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O. The 1
l I(J z, z} dt extends to a quadratic form z tt' I!;iven by ° IIln.p
z
t-+
a'(z)
= -z- + z+,
z
t-+
a(z) on F, with a gradient
= z- + zo + z+ E F.
Moreover one can show that F embeds compactly into every LP([O, 1]; R2n) for 1 ~ p < 00. We have a natural SI-action on F by phase-shift, i.e. r E R/Z acts via
(r * z)(t)
= z(t + r).
W(~
define a distinguished subgroup B of homeo(F) the homeomorphism group of /1' by saying h E B if h : F --+ F is a homeomorphism admitting the representation
,+,,- :
h(z) = exp (i-(Z))Z-
+ zO + exp (i+(Z))Z+ + K(z),
where F ~ R are continuous and SI-invariant, mapping bounded sets in F bounded sets in R. Moreover K : F --+ F is continuous and Sl--equivariant and IIUtpS bounded sets into precompact sets. It is easily verified that usual composition turns B into a group. Next we need a pseudoindex theory in the sense of Benci, 113], associated to the Fadell-Rabinowitz index, [14], in order to measure the size c)f Sl-invariant sets. This goes as follows: Given a paracompact SI-space X we IHlild a free 5 1-space X X Soo by letting SI act through the diagonal action. Here ,,",'00 = US 2n - l , with S2n-l C R 2 n ~ en. Taking the quotient with respect to 8 1 we obtain a principal 5 1 -bundle illt,o
,rile classifying map f: (X x SOO)/S'
~
Cpoo
il) 0 I card{(l,j) 11 E N*,
For example for r
j E {I, ... , n}, 1rr;1 ~
r}
~
Ie}.
= (1,1, ... ,1) we have
where [*] denotes the integer part. Moreover observe that lim dk«1,n, ... ,n)) = k7r.
n-..oo
It. has been proved in [3] that 'I'heorem 3.11 The following equations hold:
Ilere all sets are equipped with the standard structure (1. The capacities Ck enjoy n. useful representation property explained as follows. Call a compact connected Hlllooth hypersurface S to be of restricted contact type provided there exists a veetor field TJ on R2n, such that '1 is transversal to S,
L,,(1 =
(1.
W(~ have, denoting by B s the bounded component of R 2n \
s:
'I'heorem 3.12 Let S be as just described. Then there exists a sequence (Pk ) C '/)(5) and a sequence nk E N* such that
Finally let us give an application to an embedding problem: 'rheorem 3.13 Assume B 2 (1) x ... x B 2 (1) admits a symplectic embedding into "2n(r). Then r 2:: ,;n.
28
Hofer: Symplectic capacities
Let W: B2(1) x ... X B2(1) ~ B 2 n(r) be the symplectic embedding. We find, see [2], a symplectic map ~ E 'D, such that PROOF:
where ~ = ~ s and 0 < C < 1 was given. Hence, taking the n-th capacity we obtain n1rc 2
:5 7T'r 2 •
Since C < 1 was arbitrary we deduce r ~ ..;n. We had to introduce the map ~ since the C/c are by construction only invariant under symplectomorphisms in V. •
4
A capacity on Symp2n and the Weinstein conjecture
Here we present a more general construction for a symplectic capacity on arbitrary symplectic manifolds. Assume (M,w) E Symp 2n. We denote by H(M) .the set of all autonomous Hamiltonian Systems H : M -+ R such that
HI There exists a non-empty open set U C M \ 8M such that H
I U == o.
H2 There exists a compact subset K C M \ 8M and a constant m( H) that o ~ H(x) ~ m(H) for x E M
> a such
and
H(x)=m(H) forxEM\K. Given a symplectic embedding '1J : M
~
N we define a morphism
W. : H( M) -+ H( N) by
H
(w.H)(x) =
{
0
W- 1 (x)
m(H)
if x E im ('11)
if x E N \ im (w)
We call a Hamiltonian H E H(M) admissible provided every T-periodic solution of x = XH(x) for T E [0,1] is constant. We denote by ?te(M) the collection of all admissible Hamiltonians in H(M). Clearly w.Ha(M) C Ha(N) and H a(AI) f:. ¢ as one easily verifies. We define c : Symp 2n --+ r by c
(M,w)
= sup{m(H) I H
E
?ta(M,w)}.
29
Hofer: Symplectic capacities
(a(~nrly c is a covariant functor and c (M, a · w) =1 a 1·c (M, w) for a =f o. The clif-Hculty lies in proving axiom (N) which is done in [10]. Some other results listed h a is finite. Then V(S) ¢.
r
t
I) ROO F: Fix a smooth map
0 for S close to e or -e. Define a Hamiltonian
H E 1i ( ( -€, €)
X
S, d ( e t,\)
)
==
0 near zero and
Hofer: Symplectic capacities
30 by
H(x) = ep(6) for x E
is} X s.
We pick m in such a way that m > c( (-e,e) X S). Hence H is not admissible and has a non-constant periodic solution x of period T E [0,1]. We have
x(R) for some 6 and
t is a non-vanishing section of £s
•
C
{6}
d
~ dt (prs
~
0
X
S
x (t) )
8. Hence it parametrises a closed characteristic.
Let us say that a compact smooth hypersurface S in a symplectic manifold
(M,w) is of contact type provided there exists a I-form A on S such that dA = i*w where i : S ~ M is the inclusion and A(X, e) 1= 0 for non-zero elements in £s. It is not difficult to show that «-e,e) X S,d(etA» admits for some small e > 0 a symplectic embedding into (M,w) mapping {O} X 8 into S.
Definition 4.3 Let (8, A) be a compact (2n-l)-dimensional manifold equipped with a contact form. We call (S, A) embeddable into the symplectic manifold (M, w) E Symp 2n provided there exists an embedding
*w = dA. .As a corollary we obtain
Theorem 4.4 Let (S, A) be described as before and assume (S, A) is embeddable into N X B2k(R) or CP'''', where (N,w) is a compact symplectic manifold with w I 1r2(N) = O. Then V(S) 1= 4>. (Here of course we talk about codimension l-embeddings). The Weinstein conjecture is one of the key conjectures in the Existence theory for periodic solutions of Hamiltonian Systems. For more details we refer the reader to [8,9,10,12J.
5
Capacities and Instanton HOInology
In this section we describe some results presented at the symplectic year at MSRI, Berkeley. We restrict our~elves to the case of open sets in R2n. The construction
Hofer: Symplectic capacities
31
c°n.n be done in much greater generality. For this we refer the reader to a forthcoming
I)n.per, [17], in which a theory, called symplectology, is developed (Symplectology: Sylnplectic Homology). For the following we require some familarity with Floerhornology for example as described in [18,19,20]. We consider as in 3.3 the family ()f I-Iamiltonians 11. For H E 11 we have the map ~ H : F ~ R as introduced in 3.3. '1. '1 induces a map, still denoted by ~H,
It oughly speaking we study the L 2-gradient flow on (F x Soo)/ Sl. In practice we have to replace Soo by a sufficiently large compact approximation S2n-l and to show that the following construction stabilizes. From now on we shall simply ignore any tpehnical difficulty in order to present the idea. For a generic H E 1i the map .p H luts finitely many critical points. For numbers c ~ d we consider the free abelian ~roup /\~( H) defined by /\~(H) = ffixeaZx where G is the set of all x with:
{
~#(x) f) H( x)
=0
E (c, d].
We can associate to critical points of ~ H a relative Morse index J.t E Z, which can h(~ understood in terms of a Maslov class for example as in [21]. So /\~(H) has a nn.tural Z-grading. We define a boundary operator {) : /\~(H) ~ /\~(H) by
8y=
E
#<x,y>x
J1.(x)=J1.(y)-l
where # < x, Y > denotes the number of trajectories of the flow equation " = epi!(,) I'tlnning from x to y. Here these orbits carry natural orientations so that we actually (fount their signed number. We have {)2 = 0 and define a homology group Ig(H). 'I'he choice of an inner product and a positive almost complex structure with respect t.o the symplectic form is involved in studying " = epk(,). However it turns out that Ig(H 0 \J!) :k Ig(H) for all \II E 1), where 1) consists of all symplectomorphisms (»)>tained as time-I-maps for compactly supported time-dependent Hamiltonians. We define an ordering on 11 by H >!{ *=>
-
{
There exists '1J E 1) such that ~ K 0 \II(z) for all z E R2n.
