An $L_\infty$ structure on symplectic cohomology

TL;DR

This paper develops an $L_ abla$-algebraic structure on symplectic cohomology of Liouville domains, enhancing the closed-open map to an $L_ abla$-homomorphism, with novel features respecting filtrations and using a compact telescope model.

Contribution

It introduces a new $L_ abla$ structure on symplectic cohomology and an enhanced closed-open map, differing from prior constructions by respecting filtrations and employing a compact telescope model.

Findings

Constructed the $L_ abla$ structure on symplectic cohomology.

Enhanced the closed-open map to an $L_ abla$ homomorphism.

Features respect a modified action filtration and use a compact telescope model.

Abstract

We construct the $L_\infty$ structure on symplectic cohomology of a Liouville domain, together with an enhancement of the closed--open map to an $L_\infty$ homomorphism from symplectic cochains to Hochschild cochains on the wrapped Fukaya category. Features of our construction are that it respects a modified action filtration (in contrast to Pomerleano--Seidel's construction); it uses a compact telescope model (in contrast to Abouzaid--Groman--Varolgunes' construction); and it is adapted to the purposes of our follow-up work where we construct Maurer--Cartan elements in symplectic cochains which are associated to a normal-crossings compactification of the Liouville domain.