<<

. 4
( 9)



>>


+ ˜n(SJ ow-1n1,0 8)
= Sl,O
'\71,0 s s
4
E rp(A,£) and with the first D acting as a map T®r L2(A,L:) r L2(A,L:).)
for --+
$



One may check that [jjJ, V] = 0 on holomorphic sections, where DJ is the [} operator
of the holomorphic bundle £, induced by J, i.e., DJ _= ˜Ie dzk ˜k = t(l + iJ)D.
The trivial connection 8 does not commute with DJ, but neither does ", and
the two contributions cancel. The contribution from 0 arises because one meets
[˜, ˜i]' which is just the curvature of £', i.e., -iw. Both commutators are first
order differential operators.

(b): The torus

We will now describe several variations of this construction, in increasing order of
relevance. For our first example, we consider the quantization of a torus. (This
precise situa.tion a.ctually arises in the Chern-Simons theory for the gauge group
U(l).)
Witten: New results in Chern-Simons theory 79


We pick a lattice A of maximal dimension 2n in the group of affine translations of
A, integral in the sense that the action of A on A can be lifted to an action on £,
and we pick such a lift. The quotient of A by A is a torus T, which we wish to
quantize. Using the A action on L, we can push down L, to a line bundle over T
which we will also call £.

If a complex structure J is picked on A as before (compatible with wand giving
a metric on A) then the complex structure on A descends to a complex structure
on T, which thus becomes a complex torus and indeed an Abelian variety TJ. (An
Abelian variety is precisely a complex torus which admits a line bundle with the
properties of £.) The holomorphic sections 'HA,J = HO(TJ,L,A) = ('HJ)A form a
vector bundle over T as before. HO(TJ' £A) is a space of theta functions, of some
polarisation depending on A. The A invariant subspace of the bundle 'H. over 7 -
whose fiber is given by 'HA,J - will be denoted as ?-lAo

Because the action of A commutes with the connection V, V' restricts to a con-
nection on the subbundle 'HAl In this case the connection can be made flat (not
just projectively flat) by tensoring ?-f.A with a suitable line bundle with connection
over T; the theta functions as conventionally defined by the classical formulas are
covariant constant sections of 1-lA. The condition of being covariant constant is the
"heat equation" obeyed by the theta functions, which is first order in the complex
f;tructure J and second order in the variables z along the torus. This is thus the
origin, from the point of view of symplectic geometry, of part of the classical theory
of theta functions I




(c): The symplectic quotient

Somewhat closer to our interests is a situation in which the lattice A is replaced
hy a compact group 9 acting linearly and symplectically on A, with a chosen lift
of the Q action to an action on £ preserving D. In this situation, we restrict the
˜eneral discussion of quantization of affine spaces to the Q invariant subspaces.
Thus, we let Tg be the subspace of T consisting of g -invariant complex structures.
()ver 7g, we form the Hilbert space bundle 11,cJ whose fibre over J is the g invariant
˜ubspace ('HJ)Q of 'H J • Restricting to 7g and to ('H.J)Q, equation (5) gives a natural
connection on 'H Q , which of course is still projectively fiat.

III geometric invariant theory, once one picks a complex structure J, the symplectic
action of the compact group 9 on A may be extended to an action of the complex-
jfication Qc, which depends on J. (The vector fields generated by the imaginary
part of the Lie algebra are orthogonal to the level sets of the moment map, in the
rlletric determined by the complex structure and the symplectic form.) As a 8ym-
11lectic manifold, A.T IQc is independent of J; it ITolay be identified with AI/Q, the
Witten: New results in Chern-Simons theory
80


symplectic or Marsden-Weinstein quotient of A. This is defined as P- 1 (O)IQ, where
F : A ˜ Lie (Q)'II is the moment map for the Q -action_ However, AI/Q acquires a
complex structure from its identification as AJ/Qc. The line bundle £, over p-1(O)
pushes down to a unitary line bundle with connection lover AI/Q; this line bundle
is holomorphic in the complex structure that AI/g gets from its identification with
AJ Iyc (for any J). The g invariant space ('H.J)Q considered in the last paragraph
can be identified with 'H.J = HO(AJ /Qc, l). This identification is very natural from
the point of view of geometric invariant theory. The 'H.J sit inside the fixed Hilbert
space 'H.o = r L2(A/ /Q, l). The connection described in (5), restricted to the 9 in-
variant subspace and pushed down to an intrinsic expression on A/ /Q, is still given
by a second order differential equation, with the same leading symbol but more
complicated lower order terms.

(d): The moduli space of representations

Suppose G is a compact Lie group with Lie algebra 9, and :E an oriented compact
surface without boundary. We fix a principal G- bundle P ˜ ˜; A, the space of
connections on P, is an affine space modelled on nl(˜,ad(P)). The gauge group g
is nO(˜, Ad(P)), and acts on A by dA ˜ 9 dA g-1.

If a,(3 E nl(˜, ad(P)), we may form a pairing

h(0: /\,8)

{ 0:, ,8}' H ,


where we have used a G-equivariant inner product (.,-) on Lie(G). This skew-
symmetric form on nl(˜,a.d(P)) defines a natural symplectic structure on A. The
normalisation of the inner product is chosen as follows. If F is the curvature of
the universal G-bundle EG ˜ BG, then an invariant inner product (-,.) on Lie(G)
defines a class (F,AP) E H4(BG). If G is,8imple, all invariant inner products are
related by scalar multiplication; we choose the ba8ic inner product t·o be the one
such that (F,AF)/(21r) is a generator of H 4 (BG,Z) = I. (It has the property that
(0:,0:) = 2 where 0: is the highest root.)

We take this to define the basic symplectic form Wo on A; it is integral in that it
may be obtained as the curvature of a line bundle over A- The action of G on A
preserves the symplectic structure. (This situation was extensively treated in [1].)
In general, we pick an integer k (which, as it turns out, corresponds to the "level" in
the theory of representa.tions of loop groups) and consider the symplectic structure
w = kwo.

If A E A is a connection, its curvature is FA = dA + A A A E n2(˜, ad(P)). Now
the Lie algebra of the gauge group is nO(˜,ad(P)), so under the pairing given by
the symplectic form, the dual ˜ie(g)* = ffl(˜,ad(P)). The moment map for the g
Witten: New results in Chern-Simons theory 81


u.ction is A FA: thus the symplectic quotient is
t-+



AI/9 = {A: FA = 0}/9 (6)
M=
= Rep(1rl(˜) --+ G)/conjugation

M does not have a natural complex structure, but a complex structure can be
picked as follows. Any choice of complex structure J on ˜ decomposes TeA into:

°A = nO, 1(˜, g)
T 1, (7)
TO' 1A = n1, O(˜, g)

(This conventional but seemingly inverted choice is made so that the operator 8A will
be a map Lie(Qc) ˜ T 1 ,° or equivalently so that the moduli space of holomorphic
A
hundles varies holomorphically with the complex structure on ˜.) This complex
Htructure on A is compatible with the symplectic form, as the symplectic form
Ilaturally pairs (0,1)- forms on ˜ with (l,O).. forms.

