Lagrangian Floer theory of arbitrary genus

andGromov-Witten invariant

Kenji Fukaya

(Kyoto University)

at University Miami (US)

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A finite set of pairs

(relatively spin) Lagrangian submanifolds

weak bounding cochains

A cyclic unital filtered A inifinity category

set of objects

set of morphisms

F, Oh, Ohta, Ono (FOOO) (+ Abouzaid FOOO (AFOOO))2

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Hochshild cohomology

Hochshild homology

Cyclic cohomology

Cyclic homology

Open closed maps FOOO , AFOOO

Gromov-Witten invariant

Counting genus g pseudo-holomorphic maps intersecting with cycles in X

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Problem to study

or

Compute

in terms of the structures of

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structure.

relation

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is (up to sign) a Poincare duality on

cyclicity

Inner product and cyclicity

Problem to study

or

Compute

in terms of the structures of

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Answer

is determined by the structures of

in case

In general we need extra information.

I will explain those extra information below.It is Lagrangian Floer theory of higher genus (loop).

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NO we can't !

dIBL structure (differential involutive bi-Lie structure) on B

3 kinds of operations

differential

Lie bracket

co Lie Bracket

is a derivation

is a coderivation with respect to d

Jacobi

co Jacobi

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is compatible with

Involutivea

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IBL infinity structure = its homotopy everything analogue

operations :

Homotopy theory of IBL infinity structure is built (Cielibak-Fukaya-Latschev)

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chain complex with inner product

has a structure of dIBL algebra (cf. Cielibak-F-Latschev)

(dual cyclic bar complex)

basis of C

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There is a category version.

Dual cyclic bar complex has dIBL structure

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Remark: This structure does NOT (yet) use operations mk except the classical part of m1, that is the usual boundary operator.

Cyclic stucture (operations mk ) on

satisfying Maurer-Cartan equation

This is induced by a holomorphic DISK

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is given by

relation among

Theorem (Lagrangian Floer theory of arbitrary genus) (to be written up)

There exists such that BV master equation

is satisfied.

The gauge equivalence class of is well-defined.

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Note: induces

and

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induces

is obtained from moduli space of genus g bordered Riemann surface with boundary components

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Remark ：

(1): In case the target space M is a point, a kind of this theorem appeared in papers by various people including Baranikov, Costello Voronov, etc. (In Physics there is much older work by Zwieback.)

(2): Theorem itself is also expected to hold by various people including F for a long time.

(3): The most difficult part of the proof is transversality. It becomes possible by recent progress on the understanding of transversality issues. It works so far only over . It also requires machinery from homological algebra of IBL infinity structure to work out the problem related to take projective limit, in the same way as A infinity case of [FOOO]. This homological algebra is provided by Cielibak-F-Latshev.

(4): Because of all these, the novel part of the proof of this theorem isextremely technical. So I understand that it should be written up carefullybefore being really established.

(5): In that sense the novel point of this talk is the next theorem (in slide 33)which contains novel point in the statement also.

Relation to `A model Hodge structure'

Let Put

We (AFOOO) have an explicit formula to calculate it based on Cardy relation.

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We need a digression first.

Formula for

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a basis of

Theorem (AFOOO, FOOO ....)

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Let

be the operator obtained by `circle' action.

Hochshild complex

Hochshild homology

is homology of the free loop space of L.

is obtained from the S1 action on the free loop space.

Proposition

Let

be the operator obtained by `circle' action.

it implies that there exists

such that

because

if Z is non-degenerate

Hochshild complex

292nd of BV master equation

Corollary (Hodge – de Rham degeneration) (Conjectured by Kontsevitch-Soibelman)

If Z is non-degenerate then

Remark: This uses only : moduli space of annulus.

To recover

we must to use all the informations

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Remark: Why this is called `Hodge – de Rham degeneration' ?

Hodge structure uses with

One main result of Hodge theory is

we may rewrite this to is independent of u

We have

Put

is independent of u

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Table from Saito-Takahashi's paper

FROM PRIMITIVE FORMS TO FROBENIUS MANIFOLDS

Floer's boundary operator

Landau-Gizburg potential

(Similar table is also in a paper by Katzarkov-Kontsevich-Pantev)

plus paring between HH* and HH*

MainTheorem (work in progress)

The gauge equivalence class of determines Gromov-Witten invariants

for

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if Z is non-degenerate.

Idea of the proof of Main theorem

Metric Ribbon tree Bordered Riemann surface

Example: 2 loop

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1

1

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Moduli of metric ribbon graph

Moduli of genus g Riemann surface with marked points

This isomorphism was used in Kontsevich's proof of Witten conjecture

etc.

=

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is identified with moduli space of bordered Riemann surface.

Fix and consider the family

parametrised by

Consider one parameter family of moduli spaces

L

L

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Study the limit when

? ?

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etc. (various combinatorial types depending on .)

Counting

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More precisely integrating the forms

on the moduli space

by the evaluation map using the boundary marked points

gives

Counting

We obtain numbers that can be calculated from

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Actually we need to work it out more carefully.

Let

and try to compute

Need to use actually

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Forgetting defines

is identified with the total space of complex line bundle over

(the fiber of

is identified to the tangent space of the unique interior marked point.)

(The absolute value corresponds to c

the phase S1 corresponds to the extra freedom to glue.)

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is not a trivial bundle. (Its chern class is Mumford-Morita class.)

So is not homlogous to a class in the boundary.

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is homologous to a class on the boundary and

here D is Poincare dual to the c1

a class on the boundary

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Because is a union of S1 orbits,

then

Hodge to de Rham degeneration

something coming from boundary.

and

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A a class on the boundary

are determined by and

QED

and