H(z)
'I'aking a homotopy from !{0'1J to H which is increasing, say L s such that L s = !{ 0'1J ror s :::; -1 and L s = H for s ~ 1 and :sLs(x) ~ 0 we can study the flow equation
" = ~~s(1'),
32
Hofer: Symplectic capacities
where one looks for solutions running from critical points of ~ 1(0\11 to critical points of c1> H. Studying this problem for critical points of the same Morse index one obtains an induced morphism 1:(K) ~ 1:(H). One defines for a bounded subset 5 of R2n using 11(5) as defined in 3.3
1:(S) = l~ 1:(H) where the direct limit is taken over all H with H E H(S). Recall that H(S) is partially ordered. We also note that we have for c ::; d ~ e morphisms
1:(H) ~ 1~(H), which induce We define a terminal object
e by
e := lim ItOO(H)withH E 11(*). It turns out that e := Z[t] is the polynominal ring in one variable of degree 2 (recall that we have a Z-grading ). Assume '!I( S) C T for some W E V, where Sand T are bounded sets. Then we obtain a natural map 1:(T) ~ 1:(8) We define for an unbounded set U the group 1:(U) to be the inverse limit of the Ig( 8) for S running over the bounded subsets of U. The family Ig, for c :5 d, of functors is the symplectology. Using our terminal object e we have induced maps Ig~e,
such that if w(S)
c T,
'11 E V and c :5 d
e 19(T) '\,
e Taking the second diagram we see that the image of the map 19(T) ~ e is increasing in d. Studying the images of the maps - - ? e we define a non-decreasing sequence
18
33
Hofer: Symplectic capacities
{d k } as follows. The sequence {d k } consists of all points where the rank of the ilnage changes. Here we repeat a point of discontinuity according to its multiplicity, namely the net change of the rank. The first diagram shows that the numbers d k are monotonic invariants, Le. if w(S) c T then dk(S) ~ d!c(T). This Instantonhomology approach to symplectic capacities shows that symplectic capacities can he understood as numbers, where a certain classifying map into Z[t] changes its ra.nge.
References [1] M. Gromov: Psettdoholomorphic curves in symplectic manifolds, Inv. Math. , 1985 , 82, 307-347.
[2] I. Ekeland, H. Hofer: Symplectic topology and Hamiltonian Dynamics, Math. Zeit. , 1989, 200, 355-378.
[3] I. Ekeland, H. Hofer:
Symplectic topology and Hamiltonian Dynamics II,
Math. Zeit. , 1990, to appear.
[4] M. Gromov: Partial differential relations, Springer, Ergebnisse der Mathematik, 1986.
[5] Y. Eliashberg:
A Theorem on the structure of Wave Fronts,
Funct. Anal.
Appl., 1987, 21, 65-72.
[6] M. Gromov:
Soft and Hard Symplectic Geometry,
Proc. of the ICM at
Berkeley 1986, 1987, 81-98.
[7] H. Hofer: On the topological properties of symplectic maps, Proc. Royal Soc. of Edinburgh, special volume on the occasion of J. Hale's 60th birthday. [8] C. Viterbo: A proof of the Weinstein conjecture in R'n., Ann. Inst. Henri Poincare, Analyse non lineare, 1987, 4, 337-357.
[9] A. Weinstein:
On the hypotheses of Rabinowitz's periodic orbit theorems, J. Diff. Eq. , 1979 , 33,353-358.
[10] H. Hofer, E. Zehnder: A new Capacity for symplectic Manifolds, to appear in the proceedings of a conference on the occasion of J .Moser's 60th birthday. Ekeland, H. Hofer: Convex Hamiltonian Energy Surface3 and their periodic Trajectories, Comm. Math. Phy. , 1987 , 113, 419-469.
r 11] I.
/12] H. Hofer, E. Zehnder: Periodic Solution3 on Hyper3urface3 and a result by C. Viterbo, Inv. Math. , 1987 , 90, 1-9.
34
Hofer: Symplectic capacities
[13] V. Benci: On the critical Point Theory for indefinite Functionals in the Presence of Symmetries, Transactions Am. Math. Soc. , 1982 , 274 , 533-572. [14] E. Fadell, P. Rabinowitz: Generalized cohomological index theories for Lie Group Actions with an Application to Bifurcation Questions for Hamiltonian systems, Inv. Math. , 1978 , 45, 139-173. [15] A. Floer, H. Hofer, C. Viterbo: The Weinstein Conjecture in P Zeit., 1990, to appear.
X
C', Math.
[16] H. Hofer, C. Viterbo: The Weinstein Conjecture in the Presence of holomorphic Spheres, in preparation. [17] A. Floer, H. Hofer: in preparation.
[18] D. Salamon: Morse theory, the Conley Index and the Floer Homology, Bull. of the London Math. Soc., to appear. [19] D. McDuff: Elliptic Methods in symplectic geometry, Lecture notes. [20] A. Floer: Morse theory for Lagrangian intersection theory, J. DifF. Geom., 1988, 28, 513-547.
[21] D. Salamon, E. Zehnder: Floer Homology, the Maslo1J Index and periodic orbits of Hamiltonian Equations, preprint , Warwick, 1989 . [22] Y. Eliashberg, H. Hofer: Towards the definition of a symplectic boundary, in preparation.
TIle Nonlinear Maslov index A.I~.
GIVENTAL
LfiUill Institute for Physics and Chemistry, Moscow
I will present here a nonlinear generalisation of the Maslov-Amold index concept [l],and liM"
it to deduce the following theorem
.
TUI·;OREM (GIVENTAL).
llt1t Rpn-l C cpn-l be the fixed-point set of the standard anti-holomozphic involution rtf cpn-l. Then if f : cpn-l --+ cpn-I is a map which can be deformed to the identity tlll'ough a Hamiltonian isotopy, the image f(Rpn-l) intersects Rpn-l in at least n points.
= 2 ; it is evident that the equatorial circle in the 2-sphere any area bisecting circle at least twice. 'l'hiR theorem is a typical fact of symplectic topology; similar to results proved by Conley~rhnder and Floer. We shall see, I hope, that the nonlinear Maslov index provides a IU\tllral and convenient language to formulate "Arnold- type" conjectures on symplectic nx(~d points, or Langrangian intersections (see [2]). 11\ ... a simple example, take n Illt·(~ts
'I'he linear Maslov index. fly the linear Maslov index we mean the only homotopy invariant of loops in the Lagrange(: ..n.ssman manifold An, the space of n-dimensional Lagrangian linear subspaces of R 2 n. Il(ofore generalising this notion it is convenient to projectivise it. L.~t en be complex n-space , endowed. with its standard symplectic structure. The real pl'ojectivization, Rp2n-l has a standard contact structure. A point p in Rp2n-l is a I·t~nlline in en ; its' skew-orthogonal complement is a hyperplane containing this line. In ))('ojective space we get a hyperplane through p , and the tangent space of this defines t.ll(~ element of the contact structure at p. With thi~ contact structure, the Legendrian IU'ojective subspaces of Rpn-l are exactly the projectivisation~ of Lagrangian subspaces or en (if a Lagrangian subspace contains a line then it is contai~ed in the skew-orthogonal complement of the line, Le. it is tangent everywhere to the contact structure). Thus An is t.he manifold of projective Legendrian subspaces in Rp2n-l. It is a homogeneous space of the the group Gn = Sp(2n, R)/ ± 1, and its' subgroup H n = U(n)/ ± 1. A linear Maslov iudex m('Y) E Z of a loop 'Y ,in any of these three spaces, is just its' homotopy class under the canonical identification ([1])
.:~
,~.~~
:;1~
Givental: The nonlinear Maslov index
36
:~
Legendre-Grassmann manifolds. We shall deal with the following infinite-dimensional manifolds:
;0 ,j
(1) C!5 n- the identity component of the contactomorphism group of Rp2n-l. ~ (2) ..e n- the space of all embedded Legendrian submanifolds of the contact manifol~ Rp2n-l which can be obtained from the standard Rpn-l C Rp2n-1 by a Legen~ drian isotopy. We call ..en the Legendre- Grassmann manifold. It is a homogeneou~ space of C!S n . :'~~ (3) i)n the identity component of the subgroup of (!5n consisting of transformation~ which preserve the standard (U(n)-invariant) contact I-form no on Rp2n-l. '~~ '!!
One may consider the contact form ao as a pre-quantisation connection on the square~ Hopf bundle Rp2n-l ~ cpn-l , having structural group the circle T = {e it } / ± 1. Thu~ J)n is a central T-extension of the identity component of the symplectomorphism group o~ cpn-l. We call fJ n a quantomorphism group because it realises, at the group level, th~ Poisson bracket extension of the Lie algebra of Hamiltonian vector fields: :~~..;,
o-+ R
-+
coo(cpn-l)
-+
sym(cpn-l)
-+
,j
O.
~!
The finite dimensional manifold An is naturally contained in the infinite dimensional spacJ C!Sn. Similarly, Q5 n contains G n and .JJ n contains H n . The weak statement about thes4 space.s is that the linea:r Maslov index can be. exte.nded to the loops in ~hese infinite dil.'l menslonal spaces, that IS we have a commutatIve dIagram of homomorphIsms: :~
>~
cw
1t"t(A n ) ~
Z
1/
1t"1 (..en)
,~
~ 1
J
~~
'1 The stronger statement is that Arnold's geometrical definition ([1]) of the linear Maslov;j index can also be extended to these infinite dimensional spaces. ~
Discriminants. To extend Arnold's geometrical definition we define a discriminant, a subspace ~ c .cn.i Let us mark a point in ..en, for example the projectivized imaginary subspace iRn c cn..,:~ The discriminant ~ consists of all Legendrian submanifolds which intersect this marked:l one. Two Legendrian subspaces are , in general, linked so ~ is a hypersurface in ..en:~ (with singularities). We also define a discriminant in C!Sn to consist of contactomorphisms'~ 9 which fix a point of Rp2n-l, and fix the contact I-form at that point; i.e. g(x) =f x and g*(ao}Jx = aol x , for some x in Rp2n-l. Analogously we define the discriminant in'i 1)n to consist of quantomorphisms which have fixed points (in fact, circles of fixed points, ~;: fibres of the bundle Rp2n-l -+ cpn-l). THEOREM
1.