'rhe complex structure that M gets in this way has a very natural interpretation.
According to the Narasimhan-Seshadri theorem (which was interpreted by Atiyah
",ud Batt as an analogue for the infinite dimensional affine space of connections of
(Ionsiderations that we sketched earlier for symplectic quotients of finite dimensional
tt.ffine spaces), once a complex structure J is picked on ˜, M has a natural identin-
(˜8,tion with the moduli space of holomorphic principal Qc bundles on ˜. The latter
has an evident -complex structure, and this is the complex structure that M gets
hy pushing down the complex structure (7) on the space of connections.

lu fact the complex structure on M depends on the complex structure J on ˜ only
up to isotopy. (The interpretation of M as a moduli space of representations of the
fundamental group shows that diffeomorphisms isotopic to the identity act trivially
enl M; and the Hodge decomposition that gives the complex structure of M is
likewise invariant under isotopy.) Thus, we actually obtain a family of complex
Rtructures on M parametrized by Teichmuller space, which we will denote as T.
So in this case we will construct a projectively flat connection on a bundle over
'TI Just as for symplectic quotients of finite dimensional symplectic manifolds, the
fonnection form will be a second order differential operator.


2. THE GAUGE THEORY CASE

III this lecture, we will give more detail about the preceding discussion. To begin
with, we will describe more precisely how to push down the basic formula (5) for
t.he connection that arises in quantizing an affine space, to an analogous formula
(It˜scribing the quantization of a symplectic quotient.
I
82 Witten: New results in Chern-Simons theory


The Q action on A and the complex structure on the latter give natural maps
˜˜˜


˜
T(P-l(O))
Lie(Q)
T --+

Lie(Qc) ˜ TcA
Lie(Qc) ˜ T1,OA
I
TO, 1A
Lie(Qc) --+
:1
'.;˜
If we introduce an invariant metric on Lie(Q) and take its extension to Lie(Qc)
(note A already has an invariant metric from the symplectic form), we may form an 'l
operator Tz- 1 : TI,OA -+ Lie(Qc), which is zero on the orthogonal complement to '.'.˜
Im(T;e) in Tl,OA, and maps into the orthogonal complement of Ker(Tz ). We may ·It


also form K: , the operator that projects TI, 0A onto TI,O M, the orthocomplement .•˜
;g
of Im(Tz ) in TI, 0A. (One sees this because

˜
TA = Lie(g) EB JLie(Q) EB T M,
˜.˜:J
1
since the codimension of F- (O) in A is then dim Q and JLie(g) is orthogonal to
TF-l(O). Thus TeA =TcM EB Lie(Qc) and the projection onto (1,0) parts preserves
:1
this decomposition, so also T1, 0A = T1, 0M $ Im(Tz ).) From this, id = /C +TzTz 1
-
on T1,OA. "˜
.:0;-:




J
If X is a vector field in the image of T, and s is a Q invariant section, one has
Dxs = iFxs. This condition, which determines the derivatives of s in the gauge '˜
directions, permits one to express the connection form 0 = -(1/4)Di 8Ji j D j acting ;t˜

on g -invariant holomorphic sections over F-l(O), in a manner that only involves ;˜

derivatives along the T 1 , 0M directions and has the other directions projected out. j
The result is: i ;.
i
1 .. 1··
o k
+
--D(/(, 8 )8J˜3 D(/C 8 ) .'˜
-SJtJ(D·/C ·)D}( 8
= -
::˜
4 4
87 -;;; a;;;)
3 (
t


+ ˜Tr(Tz-lc5J TJ ) ˜
(8)

where the indices denote bases for T I , 0A or TI , OM. It turns out that this formula I.".
can be expressed in a way that depends only on the intrinsic Kahler geometry ofl
AJ /Qc and the f u n c t i o n ; ˜
/11
H = det'6., .˜˜
),

'1
where the operator L : Lie(Qc) Lie(Qc) is defined by
--+

= T/Tz = ˜T/Tc. ˜
D. .˜




(Here, "det" "denotes the product of the nonzero eigenvalues.) The final expression
for the connection is:
\71,0 = 61,0 - 0
83
Witten: New results in Chern-Simons theory


= ˜o, 1
\70,1 (9)
o = -˜{Di6.rjDj+c5Jij(DdOgH)Dj}

where now the indices represent a basis of T1,0 M.

'I'he appearance of log H has a natural explanation: this object appears in the
(·xpression for the curvature of the canonical bundle of M . Assume for simplicity
Ker Tz = 0. Then

HO




(AmaxLie(Qe))* ® (AmaxT1,OA)
˜ (AmaXLie(Qc))* ® (AmaxImTz ) ˜ (AmaxT1,0 M)

(AmaxLie(Qc))*®(AmaxImTz ) is isomorphic to the trivial bundle, with the norm
Now
det Tz *Tz • If Q acts trivially on AmaxTl,OA, then a "constant" section of the line
I.undle on the left hand side is Qc invariant and gives under this isomorphism a
H('ction s of (AmaxT1, 0 M)* It then has norm canst · det(Tz "'Tz ). In other words the
/l.icci tensor ( the curvature of the dual of the canonical bundle IC M ) is

= -8810gH
R (10)

where H = det(Tz "'Tz ).

