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Lagrangian Floer theory of arbitrary genus and Gromov-Witten invariant Kenji Fukaya (Kyoto University) at University Miami (US) 1 L = {(Lk , bk )} A finite set of pairs Lk (relatively spin) Lagrangian submanifolds bk ÎH odd (L;L 0 ) weak bounding cochains A cyclic unital filtered A inifinity category L set of objects set of morphisms F, Oh, Ohta, Ono (FOOO) (+ Abouzaid FOOO (AFOOO)) 2 Hochshild cohomology Hochshild homology Cyclic cohomology Cyclic homology q Open closed maps FOOO , AFOOO p 3 Gromov-Witten invariant GWg, : H(X;L)Ä ® L Counting genus g pseudo-holomorphic maps intersecting with cycles in X 4 Problem to study or Compute GWg, ( p ( x1 ) ,… , p ( x )) in terms of the structures of 5 structure. relation 6 Inner product and cyclicity is (up to sign) a Poincare duality on H(L Ç L';L0 ) mk (x1,… , xk ), x0 = ± mk (x0, x1,… , xk-1 ), xk cyclicity 8 Problem to study or Compute GWg, ( p ( x1 ) ,… , p ( x )) in terms of the structures of 9 Answer NO we can't ! GWg, ( p ( x1 ) ,… , p ( x )) is determined by the structures of in case g = 0, =3 In general we need extra information. I will explain those extra information below. It is Lagrangian Floer theory of higher genus (loop). 10 dIBL structure (differential involutive bi-Lie structure) on B 3 kinds of operations differential dd = 0 Lie bracket Jacobi co Lie Bracket { } }{ co Jacobi is a derivation is a coderivation with respect to d 11 }a{= å a1,c a2,c { } is compatible with }{ å ±a {a ,b} ± a {a ,b} + å ±b {b ,a} ± b {b ,a} 1,c 1,c 2,c 2,c 2,c 2,c 1,c 1,c +}{a,b}{= 0 Involutive a å {a a }=0 1,c 2,c 12 IBL infinity structure = its homotopy everything analogue operations : P n,m : EnB ® EmB EnB = B Ä Ä B / Sn n Homotopy theory of IBL infinity structure is built (Cielibak-Fukaya-Latschev) 13 (C, ,d) ( B= B C cyc chain complex with inner product ) (dual cyclic bar complex) * j Î( B C ) , j cyc * g st = es ,et i1 ik has a structure of dIBL algebra (cf. Cielibak-F-Latschev) = j (e , , e ) i1 ik ei basis of C (gst ) = (g st )-1 14 {j ,y }l1 }j {i1 lk+k ' -2 ik ; j1 jl = å ±gstj l1 = å ±gstj ia la-1 sla+b+1 ic ia-1 sjb yl a la+b t lc lk+k ' -2 jb-1t 15 There is a category version. Dual cyclic bar complex has dIBL structure Remark: This structure does NOT (yet) use operations mk except the classical part of m1, that is the usual boundary operator. 16 Cyclic stucture (operations mk ) on satisfying Maurer-Cartan equation 1 dM 1,0 + {M 1,0 ,M 1,0 } = 0 2 This is induced by a holomorphic DISK 17 is given by M 1,0 (x1,… , xk ) = mk-1 (x1,… , xk-1 ), xk 1 dM 1,0 + {M 1,0 ,M 1,0 } = 0 2 relation among mk 18 Theorem (Lagrangian Floer theory of arbitrary genus) (to be written up) There exists 1 dM ,g + 2 M å å 1+ 2 = ,g ÎE B {M +1 g1 +g2 =g 1,g1 such that BV master equation ,M 2 ,g2 }out +}M -1,g {+{M } =0 +1,g-1 int is satisfied. The gauge equivalence class of {M ,g } is well-defined. 19 Note: { } :B ÄB ® B induces and { }int : EnB ® En-1B {x1 xn, y1 ym }out = å ±{xi , y j }x1 xˆi xn y1 yˆ j ym i, j {x1 xn }int = å ±{xi , x j }x1 xˆi xˆ j yn i, j 20 }{: B ® B Ä B induces }{: En-1B ® EnB }x1 xn {= å ±x1 }xi { xˆ j yn i 21 M Î( E B) * ,g is obtained from moduli space of genus g bordered Riemann surface with boundary components L L g = 2, =3 L 22 1 dM ,g + 2 {M ,M 1,g1 å å 1+ 2 = +1 g1 +g2 =g } 2 ,g2 out {M 1,g1 ,M }M 2 ,g2 -1,g }out +}M { -1,g } =0 {+{M +1,g-1 int {M +1,g-1 int } 23 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 is extremely technical. So I understand that it should be written up carefully before 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. 