Perturbative Topological Field Theory

Robbert Dijkgraaf

Department of Mathematics University of Amsterdam

Plantage Muidergracht

TV Amsterdam The Netherlands

Abstract

We give a review of the application of p erturbative techniques to top ologi

cal quantum eld theories in particular threedimensional ChernSimons

Witten theory and its various generalizations To this end we give an

intro duction to graph homology and homotopy algebras and the work of

Vassiliev and Kontsevich on p erturbative knot invariants

Intro duction

In these lecture notes we will give a review of some recent mathematical devel

opments in top ological eld theory following the work of Kontsevich Axelro d and

Singer Vassiliev BarNatan Witten and others Quite remarkable these new ideas

are all related to oldfashioned p erturbative techniques in eld theory Indeed it is

an interesting comment on the development of the interaction b etween physics and

mathematics that these days pure mathematicians are calculating Feynman diagrams

whereas this skill is slowly disapp earing among a large fraction of theoretical physics

students

Our starting p oint in all this will b e mostly threedimensional ChernSimons

Witten gauge theory This top ological eld theory lends itself most conveniently

to a p erturbative formulation Actually one can argue that the mo del is even more

elegant and symmetric in p erturbation theory The reason b ehind this phenomenon

has b een p ointed out b e Witten ChernSimons is an exact lowenergy limit of an

op en string eld theory

The p erturbative asp ects of ChernSimons theory have b een studied in many

pap ers starting so on after the nonp erturbative solution of Witten in see eg

A very precise analysis aimed at mathematical rigor has b een p erformed by

Axelro d and Singer and we will follow their work closely The innovations and

generalizations we will discuss in particularly the relations with homotopy algebras

follow the b eautiful ideas of Kontsevich The applications to knot invariants center

around the work Vassiliev which has b een a great step forward in the classication

program of knots The relation of the work of Vassiliev to p erturbative eld theory

was rst p ointed out by BarNatan The p ower of this approach is mainly due to

a wonderful theorem proved by Kontsevich

It is wellknown how the partition function of ChernSimonsWitten gauge the

ory when considered on a closed threemanifold M pro duces a top ological invariant

Z M Although this quantity can b e dened nonp erturbatively and the coupling

constant k is quantized the resulting expression is analytic in and thus has

a welldened p erturbative expansion

X

n

F M Z M exp

n

n=0

The co ecients F M dene perturbative invariants that only dep end on calculations

n

of connected Feynman diagrams up to nlo ops They can therefore b e considered as

nite order invariants This p oint of view has some interesting consequences and

advantages that we will develop in these notes

Finite order invariants can b e calculated evaluating nitedimensional integrals

They therefore stand a greater chance of b eing rigorously dened This is of course

a very familiar argument for particle physicists

A p erturbative approach greatly facilitates classication since we now have

a natural ltration on the space of invariants by order in p erturbation theory If

we are lucky the space of invariants of a given order can turn out b e nite and can

b e analyzed using combinatorical means This turns out to b e the case for knot

invariants as we will see in section

A p erturbative framework can b e used to generalize the concept of a top ological

invariant If we are somewhat p edantic we can call a smo oth manifold invariant a

function Z g that is invariant under smo oth changes in the Riemannian metric g

ie

Z g g Z g or Z

where is the exterior dierential on the space of isomorphism classes of Rieman

nian manifolds B Met M Di M B Di M Stated otherwise Z is lo cally

constant function on B

0

Z H B

This p oint of view leads to the natural generalization where top ological invariants are

higher dimensional cohomology classes on the space of Riemannian structures

k

Z H B

This idea is actually wellknown from string theory where the partition function Z

of the underlying conformal eld theory on a Riemann surface is a volume form on

R

Z the mo duli space M The string partition function is then obtained as

g

M

g

Closely related to the previous remark is the fact that certain generalizations

of the action b ecome p ossible once we have obtained a gauge xed p erturbative

formulation in terms of Feynman diagrams Stated simply one can replace the cubic

vertex in ChernSimons theory by quartic quintic etc vertices These higher order

vertices typically involve interactions where elds ghosts and antields couple in a

nontrivial way This leads to algebraic structures with multilinear op erations so

called homotopy algebras These structures have recently b een describ ed in a very

elegant way using the language of op erads see eg a sub ject we will not touch

up on in these notes

Finally many of the ab ove remarks have a clear interpretation from the p oin of

view of string eld theory where they help us understand the issues of background

indep endence spacetime ghosts etc

Algebras and Feynman Diagrams

Graph Cohomology

It is often stated that one of the elegant features of string theory is that one has to

consider only one diagram in the top ological expansion a Riemann surface of given

genus this in contrast with ordinary p ointparticle eld theory with its many dierent

Feynman diagrams Of course there still remains a very complicated integral to b e

done over the mo duli space M of such a surface

g

It is less appreciated that certain typ es of eld theories can b e formulated in a very

similar avour Indeed let us consider onedimensional quantum gravity where our

spacetime is an arbitrary graph The space of metrics mo dulo dieomorphisms

on such a singular space is parametrized by an assignment of lengths l l to

1 E

all the edges of the graph In D quantum gravity we want to sum over all graphs and

integrate over the lengths with some particular weights Of course we can think of

i

the lengths as the Schwinger parameters of the Feynman diagrams of quantum eld

theory As such they are completely equivalent to the mo duli of Riemann surfaces

that app ear in string theory For example the usual eld theory propagator takes the

form

Z

2 2

l (p +m )

dl e

2 2

p m

The Schwinger parameter spaces of the individual graphs can b e glued together to

form a space that is very analogous to the mo duli space M of Riemann surfaces To

g

E

R where a p oint x c is parametrized every diagram we can asso ciate a cell c

+

as x l l These cells can b e glued together in the following fashion If one

1 E

of the lengths say l b ecomes zero the graph will change top ology If the particular

