Gluing localized mirror functors (Q6561313)

The authors [J. Geom. Phys. 136, 284--320 (2019; Zbl 1410.53083); J. Differ. Geom. 106, No. 1, 45--126 (2017; Zbl 1369.53062); Noncommutative homological mirror functor. Providence, RI: American Mathematical Society (AMS) (2021; Zbl 1476.14001)] introduced a localized mirror formalism to construct Landan-Ginzburg (LG) mirrors \(W:Y\rightarrow\Lambda\) of a symplectic manifold \(X\), and to understand homological symmetry (HMS) between them. They used a single Lagrangian immersion \(L\) as a reference to construct a LG model \(W\), as well as an \(A_{\infty}\)-functor from the Fukaya category \textrm{Fuk}\((X)\) to the matrix factorization category of \(W\).\N\NThis paper considers a collection of Lagrangians lying in the same deformation class, developing a method to glue the localized mirrors and functors constructed from these Lagrangians to obtain a global mirror LG model together with a global \(A_{\infty}\)-functor\ for the study of HMS. This is applied to punctured Riemann surfaces rigged with pair-of-pants decompositions, showing that the functor is an equivalence on the desired level.\N\NThere have been several works for a local-to-global approach to HMS [\textit{K. Fukaya} and \textit{Y.-G. Oh}, Asian J. Math. 1, No. 1, 96--180 (1997; Zbl 0938.32009); \url{https://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf}; \textit{P. Seidel}, Surv. Differ. Geom. 17, 411--425 (2012; Zbl 1382.53025); \textit{D. Nadler} and \textit{E. Zaslow}, J. Am. Math. Soc. 22, No. 1, 233--286 (2009; Zbl 1227.32019); \textit{T. Dyckerhoff}, Compos. Math. 153, No. 8, 1673--1705 (2017; Zbl 1387.18029); \textit{M. Abouzaid} and \textit{P. Seidel}, Geom. Topol. 14, No. 2, 627--718 (2010; Zbl 1195.53106); \textit{H. Lee}, ``Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions'', Preprint, \url{arXiv:1608.04473}; \textit{J. Pascaleff} and \textit{N. Sibilla}, Compos. Math. 155, No. 3, 599--644 (2019; Zbl 1421.53085); \textit{B. Gammage} and \textit{V. Shende}, Acta Math. 229, No. 2, 287--346 (2022; Zbl 1536.14030)]. The approach in this paper is distinct from the preceeding ones.\N\N\begin{itemize}\N\item[(1)] The authors deal with the global Fukaya category without decomposing it into local ones. The main difficulty of decomposing the Fukaya category into local pieces is that there are quantum corrections to the global category from global-holomorphic curves not contained in any of the local pieces. The authors' formulation automatically incorporates global pseudo-holomorphic curves, bypassing this difficulty.\N\N\item[(2)] The mirror symmetry \ and the mirror functor is geometrically constructed in a systematic way. The authors' approach is closer to [\textit{A. Strominger} et al., AMS/IP Stud. Adv. Math. 23, 333--347 (2001; Zbl 0998.81091); \textit{M. Gross} and \textit{B. Siebert}, Ann. Math. (2) 174, No. 3, 1301--1428 (2011; Zbl 1266.53074)]. \ The authors find a symplecto-geometric way to glue the local mirror spaces \(U_{i}\) to obtain a generally non-affine mirror space \(Y\), with a well-defined potential function \(W:Y\rightarrow\Lambda\). More importantly, localized mirror functors \(\mathcal{F}^{L_{i}}\) are glued as well to form a global \(A_{\infty}\)-functor\N\[\N\mathcal{F}^{\mathrm{global}}:\mathrm{Fuk}(X)\rightarrow \mathrm{MF}(W)\N\]\Nwhere \(\mathrm{MF}(W)\) is defined as a homotopy fiber product of each \(\mathrm{MF}(W_{i})\).\N\end{itemize}