On some deformations of Fukaya categories (Q2804130)

In the Strominger-Yau-Zaslow picture of mirror symmetry, the paradigmatic example of mirror manifolds is a pair \(M\), \(\check{M}\) of manifolds that arise as dual torus fibrations over a base manifold \(B\) carrying an integral affine structure. The present paper studies homological mirror symmetry for certain noncommutative deformations in this context, that is, without singular fibers. This is sufficient to study tori, and this paper is part of the author's project to understand mirror symmetry for noncommutative tori.NEWLINENEWLINEOn the complex or B-side, the author considers a complex manifold \(\check{M}\) admitting a torus fibration \(M\to B\), carrying a Poisson bivector of the form \( \theta = \sum_{i,j=1}^n \theta_{ij} \frac{\partial}{\partial x_i} \wedge \frac{\partial}{\partial y_j}, \) where \((x_1,\dots,x_n)\) denote integral affine coordinates on the base \(B\), and \((y_1,\dots,y_n)\) are the corresponding coordinates on the fiber.NEWLINENEWLINEOn the symplectic or A-side, the author considers the dual torus fibration \(M \to B\) equipped with a symplectic structure \(\omega_\theta\) that is T-dual to the above pair \((\check{M},\theta)\), and also a foliation \(\mathcal F_\theta\) constructed from \(\theta\). The author argues that this foliation structure must be taken into account in order to properly formulate homological mirror symmetry in these cases.NEWLINENEWLINEThe author begins by recalling the theory of integral affine manifolds, torus bundles, and the correspondence between line bundles with connection on \(\check{M}\) and sections of \(M\to B\). Then he introduces two curved differential graded categories, \(\mathrm{DG}_{\check{M}}\) whose objects are certain line bundles with \(\mathrm{U}(1)\)-connection, and \(\mathrm{DG}_M\) whose objects are lifts of sections \(s\) of \(M\to B\). In the former, the ``curvature'' is the \((0,2)\) part of the curvature of the connection, while in the latter, it is a certain expression that measures the failure of \(s\) to be Lagrangian. The categories \(\mathrm{DG}_{\check{M}}\) and \(\mathrm{DG}_M\) are isomorphic: there is a bijection between the objects, and in both cases the morphisms are defined in terms of differential forms in closely related ways. The author discusses a possible strategy for showing that \(DG_M(0)\) (the zero-curvature subcategory) is equivalent to a full subcategory of the Fukaya category of \(M\), following ideas of Fukaya-Oh and Kontsevich-Soibelman.NEWLINENEWLINEThe main results of the paper concern the deformation by \(\theta\). For \(\check{M}\), one considers the quantization \(\check{M}_\theta\) where the ring of functions carries the Moyal product. The author analyses which objects of \(\mathrm{DG}_{\check{M}}\) survive this deformation. There is a geometric interpretation of these objects in terms of the dual geometry \((M,\omega_\theta, \mathcal{F}_\theta)\). Based on this analysis, he defines two curved dg-categories \(\mathcal{DG}_{\check{M}_\theta}\) and \(\mathcal{DG}_{M_\theta}\) consisting of the deformed objects on the two sides, that are again isomorphic by construction. Along the same lines as before, the author argues that, for a fixed value \(W\) of the curvature, the category \(\mathcal{DG}_{M_\theta}(W)\) ought to be equivalent to a subcategory of the Fukaya category of \(M\) with the symplectic form \( \omega_\theta-(\pi_{\mathcal{F}_\theta})^\ast W\), where \(\mathcal F_\theta\) is the foliation. See the paper to make sense of this formula; the point is that the foliation \(\mathcal F_\theta\) enters into the definition of the symplectic form when the curvature is not zero.NEWLINENEWLINEFor the entire collection see [Zbl 1320.53003].