ñòð. 3 |

11hllowing Gromov (see [Gr) we say that an almost complex manifold is tame if

exists a symplectic structw"e w on X such that the form weT, JT), T E T(X)

Uu'rc

iN positive definite.

Nute that any Stein (i.e. affine) cOlnplex manifold, J(ahlerian manifold or (genuine)

(,oluplex manifold with pseudoconvex boundary is tame. Any symplectic manifold

,uhnits an ahnost cOlnplex structure ta.med by the symplectic structure.

All ahnost complex manifold )(, J is called holomorphically a8pherical if there are

tlo holomorphic embeddings Cpl -+ (X, J).

˜. FILLING BY HOLOMORPHIC DISCS

Iia 1969 E. Bishop (see [Bi]) discovered that a punctured neighborhood of an elliptic

point admits a (unique!) foliation by circles spQnninp; holomorphic discs. By simple

topological reasons these discs have to be di of a Levi-flat

--!.--Q

50 Eliashberg: Filling by holomorphic discs and its applications

3-manifold. Many attempts were made to globalize Bishop's result: How far from

the elliptic point can the family of circles spanning holomorphic discs be developed?

The first serious success in this direction was obtained by E. Bedford and B. Gaveau

in 1982 (see [BG]). They considered the following problem: Given a 2-sphere 8

embedded in C 2 with exactly two complex points (which are in this case necessarily

elliptic), does there exist a Levi-flat 3-ball in C 2 bounded by the sphere 8? Or

equivalently, does the complement of complex points in S admit a foliation by

circles spanning holomorphic discs?

Bedford and Gaveau gave a positive answer to this question under two additional

assumptions:

n c C2 ; .˜

-8 is contained in the pseudoconvex boundary of a domain :"˜ !

.˜I

-orthogonal projection of 8 onto R3 C C 2 is an embedding.

Â·f

While the first assumption looks very important and cannot be completely removed

(see Section 9 below) the second is absolutely unnecessary and even the original

-:˜

Bedford-Gaveau's proof can be easily modified to work without this assumption . !˜

(see [EI2]). The first c0111plete proof of the result belongs to M. Gromov (see [Gr]). 1;

':˜

The key point in Bedford-Gaveau's proof is Lipshitz estimates for boundaries of :;,˜

holomorphic discs while Gromov uses the more general compactness theorem which

he proves in the same paper.

In our joint paper with V. M. Harlamov we related the technique of filling by

holomorphic discs with topological problems and, in particular, with Bennequin's ',˜

).˜

.,y(

theorem [Be] \vhich was just proved. We showed that if one could prove a filling i}

.;/

result for spheres which do not belong to a pseudo-convex boundary, that would ",

imply very strong corollaries in low-dimensional topology (see Problem 10.7 below).

Unfortunately (or fortunately?) the result in that generality is false (see Section 9 1

below) but the problem is far from being understood.

Since that time there were two major breakthroughs which sufficiently extended

possibilities of the method. In 1985 Gromov wrote his famous "Pseudoholomorphic

curves in symplectic manifolds" \vhere he showed that holomorphic curves and,

in particular, holomorphic discs can be successfully used not only in complex but

Eliashberg: Filling by holomorphic discs and its applications 51

".tHO in tame almost complex manifolds. Recently E. Bedford and W. Klingenberg

(Are [BK]) found a teccnique for developing the family of holomorphic discs in the

I..'('sence of hyperbolic complex points. That allowed them to prove the result about

til(' filling of a 2-sphere in C 2 without any additional assumptions (besides pseudo-

,Â·cHlvexity).

the next section I formulate \vithout proofs a summary of results concerning

III

I'..,˜ularity and compactness properties for fa.milies of holomorphic discs. The main

"rrntlts (Theorems 3.4.1 and 3.4.2) can be proved within the ideology of [BG], [BK]

luul [Gr] but are not straightforward corollaries of results of these papers. The gen-

"I,tl.li˜a,tionof [BI(] for an almost complex case requires a fine analysis of singularities

ur .T-holomorphic curves which was recently done by D. McDuff (see [McD2]).

:i. COMPACTNESS AND REGULARITY PROPERTIES FOR FAMI-

L114˜S OF HOLOMORPHIC DISCS

:t.1 'P-admissible sets

By Morse function we mean a function with nondegenerate critical points and

H

puil'wise different critical values. By an almost Morse function we mean a function

wl,i('h has either non-degenerate or birth-death type critical points and at most

dlluhle critical values.

L.,t I( be a 3-manifold with the boundary 8I{ = M and 'P : K --+ R be a function

Wh()H(˜ restriction on /vI is ahnost Morse. For a closed subset A C K we denote by

ii' A the intersection A n 8I( and by 8"A the rest of its boundary: a˜˜=:A.

A closed subset A C I( is called tp-admissible if the following conditions AI-A5 are

˜"t.iHfi(\d:

A1. 'rhe function 'P does not have critical points in A.

A:J.. lutA is a union of some components of sets {x E IntI<, e < c.p(x) < e'} where

are critical values of <.p l/VI. in particular, <p is constant on components of 8"A.

' \ ,,I

/\:1. For any e E R all components of the level set A c = {x E Int A, <p(x) = c} are

ftilUpJy connected.

A-1. For any critical value c of'P 11\1 the intesection of closures in K of sets {<p(x) <

'I .,. > C, x E ..4} is x E A}.

C A} and {tp(x) open-closed subset of {tp =

all C,

Eliashberg: Filling by holomorphic discs and its applications

52

a"A

contains a critical point of the function ˜ 1M and

A5. Any component C of

it is either not simply connected or it does not cover the whole component of the I

1<, Ie}.

level set {x E <p( x) = 'P

HAC K is ep-admissible then one can form a graph GA,cp identifying each compo-

nent of level sets {x E A, ep(x) = c}, c E R, to a point. Vertices of G correspond to

local maxima and minima of 'P 1M and to components of aliA.

..