We' now consider the gauge theory case, in which one is trying to quantize the finite
clilnensional symplectic quotient M of an infinite dimensional affine space A, with
t.llc symplectic structure w = kwo. Since we do not have a satisfactory theory of
t.1t˜ quantization of the infinite dimensional affine space A which could a priori be
pushed down to a quantization of M, we simply take the final formula (9) that
nrises in the finite dimensional case and attempt to adapt it by hand to the gauge
t.heory problem. The Kahler geometry of the quotient M exists in this situation,
jllHt as it would in a finite dimensional case, according to the Narasimhan-Seshadri
t.tlf-'orem. Also, the pushing down of a trivial line bundle £ on A to a line bundle l
•• 11 M, which is holomorphic in each of the complex structures of M, can be carried

out. rigorously even in this infinite dimensional setting. We will explain this point
ill some detaiL Start with a trivial line bundle £, = A x c. We will describe a lift
t.f the 9 action on A to £; the required line bundle lover M is then the quotient
•• 1' [, under this action. Actually, we will lift not just g, but its semidirect product
with the mapping class group, in order to ensure that the action of the mapping
I
84 Witten: New results in Chern-Simons theory


I
E.
class group on M lifts to an action on We have exact sequences

J
r
o˜ Q Aut(P) 0
---t 0




1
II U U
o˜j
˜O
Q
o˜ fa
Auto(P)
]
where r are the diffeomorphisms of ˜, and r o those isotopic to the identity. Thus .˜
the mapping class group r Ir o ˜ Aut(P)jAuto(P). The action of any automorphism 1',˜
X E Aut(P) covering ¢ E r enables one to form a bundle over the mapping torus
˜ xq, [0,1] by gluing the bundle P using x. Given a connection A on P , one
forms a connection on this new bundle by interpolating between x*A and A. The ˜
Chern-Simon8 invariant of this connection is an element of U(l); thus one gets a j
map A x Aut(P) --+ U(l), and one may use the U(l) factor as a multiplier on ;1
I:. = A x C to lift the action of Aut(P) on A. (Restricting to 9 C Aut(P), one ˜
,I
obtains the moment map multiplier that is use? to lift the 9 action to £.) Thus
the mapping class group action on M lifts to I:. . "
;1˜
The only additional ingredient that we must define in order to adapt (9) to this
i
situation is the determinant H. In this case the map T z : Lie(Qc) ˜ T1, 0A is
I
˜
˜J ..


so
1
n CE,gc) --+ n°(˜,gc)
1,..
6 = 8A *-A = 2dA dA
- 0 ;,
8 :


!
is the Laplacian on the Riemann surface ˜, twisted by the connection A. Following

Ray and Singer, the determinant of the Laplacian can be defined by zeta functional
regularization. With this choice, we can use the formula (9) to define a connection on .;!˜
the bundle over Teichmiiller space whose fiber is HO(MJ,L). However, in contrast ;˜
to the case of quantization of the symplectic quotient of a finite dimensional affine ;j
space, in which one knows a priori that this connection form commutes with the iJ
'i
operator and is projectively flat, in the gauge theory problem we must verify these
properties. ˜˜
10˜



In verifying the projective flatness of the connection (9), or more precisely of a :oj
.f.
slightly modified version thereof, the main ingredients required, apart from general ;!
facts about Kahler geometry, are formulas for the derivatives of H which are con- ˜
.;
sequences of Quillen's local families index theorem. One important consequence of ·1
o
this theorem is the formula

+ Balog H = 2h( -iwo) = -2ih˜, (11)
R

where R is the Ricci tensor of M, h is the dual Coxeter number of the gauge group,
and Wo is the fundamental quantizable symplectic form on M.
85
Witten: New results in Chern-Simons theory


rrhe term proportional to h, which is absent in the analogous finite dimensional
formula (10), is what physicists would call an "anomaly"; it is, indeed, a somewhat
disguised form of the original Adler-Bell-Jackiw gauge theory anomaly, or more
('xactly of its two dimensional counterpart. Because of this term, when one tries
t.o verify the desired properties of the connection, one runs into trouble, and it is
1Iecessary to modify the formula (9) in a slightly ad hoc way. The modified formula





V 1,0 (12)
V O,l

where the new formula for () is the same as the old one (9). The identity (11) enters
crucially into the proof that this connection satisfies [DJ,V] = 0 on holomorphic
H(Octions and thus preserves holomorphicity.

'I'he formula (11) corresponds to the local index theorem formula for the (M,M)
('()lnponent of the curvature of the determinant line bundle, which is, however,
(I(Ofined over M x T. The local index formula completely determines the curvature:
r xplicitly,




'I'he R terms represent the curvature the determinant bundle would have had for the
cH'iginal metric without Quillen's correction factor H; 8,8 now refer to M x T. The
lrft. hand side is the local index, and the right .hand side the curvature computed
fl'()m the Quillen metric. This identity enters in determining the curvature of the
"ounection (12) over T. One finds that the (1,1) part of the curvature is central,
wit.h the explicit formula being


= 2(k k h) Cl (IndT˜).
11
(14)
+
R'

'('he (0,2) component of the curvature is trivially zero since V O,l = 6°,1, but to
Nhow that the (2,0) component is zero using techniques of the sort I have been
f11«·t.ching requires a great deal of work. (There is also a simple global argument, of
n very different flavor, which was explained in Hitchin's lecture.)


'I'liis is in contrast to the finite-dimensional case, where the vanishing of the (2,0)
(olnponent of the curvature would follow simply from the fact that the bundle 'H.
hH˜ a unitary 8tructure that is preserved by \7 (i.e. V is the unique connection
pr˜'serving the metric and the holomorphic structure on 'H). In the gauge theory
(·as{˜, we do not have such a unitary structure rigorously. Formally, there is such a
86 Witten: New results in Chern-Simons theory


unitary structure: for"¢ E 'Ji J = HO(M, c')J, we pull ;p up to a Qc -invariant section
"p E (HO)(A, £)J and define




where df-L is the formal "symplectic volume" on A. (This is the formal analogue
of the unitary structure in the case of a finite dimensional symplectic quotient.) In
the finite dimensional case, one would integrate over the We orbits to get a measure ,
on M rather than on A. In the gauge theory case, Gawedzki and Kupiainen [5]
have shown that it is miraculously possible to do this explicitly (though not quite
rigorously) for the case when E is a torus; the result is




where as before H = det Tz *Tz is a factor from the "volume"of the gauge group
orbit.

Their construction does not generalise to other ˜, for it uses the fact that 1rl (˜) is
abelian. However, one may construct an asymptotic expansion




where bo = 1 and the higher bk's are functions on M x T that can be computed by a
perturbative evaluation of the integral over the We orbits. (The required techniques
are similar to the methods that we will briefly indicated in the next lecture.) If
one can establish unitarity to some order in 11k, then obviously (\7 2 )2,0 vanishes to '0,'

the same order. But it is actually possible to show that unitarity to order 1/k 2 is 1




enough to imply that the (2,0) curvature vanishes exactly. This is an interesting
approach to proving that statement.

At this point, we can enjoy the fruits of our labors. The monodromy of the pro-
jectively flat connection that we have constructed gives a projective representation
of the mapping class group. The representations so obtained are genus 9 analogues '
of the Jones representations of the braid group. The original Jones representations
arise on setting 9 = 0 and generalizing the discussion to include marked points; the
details of the latter have not been worked out rigorously from the point of view
sketched here.

