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Published online by Cambridge University Press: 25 June 2025
A certain deformation of a mirror pair in Strominger–Yau–Zaslow mirror setting is discussed. We propose that the mirror dual of a deformation of a complex manifold by a certain (real) deformation quantization is a symplectic manifold with a foliation structure. In order to support our claim that these deformations of mirror pairs are mirror dual to each other, we construct categories associated to these deformations of complex and symplectic manifolds and discuss homological mirror symmetry between them.
1. Introduction
In this paper, we discuss a certain deformation of a mirror pair in Strominger–Yau– Zaslow mirror setting. In particular, we construct categories associated to these deformations of complex and symplectic manifolds and discuss homological mirror symmetry between them. One of our hopes is to understand how to formulate deformations of categories. This is motivated by what the homological mirror symmetry [Kontsevich 1995] is expected to reproduce: (genus zero part of ) the mirror symmetry isomorphism of Frobenius manifolds [Kontsevich 1995; Barannikov and Kontsevich 1998]. In this story, a category is believed to reproduce a Frobenius manifold as a space of deformations of the category with suitable structures on it. However, at present, there is no formulation of deformations of categories which reproduces Frobenius manifolds. Actually, it is already unclear how to obtain a manifold of the space of deformations even locally.
In order to figure out what we should do for this problem, we can study some examples. For noncommutative two-tori, we can actually construct categories associated to them, which are regarded as deformations of categories on two tori, and discuss homological mirror symmetry.
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