1

propagator connects two dierent vertices of order n and m the degenerated diagram

will have one vertex instead of two now of order n m For example in the case

of two cubic vertices we will create a quartic vertex

l lim

=

l 0

In this pro cess the numb er of lo ops will not change the numb er of edges and the

numb er of vertices are reduced by one

Another case is an edge that starts and ends at the same vertex say of order n

In the limit where the lenth of the edge tends to zero we are left with a diagram with

one lo op less and a vertex now of order n One veries that in b oth cases the

Euler numb er of the graph is reduced by one

Clearly the b oundary of the cell c will consist of a sum of lower dimensional cells

with signs asso ciated to graphs e where one propagator e is contracted to

zero

X

c c

e

edg es e

We can write this more abstractly as c c where we intro duce the b oundary of

d

a graph by

X

d e

edg es e

We can now make a mo duli space F of all Feynman diagrams with g lo ops and

g s

s external lines by gluing together the cells that b elong to graphs with only cubic

vertices with appropriate symmetry factors such that the sum of the b oundaries of

all cells is zero

X

c

F

g s

Aut

cubic

This space obviously has real dimension

dim F g s g s

We have to decide It is actually an orbifold space since graphs typically have auto

morphisms For example consider the graph

l1 l 2

l 3

Whenever we degenerate one of the l we obtain the quartic diagram

i

So we have to identify the three faces l

i

Graph cohomology can b e dened as the cohomology of this mo duli space F

Actually following Kontsevich we can formulate graph cohomology just in terms of

graphs First we build a vector space C with a basis spanned by all graphs We

will take these graphs to b e closed and with vertices of order We further

give them an orientation We write for the graph with the opp osite orientation

This immediately has imp ortant consequences b ecause some graphs are actually

isomorphic to their opp osite

If we work over the complex numb ers these graphs therefore vanish An example of

such a graph is the quartic diagram ab ove or the dumbb ell diagram

Of course the space of graphs is naturally graded

M

g

C C

g =0

g

where C are the graphs with g lo ops ie

vertices edges g

g

Now the spaces C are actually complexes We can write

3g 3

M

g

g

C C

k

k =0

where k is the numb er of edges We now have a natural b oundary map

X

g g

d C C d e

k k 1

edg e e

where we sum over the contraction of edges It is an interesting exercise to verify that

2

indeed d We can now dene chains as linear combination of graphs

X

a a a C

A closed chain will now b e a chain that satises da An example of a closed

graph is the graph

d

We can also dene homology classes by

a db da a

Similarly graph cohomology classes are linear functions C C such that d

d where we simply dene

d d

Cohomology classes thus vanish on exact graphs We should think of graph cohomol

ogy classes as very sp ecial Feynman rules that resp ect the degeneration of graphs

That is any combination of graphs that is a b oundary of another graph vanishes

according to these rules

Lie algebras

An interesting example of such a set of Feynman rules that we will meet many

times in these lectures is the one based on Lie algebras and more general homotopy

Lie algebras Recall that a Lie algebra is nothing but a vector space V with a bilinear

bracket

V V V u v u v

that satises two conditions symmetry

u v v u

and the Jacobi relation

u v w v w u w u v

If we cho ose an explicit basis e then we have structure co ecients

a

X

c

e e f e

a b ab c

c

If the Lie algebra is even b osonic and simple it has a unique invariant inner pro duct

and we can intro duce completely antisymmetric tensors f that we can use as

ab abc

vertices of Feynman graphs a

b

f abc

c

Since this vertex is antisymmetric we should orient it to make it welldened for

example using blackb oard orientation With this convention the Jacobi identity

X

f f f f f f

abe edc dae ebc ace edb e

reads pictorially

X

e

We can now write an interesting graph homology class as follows For every closed

cubic graph let I b e the weight asso ciated to it by the ab ove Feynman rules

We now dene the chain by a summation over all cubic g lo op diagrams

X

I

a

g

Aut

cubic C

g

We now claim that da This is actual a direct consequence of the Jacobi identity

g

The diagrams that app ear in da will have one and only one quartic vertex Such

g

a diagram can b e obtained as the b oundary of three dierent cubic graphs That is

to say there are three inequivalent way to resolve the quartic vertex into two cubic

vertices These are of course the s t and uchannel diagrams which we will denote

as and So the total weight asso ciated to this particular quartic diagram

s t u

will b e

I I I

s t u

which vanishes precisely by the Jacobi identity To pro duce other graph homology

classes we have to go to homotopy algebras

Homotopy Algebras

The concept of a homotopy algebra is quite general and exists in many dierent

contexts such as commutative asso ciate Lie or dierential graded algebras The

unifying principle is that the algebraic op erations are no longer restricted to b e binary

As such the most natural language to describ e homotopy structures in algebra is

the formalism of op erads or trees

However here we will take a much more down to earth p oint of view To b e as

concrete as p ossible we also restrict ourselves rst to homotopy Lie algebras We will

start from the b eginning with graded algebras That is our elements might b e either

commuting or anticommuting So a Lie algebra can also b e a sup er Lie algebra

For a homotopy Lie algebra we just intro duce generalized brackets on N elements

u u V u u V

1 N 1 N

that again satisfy two relations symmetry and Jacobi The symmetry relation now

reads

u u u u u u u u

1 i j N 1 j i N

and the generalized Jacobi identity takes the form

N

X

u u u u per m

1 k k +1 N

k =1

where we refer to the literature for the precise choices of signs The latter relation

can b e written much more clearly in terms of Feynman graphs Hereto we assume

that the algebra V also has an invariant inner pro duct by which we can raise and