3. LINK INVARIANTS

In this lecture we aim to describe more concretely the way in which the theory
constructed in the first two lectures gives rise to link invariants. Also, I want to tell
87
Witten: New results in Chern-Simons theory


you a little bit about how physicists actually think about the subject. First we shall
put the theory in context. Symplectic manifolds arise in physics in a standard way,
as phase spaces. For example, the trajectories of the classical mechanics problem

. 8V
(i=l, ... ,n)
zt=--
8x i
x at t= o. The above equation is the equation
are determined by the values of x,
for the critical trajectories of the Lagrangian


(15)

or the equivalent Lagrangian

(16)

in which the momentump (which equals ˜ for classical orbits) has been introduced
n˜ an independent variable. Classical phase space is by definition the space of
classical solutions of the equations of motion. In this case, a classical solution is
determined by the initial values of x and i or in other words of x and p; so we
("a.n think of the phase space as the symplectic manifold R2n with the symplectic
form w = dp 1\ dx. The space of critical points of such a time dependent variational
problem always has such a symplectic structure.

moduli space M is no exception. Consider an oriented three-manifold M.
()ur
Let G be a compact Lie group, P the trivial G bundle on M and A a connection
on P. The Chern·Simons invariant of A is

(17)

(It. may also be defined as the integral of Pl(FA ) over a bounding four manifold over
which A has been extended.) The condition for critical points of this functional is
o = FA = dA + A /\ A. I is used as the Lagrangian of a quantum field theory whose
fit·lds are the connections A. There are two standard ingredients in understanding
Milch a theory:

(n): Canonical quantization

separate out a "time" direction by considering the manifold M = ˜ x R; then

w(" consider the moduli space of critical points- of the Chern-Simons functional for

thi˜ M. It is the space of equivalence classes of flat connections under gauge trans-
(orInations, or of conjugacy classes of representations:

Rep(7rl(˜ x R), G)/conj = Rep(1rl(˜)' G)/conj
M=
88 Witten: New results in Chern-Simons theory


In other words, we recover our earlier moduli space, but we have a new interpretation
of the rationale for considering it: it is the phase space of critical points arising in
a three dimensional variational problem. This change in point of view about where
M comes from is the germ of the understanding of the three dimensional invariance
of the Jones polynomial.

After constructing the classical phase space associated with some Lagrangian, the
next task is to quantize it. Quantization means roughly passing from the symplectic
manifold to a quantum vector space of "functions in half the variables" on the
manifold (holomorphic sections of l ˜ M in the Chern- Simons theory; "wave ,
functions" t/J(p), or 1/J(:c), or holomorphic functions on en for some identification of
en with R2n , for classical mechanics). Quantization of the phase space M of the
Chern-Simons theory is precisely the problem that we have been discussing in the
first two lectures.

r
Often one is interested in some group of symplectic transformations of the classi-
cal phase space (the mapping class group for Chern-Simons; the symplectic group I



Sp(2n; IR) for classical mechanics). Under favorable conditions, quantization will
then give rise to a projective representation of this group on the quantum vec-
tor space. (The classical mechanics version is the metaplectic representation of
Sp(2n; R).)

The Chern-Simons function does not depend on a metric on M (or a complex struc-
ture on E): thus the association of a vector space H˜ to a surface ˜ by quantization
is purely topological, although in order to specify 'H.r, we need to introduce a com-
plex structure on ˜, as we have seen. This is analogous to trying to define the
topological invariant H1(˜, C) by picking J and identifying H 1J(E, e) as the space
of meromorphic differential forms with zero residues modulo exact forms. Here,
the analogue of our projectively flat connection is the Gauss-Manin connection: it
enables one to identify the H1 J for different J, so that one recovers the topological
invariance though it is not obvious in the definition. Of course there are other,
more manifestly invariant ways to define Hl(˜, e)! For Chern-Simons theory with
nonabelian gauge group, however, we do not at present know any other definition. I




(b): The Feynman integral approach

In canonical quantization, after constructing the "physical Hilbert space" of a the-
ory, one wishes to compute the "transition amplitudes," and for this purpose the
Feynman path integral is the most general tool. It is here that - in the case of the
Chern-Simons theory - the three dimensionality will come into play.

˜
We work over the space W of connections A on P M (M being of course a
89
Witten: New results in Chern-Simons theory


three manifold). The Feynman path integral is the "integral over the space of all
connections modulo gauge transformations"

= fw VAexp(ikI) (18)
Z(M)

Here I is the Chern-Simons invariant of the connection A, and k is required to be
an integer since I is gauge invariant only modulo 27r. (The comparison of the path
integral and Hamiltonian approaches shows that the path integral (18) is related to
quantization with the symplectic structure w = kwo.) The path integral (18), which
InRy at first come as a surprise to mathematicians but which is a very familiar sort
of object to physicists, is the basic three manifold invariant in the Chern..Simons
theory. To physicists, Z(M) is known as the "partition function" of the theory.
More generally, a physicist wishes to compute "expectation values of observables."
'These correspond to more general path integrals

= fw VAexp(ikI)O(A)
ZO(M) (19)

where O(A) is a suitable functional of the connection A.

The functional that is important in defining link invariants is the 'Wilson line'{which
was introduced in the theory of strong interactions to treat quark confinement). If
(,' is a loop in M, and R a representation of G ,we define

(20)

where HoI denotes the holonomy of the connection A around the loop C. A link
if) M is a collection of such loops Ci ; we define a link invariant by assigning a
I"t!presentation ("colouring") R i of G to each loop Ci, and taking the product
IL ORi(Ci ). This is precisely the situation considered in R. Kirby's lecture at this
fonference, for G = SU(2), and that is not accidental; the invariants he described
".re the ones obtained from the Feynman path integral, as we will discuss in more
detail later.

