Continuous and coherent actions on wrapped Fukaya categories

TL;DR

This paper develops a framework for understanding how wrapped Fukaya categories change continuously under Liouville automorphisms, connecting symplectic geometry with homotopy theory and string topology.

Contribution

It establishes the continuous functoriality of wrapped Fukaya categories with respect to Liouville automorphisms, confirming cases of Teleman's conjecture and linking automorphism groups to string topology.

Findings

Proves continuous functoriality of wrapped Fukaya categories.

Intertwines automorphism actions with local systems on cotangent bundles.

Identifies a map from Liouville automorphisms' homotopy groups to string topology.

Abstract

We establish the continuous functoriality of wrapped Fukaya categories with respect to Liouville automorphisms, yielding a way to probe the homotopy type of the automorphism group of a Liouville sector. These methods prove Liouville and monotone cases of a conjecture of Teleman from the 2014 ICM. In the case of a cotangent bundle, we show that the Abouzaid equivalence between the wrapped category and the infinity-category of local systems intertwines our action with the action of diffeomorphisms of the zero section. In particular, our methods yield a typically non-trivial map from the rational homotopy groups of Liouville automorphisms to the rational string topology algebra of the zero section.