Integration of Hochschild cohomology, derived Picard groups and uniqueness of lifts

TL;DR

This paper develops a partial integration map linking Hochschild cohomology to the derived Picard group in A-infinity categories, providing insights into the structure and uniqueness of lifts of equivalences, with applications to Fukaya categories.

Contribution

It introduces a new integration map from Hochschild cohomology to the derived Picard group and explores its properties and implications for algebraic and geometric categories.

Findings

The integration map is injective and natural.

Vanishing of the domain is necessary for lift uniqueness.

Applications to Fukaya categories and their Picard groups.

Abstract

The paper introduces a partial integration map from the first Hochschild cohomology of any cohomologically unital A-infinity category over a field of characteristic zero to its derived Picard group. We discuss useful properties such as injectivity, naturality and the relation with the Baker-Campbell-Hausdorff formula. Based on the image of the integration map we propose a candidate for the identity component of the derived Picard group in the case of finite-dimensional graded algebras. As a first application of the integration map it is shown that the vanishing of its domain is a necessary condition for the uniqueness of lifts of equivalences from the homotopy category to the A-infinity-level. The final part contains applications to derived Picard groups of wrapped and compact Fukaya categories of cotangent bundles and their plumbings and an outlook on applications to derived Picard groups of partially wrapped Fukaya categories after Haiden-Katzarkov-Kontsevich.