Givental: The nonlinear Maslov index
37
l1:n('l. of these three discriminants admits a co-orientation such that their intersection num1.,·,..-; with oriented loops define commuting homomorphisms 1rt(Hn)
1 1rt(.f)n) 1
---t
---t
---t
Z
1rl(~n)
7I"t(A n )
!:! ---t
Z
1
1
m=int. index
f.lxt.(~llding
1rl(Gn)
---t
1rt(£n)
ml
ml
Z
Z
the linear Maslov indices.
'fIlii,; theorem yields two corollaries, as we shall explain below. 1. PI, P2 be two points in i.e. embedded Legendrian submanifolds diffeomorphic to II,I)U-l. The projections of PI and P2 to cpn-l have at least n intersection points in
«!onOLLARY
'1(""
..en,
C ~1)1I-1.
2, (KLEINER-OH). 'Il/lt! standard Rpn-l C cpn-l has the least volume among all its images under Hamilto";11.11 isotopies (and, more generally, among all projections of Legendrian submanifolds in '~n (!()Il.OLLARY
).
Ilere the volume is measured with respect to the standard U(n)-invariant Riemannian 1I1r't.ric on cpn-I. A simple illustration of Corollary 2 is furnished by the theorem of Poincare which asserts that the equator in the 2-sphere has least length among all arealtiHt'eting curves. The proof of Corollary 2 is based on integral geometry. The volume of a L"'~l'angian submanifold is proportional to the average number of its intersections with the ',l'H.llslates of the standard Rpn-l by U(n). This number is not less than n, by Corollary I , and equals n if the Lagrangian submanifold is standard. 'I'he Morse Inequality. It~n("h of the manifolds .en, (!Sn, jjn is modelled on a space of smooth functions. For example, II IH'ighbourhood of any Legendrian submanifold L in a contact manifold is contactomorphic I.•• t.he I-jet space JI L of functions on L (that is, J1 L == RxT* L, with the contact structure tltt == pdq). A Legendrian submanifold GI-close to L is represented by a Legendrian section of .IlL, i.e. by the graph of a smooth function f on L , with u = f(q) , p = dqf. The ".'ction meets the zero section L at a critical point of f with zero critical value. Thus the diseriminant ~ ,near the marked point, looks like the hypersurlace in GCXJ(L) of functions with singular zero level. Consider, for example the space of polynomials in one va.riable in I,ll,.ec of GOO (L) in which the analogue of ~ is the subset of polynomials with multiple roots. F(H' polynomials of degree 4 this subvariety has the form of the "swallow tail" singularity
38
Givental: The nonlinear Maslov index
.;r. ,'I:'
f
Diagram 1 in 3-space, depicted below, and this is in fact the general model for codimension-one '; singularities in the discriminants. '0,;. To co-orient the discriminant 6 C COO(L) we introduce, for every Coo function I on L, al~ topological space ."
LJ
= {q E Llf(q) ~
o},)
and an integer bl = b*(LI, Z/2) - the sum of the Betti numbers of LI' with Z/2 co_O)I~ efficients. As I moves in the complement of 6 the boundary 1- 1 (0) remains smooth, and;:i.' b, is unchanged. When I crosses ~ at a nonsingular point a Morse bifurcation ocurs: a':~ new cell is glued to Lf and bl changes by either +1 or -1. We co-orient 6 in the direction 0.:; in which bf increases. Now we give an interpretation of the total Morse inequality in our intersection-index terms. ,; Let us consider the flow I ~ I + t in COO(L). The number, ", of intersections with 6 of :', the orbit of a Morse function I on L is, by the definition of 6 , equal to the number of critical levels of f. On the other hand the intersection index of the orbit with 6 is equal to
Givental: The nonlinear Maslov index
39
= b.(L : Z/2).
bf+oo - b/- oo
Thus the total Morse inequa1ity,~ ~ b.(L; Z/2), follows from the fact that an intersection index is never more than an intersection number. An analogous argument is used to deduce Corollary 1 from Theorem 1, at least in the case when the intersections are transverse. Intersections of ~ with a T-orbit through L E ..en eorespond to the intersections of the image of L in cpn-I with the standard Rpn-I ,just hy the definition of~. But the intersection index, being homotopy invariant, is equal to t,he linear Maslov index of T- orbits in An , that is, to n.
The Calabi-Weinstein Invariant. 'rhe Calabi-Weinstein invariant([9]) is the homomorphism
which is defined by the Lie algebra homomorphism
w: coo(cpn-l) -+ R 'U1
: h 1-+
r
}cpn-t
hdi-t,
- integration with respect to normalised Liouville measure. In fact such a homomorphism is defined for any quantomorphism group.
PROPOSITION
1. The Calabi-Weinstein invariant is proportional to the Maslov index:
'Hore precisely m = (n/1r)w. COROLLARY.
The homomorphism w takes values in (1r/n)Z C R.
If Sjn were compact one could prove Proposition 1 by integral geometry arguments. 'rhere exists a (CO, locally exact, adjoint invariant) I-form M on j)n which represents the ('ohomology class m. The value of M on a tangent vector h E Tqil n is
Mq(h) =
:E
±h(x),
q(z)=x
where q denotes the underlying symplectomorphism, and signs are defined by the co()l'ientation of a at its' intersection points with the T-coset through q. IT we could average t.his form over all translations of M we would obtain a left and right invariant form, which Hhould clearly be proportional to W.
Generating functions. '[0 prove the existence of the nonlinear Maslov index we mark in Rp2n-l another La~rangian
subspace L o = Proj(Rn) C cn. Given an ambient Hamiltonian isotopy h t of Itp2n-l ( 0 ~ t ~ 1) , we factor hI into a large number N of small isotopies, and construct a function
f : Rp2D-l
~
R , D = nNe
40
Givental: The nonlinear Maslov index
This function is a kind of finite approximation to the action function of the isotopy;;: lifted to en homogeneously. It is chosen to have the property that '~.~
1- 1 (0)
r~ .
is non-singular {:} hl(Lo) E ~.
Then we define the relative Maslov index of the path 'Y in .en formed by L t = ht(Lo) to bal = b.(f- 1 (R+); Z/2) - D, and co-orient ~ in the increasing direction of the relativ~~; index when the end of the path crosses~. A crucial point in the proof of the validity of thi~ definition is the additivity of the relative index. H 1'0 and 1'1 are, respectively, a loop and ~j path in.e n which are subdivided into No and N 1 parts, then m(,O,l) = m(,o) + m(,l)?~ Moreover, if I is the function associated to the composite 1'0,)1, and fl is associated t~ 1'1, then the space f-I(R+) is cohomologicallyequivalent to the Thom space of the s~1 of Do = nNo copies of the Mobius line bundle over 1- 1 (R+). In particular, the twq~i spaces fol(R±) are cohomologically equivalent to RPd:l:-l, where the "inertia indicesn~ d± = b*(f;I(R±); Z/2) are complementary (d+ + d_ = 2Do ) , and differ from the middl~1 dimension by twice the Maslov index of the l o o p : : ~
me,)
:~
m(/o)
= d+ -
Do = Do - d_.
;~
.::~
.f.~
A similar method can be used to co-orient the discriminant in Q5n. We decribe the gen;j erating function for this case explicitly. Let hI = hN 0 •• • 0 hI be a decomposition of th~~ isotopy hlinto N small parts - lifted into C n homogeneously. Then we take .~
f = Proj (Q - H) : (Cn)N
-+
R
,t
where Q is the quadratic generating function of the cyclic permutation in (C PROPOSITION 2.
~
According to the "Sturm theory " the variation in the signature is equal to the linear Maslov index of the Hamiltonian flow, linearised along its trajectory. This gives
i
Givental: The nonlinear Maslov index
41
3. lJonlinear Maslov index of a loop ( in
w(~ use this result, applied to the subgroup J)n , to prove Proposition 1. Averaging over All t,ra.jectories, one finds that the nonlinear Maslov index is equal to the average value UV~I' the ball of the Laplacian divgradHt of the Hamiltonian. For homogeneous H t this is ......n.l to an average of H t over the unit sphere, which proves Proposi~ion 1.