24 Relation to `A model Hodge structure' We need a digression first. Let Put Z(x1,x 2 ) = p ( x1 ) , p ( x 2 ) We (AFOOO) have an explicit formula to calculate it based on Cardy relation. 25 Formula for Z(x1,x 2 ) = p ( x1 ) , p ( x 2 ) 26 Theorem (AFOOO, FOOO ....) Z(x1,x 2 ) = p ( x1 ) , p ( x 2 ) a basis of 27 Hochshild complex Let be the operator obtained by `circle' action. Hochshild homology HH* (W(L), W(L)) is homology of the free loop space of L. B;H* (L(L)) ® H*+1 (L(L)) is obtained from the S1 action on the free loop space. 28 Hochshild complex Let be the operator obtained by `circle' action. Proposition Z(Bx1,x2 ) = (}M 1,0 {) (x1,x 2 ) it implies that there exists such that because d K + Kd = B BK = 0 if Z is non-degenerate ±}M 1,0 { = dM 2,0 + {M 2,0, M 1,0 }out 2nd of BV master equation 29 Corollary (Hodge – de Rham degeneration) (Conjectured by Kontsevitch-Soibelman) If Z is non-degenerate then Remark: This uses only To recover M 2,0 : moduli space of annulus. GWg, ( p ( x1 ) ,… , p ( x we must to use all the informations M )) ,g 30 Remark: Why this is called `Hodge – de Rham degeneration' ? Hodge structure uses ¶, ¶ with ¶¶ + ¶¶ = ¶¶ = ¶¶ = 0 Ker¶ Ker¶ Kerd d = ¶+ ¶ @ @ Im ¶ Im ¶ Im d Kerdu is independent of u we may rewrite this to d = ¶ + u¶ u H (du ) @ Im du One main result of Hodge theory is We have Put d B + Bd = BB = dd = 0 du = d + uB d K + Kd = B BK = 0 H (du ) @ Kerdu is independent of u Im du 31 Floer's boundary operator f = PO Landau-Gizburg potential Table from Saito-Takahashi's paper FROM PRIMITIVE FORMS TO FROBENIUS MANIFOLDS (Similar table is also in a paper by Katzarkov-Kontsevich-Pantev) Z plus paring between HH* and HH* 32 MainTheorem (work in progress) The gauge equivalence class of {M GWg, ( p ( x1 ) ,… , p ( x ,g } determines Gromov-Witten invariants )) for if Z is non-degenerate. 33 Idea of the proof of Main theorem Metric Ribbon tree Bordered Riemann surface Example: 2 loop 35 1 1 1 36 37 Moduli of metric ribbon graph = Moduli of genus g Riemann surface with marked points etc. This isomorphism was used in Kontsevich's proof of Witten conjecture 38 is identified with moduli space of bordered Riemann surface. Fix C1, ,C and consider the family R(c) = (cC1, , cC ) parametrised by 39 Consider one parameter family of moduli spaces L L 40 Study the limit when ? ? 41 etc. (various combinatorial types depending on .) S 42 Counting gives GW1,2 ( p ( x1 ), p ( x 2 )) More precisely integrating the forms x1, x 2,… on the moduli space by the evaluation map using the boundary marked points 43 Counting We obtain numbers that can be calculated from {M ,g } 44 Actually we need to work it out more carefully. Let and try to compute GWg,1 ( p ( x)) Need to use actually 45 Forgetting u 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.) 46 is not a trivial bundle. (Its chern class is Mumford-Morita class.) So is not homlogous to a class in the boundary. 47 is homologous to a class on the boundary -1 and (D) p here D is Poincare dual to the c1 a class on the boundary p -1 (D) 48 òp -1 -1 P (D) Because ev* (x) = something coming from boundary. p -1P -1 (D) is a union of S1 orbits, p -1P -1 (D) = S1W then ò S1W ev* (x) = ò W ev* (Bx) = ò ev* (d Kx) = ò ev* (Kx) ¶W W Hodge to de Rham degeneration ¶W = ¶1W + S W ' 1 and ò S1W ¢ ev* (Kx) = ò ev* (BKx) = 0 W¢ 49 A a class on the boundary ò * A ev (x) and ò ¶W1 are determined by ev* (Kx) {M ,g } and QED 50