Fukaya \(A_{\infty}\)-categories associated to Lefschetz fibrations. II \(\tfrac{1}{2}\)
type: Item
instance of: scholarly article
title: Fukaya \(A_{\infty}\)-categories associated to Lefschetz fibrations. II \(\tfrac{1}{2}\)
zbMATH Open document ID: 1359.53062
author: Paul Seidel
published in: Advances in Theoretical and Mathematical Physics
publication date: 2016-12-20T00:00:00Z
review text: Let \(E\to \mathbb{C}\) be a symplectic Lefschetz fibration allowing a fiberwise compactification \(\bar{E}\to\mathbb{C}\). Then, taking a bulk term \(b\in H^2(E;\mathbb{C}^\ast)\) and its deformed version \(\bar{b}\in H^2(E;\mathbb{C}q^\times)\), the twisted Fukaya \(A_\infty\)-category \(\mathcal{F}_b(\pi)\) and its twisted formal deformation \(\mathcal{F}_{q,\bar{b}}(\bar{\pi})\) are associated [\textit{K. Fukaya} et al., Sel. Math., New Ser. 17, No. 3, 609--711 (2011; Zbl 1234.53023); the author, in: Proceedings of the international congress of mathematicians, ICM'02. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 351--360 (2002; Zbl 1014.53052)]. Then the following question arises: Question. Can we choose the bulk term so that the formal deformation induced by fiberwise compactification is trivial? The author remarks that, since compactification adds a new holomorphic disc changing the \(A_\infty\)-structure, the answer to this question is negative in known relative Fukaya categories in the ordinary sense. In this paper, instead of working on the entire Fukaya category \(\mathcal{F}\), its subcategory \(\mathcal{A}\), obtained to fix a basis of Lefschetz thimbles \(\{L_1,\ldots,L_m\}\), is considered. In \S3, via considerations of Gromov-Witten invariants, a differential equation involving \(\bar{b}\) (Assumption 3.1. In Lemma 3.2, this equation is shown to always have a solution) is presented. Then the main theorem announced in this paper is the following partial answer to this question. Theorem 4.1. Suppose that \(\bar{b}\) satisfies Assumption 3.1. Then \(\mathcal{A}_{q,\bar{b}}\) is a trivial deformation. Corollary 4.2. There is a distinguished class of bulk \(\bar{b}\), all related to each other by suitable reparametrizations (an exact description is given in (2.13)), such that \(\mathcal{A}_{q,\bar{b}}\) is a trivial deformation The proof of Theorem 4.1 is only sketched. It uses Kaledin's trivialit…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 16E45
Mathematics Subject Classification ID: 18F15
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 53D40
zbMATH DE Number: 6666960
zbMATH Keywords: Lefschetz fibration
zbMATH Keywords: Fukaya category
zbMATH Keywords: Gromov-Witten invariant
zbMATH Keywords: deformation
zbMATH Keywords: bulk term
zbMATH Keywords: \(A_\infty\)-structure
MaRDI profile type: Publication
full work available at URL: atmp.2016.v20.n4.a5
OpenAlex ID: W2548744377
DOI: 10.4310/ATMP.2016.V20.N4.A5
description: scientific article; zbMATH DE number 6666960