'J'" lis. 1\. c'cnnplete the proof of our Lagrange intersection theorem for the non- transverse case one Ut~f-(IH to modify the definition of the discriminant. Let us define the cohomologicallengtk, I( X), of a subset X of a real projective space to be the least degree in which the restriction ur t,Il(~ generator of the cohomology of the projective space vanishes on X. The length is 'U4Htotone with respect to inclusion. Let us denote by r , and call a tail the subset of the tlbwl'illlinant ~ where the length of the positive subset f-l(R+) changes - rather than the n'1f,ti Hum we considered before. The tail is a hypersurface with no topological singularities III UollY finite codimension (see the diagram). Its' co-orientation by the length-increasing tUn·(·t,ion coincides with the positive T-co-orientation (monotonicity) and makes r into a 1 c'oc~ycle cohomologous to ~ (additivity). Thus the inequality between the intersection fluu"ber and index, applied to r and T-orbits, gives estimates for geometrically different '~np~l'u.ngian intersections, or fixed points of symplectomorphisms.
r
Diagram 2 ( ·OUOLLARY
3
(FORTUNE-WEINSTEIN
[5]).
:\ "y l11uTliltonian transforrnA,tjoll of cpn-l
JUtS
at least n different fixed points.
\1
Givental: The nonlinear Maslov index
42
'''f~!'
;':(" ;.~.:t
.:.:·~ ·i
!~;;~
The asymptotical Maslov index. -;~~ Let a be a contact I-form for the standard contact structure on Rp2n-l; so a = hao foti;~ some positive function h. The characteristic vector field VOl is defined byf~ ,...~
.~
vEkerda, a(v)=I,
,.,
;;J!~ •..
and this vector field generates the characteristic flow of contactomorphisms. Let us fiJGt~ on such a flow gt or, more generally, a nonautonomous flow of a time periodic characteristi~~ field. Let L E .£n and let mL.,.(t) (respectively met») be the intersection index of the patlij (g1" L) ( respectively (g1") , 0 ~ T ::; t, with the tail r c .en (respectively C!5 n). ·t~
.~
2. The following limits exist and are equal :
THEOREM
m
'1.~
= lim m(t)/t =
lim m L ( t ) / t . 1
The asymptotical Maslov index m is monotone and homogenerous of degree -1 o~ the space of contaxct I-forms; it is continuous in CO-nonn and does not change und~ conjugations in ~ n . ; ; ~ '.:-'i For example, a positive quadratic Hamiltonian in en generates a projective characteristi~ flow on Rp2n-l. Its nonlinear asymptotical Maslov index is equal to the linear one; th61 sum of frequencies m = LWi. For example ao generates the Hopf flow with m = n/1r. .~~
:,
COROLLARY 1.
n/(1rmax. h)
~m~
':1~ n/(1rInin. h ) . j
COROLLARY 2 (VITERBO). Any closed I-form on COROLLARY
3
Rp2n-l admits a closed characteristict
(GIVENTAL-GINZBURG).1
The nu~ber # of chords (i.e. characteristics starting and finishiD? on the ~ame Legendria.4 submanlfold L) of length less than or equal to t grows at least bnearly wlth t :
jl
#(L, t)
~ mt -
const.(L)
~ n[t/1rmax h]- c o n s t . ( L ) . " )'ti
Remarks.
(1) For the quantomorphism group tl n one can also define the asymptotical Calabi! Weinstein index. If a characteristic flow of a contact I-form hao (where h is ~ positive function on cpn-l) is precompact in j)n then its Calabi-Weinstein inde~1 coincides with its asymptotical Maslov index, in accordance with Proposition 1: ~\J m=(n/1r)jh-
1
.i.~ •
1
(2) In general the asymptotical Maslov index is equal, just by the definition, to the:' density of the length spectrum of closed characteristics and seems to coincide with \~~ the density of the capacity spectrum of the domain {r 2 h $ I} C en which has been?
Givental: The nonlinear Maslov index
43
defined by Ekeland-Hofer and Floer-Hofer ([7]). When applied to a precompact t»n this turns (i) into the Duutermaat.. Heckmann integration formula:
How in
(see [9]). (3) Our proof of Theorem 2 is mainly based on the monotonicity of cohomologicallength and the qUGsiatlditivity of generating functions : a homogeneous generating function for the flow {gT'} , T ~ t 1 + t 2 , coincides with the direct sum of such functions for T ~ t 1 and T ~ t 2 after their restriction to a subspace of low codimension (that is, low compared with large values of tl and t2).
Let us note finally that our intersection-theoretic interpretation of Arnold's fixed point Lagrange intersection problems can be extended to some other symplectic and contact IUltuifolds using direct methods of Floor homology ([4]), instead of singular homology and Morse theory.
..11 0 for any non-zero vector T E ~:t, X E E. We use the word "pseudo-convex" when the almost complex structure J is not specified.
ex
e
An important property of a J-convex hypersurface E is that it cannot be touched inside (according to the canonical coorientation of E) by a J-holomorphic curve.
In particular, if n is a domain in X bounded by a smooth J -convex boundary ao then all interior points of a J-holomorphic curve C c X with 8C c belong to IntO. Moreover, C is transversal to on in all regular points of its boundary
an
ac.
A hypersurface E is called Levi-flat if do: le= o. According to Frobenius' theorem this means that integrates to a. foliation of E by J-holomorphic curves.
e
46
Eliashberg: Filling by holomorphic discs and its applications
A function c.p on X is called J .. convex or pseudoconvex if it is strictly subharmonic on any J-holomorphic curve in X. All level-sets {'P = c} of a J-convex function 'P are J-convex if we oriente them as boudaries of domains {t.p ::; c}. Conversely, any function with this property can be made J-convex by a reparametrization:
articular, if M is diffeolnorphic to S2 and can be filled by holomorphic discs t.hf'1l it bounds a 3-ball in X.
:'.4
Existence of the Inaxilnal filling 3.4.1. Let X, J be a tame almost complex manifold, n c X be a domain 1U't.!I. a smooth J .. con1Jex boundary on and Mean be a closed 2-surface. Then M 2 1'1&71. be C -approximated by a surface AI which admits a unique generic maximal Jilting by holomorphic discs. 'I'IIEOREM
'I'UI':()REM
3.4.2. Let X, J and 0, be as above. Suppose that, in addition,
n
is
1&f1lcnnorphically aspherical. Let M be a closed 2-surface and ~ be an embedding
AI x I ---. tJfl.fh
an.
Then the embedding ~ can be C 2 -perturbed in such a way that
surface ~(M x t), tEl, admit3 a maximal fiZling by holomorphic discs.
If
x t), tEl, can be filled by holomorphic discs then there exist a Ilft'U.tllebody I{ and an embedding ~ : !{ x I ---. n such that ~ coincides with q, on fllld~h ,~urface ~(M
AI x I and for any tEl the image 4»(M x t) C
n is
Levi-flat.
:1." Graph of a filling
With any partial filling ( = (/(, tp, ..4 , H" F) we can associate the graph G A,
Now we apply 6.1 to prove
6.2. Let M be a closed surface in aQ. Suppose that X(M) = -lc(M)1 (comp. 5.1). Then M either can be filled by holomorphic discs (and it has to be ,(
THEOREM
diffeomorphic to the torus T 2 in this case) or is isotopic in
an
to a surface M' ,~
without elliptic points. In particular if M is a torus then either it bounds a Levi-flat solid torus in
n or it is isotopic in an to a totally real torus.
,l .J
,~ ;~
Proof. Suppose that :AI cannot be filled by holomorphic discs and let G M be the .;~
graph of a maximal filling ( = (1(,!.p, A, 11., F). In view of 6.1 it is enough to provel . :~ that the graph G can be exhausted by a sequence of elementary contractIons. In ;~ 'i) other words, that we can always find an edge whose ends correspond to elliptic and ',:~ hyperboli~ points of the same sign. Let MA be a result of cutting A out ~f.M (see ]
3.6). As In the proof of 4.1, let us denote by k+ and k_ numbers of posItIve and .~ negative hyperbolic points of M which have been substituted by elliptic points of I! .~
;~
MA. Then we have
e±(MA) = e±(.I\1") + k~, h±(fl.1) 2:: h±(MA) + k±, k+
+ k_
;:::: bo(MA )
(*)
.~
7:'
'1
o. Then X(M) = -c(M) ~ = !(X(Al)+c(Al)) = 0, d_(M) = !(X(M}-c(M)) = X(M) . j e_(.J\,I) = h_(M) + X(M). Suppose that GM admits no)!
Let us choose the orientation of M such that c(M) 2: and, therefore, d+(Al)
or e+(M) = h+(IvI), elementary contractions. This means that there is no edge whose ends are elliptic ~ and hyperbolic points of the same sign \vhich implies inequalities
Then
h+(MA)
+ h_(M) ,
h+(MA )
~
X(M)
h_(MA)
~
h+(IVI)
+ h_(MA) 2
x(1\.f)
and
+ h+(M) + h_(M)
Eliashberg: Filling by holomorphic discs and its applications
61
or
()n
the other hand, the equality c(MA) = 0 implies that
Now we get
h+(Al) + k_ - h+(MA) =
h_(M)
+ X(1\l) + k+
1
"2 X(MA)
,
1
- h_(MA) = 2X(MA) .
