Geometric quantization and mirror symmetry
نویسنده
چکیده
After the appearance of [T3] I received an e-mail from Cumrun Vafa, who recognized that the subject is closely related to that of his preprint [V]. This text started out as an e-mail “reply” to his letter. All the constructions we propose have well known “spectral curve” prototypes (see for example Friedman and other [FMW], Bershadsky and other [BJPS] and a number of others). Roughly speaking, our constructions are the spectral curve construction plus the phase geometry described in [T3]. So this text should really come before [T3], as motivation for the development of the geometry of the phase map in [T3]. 1 spLag cycles We begin by recalling the actual geometric construction for a pair L ⊂ S, where S is a smooth symplectic manifold of dimension 2n with a given tame almost complex structure I, and L ⊂ S a smooth, oriented Lagrangian submanifold (of maximal dimension dimL = n = 1 2 dimS); this construction has recently become quite popular in the set-up of Calabi–Yau threefolds. The structure on S is an almost Kähler structure, and we say for short that S is an aK manifold. Write ω for the symplectic form on S and I for the almost complex structure, so that the tangent space TSp at a point p is C n with the constant symplectic form 〈 , 〉 = ωp and the constant Euclidean metric gp, giving the Hermitian triple (ωp, Ip, gp). We now define the Lagrangian Grassmannian Λ↑p = Λ↑(TSp) to be the Grassmannian of maximal oriented Lagrangian subspaces in TSp. Taking this space over every point of S gives the oriented Lagrangian Grassmannization of TS: π : Λ↑(S) → S with π(p) = Λ↑p. (1.1)
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