition 5.5.1. A tangleis the isotopy class of a union of nonintersecting smooth curvesin R2 [0, 1], which can have endpoints only on the lines R {0} {0}and R {0} {1}.If none of the curves have endpoints, then the tangle is simply a link in R2 [0, 1].

This definition gives rigorous meaning to the pictures used above to represent morphisms

in a ribbon category. It is natural to ask whether, given two morphisms, the correspondingtangles are distinct. To see that the answer is no, we observe thatV= 1V in a generalribbon category and yet the diagram for V is

V

V

V

.=

which is clearly isotopic to the picture of 1V. We must therefore modify the notion of atangle so that distinct morphisms correspond to distinct tangles.

Definition 5.5.2.

(1) A ribbonis a homeomorphic image of a rectangle in R3 with a distinguished pair ofopposite edges and a distinguished side called the face side.

(2) A coupon is a rectangle in R3 lying in a plane parallel to R {0} R and havingedges parallel to R {0} {0} and{0} {0} R.

(3) A generalized ribbon tangle with n strands is the isotopy class of a union of nonin-tersecting ribbons and coupons in R2 [0, 1] such that the distinguished edges of theribbons lie on the lines R {0} {0} and R {0} {1}or on the edges of couponsparallel to them, and near them the ribbons have their face sides turned upwards.

Consider a generalized ribbon tangle in which each ribbon strand is directed. Label eachribbon in this generalized ribbon tangle by an object ofC , and for any fixed coupon, letV1, . . . , V m be the labels of the ribbons that end on its top edge. We write i= + if the ithribbon points towards the coupon and i = if it points away from the coupon. Then wecan form the object X=V11 Vmm where V+ =V and V =V. (If no objects endon the top edge of this coupon, then X= 1.) Similarly, we can associate an object Y to thebottom edge of a coupon, where now we must write = + for ribbons that point away fromthe coupon.

31

7/27/2019 TQFTNotes

32/34

Definition 5.5.3. AC-colored ribbon tangle is a generalized ribbon tangle in which eachstrand is directed, each ribbon is labeled by an object ofC, and each coupon is labeled bya morphism f : X Y where X and Yare the objects associated to its top and bottomedges by the above construction.

We now use the graphical calculus to associateC-colored ribbon tangles to a morphisminC. Let us first fix V1, . . . , V n C and consider the objects obtained by tensoring V1, . . . , V nin any order, possibly with repetitions, and adding an arbitrary number of left and rightstars and factors of 1. To each expression obtained in this way, we associate a sequenceF(X) of arrows and labels in the following way: To each object V, we assign the data V if the total number of stars is even and V if the total number of stars is odd. Forexample, we associate to the object

((V1 V2) V4) ((V1 1) V2) . . .the sequence

V1 V2 V4 V1 V2 . . . .Let X1 and X2 be objects ofC of the above form. Then we can consider all morphisms : X1 X2 which can be obtained as a composition of the elementary morphisms 1,1, 1,1,e,i,1 as well as a number of other morphisms ofC. To such a composition,we associate a generalized ribbon tangle T = F() in such a way that F(X1) is the top ofT and F(X2) is the bottom ofT. We do this by letting the morphisms , , , and theirinverses correspond to trivial tangles and letting F(e), F(i), F(), and F(1) correspondto the tangles of the graphical calculus, with the blackboard framing. Given two morphisms1 and 2 inC, we get F(1 2) andF(12) by combining the tangles F(1) andF(2)in the obvious way.

Proposition 5.5.4. A morphism : X1X2 as above depends only on the isotopy classof the tangle F().

This result implies that for every C-colored ribbon tangleT, the morphismF1(T) :X1X2 inC is an isotopy invariant ofT. In particular, ifT is aC-colored framed link, then wecan view F1(T) as a numerical invariant ofT.

5.6. Modular tensor categories. The last notion from category theory that we will needis the notion of a modular tensor category. LetCbe a semisimple ribbon category and let Ibe the set of equivalence classes of nonzero simple objects in

C. For each i

I, let Vi be a

representative for the equivalence class i. Then we can define numbers sij k=End1 bythe formula

sij =1i

1j Tr Vi Vj

Definition 5.6.1. LetC and Ibe as above. We say thatC is a modular tensor category ifC has the following properties:

(1)C has only finitely many isomorphism classes of simple objects (so|I|

7/27/2019 TQFTNotes

33/34

In applications to topology, the most interesting and nontrivial examples of modular tensorcategories are categories of representations of quantum groups at roots of unity. We referthe reader to [6] for more information.

In the following discussion, we will use the notation p =

iI1i di where di = dim Vi

and D =

p+p.

5.7. Invariants of 3-manifolds. Finally, let us show how modular tensor categories giverise to quantum invariants of 3-manifolds. To do this, we need to discuss how 3-manifoldscan be obtained by surgery along links. Therefore letL be a link in S3 with components1, . . . , n and denote by Ti a small tubuler neighborhood of i. Then each Ti is a solidtorus. Let T0 be a fixed solid torus and choose orientation-preserving homeomorphismsfi: TiT0. These mappings give rise to an orientation-preserving homeomorphism

f :n

i=1

Tin

i=1

T0.

Definition 5.7.1. A surgery of S3 along the link L is a 3-manifold ML,f defined by theformula

ML,f=

S3

ni=1

Ti

f

ni=1

T0

.

Proposition 5.7.2. Any connected closed three-dimensional manifold can be obtained as asurgery ofS3 along some link.

In the definition of surgery, ML,fdepends on both the link L and the attaching map f.However, ifL happens to be a ribbon link, then there is a canonical choice offand so inthis case we write ML,f=ML.

Theorem 5.7.3. Let Cbe a modular tensor category, and let L be a framed link in R3 S3.Then the number Z(ML) defined by the formula

Z(ML) =D|L|1F1(L)

p+

p

(L)/2,

where|L|is the number of components ofLand(L) is the wreath number, is an invariantof the 3-manifold ML which comes from a topological quantum field theory.

Acknowledgments

Much of the material in these notes was originally presented in a seminar at Yale Universityin the Spring semester of 2012. I thank Giovanni Faonte, Ivan Ip, Hyun Kyu Kim, and LinhuiShen for participating in the seminar and helping me learn the subject.

References

[1] Figueroa-OFarrill, J. Unpublished notes on gauge theory. 2007.[2] Baez, J. and Muniain, J.P. Gauge Fields, Knots and Gravity. World Scientific, 1994.[3] Kock, J. Frobenius Algebras and 2D Topological Quantum Field Theories. Cambridge University Press,

2003.33

7/27/2019 TQFTNotes

34/34

[4] Mac Lane, S. Categories for the Working Mathematician. Springer-Verlag, 1988.[5] Segal, G. Unpublished notes on topological quantum field theory. 1999.[6] Bakalov, B. and Kirillov, A. Lectures on Tensor Categories and Modular Functors. American Mathemat-

ical Society, 2000.

34