'rile first equality and (*) give
,u1d therefore,
bo(MA) = k+
1
+ k_ = '2 X(MA)
·
But then the second equality and (**) imply that
",lid, therefore, bo(1\IA)
= o.
But then MA is empty which is impossible if M has
(illiptic points.
Q.E.D. 6.3. Let M be a 2-s'u1jace in an 'with the Legendrian boundary L = aM t£nd p, be the homology class of Al in H 2 ( an, L). Suppose that M is in normal form 7tt:(LT aA! and either M = D2 or
'('IIEOREM
X(1\I) = -tb(LIJt) - Jr(LIp,)I. there exists an isotopy of }vI in an which i3 fixed near L and moves M to a .Iu/.1jace M without interior elliptic point3. 'I'h(~n
an
I' 7·0 o!. Let M' be any closed surface in which contains M and let ( = (1!. The first lecture will be devoted to background about canonical quantization. In the second lecture, we will focus more specifically on the problem of actual interest - quantization of the moduli space .M of representations of the fundamental group of a surface in a compact Lie group G. The third lecture describes more concretely how these constructions lead to link invariants. A detailed description of these results can be found in [2] and [15], along with more extensive references. 1. GEOMETRIC QUANTIZATION
(a) The prototype case: affine space We begin with a standard affine symplectic space A = 1R 2n with coordinates :c 1 , ••• , :c 2n and a symplectic form W = Wij d:c i A d:c j (Wij = -Wji is a real antisymmetric matrix such that W is nondegenerate.) We may regard w as a transformation T ~ T* (T is the tangent space to A) by sending a tangent vector to its interior product with w. Then the inverse w- 1 : T" -+ T will be denoted wii : thus we have
We examine the Heisenberg Lie algebra (essentially just given by the Poisson bracket) given by {:c 1 , ••• :c 2n } with the commutation relations (1)
The quantization procedure for A constructs an irreducible unitary Hilbert space representation of this algebra. In fact it gives a representation of the Heisenberg. group, an extension by U(l) of the group of affine translations of 1R 2n • By a theorem
75
Witten: New results in Chern-Simons theory
of Stone and von Neumann, there is a unique such representation up to isomorphism: moreover, the isomorphism between any two such representations is unique up to multiplication by 8 1 C C. SO any two such representations are canonically isomorphic up to a projective multiplier. To construct representations of the Heisenberg group (first a rather uninteresting reducible one and then an irreducible subrepresentation of it), we fix a unitary line bundle £ over A with a connection D such that the curvature D 2 = -iw. (Such a bundle exists whenever w /21r represents an integral cohomology class - here it it the trivial one!) The isomorphism classes of such bundles are given by H 1 (A, U(l)), which is 0; so £ is unique up to isomorphism. The isomorphism between two such bundles is unique up to a constant gauge transformation, which must be of modulus 1 to preserve the metric.
r
We define 11,0 = L 2 (A, £). The Coo functions on A act naturally on 1-£0 as follows. Given 4> E COO(A), the Hamiltonian function associated to it is V¢ = w- 1 (d¢». Then ¢ acts on rio via an operator
(2) (This represents a lifting to £ of the vector field V¢, which preserves the symplectic structure: V¢ thus acts on the space of sections of £.) The map ¢ 1--+ U 0 if X # 0).
Thus there are holomorphic linear functions zi such that w = i
2:i dz i 1\ dzi •
Let us now consider the line bundle ([" D) which was introduced and fixed before picking J. It acquires a new structure once J is picked. Indeed, as D has curvature of type (1,1) with respect to J, D and J combine to give £ a holomorphic structure.
76
Witten: New results in Chern-Simons theory .~
.~
(One defines a lJ operator on [, as DO,l; the condition that this actually defines a holomorphic structure is [P = (DD)O,2 = 101°,2 = 0.)
,i
:~ ",!
The line bundle £, can be given a particularly simple description once J is picked. Let [,0 be the trivial holomorphic line bundle on A with the Hermitian metric (for "p E coo(.Co) )
:i '~t
::~
and the connection compa.tible with this metric a.nd the standard holomorphic structure" The curvature of this connection is
':~
.~~
"~ :i ~
:~,'t
it This is the formula that characterizes £', so £, is isomorphic to [,0, by an isomorphism that is unique up to a projective ambiguity"
, .~~
~
::~
::.~
Now that J has been picked, the Heisenberg algebra that we are seeking to represent :~
.~
can be written
~\
(3) ,~;1
[Zi, zi] [zi, Z3]
~~
=
(1 ,'~
'i
We can represent this algebra. on the subspace
,~
i
':j of our original Hilbert space consisting L2 sections that are also holomorphic" This .~ representation p is j :~ ')~
p( zi) " 1/J = zi"p p( zi) "1/J = 8~' 1/J
(4)
1 ,:1 ~
,~
The commutation relations are preserved, as
8· 8z" To verify unitarity, note that
..
-.(zJ"p) = S'3"p
But
~
.8 + Z3_."p"
8z'
y
Witten: New results in Chern-Simons theory
77
so upon integrating by parts and using the holomorphicity of "p, we find that this equals
Jd"zd"z,¢ exp(- :~:>kzl')ix = (.,p,p(z')X)' Ie
This representation of the Lie algebra in fact exponentiates to a representation of the (complexified ) Heisenberg group (see for instance the discussion in [8], p. 188). It is irreducible, because holomorphic functions can be approximated by polynomials. Thus we have found the unique irreducible representation of the Heisenberg group described by the Stone-von Neumann theorem. We now want to let J vary, fixing its key properties (translation invariance and the fact that w is positive of type (1,1)). The space T of all complex structures on A with these properties is the Siegel upper half plane of complex symmetric n X n matrices with positive imaginary parts. Over T we get a Hilbert space bundle 1-£ whose fibre over JET is /1,J. The uniqueness theorem for representations of the Heisenberg group implies that there is a natural way to identify the fibres of this bundle, i.e., 'H, has a natural projectively flat connection. We would like to identify this connection explicitly. To this end, we observe that if xi ~ ii is any representation of (1), then the objects
themselves obey a Lie algebra, which can be worked out explicitly by expanding the commutators, i.e.,
[Dij, D lel ] = _i(Dilw jle + Dikwil
+ D lj w i1c + Dkjw il )
This is just the Lie algebra of Sp(2n, R), if we identify A with R2n ( it is given by the algebra of homogeneous quadratic polynomials under Poisson bracket). Thus, the Lie algebra of Sp(2n, R) acts in any representation of the Heisenberg algebra.. Actually, the group Sp(2n, R) acts on the Heisenberg algebra by outer automor.. phisms, and so conjugates the representation ?-lJ of the Heisenberg group to another representation. By the uniqueness of this irreducible representation, Sp(2n, R) must act at least projectively on fiJ. The Dij give the action explicitly at the Lie algebra level. The zi act on /1,J as zeroth order differential operators, while the zi act as first order operators. The Dij are thus at most second order differential operators. The Dij, which generate vector fields that act transitively on T, essentially define a connection on 'H, and this connection is given by a second order differential operator since the D's have this property. (Actually, the full group of symplectic diffeomorphisms of A has a natural 'loprequantum" action on the big Hilbert space 'H,o. This gives
78
Witten: New results in Chern-Simons theory
a representation of the Lie algebra of Sp(2n,R) by first order operators Dij '. The connection is really the difference between the D ij and the D ij '.) The connection can be described explicitly as follows. Let 6 be the trivial connection on bundle A x 'H o, regarded as a trivial Hilbert space bundle over A. 6 does not respect the subspace rtJ C 1-{,o. A connection which does respect it can be described by adding to the trivial connection a certain second-order differential operator. In describing it, note that the Siegel upper half plane has a natural complex structure, so the connection decomposes into (1,0) and (0,1) parts. Let T denote the tangent space to A; then a tangent vector to T corresponds to a deformation of the complex structure of A which is given explicitly by an endomorphism 6J : T ~ T (which obeys J 6J + 6J J = 0). Associated with such an endomorphism is the object 6J 0 w- 1 E T ® T, i.e.,. 6J 0 w- 1 : T* ~ T, which in fact lies in 8ym2 T; indeed, it lies in Sym2 T 1 ,O €I:) Sym2 TO,1. The (2,0) part of this (which is the (1,0) part of aJ with respect to the complex structure on T ) will be denoted aJij. Then the connection on 'Ii can be written: V O,l \71,0
0
= SO,1 = Sl,O - 0 D.SJij~ = -~4 i,j=l Dz" DZ3
(5)
t
tzi
where is the covariant derivative acting on sections of £. As promised, this connection is given by a second-order differential operator on sections of £'. In more invariant notation we have
'\71,0 for
$
s
= Sl,O
s
+ ~n(SJ ow-1n1,0 8) 4
E rp(A,£) and with the first D acting as a map T®r L2(A,L:)
--+
r L2(A,L:).)