lower indices We can than dene structure co ecients

X

b

f f

a a a a b a a

0 1 0 1

N N

b

with

X

b

e e f e

a a a a a

1 1

N N

b

We can use these fully graded symmetric tensors as vertices of our Feynman rules

For a vertex of order n we write symb olically

f

n

a a a

0 1

N

The Jacobi identity then takes the form

X

n m

m+n=N

It is now clear how to employ these homotopy algebras to pro duce graph cycles We

simply sum over al l graph of a given numb er of lo ops not necessarily only the cubic

ones and asso ciate to each graph the ab ove weight I The classes

X

I

a

g

Aut

C

g

are easily seen to b e closed The ab ove algebraic structure can b e formulated a bit

more elegantly as follows We will consider an algebra A with a multiplication

m HomA A A

that satisfy a quadratic relation that we will write symb olically as

m m

with m m HomA A A A Familiar examples are Asso ciative algebras

with a multiplication that we will write as

ma b a b

and that satises the quadratic relation

m ma b c a b c a b c

Commutative asso ciative algebras with the further constrained

ma b mb a

Lie algebras where

ma b a b b a

and the quadratic condition is the Jacobi identity

m ma b c a b c b c a c a b

We can now try to generalize the algebraic op eration m to an op eration on n elements

of A with n

The rst nontrivial generalizations are dierential graded algebras DGAs such

as the familiar example of the space M of dierential forms on a manifold M

This is a graded commutative asso ciative algebra with multiplication the wedge

pro duct

m n m+n ab

a b a b b a a b

This algebra also has a derivation d the exterior dierential

n n+1 a

d da b da b a db

2

that is nilp otent d Generally a DGA has by denition a nilp otent linear

op erator of degree one

2

d HomA A d

that is a derivation

a

dma b mda b ma db

Conditions and can b e combined in writing

where is now the nonhomogeneous linear combination

d m HomA A HomA A A

and the op eration is dened in the obvious way So we see that DGAs t in the

same framework as ordinary algebras if we generalize the multiplication op eration

to involve b oth one and two elements

Of course we can also take more than two elements which leads us to homotopy

algebras with an algebraic op eration

M X

k

HomA A

k

k 0 k

with relation

For example d m for dierential graded algebras

1 2

The relations enco ded in the ab ove relation actually make more sense if we write

them in comp onents To b e concrete we will write them for homotopy Lie algebras

where is denoted by the k fold bracket Of course we start in degree one with the

k

2

map d A A that satises d Then in degree two we nd the

1

usual Lie bracket A A A and the relation

2

da b da b a db

that expresses that d is a derivation of the Lie algebra At degree three we see the

3

rst o ccurrence of the ternary bracket A A It actually mo dies

3

the Jacobi relation by

a b c b c a c a b da b c da b c a db c a b dc

So Jacobi only holds up to dexact terms This is the reason one calls these algebras

homotopy algebras At the following degree we nd a relation involving the and

brackets that is again zero up to exact terms involving the bracket To nd an

application of all this we will now turn to ChernSimonsWitten theory

ChernSimonsWitten Theory

We will now apply the ab ove ideas to ChernSimons gauge theory In this mo del

the fundamental eld is a connection A on a threemanifold M We cho ose a Lie group

G with Lie algebra g and consider the gauge eld as a Lie algebra valued oneform

We will write

1 1

A M g

Under gauge transformation A transforms as

1 1

A g dg g Ag g M G

or innitesimally

A d d A

A

where is an element of the Lie algebra of the group of gauge transformations

0 0

M g

ChernSimons theory is concerned with at connections that is connections for which

the curvature

2 2 2

d F dA A

A

vanishes It is a particular feature of three dimensions that the equation of motion

F can b e seen as the variation of an action namely

Z

3

S A Tr AdA

M

This action is almost gauge invariant Under gauge transformations it picks up a

term

Z

1 3

S S Tr g dg

M

This last term is for top ological reasons always times an integer

The quantum eld theory is dened through the pathintegral

Z

ik S

Z dA e

where one integrates over equivalence classes of connections The coupling constant k

which plays the role of is required to b e integer in order to make the pathintegral

welldened

We can now consider this quantum eld theory in an asymptotic expansion in

p

k we have Plancks constant k Indeed after a scaling A A

Z

3

p

A Tr AdA k S

M

k

so this corresp onds with the usual expansion of a eld theory with cubic interaction

the For the expansion we pick a classical solution A satisfying with d d

0 0 A

0

classical equation of motion

2

d

0

and write

Z

3

S A A A Tr Ad A

0 0

M

The discussion b elow greatly simplies if we assume not very realistically that the

complex d is acyclic ie all cohomology groups vanish

0

H M g

This eliminates in particular the p ossibility of zero mo des ie a family of classical so

lutions since deformations of a classical solution are innitesimally given by solutions

to the equation

d a

0

mo dulo gauge transformations

a d

0

The gauge xed action can b e obtained very directly if one follows the BV quantization

scheme Here we need three ingredients

A njndimensional X of elds and antields with an o dd symplectic form

This makes the function space H C X into a socalled BatalinVilkovisky

algebra This is a generalization of a Poisson algebra H is a graded commutative

algebra and a Lie algebra with an o dd Lie bracket the socalled antibracket

n m n+m1

f g H H H

which reads in lo cal co ordinates

X

b a

ij

fa bg

i j

x x

ij

It satises the relations

(a1)(b1)

fa bg fb ag

(a1)(b1)

fa fb cgg ffa bg cg fb fa cgg

a(b1)

fa bcg fa bgc afb cg

Furthermore there is an op erator the BVLaplacian dened as

2

X

ij n n1

H H

i j

x x

ij

that is nilp otent of order two

2

and satises the compatibility relation

a

fa bg ab ab ab

Actually this last relation denes the bracket in terms of the Laplacian and can b e

taken as the denition of the bracket

An action S H R that satises the master equation

1

fS S g S

2

In the case of ChernSimons theory the space of eldsantields is taken to b e the

space of functions on the total space of Lie algebravalued dierential forms

X M g

n

Note that we consider the co ordinates f to b e o ddeven dep ending on whether

n is eveno dd A general eld has decomp osition

3

X

(n) (n) n

n=0

(1)

