Presentation on theme: "Lagrangian Floer theory of arbitrary genus and Gromov-Witten invariant Kenji Fukaya (Kyoto University) at University Miami (US)"— Presentation transcript:
1 Lagrangian Floer theory of arbitrary genus and Gromov-Witten invariant Kenji Fukaya (Kyoto University) at University Miami (US)
2 A finite set of pairs(relatively spin) Lagrangian submanifoldsweak bounding cochainsA cyclic unital filtered A inifinity categoryset of objectsset of morphismsF, Oh, Ohta, Ono (FOOO) (+ Abouzaid FOOO (AFOOO))
8 cyclicity Inner product and cyclicity is (up to sign) a Poincare duality oncyclicity
9 Problem to studyorComputein terms of the structures of
10 NO we can't ! 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).
11 dIBL structure (differential involutive bi-Lie structure) on B 3 kinds of operationsdifferentialLie bracketJacobico Lie Bracketco Jacobiis a derivationwith respect to dis a coderivation
24 Remark：(1): In case the target space M is a point, a kind of this theorem appearedin papers by various people including Baranikov, Costello Voronov, etc. (InPhysics there is much older work by Zwieback.)(2): Theorem itself is also expected to hold by various people including F fora long time.(3): The most difficult part of the proof is transversality. It becomespossible by recent progress on the understanding of transversalityissues. It works so far only over It also requires machinery fromhomological algebra of IBL infinity structure to work out theproblem related to take projective limit, in the same way as A infinity caseof [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.
25 Relation to `A model Hodge structure' We need a digression first.LetPutWe (AFOOO) have an explicit formula to calculate it based on Cardy relation.
28 Hochshild complexLetbe the operator obtained by `circle' action.Hochshild homologyis homology of the free loop space of L.is obtained from the S1 action on the free loop space.
29 Hochshild complexLetbe the operator obtained by `circle' action.Propositionit implies that there existssuch thatif Z is non-degeneratebecause2nd of BV master equation
30 Corollary (Hodge – de Rham degeneration) (Conjectured by Kontsevitch-Soibelman) If Z is non-degenerate thenRemark: This uses only: moduli space of annulus.To recoverwe must to use all the informations
31 Remark: Why this is called `Hodge – de Rham degeneration' ? Hodge structure useswithOne main result of Hodge theory iswe may rewrite this tois independent of uWe havePutis independent of u
32 Floer's boundary operator Landau-Gizburg potentialTable fromSaito-Takahashi'spaperFROM PRIMITIVE FORMSTO FROBENIUS MANIFOLDS(Similar table is also in a paperby Katzarkov-Kontsevich-Pantev)plus paringbetween HH* and HH*
33 MainTheorem (work in progress) The gauge equivalence class ofdetermines Gromov-Witten invariantsforif Z is non-degenerate.
34 The story offits naturally to the on going projectto prove homological Mirror symmetry by family Floer homologies.So we can expect that it can be used to enhancehomological Mirror symmetry and classical Mirror symmetryto include arbitrary genus.(B sides should be quantum Kodaira-Spencer theory. (BCOV).)
35 Idea of the proof of Main theorem Metric Ribbon treeBordered Riemann surfaceExample: 2 loop
43 Counting gives More precisely integrating the forms on the moduli spaceby the evaluation map using the boundary marked points
44 CountingWe obtain numbers that can be calculated from
45 Actually we need to work it out more carefully. Letand try to computeNeed to use actually
46 Forgettingdefinesis identified with the total space of complex line bundle over(the fiber ofis identified to the tangent space of the uniqueinterior marked point.)(The absolute value corresponds to cthe phase S1 corresponds to the extra freedom to glue.)
47 is not a trivial bundle. (Its chern class is Mumford-Morita class.) Sois not homlogous to a class inthe boundary.
48 is homologous to a class on the boundary andhere D is Poincare dual to the c1a class on the boundary
49 something coming from boundary. Becauseis a union of S1 orbits,thenHodge to de Rham degenerationand
50 A a class on the boundary andandare determined byQED