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 D J, 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 connection 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 complexjfication 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 8ym11lectic manifold, A.T IQc is independent of J; it ITolay be identified with AI/ Q, the
80
Witten: New results in Chern-Simons theory
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 invariant 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 { 0:, ,8}' H
4~
h( /\,8) 0:,
where we have used a G-equivariant inner product (.,-) on Lie(G). This skewsymmetric 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 u.ction is A
t-+
81
FA: thus the symplectic quotient is M =
AI/9
= {A: FA = 0}/9 = Rep(1rl(~) --+ G)/conjugation
(6)
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:
°
T 1, A = nO, 1(~, g) TO' 1 A = n1, O(~, g)
(7)
(This conventional but seemingly inverted choice is made so that the operator 8A will be a map Lie(Qc) ~ T 1 , A or equivalently so that the moduli space of holomorphic 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 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. up
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.
82
I
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(Qc) ~ Tc A Lie(Qc) ~ T1,OA Lie(Q)
T
--+
Lie(Qc)
--+
I
TO, 1 A
If we introduce an invariant metric on Lie(Q) and take its extension to Lie(Qc)
:1'.;~
(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 also form K: , the operator that projects TI, 0A onto TI,O M, the orthocomplement .•~ of Im( Tz ) in TI, 0 A. (One sees this because ;g ·It
TA = Lie(g) EB JLie(Q) EB T M, since the codimension of F- 1 (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 this decomposition, so also T1, 0A = T1, 0 M $ Im(Tz ).) From this, id = /C + TzTz- 1 on T1,OA.
~ ~.~:J
:1 "~ .: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 on g -invariant holomorphic sections over F-l(O), in a manner that only involves ;~ :~ derivatives along the T 1 , 0 M directions and has the other directions projected out. j The result is: i ;t~
o
=
1 4
--D(/(,
..
8
87
)8J~3 D(/C
8
-;;;
) -
1·· 4
;.
k
-SJtJ(D·/C ·)D}( t
+ ~Tr(Tz-lc5J TJ )
3
(
i
+ a;;;)
.'~
8
::~
(8)
~
where the indices denote bases for T I , 0 A 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., .~~
),
where the operator L : Lie(Qc)
--+
D.
Lie(Qc) is defined by
= T/Tz = ~T/Tc.
(Here, "det" "denotes the product of the nonzero eigenvalues.) The final expression for the connection is: \71,0 = 61,0 - 0
'1 .~ ~
.~
Witten: New results in Chern-Simons theory
\70,1
=
~o, 1
83
(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) Now
(AmaxLie(Qc))*®(AmaxImTz ) is isomorphic to the trivial bundle, with the norm
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
R
= -8810gH
(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
84
I I
Witten: New results in Chern-Simons theory
class group on M lifts to an action on o~ o~
Q
II
Q
E.
We have exact sequences
Aut(P) U Auto(P)
r
---t
J
0
0
1
U
fa
o~j
~O
]
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 ~ 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:. . "
,I
The only additional ingredient that we must define in order to adapt (9) to this situation is the determinant H. In this case the map T z : Lie(Qc) ~ T1, 0 A is
;1~
i
I ~
~J ..
so 1,.. d 6 = 8- A *8A = 2dA A
:
1;,
n CE,gc) --+ n°(~,gc) 0
!
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 operator and is projectively flat, in the gauge theory problem we must verify these ~~ properties.
.~
'i 10~
J In verifying the projective flatness of the connection (9), or more precisely of a slightly modified version thereof, the main ingredients required, apart from general facts about Kahler geometry, are formulas for the derivatives of H which are consequences of Quillen's local families index theorem. One important consequence of this theorem is the formula
R
+ Balog H = 2h( -iwo) = -2ih~,
(11)
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.
~
:oj .f.
;! ~ .; ·1
o
Witten: New results in Chern-Simons theory
85
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 I~
V 1,0
(12)
V O,l
where the new formula for () is the same as the old one (9). The identity (11) enters [DJ,V] = 0 on holomorphic H(Octions and thus preserves holomorphicity.
crucially into the proof that this connection satisfies
'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 xplici tly,
'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 11
R'
= 2(k k+ h) Cl (IndT~).
(14)
'('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 E (HO)(A, £)J and define
;p up to a Qc -invariant section
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 abelian. However, one may construct an asymptotic expansion
1rl (~)
is
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 the same order. But it is actually possible to show that unitarity to order 1/ k 2 is enough to imply that the (2,0) curvature vanishes exactly. This is an interesting approach to proving that statement.
'0,' 1
At this point, we can enjoy the fruits of our labors. The monodromy of the projectively 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
Witten: New results in Chern-Simons theory
87
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 8x i
zt=--
(i=l, ... ,n)
x at t = o. The above equation is the equation the critical trajectories of the Lagrangian
are determined by the values of x, for
(15) or the equivalent Lagrangian
(16) in which the momentump (which equals ~ for classical orbits) has been introduced 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. n~
moduli space M is no exception. Consider an oriented three-manifold M. 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 ()ur
Let G
(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 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: w~
w("
M =
Rep(7rl(~
x R), G)/conj = Rep(1rl(~)' G)/conj
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 classical phase space (the mapping class group for Chern-Simons; the symplectic group Sp(2n; IR) for classical mechanics). Under favorable conditions, quantization will then give rise to a projective representation of this group on the quantum vector space. (The classical mechanics version is the metaplectic representation of
I
Sp(2n; R).)
The Chern-Simons function does not depend on a metric on M (or a complex structure 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 complex 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.
(b): The Feynman integral approach In canonical quantization, after constructing the "physical Hilbert space" of a theory, 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
I
Witten: New results in Chern-Simons theory
89
three manifold). The Feynman path integral is the "integral over the space of all connections modulo gauge transformations"
Z(M)
= fw VAexp(ikI)
(18)
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
ZO(M)
= fw VAexp(ikI)O(A)
(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) w here 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 mani 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 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 gain some intuition about what Feynman path integrals mean, it is essential to 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 Chern-Simons problem, even though it is exactly soluble by other methods, the asymptotic expansion gives results that are significant in their own right. To understand the construction of the asymptotic expansion, we consider first an analogous problem for finite dimensional integrals. To evaluate the integral
Jexp(ikf(z ))d"'z
.~
:~
"
j '~
J
:t ~
~;
'I ;1
.~
~~ !:i~
~
~~ .~
.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 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): ill
n/2'" {"kf( .)}exp{~signHpif} 1r L.t exp 1, P", Id H 111/2
et
Pi
~
(21)
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 connections, the analogous expression in Chern..Simons gauge theory is:
Z(M) =
2: eiklcs{Aa) [TRRS (A
a
)]l/2 • ei1rfJ (O)/2
• eihlcs{Aa}
.~
;1 ~'l
(22) j:(,
Q
.\. s~
.:~
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. At each A a , this leading term is multiplied by an asymptotic expansion 1+
f: 'bn~:). 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
I .~ :~
.]
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 exp
(
t.
2k
_
I: nan" ~ dz· ~ a,b J~ECo. Ji/EC
-))
_ x-y (dy x ,- ~3 X - Y L
(23)
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 dimension m, one gets a contribution k m / 2 : for instance one sees this behaviour ill the explicit formula for Z(S3) for G = SU(2), which is "V
Z(S 3) =
J
2
k+2
· s1n
_1r_
rv
k+2
k- 3 / 2 ·
(24)
S3, since it has a three dimensional stabilizer, corresponds to a component of the moduli space of flat (Oounections of virtual dimension -3. '("'he exponent reflects the fact that the trivial connection on
(c): Putting these approaches together We now discuss how to fit together the path integral and quantization approaches. connection on ~. Define a functional \}1 on connections AI: by integrating over those connections on M that rot'strict to A I:: t w(A ) = L.:AI1J=A1J VA exp{ikI(A)} (25)
Ir M is a manifold with boundary E, let AI: be a G
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
for some section tP, with ~8 the 8 Laplacian on M. As T ~ 00 this Laplacian (.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: :~-
Z(M)
= =
=
J
VAexp{ikI(A)}
r
JA!}E
J
A
(1:)
!>A1:
r
JALE
;
A
(ML)
VALexp{ikI(A L )}·
·JARE r A (MR) 1)AR exp{ikI(AR )}
VA!: '1I L *(AI:) '1IR(AI:)
f :i 1
(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 ~ spaces of holomorphic and antiholomorphic sections of f. , are canonically dual; the integral over AI: formally defines this pairing, i.e.,
J
-i (This is similar to the way invariants are constructed in the topological quantum field theory for Floer-Donaldson theory. The difference between these situations 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 quantummechanical nature of the Chern-Simons theory.)
M(U)1/Ji
= EM(u)i{tPj. j
j
'I i
-:1~.!• •~.I'
Witten: New results in Chern-Simons theory
93
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 interpretations, 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 represent.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
Z(Mu;[L,{k}])
= (WL,M(u)1Pl)
= 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: Z(Mu ; [L, {k})
= L M(U)ljZ(M; [L, {k}], [C,j)). j
94
Witten: New results in Chern-Simons theory
(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 Mathematical aspect8 of 8tring theory, edt S.T.
):~au,
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 .surface, Funct. Anal. Appl. 19 (1985) 31-34.
