Floer Homology Language TANAKA Akio Note 8 Discreteness of LanguageFlux Conjecture(Lalonde-McDuff-Polterovich 1998)Image of Flux homomorphism is discrete at H1(M; R).Lemma1Next two are equivalent.(i) Flux conjecture is correct.(ii) All the complete symplectic homeomorphism is C1 topological closed at symplectic transformation group.Lemma 2Next two are equivalent.(1) Flux conjecture is correct.(ii) Diagonal set MM×M is stable by the next definition.DefinitionL is stable at the next condition.(i) There exist differential 1 form u1, u2 over L that is sufficiently small.(ii) When sup|u1|, sup|u2| is Lu1Lu2 for u1, u2 ,there existsf that satisfies u1 - u2 = df .Explanation0 <For conjecture and lemmas> is de Rham cohomology class.Symplectic manifold (M, w)Group's connected component of complete homeomorphism Ham (M, w)Flux isomorphism Flux: π1(Ham(M, w) )→ RRoad of Ham (M, w) γ(t)δγ / δt = Xu(t) that is defined bu closed differential form Utover MExplanation1Symplectic manifold Mn-dimensional submanifold L ML that satisfies next condition is called special Lagrangian submanifold.Ω's restriction to L is L's volume. 2M's special Lagrangian submanifold LFlat complex line bundle LLAGsp(M) (L, L)3Complex manifold M†p M†Sheaf over M† fpfp (U) = C ( pU)fp (U) = 0 ( p U)4Special Lagrangian fiber bundle π : M → NComplementary dimension 2's submanifold S(N) Nπ-1 (p) = LPPair (Lp, Lp)pN-S(N)Lp Complex flat line bundleAll the pair (Lp, Lp) s is M0† .5(Geometric mirror symmetry conjecture Strominger-Yau-Zaslow 1996)Mirror of M is diffeomorphic with compactification of M0† .6 Pairs of Lagrangian submanifold of M and flat U(1) over the submanifold (L1, L1), (L2, L2)(L1, L1) (L2, L2) means the next.There exists complete symplectic homeomorphism that is ψ(L2 ) = L2andψ*L2 is isomorphic with L1.ImpressionDiscreteness of language is possible by Flux conjecture 1998.[References]Quantization of Language / Floer Homology Language / Note 7 / June 24, 2009For WITTGENSTEIN Ludwig / Position of Language / Tokyo December 10, 2005To be continuedTokyo July 19, 2009Sekinan Research Field of Language