Let 1r be a projection A -+ GA.",. For a subgraph r c GA,cp we will denote by Ar

the inverse image 1r- 1 (r) and by Mr the intersection Ar n M = ()'Ar. Note that

= r. r

if the set Ar is ˜-admissible then G Ar.cp We will say in this case that is an

admi8sible subgraph of G.

The function 'P factorizes to a function <j; : G A,cp ˜ R with <p = 1r 0 tj;. The function

rp is non-degenerate on each edge of GA,cp and, therefore, defines an orientation of

the graph GA,'P.

3.2 Partial and maximal fillings

Let M now be an oriented closed 2-surface in an almost complex manifold X, J.

By a partial filling of M by holomorphic discs we mean a quintuple

, = (1<, cp, A, rt, F) which consists of:

.J

-a 3-manifold J( bounded by M, ;

1M' .˜

-a function <p : I( --+ R with an almost Morse restriction <p

.˜

-a cp-admissible set A , ' .˜

-a family 1Â£ of complex structures on levels Ae = {x E Int A, <p(x) = c} which .˜

˜

depends smoothly on c E R,

I

-a smooth map F : A -+ X which is holomorphic on levels Ae , c E H, and coincides

with the inclusion 8'A ˜ )( on a'A = An M. /˜

'˜

Note that for any component ˜ of Ae , c E R, the image F(˜) is a holomorphiej

disc in X and its boundary is contained in M. For any partial filling ( all critical

points of ˜ 11\/ from A are, necessarily, complex points of M and elliptic points '

correspond to local maxima and minima of <p 1M' We will require, in addition,

that local maxima and minima of c.p 1M correspond to negative and positive elliptic I

points, respectively. With this choice of the orientation of M the orientation of any

Eliashberg: Filling by holomorphic discs and its applications 53

˜ C,

x E A'A = MnA} x E {)' A},

{<p(x) = {cp(x)

as the boundary of the set

.Â·ul've C,

.l (: R, coincides with its orientation as the boundary of a corresponding holomorphic

F(˜).

.tiNe

A partial filling is called maximal if all elliptic points of M belong to A and are,

UH˜rcfore, local maxima and minima of cp 1M. A partial filling is called generic if

V' IM is a Morse function.

WtÂ· Hay that M can be filled by holomorphic discs if it admits a maximal filling with

Ii. Note that in this case]( must be a handlebody bounded by the surface M.

" .::=

II. l>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

'I'IIEOREM 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.

n

3.4.2. Let X, J and 0, be as above. Suppose that, in addition, is

'I'UI':()REM

1&f1lcnnorphically aspherical. Let M be a closed 2-surface and ˜ be an embedding

an. Then the embedding ˜ can be C 2 -perturbed in such a way that

AI x I ---.

If

surface ˜(M x t), tEl, admit3 a maximal fiZling by holomorphic discs.

tJfl.fh

x t), tEl, can be filled by holomorphic discs then there exist a

fllld˜h ,˜urface ˜(M

Ilft'U.tllebody I{ and an embedding ˜ : !{ x I ---. n such that ˜ coincides with q, on

n is

AI x I and for any tEl the image 4Â»(M x t) C Levi-flat.

:1." Graph of a filling

With any partial filling ( = (/(, tp, ..4 , H" F) we can associate the graph GA,<p (see

:1.1). If ( is a maxhnal filling of M then \ve ",{ill write GM instead of GA,cp. Vertices

ttl' (,t 1\1 correspond to all elliptic and some hyperbolic points of M. Elliptic points

un' ('ud-points of GM. The orientation of GA1 has been chosen in such a way that

ftcll1:(˜H start at negative elliptic points and end at positive ones. H M is filled by

JIt.ltHllorphic discs then all end-points of GAl correspond to elliptic points of M.

If this is not so then any connected cOlnponent of GM must contain an end-point

˜

j

Eliashberg: Filling by holomorphic discs and its applications

54 ':˜

,˜

1

which is a hyperbolic point of M. Note that edges can start at positive hyperbolic

.˜

points and end at negative ones.

˜

1

3.6 Cutting A out of M

l

Let ( = (1(, '1', A, 11" F) be a partial filling of M and G = GA,tp be the graph of

the filling. The boundary of A is a piecewise smooth surface which consists of two ˜

˜j,˜

parts: 8A = (f Au{)" A. It can be sInoothed in such a way that the new surface MA ˜.

.

is filled by holomorphic discs. The graph GM A is isomorphic to G. End-points of ':','1;,

G which correspond to hyperbolic points of M are replaced in GMA by end-points

:˜::::s;:nsd;˜l::::::::;˜LLEDBY HOLOMORPHIC DIscsl

n de- .˜

Through the rest of the paper )[, J denotes a tame almost complex manifold,

notes a domain in X \vith a smooth J-convex boundary and M denotes a connectedJ1 ',

2-surface (sometinles closed, sOluetilnes \vith a boundary) which is contained in an.

I

For purposes of our applications nothing changes with a small C 2 -perturbation. So

we will speak about the filling of a stuface !vI Ineaning that it can be done after,

'j

probably, some small C 2 -perturbation.

'˜i

an different/rom 52 satisfies the inequality

4.1. Any closed sur/ace A1 C

THEOREM

X(M) ::; -lc(M)1 .