Of course A is the original physical gauge eld We will see that furthermore

(0) (2) (3)

c is the usual ghost A is the antield of the connection and c

is the antield of the ghost not to b e confused with the antighost

On H we dene the antibracket as follows With f g we dene their anti

bracket as

Z

ff g g Tr f g

The action is taken to b e the original ChernSimons action

Z

3

S Tr d

0

M

where is now an arbitrary dierential form The action S reduces to the classical

1

action we restrict One checks that S satises separately the classical master

action

Z Z

2 2 3

fS S g Tr d d d

0 0 0

and the quantum correction S The recip e is now completed by putting half

of the variables to zero Hereto we imp ose the Lorentz gauge condition We cho ose a

metric on M this gives in particular a Ho dge star

k 3k

and an adjoint

2

d d d

0

0 o

The gauge condition is now

d

0

In case all cohomology vanishes no harmonic forms this precisely eliminates the

required degrees of freedom

This description can also b e derived using more conventional techniques for ex

ample as done by Axelro d and Singer If one cho oses again Lorentz gauge

d A

0

we now have to intro duce ghost antighosts a Lagrange multiplier and a BRST

op erator satisfying the usual relations

1

c c Q QA D c Qc c b Qb

2

The gauge xed action now reads

Z

S S A A Q Tr cd A

g f 0

0

If we redene variables as

2

c B d b g A d

0 0

these twoforms satisfy by denition

d C d B

0 0

If we intro duce the combination

c A A c g

then we recover precisely the gaugexed action as written ab ove

1

The propagator d as can now b e dened on ker d

0

0

1

d d

0

0 0

with

d d

0 0

0

The propagator has a kernel

Z

1

x d Lx y y

0

y M

that is a dierential form of total degree on M M

2 2

Lx y M g g

and that satises

3

d Lx y x y

0

This kernel has the nilp otency prop erty

Z

2

d Lx y Ly z

0

y M

and symmetry

ab ba

L x y L y x

The interactions are given by the threep oints vertex

V R

with

Z

X

a b c

V A B C f A B C

abc abc

Let us now present the argument of Axelro d and Singer why the nlo op contributions

to the partition function are top ological invariants The contribution will have a

typical form

X

I

F

g

Aut

where we sum over all connected cubic diagrams with g lo ops and I is the

weight of the diagram This consists of two contributions a spacetime factor I

and a group theory factor I In case that the at bundle is trivial these two

contributions factorizes

I I I

which we will now assume for the sake of simplicity We rst concentrate on the

spacetime factor It has the general form

Z

E

I L

V

M

where V g is the numb er of vertices and E g the numb er of edges

Let us now consider a variation g of the metric used in the gauge xing and

denition of the propagator L Since the propagator satises by denition

3

d L x y

0

and the deltafunction do es not dep end on the choice of metric the variation of the

propagator L will satisfy

d L L d K

0 0

where we assumed acyclicity again We can therefore write following Axelro d and

Singer

Z

E

I L

V

M

Z

E 1

E L L

V

M

Z

E 1

L d K E

0

V

M

Z

E 2

E E L d LK

0

V

M

Z

E 2

E E L K

V 1 M

3

where in the last line we have used that d L x y This eectively shrinks the

0

propagator to zero and so reduces our cubic Feynman diagram to a diagram with

one quartic vertex precisely as in our discussion of graph homology Actually we can

write the ab ove manipulation as

I I d

g

Of course to complete the argument we have to add the group theory factors I

But these are precisely the rules related to Lie algebras we were discussing in the

previous section That is the g lo op contribution to the ChernSimons partition

function is

F I a

g g

with a the class

g

X

I

a

g

Aut

C cubic

g

Therefore we now have the following elegant one line pro of of the top ological prop er

ties of p erturbative ChernSimons theory

da F I da

g g g

In light of our discussion ab ove it is now obvious how to generalize this prop erty

Instead of cubic graphs weighted with Lie algebra symmetry factors we now take

arbitrary graphs weighted with homotopy Lie algebra Feynman rules However things

are a bit more complicated In particular cubic graphs are sp ecial in the following

resp ect The numb er of edges is E g and the numb er of vertices V g

2 3g 3

So since L is a dierential form on M of total degree the integrand L is a

2g 2

dierential form on M of degree g This is a top degree form and can thus

b e integrated to give a numb er I

This is clearly a very sp ecial prop erty of cubic vertices and threedimensional

P

manifolds In the general case the graph has E edges and V v vertices

k

k =3

where v denotes the numb er of vertices of valency k If the spacetime manifold M

k

has dimension d the degree of the weight I will formally b e

X

d

k dv k d E dV

k

k =3

So how to interpret a weight that is not a numb er but a dierential form of p ositive

degree The idea is to consider not one manifold M but a family of manifolds M t

where t ranges over some subspace C B the space of Riemannian manifolds The

weight I now has a natural interpretation as a dierential form on B of degree k

Furthermore if we rep eat our calculation of the metric dep endence of I we nd again

I I d

where now has the interpretation of b eing the exterior dierential on B The g lo op

partition function is now closed and thus gives a cohomology class on B

k

F H B

g g

Perturbative Knot Invariants

It is quite natural to extend the ab ove ideas of p erturbative top ological gauge

theories to include observables in particular Wilson lo ops This is already very

interesting for the usual ChernSimonsWitten mo del and we will consequently not

discuss generalizations to homotopy structures

1

For an emb edding K S M and a representation R Wilson lo op averages are

dened as

Z I

ik S (A)

hW K i dA e Tr P exp A

R R

Z M

K

As is wellknown these exp ectation values lead directly to the famous knot p olyno

mials of Jones and their generalizations These invariants again are analytic

in k h and consequently have a p erturbative expansion

X

n

a K hW K i

n R

n=0

In this way we are naturally led to also consider knot theory in p erturbation theory

