Continuous and coherent actions on wrapped Fukaya categories
Yong-Geun Oh, Hiro Lee Tanaka
TL;DR
This work proves that wrapped Fukaya categories admit a continuous and coherent action by Liouville automorphisms, realized as a map from the automorphism space to the automorphism space of the wrapped category via localizations in A∞-categories. The authors develop a bundle-level framework: constructing a local system of wrapped categories over the classifying space of automorphisms and proving these localizations yield a well-defined, continuous action; they further connect this action to Hochschild-type invariants and the Abouzaid map in the cotangent bundle case. In T^*Q, Abouzaid’s equivalence between the wrapped category and local systems on Q is enhanced to be Diff(Q)-equivariant, linking higher homotopy of diffeomorphism groups to string topology via Hochschild cohomology and loop-space Algebras. The paper also outlines spectral refinements, relations to Teleman and Lekili-Evans, and future directions for filtered automorphism data and alternative automorphism notions, suggesting deep connections between symplectic topology, higher algebra, and geometric 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.