If M is diffeomorphic to 52 then it can be filled by holomorphic discs. In particular, '

n.

it bounds an embedded ball B 3 C :1

iJ

Proof. IT M is not 52 but can be filled by holomorphic discs then c(M) = 0 and the ˜l.˜ ,

inequality is autonlatically satisfied. Hence it is enough to consider'the case when :1

1

M cannot be filled by holomorphic discs. Let ( = (K, cp, A, 'H., F) be a maximal

filling of M. Then a'A conta.ins no closed components. Cutting A out of M we get :t

a surface M' = AlA which can be filled by holomorphic discs. The graph GM' of

this filling coincides with G A1. Let us denote by k+ and k_ numbers of positive and

negative hyperbolic points of M \vhich have been substituted by elliptic points of

MA. Then

eÂ±(Al') = eÂ±(1\l) + k=f

hÂ±(Al) 2 hÂ±(J..l') + kÂ±

Eliashberg: Filling by holomorphic discs and its applications 55

Note that if bo = bo(M') is the number of components of M' then k+ + k_ ˜ boo

Now we have c(M') = 0 because M' is filled by holomorphic discs. Therefore,

qllalities 1.4.1 imply that

11

11.'uee,

dÂ±(M) = eÂ±(M) - hÂ±(M)

+ hÂ±(l\1') - hÂ±(M)

˜ bo - k˜

˜˜=F - kÂ± = b - (k - + k+) ::; 0

::; bo -

But

1

dÂ±(A1) = 2(X(l\1) Â± c(M))

which implies the required inequa.lity

X(1\1) ˜ -lc(1\I)1

Q.E.D.

that Theorem 4.1 provides an obstruction for a 4-manifold to carry a structure

Not.e

flf a tame almost complex Inanifold ,vith the J-convex boundary. For example,

( ˜()HOLLARY 4.2 S2 x D 2 has no tame almost complex structure with the J -convex

bO'l/,'fI,dary.

Illdc\ed, a sphere S2 x p,p E 8D2 does not hound any ball in S2 x D 2 which

4.1.

Hltradicts

fie

A eontact structure on a 3-n1anifold V is called fillable if it can arise as the dis-

eof complex tangencies to the J-convex boundary V = an of a domain

tribution

X, J. Note that the first Chern class Cl(X)

\l ill a tame aln10st cOlnplex Inanifold

e.

Je'stricted to V coincides with the Euler class e(e) of the bundle Let us denote

Ily Fill(V) the subset of H 2 (V) which consists of classes e(e) for all fillable contact

eon

!1t.1't1(˜tures V.

56 Eliashberg: Filling by holomorphic discs and its applications

4.3 The set Fill(V) is finite.

COROLLARY

E H 2 (V) can be realized by a smooth surface MeV.

Proof. Any homology class J.L ;J

According to 4.1 we have le(e)[Jlll = te(M)1 ˜ -X(M). Thus e(e) can take only :˜

finite many values on any homology class from H 2 (V).

on be a 2-surface with the boundary:::

Now we consider the relative case. Let AI C

eof

aM complex tangencies to an. Let us choose ˜

transversal to the distribution

the orientation of M which induces the positive orientation of 8M (see 1.2). ;

equivalently, X(M) $ e(M).

4.4. If lv/ is a.'.f above then d_ $ 0

THEOREM OT,

This is a generalized Bennequin's theorenl. In [Be] he proved the inequality for the ,I

= C2 â€¢

n is a ,˜

case when round ball in .'\.

˜˜

"1

Mwhich arises as the boundary'

Proof. Let us double M: consider a closed surface

an.

of a tubular neighborhood of M in We can consider the oriented surface M to)

be contained in Iv! as a half of M. Let G= Gif be the graph of the maximal filling 'j

va

be the set of those vertices of G\vhich correspond to negative elliptic ˜

of M. Let

points of M which belong to A([. Let Go be the least admissible (see 3.1) subgraph ':˜

of Gwhich contains Vo. Let Ao = AGo and Afo = Mao be parts of A and Mwhich ˜

correspond to Go (see 3.1). Note that 1\10 is contained in IntM. Indeed, if it is not:

so then there exists a hololnorphic disc 6. whose boundary {)˜ C M is tangent to '

a6. ;

8M at a point p E 8A1. By our choice of orientation of M the orientation of

(as the boundary of the hololnorphic disc ˜) at p coincides with the orientation

of oM as the boundary of AI. But the complex orientation of a˜ is opposite to :

at::. induced by the orientation of the set {ep ::; c} (see 3.2). This

the orientation of

M we get

c l11t AI. Cutting Ao out of

contradiction proves that M o a sunace

MAo

Mo = (see 3.6). Then, arguing as in 4.1, we get

o= c(Mo) = d+(Mo) - d-(Mo), d+(Alo) + d_(Mo) = X(Mo)

= e_(M)

e_(Mo)

h-(Mo) ˜ h_(M) - bo(Ma)

Eliashberg: Filling by holomorphic discs and its applications

- 1 ---- ..-.

e_(M) = e_(Mo) = 2X(Mo) + h_(Mo)

1 ..-.

'2 X(Mo) + h_(M) - bo(Mo)

$

= h_(M) - bl(˜O) ::5 h_(M)

o. Q.E.D.

which means that d_(M) = e_(.1\l) - h_(M) :5

=L

4.5. Let Mean be a 8urface with a Legendrian boundary aM

(˜()ROLLARY

It be the homology class of M in H 2 (an; L). Then

tÂ£11.d

X(M) ;5 tb(Llp) -lr(Llp)lÂ·

prove 4.5 we can consequently approximate M by a surface whose boundary is

'('0

eand apply 4.4 and 1.4.2 ( see[Be] for similar

n. negative or positive transversal to

uol'guments).

IS. MANIFOLDS BOUNDED BY A J-CONVEX SPHERE

'l'he theorem of this section indicates that the topology of the J-convex boundary

itllposes very strong restrictions on the topology of the domain.

5.1. let X and n be as above and suppose, in addition, that n is holo-

'I'IIEOREM

1norphically aspherical. If an is diffeomorphic to S3 then n i8 diffeomorphic to the

"nIl B 4 â€¢ Moreover, n admits a J -con'vex function which is con8tant on an and has

(::J;(J,ctly one critical point-the minim'lI,m-in O.