Remarkably this p erturbative approach to knot invariants was develop ed more

or less indep endent from physics by Vassiliev In our exp osition we will follow

mainly the b eautiful and extensive review of BarNatan and the seminal pap er by

Kontsevich See also the interesting pap ers

For our purp oses a knot will b e an emb edding of a circle in Euclidean threespace

1 3

K S R

and more generally a link will b e an emb edding of a collection of circles

1 1 3

L S S R

The imp ortant restriction is of course the absence of selfintersection in these maps

Two knots K K are said to have the same knot typ e K K if there exists

1 2 1 2

3

an isotopy of R mapping K into K The fundamental problem in knot theory is to

1 2

make sense of the classication of knot typ es A knot or link invariant K dep ends

by denition only on the knot typ e

K K K K

1 2 1 2

It is clear that a classications of knot invariants is dual to a classication of knots

Vassiliev invariants

A fruitful way to think ab out knot invariants is to consider the space B of all

emb eddings This is a very nontrivial innitedimensional space with the top olog

ical complications coming from the fact that we have to excluded selfintersections

In particular B is not connected and the connected comp onents of B corresp ond

precisely to the dierent knot classes From this p oint of view a knot invariant is a

lo cally constant function on B

0

H B

1 3

We can think of B as a subspace of the space of all smo oth maps S R In this

space we have dierent strata B of co dimension n where we allow n distinct normal

n

intersections and B B We can inductively extend to these singular knots in

0

B through the lo cal denition

n

We will also write

r

This denes the derivative of a knot invariant Notice it is a partial derivative since

its denition dep ends on which crossing we have picked in a particular blackb oard

pro jection of the knot It is the wonderful idea of Vassiliev that this p oint of view

pro duces a natural ltration on the space of knot invariants This is b est explained

by an analogy

Consider the space A of analytic functions in z We have a natural ltration

P P P A

0 1 n

where P is the space of p olynomials of degree n

n

n

z

a z a a C p P pz a

1 0 i n n

n

With r ddz we have

n+1

p P r p

n

Furthermore the quotient spaces P P C are onedimensional and corresp ond

n n1

to the leading co ecient of p P

n

n

r p a P P

n n n1

With this analogy in mind we now try to dene p olynomial knot invariants not to

k

b e confused with knot p olynomials These will b e the invariants satisfying r

for large enough k

More precisely we call a knot invariant a Vassiliev invariant of order n and we

will write V if vanishes on B and consequently on all B with k n

n n+1 k

Stated otherwise is a Vassiliev invariant of order n if al l n th derivatives

vanish

n+1

r

or equivalently

B C

B C

B C

A

z

n+1

We will see in a moment that

dim V V

n n1

and that the leading co ecient

n

r V V

n n1

has a natural interpretation in terms of Feynman diagrams Actually this relation

makes the description of the spaces V algorithmically sp eaking completely straight

n

forward There is now a universal classication of Vassiliev invariants The classi

cation of all knot invariants can only dier from this by nonp erturbative terms

The simplest examples of Vassiliev invariants are the co ecients of the Conway

Alexander p olynomial cK Recall that this famous knot invariant asso ciates to

each knot a p olynomial in the formal variable z

n

cK c K c K z c K z

0 1 n

dened recursively through

c

and the skein relation

c z c c

The rst few nontrivial examples are

B C

B C

c B C if n

A

z

n

c z

B C

B C

2

B c C z

A

C B

2

C B

z c

A

It is an easy exercise to verify that the co ecients c are Vassiliev invariants of order

n

n

c V

n n

We simply use the skein relation

rc z c

n times which implies that

n+1 n+1

r c z

and consequently

n+1

r c

n

The invariants c and c have an elementary interpretation

0 1

if K has one comp onent

c K

0

otherwise

and

l k K if K has two comp onents

c K

1

otherwise

where l k is the linking numb er The higher order invariants are much more mysterious

Analogous results hold for the Jones HOMFLY and Kauman p olynomials For

example the HOMFLY p olynomial H n Z is dened through the skein relation

n

n+1 n+1

1 1

2 2 2 2

t H H t H t t

n n n

z

The Jones invariant is given by n If we substitute t e and expand

X

k

H h z

n nk

one veries that Vassiliev invariants are pro duced h V

nk k

Weight systems and chord diagrams

The relation with p erturbative eld theory emerges in the following remarkable

way To each Vassiliev invariant we can asso ciate a socalled weight system Weight

systems are basically generalizations of Feynman rules

n

Consider a Vassiliev invariant V and its derivative r The latter is a knot

n

invariant for knots with n distinct normal selfintersections It has the imp ortant

prop erty that it vanishes as so on as we create another intersection since by denition

n+1

r So we can assume that this knot is unknotted except for the n self

intersections An example would b e the following knot with three selfintersections