[10] T.R. Ramadas, I.M. Singer, and J. Weitsman, Some comments on ChernSimons 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 geometric 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 quant.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
V = HO(M;L)
= kerVO,l : OO(L) ~ OO,I(L).
'I'he corresponding projective space P(V) is the quantization relative to the polarization. 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
(1
+ iI)Vs = O.
Differentiating with respect to a parameter t (i.e. where X is onc-dimensional) we obtain
iiVs + (1 Here VO,l s
i
E
n0
11
+ iI)Vs = o.
(T) is a I-form with values in the holomorphic tangent bundle and since
= 0 we can write this equation as iiv 1,0 s
+ VO,l S = o.
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 ,o s + 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
d,(u,D)
= (au + (-I)P- 1 Ds,8D).
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 preserving 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 0
--+
o ' ---+
1)l(L)
1,
---+
1)2(L)
--+
S'2T
---+
0
L
"'"
L
----+
10
---+
0
1,
iT
(**)
Hitchin: Geometric quantization of spaces of connections
99
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 pararnetrized 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
l
represents a class in H (T) which lifts to one in III (1)1 (L )). Identifying this class
100
Hitchin: Geometric quantization of spaces of connectIons
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 atA
--dAs=O
where A = 1, ... , 3g - 3. This point of view leads to the flatness of the connection, for
is a globally defined holomorphic differential operator on L. can be written as
86. {}6. --a + -a + [LlA' ~B t t B
A
A
Consid~~ing it locally, it
]
•
B
However, as shown in [3], the symbols of 6. A and ~B Poisson-commute when considered 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 ChernSimons 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 Turaev [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 version 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. We adopt the notation e( a) = e21ria , s = e( 21r)' t = e( 41r)' (so that q = S2 = t 4 ), and
[k]
=
Sk, _ Ski
_ s-s
sin 'Irk = __ r_ sin;' ·
I)EFINITION: Let
(4)
Tr(M) =
O!L
L:[k]JL,k 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
(5)
a1 J
= I)n c =
U'deC
(f;.r
7r)n (e (-3(r-2»))U'
SIn -
r
8r
102
Kirby and Melvin: Evaluations of the 3-manifold invariants
'(i;;,
and
.~~
1 ~
n
(6)
.J~
[k]
The sum is over all colorings k of L.
= II[ki].
I
;=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
±1 full twists
...
I
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 J L,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 0 and -1 if a < O. It follows that T6(M) is determined by a and the Witt class of the quadratic form Q of K. Thus, for odd a, or a = 0, T6(M) is determined by H1 (M; Z) (with its torsion linking form, needed to determine the sign of a when a is divisible by 3). For even a, one also needs to know H1(M; Z) with its torsion linking form (which determines the Witt class of Q) where M is the canonical 2-fold cover of M. We are especially grateful to N. Yu. Reshetikhin for his lectures and conversations on [RTl] and [RT2], and to Vaughn Jones, Greg Kuperberg and Antony Wasserman for valuable insights into quantum groups.
REFERENCES
[D] V. G. Drinfel'd, Quantum groups, Proc. Int. Congo Math. 1986 (Amer. Math. Soc. 1987), 798-820. [FR] R. Fenn and C. Rourke, On Kirby's calculus of links, Topology 18 (1979), 1-15. [J] M. Jimbo, A q-difference analogue of U(Q) and the Yang-Baxter equation, Letters in Math. Phys. 10 (1985), 63-69. [Kl] R. C. Kirby, A calculus for framed links in S3, Invent. Math. 45 (1978), 35-56. [K2] , "The Topology of 4-Manifolds," Lect. Notes in Math., v. 1374, Springer, N~w York, 1989. [KM] R. C. Kirby and P. M. Melvin, Evaluations of new 3-manifold invariants, Not. Amer. Math. Soc., 10 (1989), p. 491, Abstract 89T-57-254. [KS] H. K. Ko and L. Smolinsky, A combinatorial matrix in 3-manifold theory, to appear Pacific J. Math.. [Ll] W. B. R. Lickorish, A representation of orientable, combinatorial 3-manifolds, Ann. Math. 76 (1962), 531-540. [L2] , Polynomials for links, Bull. London Math. Soc. 20 (1988), 558-588.
[L3]
, Invariants for 3-manifolds from the combinatorics of the Jones polynomial, to appea.r Pacific J. Math.
114
Kirby and Melvin: Evaluations of the 3-manifold invariants
[LM1] W. B. R. Lickorish and K. C. Millett, Some evaluations of link polynomials, Comment. Math. Helv. 61 (1986), 349-359. [Lip] A. S. Lipson, An evaluation of a link polynomial, Math. Proc. Camb. Phil. Soc. 100 (1986), 361-364. [Mur] H. Murakami, A recursive calculation of the Arf invariant of a link, J. Math. Soc. Japan 38 (1986), 335-338. [RT1] N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, MSRI preprint (1989). [RT2] , Invariants of 3-manifolds via link polynomials and quantum groups, to appear Invent. Math. [Wa] A. H. Wallace, Modifications and cobounding manifolds, Can. J. Math. 12 (1960), 503-528. [W] Ed Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399. Department of Mathematics University of California Berkeley, CA 94720
Department of Mathematics Bryn Mawr College Bryn Mawr, PA 19010
Representations of Braid Groups Lecture by
M.F.ATIYAH
The Mathematical Institute, Oxford Notes by
S.K.DONALDSON
The Mathematical Institute, Oxford
In this lecture we will review the theory of Heeke algebra representations of braid 1!;roups and invariants of links in 3-space, and then describe some of the results obtained recently by R.J. Lawrence in her Oxford D.Phil. Thesis [3].
(a)Braid group representations, Heeke algebras and link invariants. We begin by recalling the definition of the braid groups, and their significance for the theory of links. The braid group on n strands, B n , can be defined as the fundamental p;roup of the configuration space of n distinct points in the plane. Thus is t.he quotient:
en
en
(1)
en
where = {(Xl, ... ,X n ) E (R 2 )nl xi -I Xj for i -# j}, and the symmetric group S'n acts on Cn in the obvious way. Elements of the braid group ("braids" ) can be (lescribed by their graphs in R 2 X [0,1] C R3, as in the diagram. 'I'he action of the braid on its' endpoints defines a homomorphism from B n to Sn, and this is just the homomorphism corresponding to the Galois covering (1). IleH' any braid /3 we can construct a link /3' in the 3-manifold 8 1 X R2 by identifying the top and bottom slices in the graph. The link has an obvious monotonicity l»l'Operty: the projection from the link to Sl has no critical points. It is easy to see tha.t if a is another braid then (afja- 1 )' is isotopic to fj', and then to show that I.Ilis construction sets up a 1-1 correspondence
U Conjugacy cbu..;:
,
Atiyah: Representations of braid groups
116
1
.~
1
_
o~-----
1
The graph of a braid Thus an invariant of monotone links in Sl x R 2 is just the same thing as a set of class functions on the braid groups. In particular, for any finite dimensional linear representation p of B n we obtain 8Jl 51 x R 2 link-invariant pi through the character
p'(f3')
= Tr p((3).
P
It is also possible to associate a link in 53 to a braid (3, using the standard embedding 8 1 X R 2 C R 3 • All links in the 3- sphere are obtained in this way and the isotopy classes of links in 53 can be regarded as obtained from the braids by imposing an equivalence relation generated by certain "Markov moves ". We get invariants of links in the 3-sphere from representations of the braid groups which ar "consistent" with these moves.
r
The representations of the braid groups which we will discuss have the feature that they depend on a continuous parameter q E C. When q = 1 the representations are just those coming from the symmetric group and in general they factor through an intermediate object; the Heeke algebra Hn(q). The idea of using Hecke algebra representations to obtain link invariants is due to Vaughan Jones and we refer to the extremely readable Annals of Mathematics paper by Jones [2] for a beautiful account of the more algebraic approach to his theory. Here we will just sun1nlarise the basic facts and definitions. To define the algebra Hn(q) we recall that there is a standard system of generators CTi i = 1, ... ,n - 1 for the braid group En which are lifts of transpositions in 5 n , as pictured in the diagram. The Hecke algebra Hn(q) is the quotient of the group algebra C[B n] :
Hn(q)
= C[B n]/ < (Ui -
l)(O"i
+ q) = 0 ; i = 1, ... , n -
1> .
In fact the braid group can be described concretely as the group generated by the Uj subject to the relations
Atiyah: Representations of braid groups
0
:
117
0 0
~
------
0
0
0
0
0
The braid
(J*2
O'i+lO"jO"i+l
=
O'iO"i+lO"j
UiUj
=
UjO"i
for
Ii - jl > 1.