Take a Morse function h: an ˜ R with exactly two critical points. All non-

1"1'0of.

c"riticallevels of h are diffeomorphic to S2. By a small C 2 -perturbation of 8W near

critical points of h we can arrange that level-sets of h which are close to critical

IH)ints are filled by holomorphic discs. This defines a foliation of a neighborhood

of critical points in n by Levi-flat 3-balls. By an additional C 2 -perturbation of

as! we can arrange (via 3.4.2) that all other levels also admit maximal filling by

lu)lornorphic discs. But all these levels are diffeomorphic to 8 2 and, therefore,

j

'˜

58 Eliashberg: Filling by holomorphic discs and its applications

I˜I

according to Theorem 4.1 they are actually filled by holomorphic discs. Hence ,.i

1

an

each 2-sphere {h = c} c bounds a Levi-flat 3-ball Be. The family of these

balls depends smoothly on c E R (see 3.4.2) and are pairwise disjoint. Indeed, if ˜

;˜

nBc' =F 0 for c :/= c' then there exist holomorphic discs ˜c and ˜e' with aA e E :.%

Be

Â·iÂ·˜

aBe, at1 c E aBel and ˜e n ˜Cl =F 0. The intersection index of two holomorphic ::˜

l

˜e' is positive (see [McD2] for the non-integrable case) but it has to be :J,

discs A c and

zero because their boundaries aA c c aB˜ and. aA c: C aBcl bo˜d discs in disjoint,)

n IS folIated by LeVI-fiat balls Be and, ;:!

spheres aBc and aBel. Hence the domaIn

J

therefore, ˜eomorphic.to B4. To prove the .second part of the .theorem, denote by

H the functIon on 0 whIch extends h and whIch has balls He as Its level-sets. There jJ

exists a diffeomorphism c.p: n - n which is fixed at the boundary an and such that :˜

,!J.ll

all level-sets of the function HI = H 0 t.p are J-convex. As it was explained in 1.1 ".j

there exists a function i: R ˜ R such that the composition i 0 HI is a J-convex ˜˜

function. Let M = maxo H2, m = minn H2. Let H3 be a J-convex function which ,˜

is defined in a tubular neighborhood U :> an and which is constant on an and au.:˜;

˜˜

We can arrange that H a Ian> M and H 3 launn< m. Now extend somehow H a on.˜

n \ U with maxH3 Ifl\U< m and let H 4 = max(H3 , H 2 ). Smoothing H 4 we get the!

required J-convex function H s : n ˜ R. :˜

QED :;J

... ˜

:˜

D. McDuff's study of blowing down in tame almost complex manifolds allows us to:˜;j

j

understand what is going on when n is not holomorphically aspherical.

,{˜

,:rQ

5.2. Let X, J be a tame symplectic manifold and 0 c X be a domain::˜

THEOREM

Â·˜:˜S

bounded by a J .. convex 3..sphere. Then fOT an almost complex structure J' which i,,˜i˜

',1j:;

Coo -close to J the manifold (Q, JI) is a 4..ball up to blowing up a few points. In;:˜˜

.˜

particular, n is diffeomorphic to B4#kCP2.

.˜

,;;;,

ietl

Proof. According to [McDl] there exists an almost complex structure J' which

';˜

COO-close to J and such that all elnbedded J-holomorphic spheres in n are disjoint,!˜

non-singular and have self-intersection number (-1). These spheres can be blown:1

down to give a non-singular hololnorphically aspherical almost complex manifold:˜

0' with the same boundary an. Thus we can apply 5.1 to conclude that Q' is:˜

diffeomorphic to B4 and, therefore, n is diffeomorphic to B 4 #kCP2.

Eliashberg: Filling by holomorphic discs and its applications 59

eon

5.3. Let X, J and n be as in 5.2. Then the contact structure

(;OROLLARY

O!! (formed by complex tangencies to an) is isomorphic to the tJtandard contact

.' trhlJ,cture on S3 (i. e. 3tructure induced on a unit sphere in C 2 .)

/',"oo!. Perturbation of J does not change the contact structure. So using 5.2 and

tâ€¢. l we can get a ball n' bounded by an and a J-convex function H on n whic˜

1M (˜onstant on 8n = an' and has exactly one (non-degenerate) critical point: the

luillimum. All non-singular level sets of Hare J-convex and diffeomorphic to 8 3.

IIrllee all of them have contact structure and, therefore, all these contact structures

urn iHomorphic (see 1.2). But near the critical point the almost complex structure

."Ul he made integrable and level sets can be made biholomorphically equivalent to

Ulf' round sphere in C 2 and, therefore, the contact structure on all level-sets is the

Mtn.ll<!ard one.

o. I(ILLING OF ELLIPTIC COMPLEX POINTS

surface At c X has hyperbolic and elliptic points p, q E M of the same sign

If Il.

t.1U\1l one can easily CO-perturb M along a path connecting p and q to kill these 2

IJuilll.s (see [HE]). If M is contained in a J-convex boundary an of a domain n c X

an.

t.lU'1i it would be itnportant to be able to kill p a.nd q by an isotopy of M in8ide

tl'laiH is impossible to do in a neighborhood of a path connecting p and q (it would

11tlllt.l'H.dict the sY111plectic rigidity) but as it follows fronl Theorem 6.1 below this is

pu,mihle when enough room is provided.

VVC' Htart with the following definition. Let G be the graph of a maximal filling

Itt' A'I. An elementary contraction C, of G is the removal of an edge I with ends

whi.1b eorrespond to elliptic and hyperbolic points of the same sign. The following

11"'01'('111 which will be proved in [EI3] sho\vs that any elementary contraction can

lUI nÂ·ali7.cd by an isotopy of !vI inside an.

Mean, be a closed surface, G be a graph of a generic

6.1. Let M,

'I'IIEOHEM

1'" "lzt/,I filling and I be an edge of G with ends which correspond to elliptic and

'''JI,,Â·.,.!Jolic point3 of the ..˜ame sign. Let All be a part of M which is covered by

,.,,,, "tln1'i c., of holo m01Â·phic discs 'which correspond to points of 1. Then there ezist8

an an, t E [0, 1], ..fntch that h o = id, h t is fixed outside a

" ,h'l'cot()py ht. : -+

Eliashberg: Filling by holomorphic discs and its applications

60

neighborhood of M, for all t E [0,1] and the surface h 1 (M) admits a maximal filling

whose graph is the result of the contraction C, of G. In particular hI (M) has two '

complex points less than M.