K

To such a conguration we can asso ciate in a unique way a graph a socalled chord

diagram For convenience let us assume that our knot has only one comp onent The

1 3

map K S R is onetoone apart from n sp ecial p oints on the circle where the

1

knot intersects itself Let us lab el these p oints consecutively P P S The

1 2n

3

p oints P and P are pairwise related if K P K P R In the corresp onding

j j i j

diagram the p oints P and P will b e connected with a dashed line A diagram

i j

with n of these internal lines will b e called a chord diagram of order n In our example

we obtain the graph

where the intersections of the dashed lines are meaningless

In physical terminology chord diagrams are just fermion lo ops with exchange of

gauge b osons These interactions are ab elian we ignore the selfinteractions of the

b osons for the moment In this intuitive physical picture the ab ove pro cedure can

b e regarded as replacing an eective fourfermion self interaction by a propagating

intermediate gauge b oson

y 2 y 2

A A

n

Clearly a knot invariant V leads to a function r on the space of chord

n

n

diagrams The question is now what are the conditions that the functions r that

come from knot invariants satisfy There are two rather obvious constraints that we

will discuss in a moment but it is a p owerful result of Kontsevich which we will

describ e in the next section that these two conditions are suciently characterize all

the Vassiliev invariants

The rst condition states that vanishes whenever there is an isolated chord

B C

B C

condition

A

This relation is easily proved by insp ection of the following gure

-

since the two terms on the RHS are simply related by a rotation of one of the two

blobs We do remark though that in the case of knot invariants that are framing

dep endent such as is the case with many invariants coming from conformal eld

theory this identity will not b e satised That is why in actual quantum eld theory

realizations we will sometimes discard condition

Condition can b e explained b est by considering a diagram with an internal

vertex of valency three

where we did not indicate all other chords that are of the usual typ e ie just con

necting two b oundary p oints Note that at present this is not a welldened chord

diagram We can now resolve this third order vertex using the rule

This resolution can now b e done in three inequivalent ways dep ending on which of

the three edges emanating from the vertex we cho ose Condition says that all

three are equivalent Diagrammatically this gives a relation with four terms

ondition c - = -

and therefore this relation has b een baptized the fourterm relation The top olog

ical pro of of this relation follows quite naturally by considering a degenerated self

1 3

intersection where three p oints P P P S are mapp ed to one p oint in R

i j k

K P K P K P

i j k

and the various resolutions in two distinct crossings see

Before we continue our discussion of knot invariants let us rst make some further

remarks ab out the space of chord diagrams We will write

M

C C

n

n=0

for the free graded mo dule of all chord diagrams factored by the relations and

That is we consider linear combinations of chord diagrams identify combinations

through the fourterm relation and put all diagrams with an isolated chords identically

to zero A weight system is now simply a map C C One should think of such a

n

map as a very sp ecial set of Feynman rules To every diagram we assign a numb er

resp ecting the relations and The b eautiful result of Kontsevich is that

C V V

n n1

n

That is every chord diagram of order n determines a Vassiliev invariant of order n

and do es so uniquely up to terms of lower order This is a very nontrivial result In

the original work of Vassiliev it was not immediately clear that no other conditions

than and would emerge by considerations of higher co dimension In particular

the space V is nitedimensional

n

It is actually more convenient to temp orarily forget ab out condition and only

restrict ourselves to condition the term relation Let

M

F F

n

n=0

denote the mo dule of chord diagrams where we only imp osed the condition Re

markably the space of such diagrams forms a Hopf algebra That is we can b oth

multiply and comultiply such diagrams The multiplication

F F F

is essentially the connected sum of the two graphs If F and F then

a n b m

F is dened as

a b n+m

a . b = a b

It is a remarkable application of condition that this connected sum is indep endent

of the p oints on the graphs and that we cho ose to connected the two The

a b

comultiplication is a map

F F F

dened as

X

a b c

bc=a

where we partition the set of chords a into two subsets b and c For example

With unit O and counit one shows that the ab ove denes a cocommutative

O

coasso ciative Hopf algebra Such simple Hopf algebras have a very canonical struc

ture They are always a free p olynomial algebra generated by the primitive elements

A primitive element x satises by denition

x x x

If we denote these elements as x x then we consequently have

1 2

F C x x x

1 2 3

The most imp ortant example of a primitive element is the generator

F x

1 1

This is typically the diagram that we should put to zero if we also imp ose condition

ie go from F to C Actually it is enough to just put x in the ring generated

1

by the primitive elements of F That is we have

C C x x

2 3

The comultiplicative structure on the space of chord diagrams is also interesting

from the p oint of Feynman rules or knot invariants Since we can comultiply diagrams

we can multiply invariants using

X

a F F

1 2 1 2 a 1 b 2 c

bc=a

Lie algebras

There are a natural set of Feynman rules coming from Lie algebras that realize

the weight systems in F Let g b e a semisimple Lie algebra We will write e for

a

an orthonormal basis for g and have structure co ecients

X

e e f e

a b abc c

c

with f fully antisymmetric Let R b e a representation R g End V We write

abc

R R e and clearly have by denition

a a

X

R R f R

a b abc c

c

If we now intro duce the Feynman rules

i j a b

ij ab

i b

a ij a

R f

abc a

j c

then equation corresp onds precisely to the relation Condition is

therefore satised

In general condition will not b e satised in these Lie algebra Feynman rules

In fact one easily computes that

c R

with c the value of the quadratic Casimir

R

X

C e e

a a

a

in the representation R We obtain real framing indep endent knot invariants from

these rules by factoring out the ideal generated by x One may wonder whether all

1

weight systems are related to classical Lie algebras Numerical calculations have

not yet lead to the opp osite conclusion

Actually the structure of Lie algebras with their cubic vertex is quite naturally

as b ecomes clear from a third description of the space F that is much closer related

to ChernSimons p erturbation theory and that also includes the cubic gauge b oson

vertex Let

M

D D

n

n=0

b e the space of chord diagrams where we also allow cubic interactions of the gauge

elds That is we have one fermion lo op a fermionb oson interaction plus the cubic

vertex We further imp ose the relation One can now show that D F

and the graphical representations of antisymmetry of the structure co ecients

and the Jacobi relation

Of course to show that Lie algebras exhaust the set of knot invariants it remains to

b e shown that the weight systems in D are actually Feynman rules ie based on

the comp osition principle where general weights are built out of elementary weights

of the individual trivalent vertices

The space D is a Hopf algebra to o In this formulation there is a simple criterion

