The Fukaya Category of the Elliptic Curve as an Algebra over the Feynman Transform.

摘要

In [4] Barannikov proves the equivalence between the existence of a morphism of twisted modular operads FDSt →EV &parl0;F and EV as defined in [17]), and certain tensors of EV satisfying the quantum master equation of Batalin-Vilkovisky geometry of an affine S [t]-manifold. He then suggests the possibility of generalizing this morphism to the categorical case by replacing EV with a twisted modular operad, referred to here has EL , constructed from the Lagrangian submanifolds of a fixed symplectic manifold.;The main result of this thesis is the construction of an explicit example of such a morphism in the case that the symplectic manifold is an elliptic curve.;Given a symplectic manifold X, one constructs a precategory closely related to the Fukaya Category, denoted Fuk(X), whose objects are the Lagrangian submanifolds L ⊂ X, and whose morphism spaces Hom(Li, Lj) are finite dimensional modules over the Novikov ring, generated by the points of Li ∩ Lj. There is a nondegenerate bilinear pairing B:HomLi,L j⊗Hom&parl0; Lj,Li&parr0;→ C, which is degree -1 in the elliptic curve case.;Given a finite sequence of cyclic chains of Lagrangian submanifolds Li0,&ldots;,Lidi b-2g+1i=1 in the elliptic curve, we construct elements md,b&parl0; s1&cdots;sb-2g+1&parr0;∈ ⨂i=1b-2g+1 ⨂j=1di Hom&parl0;Lij,Li &parl0;j-1&parr0;&parr0;⊗Hom&parl0; Li0,Lidi &parr0;&parl0;0.1&parr0; of degree (d + 1) - (2 - 2b), defined by summing over zero-dimensional tropical Morse graphs G with dim H1( G ) = b, where d + 1 = i=1b+1 (di + 1) and b - 2g + 1 is the number of cycles.;The tensors (0.1) define the twisted modular operad EL&parl0;&parl0;d +1,b&parr0;&parr0; :=⨂i=1 b-2g+1 ⨂j=1 diHom&parl0;Lij ,Li&parl0;j-1&parr0;&parr0;⊗ Hom&parl0;Li0,Lid i&parr0;, whose contraction maps mEL G are given by contraction via the bilinear form B.;We construct a morphism of twisted modular operads from the Feynman transform of a twist of S&d5; [t] to EL , where S&d5; [t] is an untwisted version of S [t], by mapping the generators {sigma1···sigma b-2g+1} of S&d5; [t] to the elements {m¯d,b(sigma 1···sigmab- 2g+1)}. This algebra structure is equivalent to the set {m¯d,b(sigma 1···sigmab- 2g+1)} being a solution to the quantum master equation of [4], or, equivalently, to {m¯ d,b(sigma1···sigma b-2g+1)} satisfying what will be referred to here as the quantum Ainfinity -relations. The usual Ainfinity-relations on Fuk(X) are recovered by setting b = 0.