Now we apply 6.1 to prove

6.2. Let M be a closed surface in aQ. Suppose that X(M) = -lc(M)1

THEOREM

(comp. 5.1). Then M either can be filled by holomorphic discs (and it has to be ,(

diffeomorphic to the torus T 2 in this case) or is isotopic in an to a surface M' ,˜

,l

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. .J

,˜

;˜

Proof. Suppose that :AI cannot be filled by holomorphic discs and let GM 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

.˜

+ k_

eÂ±(MA) = eÂ±(.I\1") + k˜, hÂ±(fl.1) 2:: hÂ±(MA) + kÂ±, k+ ;:::: bo(MA ) (*) 7:'

'1

o. Then X(M) = -c(M) ˜

Let us choose the orientation of M such that c(M) 2:

0, d_(M) = !(X(M}-c(M)) = X(M) . j

= !(X(Al)+c(Al)) =

and, therefore, d+(Al)

e_(.J\,I) = h_(M) + X(M).

or e+(M) = h+(IvI), Suppose that GM admits no)!

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_(M) ,

˜

h+(MA ) X(M)

˜

h_(MA) h+(IVI) and

+ h+(M) + h_(M)

+ h_(MA) 2

h+(MA) x(1\.f)

Eliashberg: Filling by holomorphic discs and its applications 61

or

the other hand, the equality c(MA) = 0 implies that

()n

Now we get

1

"2 X(MA)

h+(Al) + k_ - h+(MA) = ,

1

+ X(1\l) + k+

h_(M) - h_(MA) = 2X(MA) .

'rile first equality and (*) give

,u1d therefore,

1

+ k_ = '2 X(MA)

bo(MA) = k+ Â·

But then the second equality and (**) imply that

= o. But then MA is empty which is impossible if M has

",lid, therefore, bo(1\IA)

(illiptic points.

Q.E.D.

6.3. Let M be a 2-s'u1jace in an 'with the Legendrian boundary L = aM

'('IIEOREM

tÂ£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

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

'I'h(˜n

.Iu/.1jace M without interior elliptic point3.

an

which contains M and let ( = (1<, cp, A, 1t, F)

I'7Â·0o!. Let M' be any closed surface in

I)(˜ its maximal filling by hololnorphic discs. Let r be the minimal admissible (see

:1.1) subgraph of the graph of ( which contains all interior elliptic points of M.

Eliashberg: Filling by holomorphic discs and its applications

62

Let :Alr = 8'Ar = M' n Ar be the corresponding part of M'. I claim that Mr is

contained in IntM. Indeed, if it is not then there must exist a holomorphic disc ˜ '!˜

whose boundary a˜ is tangent to L = 8M. But that is impossible either in the .˜

totally real part of L or in a complex elliptic point which belong to L. The rest of:˜

the proof repeats the proof of 6.1 \vhere, of course we need to use the formula from :˜

,˜˜

1.4.2 insteed of 1.4.1. J

.\

Q.E.D..}

. 'Iii

:.˜

˜

,

.˜t

7. LEGENDRIAN KNOTS .:˜

';)

'iJI

:ii

eo

be the standard contact structure ,˜

2

Let S3 be a boundary of a. unit ball in C and

on S3 (formed by complex tangencies). By a standard Legendrian curve in S3 .˜

we will mean an intersection L o of S3 with a Lagrangian plane P in C 2 or any ':˜1

Legendrian curve \vhich is Legendrian isotopic to this intersection. Note that if J˜

L o is a standard Legendrian curve in S3 then r(L o) = 0 and tb(L o) = -1. Any'::;˜

contact 3-manifold M, eis locally isolTIorphic to S3 \ p (Darboux chart) with the ˜ ;˜

'.,˜

standard contact structure. 'Ve will call a Legendrian curve L C M standard if it ': :˜ ˜

is Legendrian isotopic to a Legendrian cW've \vhich is standard in a Darboux chart...',˜

"J:

With-˜

The curve L o bounds in S3 a hemisphere V o which is the intersection of S3

a hyperplane which contains the Lagrangian plane P. The hemisphere V o has no;'1

interior complex points and has exactly two elliptic points on the boundary Lo.;˜

That implies that leaves of the characteristic foliation (V o)eo are Legendrian arcs,';˜f˜

which connect the t,vo elliptic points on the boundary. Note that using these arcs \f

Â£':1'

one can easily construct a Legel1dl'ian isotopy in an arbitrary small neighborhood"˜:!:

I â€¢.!J.

of V o into a small standard Legcndrian curve. ':';\;

n be an di8C'\;;˜

7.1. let ..Y , J and a.s in 6.3 and V C be an embedded

-I.':

THEOREM

= L. = 0 and tb(OV) =

bounded by a Legendrian curve OV Suppose that r(OV)

:?˜

L is standard.

Then the Legendrian C'ltrve

Proof. The disc V can be deformed (without cha.nging the boundary) to a disc 1)'

xCV') = -tb(L) - r(L) and ,i˜

which is in norlnal for111 near L. The equality 1 =

Theorem 6.3 guarantee a possibility of finding a disc V" which coincides with V'

near the boundary and which does not have interior elliptic points. But according "

˜ i˜

Eliashberg: Filling by holomorphic discs and its applications 63

'De

= o.

to 1.4.2 we have dÂ± Therefore, has no interior complex points at all and

two elliptic complex points p and q at the boundary. Then the characteristic

llltH

eVil on V" is a foliation by Legendrian arcs connecting p and q.

fc)lintion But this

huplies that the pair V", {)T)" is contactomorphic to 'Do, L o and, therefore, L can

IJr eontracted to a small standard Legendrian curve through a Legendrian isotopy

ill an arbitrary small neighborhood of V".

Q.E.D.