III particular, the original Jones polynomial arises in this framework if one takes
M = S3 and one labels all links with the 2-dimensional representation of SU(2);
t.lle HOMFLY polynomial arises from the N dimensional representation of SU(N).
(H.her representations yield generalised link invariants that have been obtained by
('( Hlsidering quantum groups; however, the Chern-Simons construction gives a man-
i f(:tlstly three-dimensional explanation for their origin. These invariants are link
Upolynomials" in the sense that for M = 8 3 (but not arbitrary M) they can be
,;hown, at least in the HOMFLY case, to be Laurent polynomials in

q = exp{21T"i/(h + k)}.
˜i

˜
90 Witten: New results in Chern-Simons theory
˜˜
OJ


The Chern-Simons quantum field theory that we are discussing here is atypical in

that it is exactly soluble. The arguments that give the exact solution (such as the "
j
rigorous treatment of the canonical quantization sketched in the last lecture) are

somewhat atypical of what one is usually able to do in quantum field theory. To
J
gain some intuition about what Feynman path integrals mean, it is essential to
:t
attempt a direct assault using general methods that are applicable regardless of the
˜
choice of a particular Lagrangian. The most basic such method is the construction
of an asymptotic expansion for small values of the "coupling constant˜'l/k. In the ˜;
'I
Chern-Simons problem, even though it is exactly soluble by other methods, the
asymptotic expansion gives results that are significant in their own right. ;1


˜˜
To understand the construction of the asymptotic expansion, we consider first an !:i˜




˜
analogous problem for finite dimensional integrals. To evaluate the integral

Jexp(ikf(z))d"'z ˜˜

.1
for large k, one observes that the integrand will oscillate wildly and thus contribute .˜
very little, except near critical points Pi (in the sense of Morse theory) where dl(Pi) = ':J ill

O. The leading order contribution from such points is what we get by approximating 1
f by a quadratic function near Pi and performing the Gaussian integral (suitably ':˜!
regularized):
˜
{"kf( .)}exp{˜signHpif} ]
n/2'"
(21)
Id H 111/2
L.t exp 1, P",
1r
et
Pi Pi .1:
;l
where Hpif is the Hessian.

˜

::i
If we assume there are only finitely m&.ny gauge equivalence classes A a of fiat con-

nections, the analogous expression in Chern..Simons gauge theory is:
;1
2: eiklcs{Aa) [TRRS (A ˜'l
Z(M) = )]l/2 • ei1rfJ (O)/2 • eihlcs{Aa} (22) j:(,
a
Q .\.

.:˜

Here, TRRS is the Reidemeister..Ray.. Singer torsion [11] of the flat connection A a : ˜.˜
it is a ratio of regularised determinants of Laplacians of dA , and results from the j
formal analogue of the determinant of the Hessian. 11(0) is the eta invariant of 'J
the trivial connection: the eta invariant is a way to regularise the signature of a
self-adjoint operator that is not positive. h is, as before, the dual Coxeter number.
I

At each A a , this leading term is multiplied by an asymptotic expansion :˜


.]
f:'bn˜:).
1+
n=l


(In quantum field theory, such asymptotic expansions are usually not convergent.)
Each bn { a) is a topological invariant, capturing global information about M. The
91
Witten: New results in Chern-Simons theory


bn ( a) are constructed from Green's functions, which are integral kernels giving the
formal inverses of operators such as *d A : Ol(M,ad(P)) ˜ 01(M,ad(P)).

One may also expand the integral for ZO(M) by this method. The leading term
is the Gauss linking number: for G = U(l), links indexed by a and representations
indexed by integers n a , this is

-))
( x-y
t. _ _ (-
˜ ˜ dy x ,- ˜3
I: nan" J˜ECo. dz· Ji/ECL (23)
exp 2k
Y
a,b X-


I-Tigher order terms in the asymptotic expansions give multiple integrals of the Gauss
linking number. If the stabiliser of a flat connection A in the gauge group has virtual
k m / 2 : for instance one sees this behaviour
dimension m, one gets a contribution "V


ill the explicit formula for Z(S3) for G = SU(2), which is

J2
Z(S 3) = k- 3 / 2
· (24)
s1n _1r_ rv
k+2 k+2 ·

S3, since it has a
'("'he exponent reflects the fact that the trivial connection on
three dimensional stabilizer, corresponds to a component of the moduli space of flat
(Oounections of virtual dimension -3.

(c): Putting these approaches together

We now discuss how to fit together the path integral and quantization approaches.
Ir M is a manifold with boundary E, let AI: be a G connection on ˜. Define a
functional \}1 on connections AI: by integrating over those connections on M that
rot'strict to AI::
t
w(A ) = L.:AI1J=A1J VA exp{ikI(A)} (25)

Il(Ocause of the behavior under gauge transformations on E, this integral does not
.If-fine a function on the space of connections but a section of a line bundle. In fact,
it. defines a Q(E) invariant section '1t(AI:) E r(A,£k). In fact this is a holomorphic
˜rrtion, i.e., '11(At) E HE. (Holomorphicity may be formally proved by writing the
JHl.t.h integral in the form




some section tP, with ˜8 the 8 Laplacian on M. As T ˜ 00 this Laplacian
for
(.I tviously projects on holomorphic sections. See [10] or (4] for a derivation.) As l}1
I˜, a holomorphic section over M , it corresponds to a Qc invariant section of £ over
T·1.
92 Witten: New results in Chern-Simons theory
J
˜
To calculate the invariants ZO(M) , we split the three-˜anifold M into two pieces ';˜
M L , M R with a common boundary li. Then we split the path integral into path :I˜
integrals ov˜r connections on the two halves: :˜-


JVAexp{ikI(A)}
=
Z(M) ;

r r
A A
= !>A1: VALexp{ikI(A L )}·
(1:)
JA!}E JALE (ML)

·JARE A (MR) 1)AR exp{ikI(AR )}
r
f
JVA!:
= '1I L *(AI:) '1IR(AI:)

:i
(Since the boundary f; of M L has opposite orientation to that, li, of M R, '1I L *(AI:)
is an antiholomorphic section of I ˜ M.) The vector spaces 'HI:, and 'H.˜, being ˜
1
spaces of holomorphic and antiholomorphic sections of f. , are canonically dual; the
J
integral over AI: formally defines this pairing, i.e.,

-i
j
(This is similar to the way invariants are constructed in the topological quantum
field theory for Floer-Donaldson theory. The difference between these situations
'I
i
is that the vector spaces in the Chern-Simons theory have a unitary structure,
unlike those in the Floer-Donaldson theory. This reflects the more truly quantum- -:1-
˜.!•
•˜.I'


mechanical nature of the Chern-Simons theory.)




= EM(u)i{tPj.
M(U)1/Ji
j
93
Witten: New results in Chern-Simons theory


For instance, for


we have
M(T)ij = bij exp 21ri(hi
c/24), -

where the hi are certain quadratic expressions in i, the conformal weightJ of the
representations indexed by i, and the central charge c is a constant depending on k
and G. For


we have

Jk: 2 sin(k: 2) [ij]
M(S)ij =
(for G = SU(2)), in the notation used by Kirby. This formula can be obtained by
explicitly integrating the flat connection that we constructed in the second lecture.
(1 t was originally obtained, with different physical and mathematical interpreta-
tions, from the Weyl-Kac character formula for loop groups.) As is well known, the
(˜lements Sand T generate S£(2,1), so the above formulas determine the represen-
t.ation.