whether an element is primitive or not a D is primitive i a is connected This

allows us to write down easily the rst three primitive elements

= -

= - - +

This shows in particular

dim C dim C

0 1

dim C dim C

2 3

We are now left with the task to present the pro of of Kontsevich that the space of

chord diagrams C is dual to the quotient space V V Hereto we have to integrate

n n n1

n

a weight system r C to a knot invariant V The idea is based on

n

n

our exp erience with conformal eld theory or if one wishes with the Hamiltonian

formulation of ChernSimons theory

The KnizhnikZamolo dchikov connection

In conformal eld theory knot invariant naturally are constructed out of braid

group representation Recall that the braid group B is generated by the elements

N

with the relations

1 N 1

ji j j

i j j i

i i+1 i i+1 i i+1

Given a braid on N strands b B we can pro duce a knot out of it by identifying N

the ends

b B K Tr b

N

Given a braid group representation

R B End V

N

we can try to dene a knot invariant by

Tr b Tr R b

This knot invariant is only welldened ie only dep ends on the knot typ e if the

trace is invariant under the second Markov move

Tr R b Tr R b b B B

N N N N +1

The rst Markov move Tr R b b Tr R b b is obvious The ab ove prop erty

1 2 2 1

is indeed satised by the braid matrices of conformal eld theory A go o d concep

tual explanation of this remarkable fact was given by Wittens pro of that the CFT

braid representations are identical to the representations that app ear in the canoni

cal quantization of ChernSimons theory which is a truly covariant threedimensional

relativistic eld theory

The CFT braid matrices are typically obtained as holonomies of a connection

the KhnizhikZamolo dchikov connection on the conguration space C of N

N

distinct unmarked p oints z z in the complex plane Let us briey recall the

1 N

origin of the KhnizhikZamolo dchikov equation in conformal eld theory

The WessZuminoWitten mo del for compact Lie group G with Lie algebra g has

chiral primary elds

z V

that carries a representation R g End V Consider the chiral correlation func

tions or holomorphic blo cks

N

hz z i V

1 N

As is wellknown these are holomorphic sections of a holomorphic vector bundle over

conguration space They have an alternative interpretation as the wavefunctions of

ChernSimons theory in the presence of external charges

We denote such a correlator or wave function by a graphical representation

. . . . .

z z

1 N

z 1 z N

The holomorphic blo cks satisfy the famous KnizhnikZamolo dchikov equation which

can b e elegantly written as

d

with

X

dz d

i

z

i

i

and the KZ connection

X

dz dz

i j

C

ij

z z

i j

i=j

Here

X

(i) (j )

C R R

ij

a a

a

(i) th N

and R denotes the action of g on the i factor in V We will write symb olically

......

C ij

i j

Let us briey consider the derivation of this equation One starts with an insertion

of the stress tensor

I

dz

hT z z z i

1 N

i

z j

then uses the OPE

h w w

R

T z w

2

z w z w

and

R w

a

J z w

a

z w

together with the Sugawara construction

X

T J J

a a

k h

a

In this way one nds

I

dz

hz z i hT z z z i

1 N 1 N

z i

z

j

j

I

X

dz

h J z J z z z i

a a 1 N

i

z

j

a

X

C

ij

hz z i

1 N

z z

i j

i=j

n n (i)

p C In the ab elian case g C all representations are onedimensional R

i

a

and the sections are given by the familiar vertex op erator correlation functions

Y

~(p p )

i j

z z

i j

ij

One of the b eautiful characteristic prop erties of the KZ connection D d is

2

that it is at for all values of More precisely one has D or

d

For the pro of of this statement it is convenient to write

X

ij

ij

with

dz dz

i j

C C d logz z

ij ij ij i j

z z

i j

no summation We clearly have by the ab ove relation d so we only have to

ij

prove Hereto we have to distinguish three separate cases

First the obvious result

ij ij

with corresp onding picture

=

i j i j

Secondly the equally obvious case

i j k l

ij k l

with a pictorial interpretation

=

i j k l i j k l

The interesting case is however

dz dz dz dz

j k i j

C C

ij j k ij j k

z z z z

i j j k

dz dz dz dz

i j j k

C

ij k

z z z z

i j j k

The imp ortant observation is now that the prefactor

X

(j )

abc (i) (k )

C f R R R

ij k

a c

b

abc

is completely antisymmetric in i j k Therefore we can use Arnolds identity

dz dz dz dz

j k i j

cy cl ic

z z z z

i j j k

to show that

cy cl ic

ij j k

Of course again we can draw a picture as symmetric as p ossible

- = j

i j k i j k

i k

From this last picture it will b e clear that the abstract KZ connection is at precisely

when we consider weight diagrams that satisfy the fourterm relation Indeed the

only thing that we needed were the relations

C C i j k l

ij k l

C C C

ij ik j k

which are the innitesimal pure braid relations of Kohno The last relation is

certainly equivalent to the fourterm relation now applied to strands instead of knots

and so will b e valid in our formalism of chord diagrams

Given the integrable KZ connection the braid representations are obtained as

follows For a closed path in s C we dene the holonomy

N

Z

1

ds b b P exp

0

by the equation

db

b

dt

We write t as an abbreviation for s st This can b e integrated p erturba

tively using Dysons formula as

Z

X

dt dt t t b

1 m 1 m

m=1

0t t 1

m

1

We will now take this denition of knot invariants and generalize it the setting of

chord diagrams More precisely if K Tr b is a knot obtained from a braid we can

dene an nth order knot invariant as

Z

n

K dt dt t t C

1 n 1 n

0t t 1

n 1

with the diagrammatic KZ connection

......