II. l>rilne decomposition of l11anifolds with a pseudo-convex boundary

111 t.his section we consider talne ahnost complex manifolds il, J with a J-convex

an to be defined up to a deforlnation of an which keeps it J-convex.

bouudary

lH VPll a Levi-flat embedding h : (B, aB) --+ (0, an) of a 3- ball B one can extend it

tu 'Ul embedding H : B x [-1,1] --+ n with HIBxO = h, H(8B x [-1,1]) C an and

˜tlcth that balls H(B x 1) and H(B x (-1) are J-concave (with the orientation wich

,˜ illcluced 011 the boundary of the dOlnain H(B x [-1,1]). The boundary of the

tt.ullplement domain n \ H(Bx] -1, 1[) can be canonically smoothed to become J-

.˜cJllvex. We say that the new Inanifold 0' is the result of cutting 0 along a Levi-flat

,,,,11 h( B).

(Hvctu two tame almost complex rnanifolds 0 1 , Jl and O2 , J2 with pseudo-convex

IItt1I1Hlaries one ca.n canonica.lly define the boundary connected sum 0 1 #0 2 , J 1 #J2

ant #80,2.

f'˜ It t.alne almost cOlnplex 111anifold \vith the pseudo-convex boundary

tl'llc' manifold can be characterized by the property that cutting it along a Levi-

IH'W

""t. ball we can get the disjoint union of original manifolds n1 , J1 and O2 , J2 â€¢

I,,'t. lt˜ eall a manifold n, J prime if it cannot be presented as the boundary connected

12 1#n 2 ,J1 #J2 of two tame allnost complex manifolds with pseudo-convex

lttlill

t."Huda.ries where ill and n2 are not diffeomorphic to the 3-ball B. We say that

\',.J iH almost prime if nt and Q2 are not difFeolnorphic to B#kCp2.

r1I1';() It E tvl n, J 1]Jith a J -convex boundary is almost prime if and

8.1. A ma nifoltl

an is a p1'ime 3-manifold.

u"iJl if 1:t.'J boundary

64 Eliashberg: Filling by holomorphic discs and its applications

9. EXAMPLE OF A FAILURE OF THE FILLING PROPERTY

n Jut B 3 , t E] - 1, 1[, are holomorphic;

F n restricted to open discs V t = Pt

Eliashberg: Filling by holomorphic discs and its applications 65

Suppose that F n with these properties exists for all n = 1,.... First of all the

tuutlysis of the local behavior of hololnorphic discs near a given disc (see, for eXaIn-

pl(˜, [Gr]) shows that F n 152: 8 2 --. ˜n(82) has to be an immersion and, therefore,

elnbedding. Then the boundary of a holomorphic disc Fn(V t ), t e] - 1,1[, di-

hÂ·1l

vicles the sphere <Pn(S2) into two henlispheres S˜t and S;,t. Let w be the standard

˜Ylnplectic form in C 2 â€¢ Then, according to Stokes' theorem,

J Jwi

= I

w= $ Area S:.t $ Area rt'n(S2) Â·

Area Fn(Vt )

S˜,t

Fn ('Dr )

'rllis means that a.reas of all hololllorphic discs are uniformly bounded and according

to Gromov's cOlnpactness theoreln (see [Gr]) for each t E] - 1,1[ the sequence

II˜, Iv&, n = 1, ... , has a converging subsequence. (It is easy to check that cusp-

.1'Â·˜(˜llerations cannot occur.) But then each point of the immersed sphere <,0(82 ) C

l 2

H has to belong to a hololnorphic disc. But it is not so for points from

CC

.,,('Do ) C cp(S2). Q.E.D.

10. OPEN QUESTIONS AND CONJECTURES

10.1 Which prime 3-manifolds (LrC pseudo-convex boundaries of compact domains

tit 1.lJ,',ne almo..qt complex manifold˜?

Wr' will call such 3-Inanifolds fillable. No one example of a nonfillable manifold

i˜1(llown. Note that without the condition "talne" all 3-manifolds are, obviously,

UIII1.I)le.

111. I.A If one drops the word "almost" in 10.1 does it change the class of fillable

:J uULnifolds ˜

N. If,f' that results of [Ell] ra.ise the hope tha.t the clas of fillable (in both senses)

:1 Illu.nifolds is pretty large.

10.2 How many different (lÂ£P to blowing up) 4-manifolds can have the same 3-

'" fUI.7!()ld as a pseudo-convex boundary f

'I'll(' sphere S3 has the unique filling (see 5.1) but it is possible that the same M 3

1)(' filled by a few different 4-ll1al1ifolds (for example, the lense space L(4, 1) can

.. " II

I..Â· liJhÂ·<t at least by t\VO different lunnifolds, see [MeDl]).

66 Eliashberg: Filling by holomorphic discs and its applications

10.3 How many different fillable structures can arise on the same 3-manifolds?

Conjecture: A fillable contact structure on a 3-manifold is uniquely determined by

its Euler class (see Section 4). Together with 4.3 it would imply the existence of

only finite different fillable contact structures on a given 3-manifold. Theorem 5.3

proves the conjecture for 1'.1 = 8 3 â€¢

an

n be a compact domain with the J-convex boundary

10.4 Let in a tame

almost complex manifold X, J. Does n admit a J -convex function h : n --+ R which

an

f The answer is kno\vn to be negative if one does not allow certain

is constant on

degeneration of h.

Conjecture: There exists a s'ubset E clot n which is a union of J -holomorphic

n admits a J -convez function <p : n --+

curves and Levi-flat hypersurfaces such that

;˜

and J -con'vex in n\ E. If J is integrable than this is true '.'˜

IR which is constant on E

::˜

(even without Levi-flat components of E) according to a Grauert-Rossi's theorem. ,Ii

;˜

10.5 Let L be a Legendrian knot in the standard contact 53. Let top(L) be L itself

considered as a topological knot, 7"( L) and tb( L) be rotation number and Thurston- '˜

Bennequin's invariant of L (see 1.3). Does the triple (top(L), r(L), tb(LÂ») define L ;'˜

J

up to a Legendrian isotopy?

˜j

I think that the answer is positive \vhen top(L) is trivial (compare 7.1) and negative ˜:˜

˜˜

in the general case.