We will now explain a formula describing how the quantum field theory behaves
under surgery. We wish to comput.e the invariant Z[M; [L, {k}]) for a three manifold
M containing a link L labeled by representations {k}. Let G be a knot in a three
tllanifold M, disjoint from L. Let Mn be a tubular neighborhood of C (also disjoint
from L), and M L the complement of M R in M. The path integral on M L or M R
determines a vector q,L or q,R in the Hilbert space associated with quantization
of˜. The element WR is the path integral over the tubular neighbourhood with
II 0 Wilson lines, i.e., with the trivial representation labelling C: WR = "pt. So the

(tuantum field theory invariant is



Surgery on C corresponds to the action of some u in the mapping class group
."1 L(2, Z). We act on E R by u and glue M R back to form a new manifold M u : in
f,crorms of the representation of the mapping class group this says
= (WL,M(u)1Pl)
Z(Mu;[L,{k}]) = LM(u)lj(WL,,,pj).

II) other words, for the purpose of evaluating link invariants we may replace the
:---llrgery curve C by a Wilson line along C and sum over representation labels on C
w(-"i˜hted by the representation of the mapping class group:

= L M(U)ljZ(M; [L, {k}], [C,j)).
Z(Mu ; [L, {k})
j
Witten: New results in Chern-Simons theory
94


(Here Z(M; [L, {k}, [C,j]) is the path integral for the three manifold M containing
the link L labeled by {k} and an additional circle C labeled by j.) One thus reduces
to simpler manifolds (and eventually to 53) by adding more links; so eventually one
obtains the invariant of any manifold M as a sum over "colourings" of links in 53.
This is the formula that was to be explained.



References
[1] M.F. Atiyah and R. Bott, The Yang-Mills EquationJ over Riemann Surface.s,
Phil. Trans. R. Soc. London A308 (1982) 523.

[2] S. Axelrod, S. Della Pietra and E. Witten, ,Geometric Quantization of Chern
Simons Gauge Theory, preprint IASSNS-HEP-89/57 (1989).

[3] J. Bismut and D. Freed, The analysi8 of elliptic familie4.'i I, Commun. Math.
Phys. 106 (1986) 159˜176; D. Freed, On determinant line bundles, in Mathe-
):˜au,
matical aspect8 of 8tring theory, edt S.T. World Scientific (1987) p. 189.

[4J P. Braam, First Step8 in Jones- Witten Theory, Univ. of Utah lecture notes,
1989.

[5] K. Gawedzki and A. Kupiainen, Coset Construction from functional integrals,
Nucl Phys. B320, 625-668 (1989).

[6] N. Hitchin, Flat Connections And Geometric Quantization, Oxford University
preprint (1989).

[7] B. Kostant, Quantization And Unitary Representations, Lecture Notes in
Math. 170 (Springer-Verlag, 1970) 87.

[8] A. Pressley and G. Segal, Loop Groups, Oxford University Press, Oxford, 1988.

[9] D. Quillen, Determinants of Cauchy..Riemann operators over a Riemann .sur-
face, Funct. Anal. Appl. 19 (1985) 31-34.

[10] T.R. Ramadas, I.M. Singer, and J. Weitsman, Some comments on Chern-
Simons gauge theory, Commun. Math. Phys. 126,409-430 (1989).

[11] D. Ray and I. Singer, R-Torsion and the Laplacian on Riemannian manifolds,
Adv. Math 7 (1971) 145-210.

[12] G. Segal, Two Dimensional Conformal Field Theories And Modular Functors,
in IXth International Congress on Mathematical Physics, eds. B. Simon, A.
Truman, and I. M. Davies (Adam Hilger, 1989) 22-37.
Witten: New results in C˜ern-Simons theory 95

[13] J...M. Souriau, Quantification Geometrique, Comm. Math. Phys.. l (1966) 374.

[14] A. Tsuchiya and Kanie, Vertea: Operators In Conformal Field Theory on pI
And Monodromy Representations Of Braid Groups, Adv. Studies in Pure Math.
16 (1988) 297.

[15] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math.
Phys. 121 (1989) 351-399.
Geometric quantization of spaces of connections
N.J. HITCHIN




Witten's three manifold invariants require, in the Hamiltonian approach, the geo-
metric quantization of spaces of flat connections on a compact surface E of genus g.
Itecall that if G is a Lie group with a biinvariant metric, then the set of smooth points
M' of
M = Hom(1rl (E); G)jG
a.cquires the structure of a symplectic manifold. This can be observed most clearly in
the approach of Atiyah and Bott [1] which views AI, the space of gauge equivalence
classes of flat G-connections, as a symplectic quotient of the space of all connections.
The symplectic manifold M' clearly depends only on the topology of E. To quan-
t.ize it, we require a projective space with the same property. However, the method of
p;eometric quantization requires first the choice of a polarization, the most tractable
case of which is a [(iihler polarization.
Briefly, if (M,w) is a symplectic manifold with 2˜[W] E H2(M; R) an integral class,
t.hen we can find a line bundle L with unitary connection whose curvature is w. If
we additionally choose a complex structure on M for which w is a Kahler form (this
is what a Kahler polarization means) then the (0,1) part of the" covariant derivative
\l of this connection defines a holomorphic structure on L and we may consider the
space of global holomorphic sections
= kerVO,l : OO(L) ˜ OO,I(L).
V = HO(M;L)
'I'he corresponding projective space P(V) is the quantization relative to the polariza-
tion. What is required to make this a successful geometric quantization is to prove
t.hat P(V) is, in a suitable sense, independent of the choice of polarization.
One way of approaching this question is to pass to the infinitesimal description of
illvariance. If X is a smooth family of Kahler polarizations of M, then an identification
of P(˜) and P(˜) for x,y E X can be thought of as parallel translation of a
ronnection, defined up to a projective factor, on a vector bundle V over X. To
sa.y that the identification is independent of the path between x and y is to say that
th(˜ connection is flat (up to a scalar factor). One seeks this ,vay a flat connection on
the kernel of V-0,l as W(˜ vary the cOlnplex structure.
Hitchin: Geometric quantization of spaces of connections
98


For a complex structure I, the space V consists of the solutions to the equation
+ iI)Vs = O.
(1
Differentiating with respect to a parameter t (i.e. where X is onc-dimensional) we
obtain
+ iI)Vs = o.
iiVs + (1
n0
i
E 11 (T) is a I-form with values in the holomorphic tangent bundle and since
Here
=
0 we can write this equation as
VO,l s