X

dz dz

i j

z z

i j

ij N

1 i j

and with the obvious graphical denition of the trace as in Equivalently

Z

n

X

dw dw

i

i

K C dt dt

P n 1 n

w w

i

i

i=1

pair ing s P

0t t 1

n

1

where we sum over all choices of pairs w w fz z g and where is the chord

i 1 N P

i

diagram that corresp onds to the pairing P Proving that this denes a knot invariant

uses of course the fact that the KZ connection is at Therefore we can smo othly

deform the braid

The ab ove denition is however not general enough since we can also have knots

that are not traces of braids Every knot class can b e written as a trace of a braid

but that is not the same In that case we cannot a priori apply the ab ove prescription

Kontsevichs formula basically takes a leap of faith and applies it anyhow To give

the formula we have to use a time direction and parametrize the spacetime p oints

3

x R as x t z R C t b eing time and z space Now our knot K we will b e

C created at time t a and disapp ear at t b ie the time slices X ft sg

s

satisfy K X s a b Now for such a time t a b the set K X will

s t

consist of a numb er of p oints We can dene with the following picture in mind

t

w ' 2 w ' w1 2 w2 w1 ' w1 w' w2 1

knot chord diagram

the p erturbative knot invariant

Z

n

X

dw dw

i

i

#

b

K dt dt C

1 n P n

w w

i

i

i=1

pair ing s P

at t b

n

1

Here indicates the numb er of downgoing strands after a choice of orientation of

the knot A new feature that we didnt have to deal with in the case of braid knots

It turns out that as it stands this is not yet a true knot invariant There is

a correction due to the fact that a Morse knot can have critical p oints and the

deformation argument that follows from the atness of the KZ connection can never

change the numb er of critical p oints This eect is already observed for the gure

eight unknot which gives the expression

b

This gives a corrective factor for each critical p oint So the nal formula for the

Universal Vassiliev Invariant of the knot K with c critical p oints reads

b

K

C K

n c

1

b

2

This is a universal invariant in the sense that it takes value in the mo dule of chord

diagrams We can thus pair it with a weight system C to pro duce a numb er

n

h k i C

We will not b e in a p osition to pro of here in full detail the fact that this invariant

is welldened and indeed indep endent of smo oth deformations see eg It is also

clear that many questions ab ound How to generalize these invariants along the lines

of the ChernSimonsWitten theory How to describ e nonp erturbative eects Are

Lie algebras or even just the classical Lie algebras sucient to exhaust the Vassiliev

invariants if anything I hop e to have given the reader at least the idea that these

questions might not b e as hop eless as one might think at rst sight

References

E Witten ChernSimons gauge theory as a string theory IASSNSHEP

hepth

E Witten Quantum eld theory and the Jones polynomial Commun Math

Phys

E Guadagnini M Martellini M Mintchev Perturbative aspects of the Chern

Simons eld theory Phys Lett B

E Guadagnini M Martellini M Mintchev Link invariants from ChernSimons

theory NuclPhysB Pro c Suppl B

Wei Chen GW Semeno YongShi Wu Probing topological features in pertur

bative ChernSimons gauge theory Mo d Phys Lett A

L AlvarezGaume JMF Labastida AV Ramallo A note on perturbative

ChernSimons theory Nucl Phys B

D BarNatan E Witten Perturbative expansion of ChernSimons theory with

noncompact gauge group Commun Math Phys

G Leibbrandt CP Martin Perturbative ChernSimons theory in the light cone

gauge Nucl Phys B

JFWH van de Wetering Knot invariants and universal R matrices from per

turbative ChernSimons theory in the almost axial gauge Nucl Phys B

S Axelro d IM Singer ChernSimons perturbation theory I II Pro c XXth

DGM Conference New York S Catto and A Ro cha eds World Scien

tic and MIT preprint hepth

M Kontsevich Feynman diagrams and lowdimensional topology and Graphs

homotopical algebra and lowdimensional topology Bonn preprints

VA Vassiliev Cohomology of knot spaces in Theory of Singularities and its

Applications Ed VI Arnold Adv Sov Math Complements of

discriminants of smooth maps topology and applications Translations of Math

ematical Monographs Amer Math So c Providence

D BarNatan On Vassiliev knot invariants Harvard Preprint to app ear

in Top ology

M Kontsevich Vassilievs knot invariants Preprint to app ear in Adv Sov

Math

T Lada J Stashe Introduction to sh Lie algebras for physicists

IntJTheorPhys hepth

T Kimura J Stashe AA Voronov On operad structures of moduli spaces and

string theory hepth

E Getzler and JDS Jones Operads homotopy algebra and iterated integrals

for double loop spaces hepth

I A Batalin and G A Vilkovisky Phys Rev D

M Henneaux and C Teitelb oim Quantization of Gauge Systems Princeton

University Press Princeton New Jersey

E Getzler BatalinVilkovisky algebras and twodimensional topological eld the

ories CommunMathPhys

JS Birman and XS Lin Knot polynomials and Vassilievs invariants Inv

Math

S Piunikhin Combinatorial expression for universal Vassiliev link invariant

Harvard preprint hepth

Le Tu Quo c Thang and Jun Murakami The universal VassilievKontsevich in

variant for framed oriented links MPI fur Mathematik preprint MPI

hepth

D Altschuler and L Freidel On Universal Vassiliev Invariants Preprint

ENSLAPPL ETHTH hepth

JH Conway An enumeration of knots and links and some of their algebraic

properties in Computation Problems in Abstract Algebra Pergamon New

York

VFR Jones A polynomial invariant for knots via von Neumann algebras Bull

Amer Math so c

J Hoste A Ocneanu K Millett P Freyd WBR Lickorish and D Yetter A

new polynomial invariant of knots and links Bull Amer Math So c

VG Knizhnik and AB Zamolo dchikov Current algebra and WessZumino

model in two dimensions Nucl Phys B

T Kohno Monodromy representations of braid groups and YangBaxter equa

tions Ann Inst Fourier

VI Arnold The cohomology ring of the colored braid group Mat Zametki