'$

˜

10.6 J-convex h-cobordism problem. Let 11 be a domain in a tame almost .â€¢

j

complex (X, J) which is diffeomorphic to M x I for a 3-manifold M. Let h : n -+

\.(

n admit a ':i˜:

1. Does

I be a J-convex function such that h IMxO= 0, h IMxl=

!

nondegenerate J-convex function with this property? .;!,.t,

˜i;

t˜

10.7 Prove the conjecture from [HE] which gives 4-dimensional generalization of ˜

n be a domain with J-convex boundary in a .:˜

Bennequin's inequality (see 4.2). Let

nwhose boundary is contained

tame (almost) complex (X, J) and ˜1 be a surface in

in an and is transversal to the conta.ct structure eon an. Let us orient M in such a

aM.

way that it induces the positive (see 1.2) orienta.tion on the transversal Then

:5 c(lvI).

˜0

d_(M) or, equivalently, X(lvI)

Eliashberg: Filling by holomorphic discs and its applications 67

it. is shown in section 9, the procedure of filling by holomorphic discs which

an (see 4.2) fails in the general case but one

c

)ves the inequality for the case 1\1

hope that the result is, nevertheless, true.

1

REFERENCES

D. Bennequin, Entrelacements et equations de Pfaff, Asterique 107-108 (1983),

83-161.

E. Bishop, Differentiable Inallifolds in cOlnplex Euclidean space, Duke Math. J., 32

(1965), 1-22.

E. Bedford and B. Gaveau, Envelopes of holomorphy of certain 2-spheres in C 2 ,

Amer. Journal of Math., 105 (1983), 975-1009.

E. Bedford and W. I(lingehbcl'g, On the envelope of holomorphy of a 2-sphere in

C 2 , preprint, 1989.

Ya. Eliashberg, Topological cha.ra.cterization of Stein ma.nifolds of dimension> 2,

preprint, 1989, to appear in International Journal of Math.

Va. Eliashberg, Three lectures synlplectic topology, Proceedings of the Confer..

011

t:nce on Diff. Geometry in Gala Gonone, 1988.

Va. Eliashberg, !(illing of elliptic cOlnplex points and the Legendrian isotopy, in

I>reparation.

M. Gromov, Pseudo-hololnorphic curves in sYlnplectic ma.nifolds, In'll. Math., 82

(1985),307-347.

V. M. Harlanl0v and Va. Eliashberg, On the nUlllber of complex points of a real

Proc(˜etlings

surface in a compex sluface, of Leningrad Int. Topology Conference,

1082, 143-148.

I). McDuff, The structure of rational nnd ruled sYlIlplectic manifolds, preprint, 1989.

(). Mcduff, The loca.l behavior of holo1l10rphic curves in alnlost complex 4-manifolds,

preprint, 1989.

PART 2

JONES/WITTEN THEORY

71

Introduction

The discovery by Vaughan Jones of a new polynomial invariant of links in the

a-sphere was an important breakthrough which has lead to the introduction of a

whole range of new techniques in 3-dimensional topology. The original Jones invari-

hut, a Laurent polynomial in one variable, was obtained via a "braid" description

c)f a link, utilising the remarkable properties of some representations of the braid

KI'OUP which arose in the theory of Von Neumann algebras. Early developments

were largely combinatorial, leading to alternative definitions of the invariant and

tC) generalisations, including a 2-variable polynomial which specialises to both the

.'()lles polynomial and the classical Alexander polynomial after appropriate substi-

ttlt,ions. The new invariants are comparatively easy to calculate and have had many

.'C)11Crete applications but for some time no really satisfactory conceptual definition

c)f the invariants was known: one not relying on special combinatorial presentations

c.r u.link. It was not clear, for example, whether such invariants could be defined for

links in other 3-manifolds. While there were many intriguing connections between

t.ll.˜ Jones theory and statistical mechanics, for example through the Yang-Baxter

f'cl'lations and the newly developed theory of quantum groups, it was a major prob-

ItÂ·lll to find the correct geometrical setting for the Jones theory. We refer to [L] for

˜\ Httrvey of this phase of the theory.

his lecture at the International Congress of Mathematical Physicists in Swansea,

III

.J uly 1988, Witten proposed a scheme which largely resolved this problem. He

Mhowed that the invariants should be obtained from a quantum field theory with

14n.˜rangian involving the Chern-Simons invariant of connections. This scheme pro-

vi.led a truly natural definition of the invariants, and indeed allowed considerable

"Pll(˜ralisations, to links in arbitrary 3-manifolds. Taking, in particular, the empty

llllk, Witten obtained new invariants of closed 3-manifolds. The challenge in this

h,pproach arose from the notorious difficulties of quantum field theory, in attach-

illS( real meaning to the functional integrals over the space of connections which

w..r.˜ involved. It is, to a mathematician, a striking fact that, despite these serious

foundational difficulties, Witten's theory made concrete predictions which could be

vt˜l'ified on a more elementary level. The new invariants appear to have great poten-

Un.l, Hnd it seems possible that they might even be able to detect counter examples

t,CJ t.he Poincare conjecture, should any exist.

'n) develop the quantum field theory of the Chern-Simons function on a more rigor-

basis, Witten has also given a Hamiltonian approach, which unites these ideas

CUlM

wit.h conformal field theory in two dimensions. This approach starts by quantising

n ct('rtain symplectic manifold naturally associated to a closed surface E, the moduli

f.1IHl.Ce of representations of ?rI(E) in a given compact Lie group. This leads to the

.'C )Ilst.ruction of a projectively fiat connection on a certain bundle over the moduli

NIUl.(Â·(˜ of complex structures on E, whose monodromy defines a representation of the

u,,,pping class group. More generally, one should take a Riemann surface with a set

of luarked points (corresponding to a link in a 3-manifold), and in the case when

t.I,â€¢. surface is the Riemann sphere one gets representations of the braid group.

I

Introduction

72

Meanwhile Reshetikhin and Turaev have defined new invariants, from a more com- :˜

binatorial and algebraic point of view, which it seems should agree with the Witten 1˜

invariants. They use a Dehn surgery description of a 3-manifold, and obtain a J

manifold invariant from a link invariant of Jones type. This approach by-passes the .˜

analytical difficulties in the quantum field theory, although the conceptual geometric :;i

:1

picture is less clear.