+VO,lS= o.
iiv 1,0 s
A connection will be defined by a section A(s,i) E nO(L), depending bilinearly on s
and i and such that
iiv 1 ,os + VO,l A(s, i) = o. (*)
Now iiv1 ,0 is a (1,0)-form with values in V 1 (L), the holomorphic vector bundle of
first-order differential operators on L. The equation (*) can be given a cohomological
interpretation if we introduce the complex
CP = OO,P-l(L) E9 00,P(1)1(L))
and the differential
= (au + (-I)P- 1 Ds,8D).
d,(u,D)
From the integrability of the complex structure I and the compatibility with the fixed
symplectic structure, the (0, I)-form iivl,O E {1011(1' 1 (L)) is 8. this, together with the
fact that s is holomorphic, shows that a solution A to equation (*) gives a l-cochain
in this complex:

and hence a cohomology class.
Under the fairly mild hypothesis that there are no holomorphic vector fields pre-
serving the line bundle L, we obtain the following:

Proposition: A connection on the projective bundle P(IIO(˜f;L)) over X is
determined by a cohomology class in H˜(1Jl(L)) - the first cohonlology group of the
complex above.

(This point of view was emphasised by Welters in his paper on abelian varieties
[6].)

One way of obtaining such classes is to consider the sheaf sequence
1)2(L) S'2T
1)l(L) iT
0
0 --+ --+ ---+
---+

1, 1, 1 (**)
o ' ---+ L
L 0 0
"'" ---+
----+
99
Hitchin: Geometric quantization of spaces of connections

where 1)2(L) is the sheaf of second-order differential operators on L, (1 is the symbol
map and the vertical arrows consist of evaluation on the section s. The complex
CP defined above actually gives the Dolbeault version of the hypercohomology of the
complex sheaves Vl(L)˜L. In the exact cohomology sequence of (**), there is a
coboundary map


so that we can obtain a class in the required cohomology group for each holomorphic
symmetry tensor on M.
In fact, the spaces of fiat connections we are considering have many such tensors.
For this we have to introduce on M a Kahler polarization for each complex structure
on the surface E. The theorem of Narasimhan and Seshadri [5] then describes the
complex structure of m, the space of flat U(n) connections - it is the moduli space
of stable holomorphic bundles of rank n on the Riemann surface E.
The tangent space at a point represented by a holomorphic bundle E is the
sheaf cohomology group H 1 (E; End E) and by Serre duality the cotangent bundle
is HO(˜; End E EB !(), with 1< the canonical bundle of˜. The cup product map



t.hen defines for each vector in the (39 - 3)-dimensional vector space HI (˜; 1(-1) a
global holomorphic section G of S2T on M and hence a class S(G) in H˜('Dl(L)).
This effectively defines a connection over the space of polarizations lof M para-
rnetrized by the space of equivalence classes of complex structures on ˜ - namely
'reichmiiller space. The minor (but universally occurring) feature in all of this is the
fact that each symmetric tensor G arises in (* **) {roin a vector in HI (˜; K- 1 ) which
is naturally the tangent space of Teichmiiller space, but also froln the exact sequence

O---+O˜1)l(L)˜T---+O


t.he class 6(G) defines a cohomology class a6(G) in Hl(T). This is the tangent space
of Teichmiiller space. The composition of these t\VO Inaps is not however the identity
hut is the factor 1/2(k +1) where k is essentially the degree of L and I is a universal
invariant of the Lie group.
The above procedure defines naturally a (holomorphic) connection. To see that it
is flat, one spells out the cohomology and coboundary maps in explicit terms using a
(˜ech covering {Ua } of M. Given the symmetric tensor G, we choose on each open set
fIr< a second-order differential operator ˜Q with synlbol (1/2(k + l))G. On Ua n Up,
˜o· - ˜p is first-order since GOt = Gp and this defines a class in H 1 (V 1 (L)). On
the other hand, if t is a deformation p'arameter, the I(odaira-Spencer class of the
-
ddormation has a similar representative. The vector field ˜L ˜t˜ is tangent to M
.tll d represents a class in H l (T) which lifts to one in III (1)1 (L )). Identifying this class
Hitchin: Geometric quantization of spaces of connectIons
100


from these two points of view gives a globally defined heat operator -lt -˜. Covariant
constant sections of the connection are then solutions to the heat equations
8s
--dAs=O
atA
where A = 1, ... , 3g - 3.
This point of view leads to the flatness of the connection, for




Consid˜˜ing it locally, it
is a globally defined holomorphic differential operator on L.
can be written as
86. {}6. ]
A
--a + -a + [LlA' ˜B
B

t t B
A

However, as shown in [3], the symbols of 6. A and ˜B Poisson-commute when consid-
ered as functions on the cotangent bundle of M. This means that [6. A , t1 B ] is, like
the two derivative terms, a second-order differential operator. When, as happened
in our situation, the map HO( S2T) -+ HI ('V 1 (L)) is injective, then the cohomology
sequence of (**) tells us that every second-order operator on L is first-order. On the
other hand the hypothesis of the proposition, that no vector field preserves L, tells
us that it must be zero-order and by compactness of Al a constant.
The connection is thus projectively fiat as required.
(The necessary hypothesis is satisfied for the Jacobian and automatically satisfied
for non-abelian moduli spaces which have no globa.l holomorphic vector fields [3].)
The details of the above outline of the connection may be found in [4]. The
appearance of the heat equation in the context of symplectic quotients of affine spaces
is treated by Axelrod, Della Pietra and Witten [2] where a direct computation of the
curvature appears.

REFERENCES
[1] M.F. Atiyah and R. Bott, The Yang-Mills equation over Riemann surfaces, Phil.
Trans. R. Soc. Lond. A 308 (1982), 523-615.

[2] S. Axelrod, S. Della Pietra and E. Witten, Geometric quantization of Chern-
Simons gauge theory, preprint IASSNS-HEP-89/57.

[3] N.J. Hitchin, Stable bundles and integrable systems, Duke Alath. Journal 54
(1987), 91-114.

