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Lagrangian Floer theory of arbitrary genus
andGromov-Witten invariant
Kenji Fukaya
(Kyoto University)
at University Miami (US)
1
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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8
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
33
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
1
37
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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.
48
is homologous to a class on the boundary and
here D is Poincare dual to the c1
a class on the boundary
49
Because is a union of S1 orbits,
then
Hodge to de Rham degeneration
something coming from boundary.
and
50
A a class on the boundary
are determined by and
QED
and