So Hn(q) is the algebra generated by the aj subject to these relations and the further conditions (ai - l)(ai + q) = O. If q = 1 we obtain the relations =1 Hatisfied by the transpositions in Sn, and it is easy to check that the algebra Hn(l) is canonically isomorphic to the group algebra C[Sn]. For any q, a representation of Hn (q) gives a representation of the group algebra C[ En] and hence a representation of En and from the discussion above we see that when q = 1 these are indeed just the representations which factor through the symmetric group. 'rhese Hecke algebras arise in many different areas in mathematics, and the param(~t,er q can play quite different roles. In algebra we take q to be a prime power and let F be the field with q elements; then we obtain Hn(q) as the double coset algebra of the group G = SL(n, F) with respect to the subgroup B of upper-triangular luatrices. This is the sub-algebra of C[G) generated by the elements
ar
TD=
LX xED
fen' D E B\G/B. On the other hand in physical applications one should think of q
e ih where h is Planck's constant; the limit q lilnit of a quantum mechanical situation. ItS
--+
1 then appears as the classical
'I'llere is an intimate relationship between the representations of the Hecke algebra for general q and the representations of the symmetric group, i.e. of Hn(l). This n'lationship can be obtained abstractly using the fact that Hn(l) - the group algeIJra of the symmetric group - is semi-simple. We then appeal to a general "rigidity" I »rperty: a small deformation of a semi-simple algebra does not change the isomorpl,isln class of the algebra. Hence the Hn(q) are isomorphic, as abstract algebras,
118
Atiyah: Representations of braid groups
to C[Snl for all q sufficiently close to 1 - in fact this is true for all values of q except a finite set E of roots of unity. So for these generic values of q the representations of Hn(q) can be identified with those of Sn and we obtain a family of representations Pq,A of B n , with characters Xqt A, indexed by the irreducible representations A of Sn and a complex number q E C \ E. These are the representations whose characters are used to obtain the new link invariants. More precisely, the two-variable "HOMFLY" polynomial invariant of links can be obtained from a weighted sum of characters of the form:
X(q,z) = LaA(q,z)Xq,A' A
for certain rational functions aA of the variables q and z. The earlier I-variable Jones polynomial V(q) is obtained from X(q, z) by setting q = z. We will now recall some of the rudiments of the representation theory of symmetric groups, and the connection with the representations of unitary groups. The irreducible representations of Sn are labelled by "Young Diagrams" , or equivalently by partitions n = PI + ... + Pr, with Pt ;::: ... ~ Pr > o. For example the trivial representation corresponds to the "I-row" diagram, or partition n = n, the 1- dimensional parity representation to the I-column diagram, or partition n = 1+·· ·+1. Now let Y be the standard I dimensional representation space of the unitary group U(l). There are natural commuting actions of U(l) and Sn on the tensor product v®n = V 0 ... ~ V, so there is a joint decomposition: y®n =
EBAA 0 B~, ~
where AA is a representation space of U(l) and B A is a represenation space of Sn. The index A runs over the irreducible representation spaces of Sn, i.e. over the Young diagrams. So these Young diagrams also label certain representations AA of the unitary group. It is a fundamental result that the AA are zero except when the diagram has 1 rows or fewer ( that is, for partitions with at most 1 terms), in particular the irreducible representations of U(2) are labelled precisely by the 1 and 2-row diagrams. The co-efficients aA(q,z) have the property that they vanish when q = z for all diagrams A with more than 2 rows. Thus the I-variable polynomial V(q) uses only the representations of the Hecke algebra associated to the 1- and 2- row diagrams. These are the diagrams which label the representations of U(2) and this ties in with the quantum-field theory approach of Witten, in which the V-polynomial is obtained in the framework of a gauge theory with structure group U(2)( or rather SU(2)). The more general X polynomial is obtained from gauge theory using structure groups SUe 1), for all different ~lues of 1.
119
Atiyah: Representations of braid groups
(b) Geometric constructions of representations. It is natural to ask for direct geometric constructions of these representations of the braid group. We now change our point of view slightly: the braid group B n is the fundamental group of the configuration space Cn, so linear representations of B n are equivalent to fiat vector bundles over Cn. Thus we seek flat vector bundles whose monodromy yields the representations Pq,A, and we pay particular attention to the 2- row diagrams which appear in the V-polynomial. Constructions of these fiat bundles are already known in the context of conforInal field theory [4], using complex analysis, and these tie in well with Witten's quantum field theory interpretation of the Jones' invariants. In her thesis, Ruth Lawrence developed more elementary constructions which used only standard topological notions, specifically homology theory with twisted co-efficients. To describe her construction we begin with a fundamental example which yields the "Burau" representation and the Alexander polynomial of a link.
en.
Let X = {Xl' ... ' x n } be a point in the configuration space The complement R 2 \ X retracts on to a wedge of circles, so its' first homology is zn and there iH an n-dimensional family of flat complex line bundles over the complement , i.e representations 71"1 (R2 \ X) --+ C*. There is a preferred 1- dimensional family of representations v q , which send each of the standard generators of 1rl to the same complex number q. These are preserved by the action of the diffeomeorpmsm group of R2 \ X, and thus extend to families, as X varies in More precisely, let W n I)c the space
en.
It is not hard to see that H 1 (Wn ) is Z2, generated by a loop in which the point y t'llcircies one of the points of X and a loop in which one of the points of X encircles "'Bother. We consider the I-parameter family of representations vq : '1rl(Wn ) --+ C* which map the first generator to q and the second generator to 1. These restrict to l,ll(~ representations lJ q over the punctured planes R 2 \ X, regarded as the fibres of the natural map
We let Lq be the flat complex line bundle over W n associated to the representation 1'1/·
((t'call now the following general construction. If f : E ~ B is a fibration and /\It is a local-coefficient system over. E then for fixed r and for each b E B we ('all obtain a vector space Vb = Hr(f-l(b); M). The spaces Vb fit together to d.·fine a vector bundle over B, and this bundle has a natural flat connection, since Ilolllology is a homotopy invariant. The monodromy of this connection then gives Il n~presentation of 1rl (B), the action on the cohomology of the fibres. We apply l.his in the situation above with the map p and the co-efficient system £q (or, more
Atiyah: Representations of braid groups
120
precisely, the sheaf of locally constant sections of the flat bundle L,q) taking r = 1. This gives us a flat bundle over with fibres Hl(R2' \ X;.c q ). (Note that we could consider a 2-parameter family of representations of 1rl (Wn ), using the extra generator for H l (Wn ), but this would give no great gain in generality, since it would just correspond to taking the tensor product with the I-dimensional representations of the braid group.) To identify the representation which is obtained in this way we begin by looking at the twisted cohomology of R 2 \ X. We can replace this punctured plane by a wedge of n circles, and use the corresponding cellular cochains :
en,
C1 =
CO = Z ,
zn,
with twisted co-boundary map h': Co -. Cl given by «5 = «l-q),(l-q), ... ,(1q». For q 1= 1 the I-dimensional twisted cohomology has dimension n - 1. It is not hard to identify the n - 1 dimensional representation of the braid group which this leads to. It is obtained from a representation on en by restricting to the subspace of vectors whose entries sum to o. The standard generator qi of En acts on en by fixing all the basis vectors el, ... en except for ei, ei+l and acting by the matrix
(1 ~ 6) q
on the subspace spanned by ei, ei+l. The representation on the vectors whose entries sum to zero is the reduced Burau representation, and this is in fact the representation P>..,q obtained from the partition n = (n - 1) + 1 . (There is some choice in sign convention in here: the automorphism Gi t-+ -(1i of Hn(q) switches rows and columns in our labelling of representations by Young diagrams.) The representation is clearly a deformation of the reduced permutation representation of Sn, which is obtained by taking q = 1. The Alexander polynomial appears in the following way: if (3 is a braid and tPfj is the matrix given by the Burau representation then det(l -1/Jfj(q» = (1
+ q +... + qn-l )~,8(q),
where ~iJ is the Alexander polynomial of the knot /3 in the 3-sphere. Lawrence extends this idea to obtain other representations of the braid group. The extension involves iteration of the configuration space construction. Let Cn,m be the space : Cn,m
= { ({Xl, ... , Xn }, {Yl' ... ' Yrn})
E C n X CmlXi ~ Yj for any i,j }.
There is an obvious fibration Pn,m : Cn,m -. Cn . Notice that On,1 is just the space W n which we considered before, and Pn,l = p. In general we will obtain representations of the braid group from the twisted cohomology of the fibres of Pn,m·
Atiyah: Representations of braid groups
121
For m > 1 the group H 1 (Cn,m) is Z3, with generators represented by loops in which (1) one of the points Yj encircles one of the points Xi, (2) one of the points Yj encircles another, (3) one of the points of Xi encircles another. For the same reason as before we may restrict attention to representations which are trivial on the third generator. Thus we consider a 2-parameter family of representations iiq,a of '1rl(Cn ,m), which map the first generator to a and the second to q. We let L,q,a be the corresponding flat line bundle over Cn,m, then for each a and q we have a representation ¢Jm,q,o: of the braid group B n on the middle cohomology of the fibre:
Hm(p;;,lm(X) ; £q,a). Lawrence proves that these yield all the representations corresponding to 2-row Young diagrams. More precisely, we have: THEOREM ,[3].
Suppose 2m ~ n and let ,\ be the representation of the symmetric group corresponding to the partition n = (n - m) + m (a 2-row Young diagram). If a = q-2 t,he representation