[4] N.J. Hitchin, Flat connections and geometric quantization, preprint, Oxford,
(1989).
Evaluations of the 3-Manifold Invariants of
Witten and Reshetikhin-Turaev for sl(2, C)
RoBION KIRBY AND PAUL MELVIN




In 1988 Witten [W] defined new invariants of oriented 3-manifolds using the
Chern-Simons action and path integrals. Shortly thereafter, Reshetikhin and Tu-
raev [RTl] [RT2] defined closely related invariants using representations of certain
Hopf algebras A associated to the Lie algebra sl(2, C) and an r th root of unity,
q = e 2 '1rim/r. We briefly describe here a variant T r of the Reshetikhin-Turaev ver-
sion for q = e21ri / r , giving a cabling formula, a symmetry principle, and evaluations
at r = 3, 4 and 6; details will appear elsewhere.
Fix an integer r > 1. The 3-manifold invariant T r assigns a complex number
Tr(M) to each oriented, closed, connected 3-manifold M and satisfies:
(1) (multiplicativity) Tr(M#N) = Tr(M) · Tr(N)
(2) (orientation) T r(-M) = Tr(M)
(3) (normalization) T r (S3) = 1
Tr(M) is defined as a weighted average of colored, framed link invariants JL,k
(defined in [RT1]) ˜f a framed link L for M, where a coloring of L is an assignment
of integers ki, 0 < ki < r, to the components Li of L. The ki denote representations
of A of dimension k i , and JL,k is a generalization of the Jones polynomial of L at
q.
= e21ria , s = = e( 41r)'
e( 21r)' t = t 4 ),
We adopt the notation e(a) (so that q = S2
and
sin 'Irk
Sk, _ Ski
= __r_
= _
[k]
s-s sin;' ·
I)EFINITION: Let

L:[k]JL,k
(4) Tr(M) = O!L
k

where aL is a constant that depends only on r, the number n of components of L,
and the signature (7 of the linking matrix of L, namely

(f;. 7r)n (e (-3(r-2»))U'
= In c =
) -
U'deC
(5) a1 SIn -
r r 8r
J
102 Kirby and Melvin: Evaluations of the 3-manifold invariants
.J˜
'(i;,
;
and .˜˜



1
n
˜
= II[ki].
I
(6) [k]

The sum is over all colorings k of L. ;=1

Remark: The invariant in [RT2] also contains the multiplicative factor ell where ,J
v is the rank of H}(M; Z) (equivalently, the nullity of.the linking matrix). If this ]

factor is included, then (2) above does not hold, so for this reason and simplicity ,˜
˜
we prefer the definition in (4).
Recall that every closed, oriented, connected 3-manifold M can be described by j
surgery on a framed link L in 8 3 , denoted by ML [Ll] [Wa). Adding 2-handles to ;˜
˜
the 4-ball along L produces an oriented 4-manifold WL for which aWL = M L , and 1
the intersection form (denoted by x · y) on H2 (WL; Z) is the same as the linking ˜

matrix for L so that er is the index of WL. Also recall that if ML = ML', then one
can pass from L to L' by a sequence of [(-moves [Kl] [F-R] of the form


I I
±1 full twists

...
QCQ±1
...

L'·L
Figure 1

where L˜ · L˜ = Li . Li -t: (Li . [{))2[( · K.
The c?nstants aL and [k] in (4) are chosen so that Tr(M) does not depend on the
choice of L, i.e. Tr(M) does not change under K-moves. In fact, one defines JL,k
(below), postulates an invariant of the form of (4), and then uses the K-move for
one strand only to ˜olve uniquely for aL and [k]. It is then a theorem [RT2] that
Tr(M) is invariant under many stranded K-moves.
To describe JL,k, begin by orienting L and projecting L onto the plane so that
for each component Li, the sum of the self-crossings is equal to the fra.ming Li . Li.
Kirby and Melvin: Evaluations of the 3-manifold invariants 103




c




c


Figure 2

Removing the maxima and minima, assign a vector space V ki to each downward
oriented arc of Li' and its dual Vki to each upward oriented arc as in Figure 2.
Each horizontal line A which misses crossings and extrema hits L in a collection of
points labeled by the V k • and their duals, so we associate to A the tensor products
of the vector spaces in order. To each extreme point and to each crossing, we assign
a.n operator from the vector space just below to the vector space just above. The
composition is a (scalar) operator from C to C, and the scalar is JL,k. The vector
spaces and operators are provided by representations of A.
To motivate A, recall that the universal enveloping algebra U of sl(2, C) is a 3-
<limensional complex vector space with preferred basis X, Y, H and a multiplication
with relations HX - XB = 2X, BY - YH = -2Y and XY - YX = H. To
(luantize, U, consider the algebra Uh of formal power series in a variable h with
. h hH
coefficients in U, with the same relations as above except that XY - Y X = ˜h =
SIR "'2

1/ + H˜4H h 2 +... . Setting q = e h , and then by analogy with the above notation,
8:=:H,
. . = e h / 2 , t = eh / 4 , S = e- h / 2 , and [H] = the relations can be written
HX = X(H + 2), HY = Y(H - 2), XY - Y X = [H]. It is convenient to introduce
the element I( = ttl = e¥ and f< = t h . Note that f< = I{-l, l(X = sXK,
104 Hitchin: Geometric quantization of spaces of connections


«:=:'
1{Y = sYK and XY - YX = = [H].
2;i
We want to specialize Uk at h = (so q = e27ri / r ) and look for complex rep-
resentations, but there are difficulties with divergent power series. It seems easiest
to truncate, and define A to be the finite dimensional algebra over C generated by .˜
X, Y, K, K with the above relations'˜

˜
= 1 = K1(
K1?
'f
KX =sX1( j

!(Y = sYK
1
K2_˜2
XY-YX=[H]=
(7) ,t˜
s-s
˜
:i
as well as 'l
˜
J
'/'.

=y r = a
Xr
K 4r = 1.
A is a complex Hopf algebra with comultiplication ˜, antipode 5 and counit e
given by

X ® 1( + 1? ® J1(
˜X =
Y ® 1( + f( ® Y
˜y =
= H ® 1 + 1 ® H)
˜K = 1( ˜ [( (˜H

5X=-sX
(8)
= -sY
5Y
51< = Ie = -H)
(SH
=0
e(X) = e(Y)
e(1() = 1.

There are representations V k of A in each dimension k > 0 given by

= [m + j + l]ej+l
Xej
+ l]ej-1
(9) Yej = [m - j
1(ej = sjej

where V k has basis em, em-I, ... , e_ m .for m = k;l. The relations in A are easily
verified using the identit;y [aJ[b] - fa + IJ[b -1] = [a - b+ 1). For example, the 2 and
Kirby and Melvin: Evaluations of the 3-manifold invariants 105


3 dimensional representations are




respectively.
It is useful to represent V k by a graph in the plane with one vertex at height j for
each basis vector ej, and with oriented edges from ej to ej±l labeled by [m ±j + 1]
t:
if [m ± j + 1] [r] = 0, indicating the actions of X and Y on V k • Figure 3 gives
some examples, using the identities [j] = [r - j] = -[r + j].




• •
[2m]I![2] -[I]I![I]
• •
[2m-I] f![!] ![2J
• •

[2] f![1] [IJ f![2]

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