We were fortunate to have a number of lectures in Durham on these recent develop-.˜˜

ments, leading to the four articles below. The paper of Hitchin and Jeffrey's notes ;˜˜ ..˜

on Witten's three lectures concentrate on the geometric quantisation of the repre- .:˜

sentation spaces for surfaces; the paper of Kirby and Melvin gives the definition of .:˜

the Reshetikhin-Turaev invariants and evaluates them in some special cases. Jef- :.t

frey's notes also give Witten's explanation of how the Reshetikhin-Turaev formulae i˜

should be obtained in the Chern-Simons picture. The lecture of Atiyah lies closer to . ˜˜

the earlier work in Jones theory, involving the Hecke algebras; the paper describes .:˜

new topological constructions of braid group representations, due to Lawrence. .::1

11

:˜

[L] Lickorish, W.B.R. Polynomials for links Bull. London Math. Soc. 20 (1988)

1

˜˜˜8

,˜

n

:f1

New Results in Chern-Simons

Theory

Lectures by

Edward Witten

Institute for Advanced Study, Princeton, NJ, US.A. 08540

Notes by

Lisa Jeffreyl

Mathematical Institute, 24-29 St. Giles, Oxford OXl 3LB, UK

In these lectures, I will describe some aspects of an approach to the Jones polynomial

of knots, and its generalizations, that is based on a three-dimensional quantum

Y-ang-Mills theory in which the usual Yang-Mills Lagrangian is replaced by a Chern-

Simons action. This approach gives a manifestly three-dimensional approach to the

subject, but some of the key aspects of the story, involving the Feynman path

integral, are somewhat beyond the reach of present rigorous understanding. In

the first two lectures, I will describe aspects of the subject that can be developed

rigorously at present. This basically consists of a gauge theory approach to the

.lones representations of the braid group and their generalizations. In the last

lecture, I will describe the more ambitious Feynman path integral approach, which

is an essential part of the way that physicists actually think about problems such

as this one, and which gives the most far-reaching results.

'fhe first two lectures describe joint work with S. Della Pietra and S. Axelrod. In

this work, methods of symplectic geometry and canonical quantization are used to

associate a vector space 'HE to every oriented surface 1:. In this approach, we will

have to pick a complex structure J on ˜ as an auxiliary tool in constructing the 'Hr"

a.nd much of the effort will go into constructing a projectively flat. connection whose

lSupported ill part by all NSF Graduate Fellowship.

Witten: New results in Chern-Simons theory

74

existence shows that in a suitable sense, the 1-lt are independent of J. There are

several other approaches to this connection, due to Tsuchiya, Dena, and Yamada,

Segal, and Hitchin; Hitchin's approach, which is based on algebraic geometry, has

been presented at this meeting.

Once one has understood the association :E -+ 1-lt, one can use it to construct

topological invariants of three-manifolds (and, more generally, of links in three..

manifolds). To give a simple example, since the association :E -+ 'fit is natural,

a diffeomorphism f of ˜ gives rise to an endomorphism 9/ of 'Ht. If one forms a

three-manifold M f by taking the mapping cylinder of f (that is, by multiplying :E

by a unit interval and gluing the ends using f), then the invariant one associates to

M J is the trace of q>!.

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 , â€¢â€¢â€¢ ,

: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 bra-

cket) given by {:c 1 , â€¢â€¢â€¢ :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 â€¢ By a theorem

75

Witten: New results in Chern-Simons theory

of Stone and von Neumann, there is a unique such representation up to isomor-

phism: 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 Â£ 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 Â£ 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, Â£). The Coo functions on A act naturally on 1-Â£0 as follows.

Given 4> E COO(A), the Hamiltonian function associated to it is VÂ¢ = w- 1 (dÂ¢Â». Then

Â¢ acts on rio via an operator

(2)

(This represents a lifting to Â£ of the vector field VÂ¢, which preserves the symplectic

structure: VÂ¢ thus acts on the space of sections of Â£.) The map Â¢ 1--+ U</J is a

homomorphism of algebras, i.e.,

so this gives a representation of the Lie algebra of all Hamiltonian vector fields,

including those, the infinitesimal translations, comprising the Heisenberg algebra.

To get an irreducible subrepresentation of this, we pick a translationally invariant

complex structure J on A ( i.e., a constant endomorphism of T whose square is -1 )

such that:

(i) w is compatible ˜ith J in the sense that w(JX, JY) = w(X, Y) , or in other

words, in the complex structure J, w a form of type (1,1).

> 0 if X # 0).

(ii) w is positive with respect to J (i.e., w(X, J X)

2:i dz i 1\ dzi â€¢

Thus there are holomorphic linear functions zi such that w = i

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 Â£ 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 ,i

:˜

holomorphic structure is [P = (DD)O,2 =101Â°,2 = 0.)

",!

The line bundle Â£, 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 struc-

"˜

ture" The curvature of this connection is

:i

˜

:˜

,'t

it

,

.˜˜

This is the formula that characterizes Â£', so Â£, 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 (4)

1

p(zi) "1/J = 8˜' 1/J ,:1

˜

,˜

The commutation relations are preserved, as ˜

y

8Â· .. .8

-.(zJ"p) = S'3"p + Z3_."p"

8z" 8z'

To verify unitarity, note that

But

77

Witten: New results in Chern-Simons theory

so upon integrating by parts and using the holomorphicity of "p, we find that this

equals

Jd"zd"z,Â¢ 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-Â£

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 connec-

tion 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 Â® T, i.e.,. 6J 0 w- 1 : T* ˜ T, which in fact lies in 8ym2 T; indeed,

it lies in Sym2 T 1 ,O â‚¬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

= SO,1

\71,0

= Sl,O - 0 (5)

t

= -˜ i,j=l Dz"

D.SJij˜

0

DZ3

4

tzi

is the covariant derivative acting on sections of Â£. As promised, this

where

connection is given by a second-order differential operator on sections of Â£'. In more

invariant notation we have

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