The cyclic Deligne conjecture and Calabi-Yau structures

TL;DR

This paper proves the cyclic Deligne conjecture, showing that Hochschild cochains of Calabi-Yau structured categories have a circle-equivariant algebraic structure, with applications to string topology on manifolds.

Contribution

It establishes the cyclic Deligne conjecture for a broad class of linear categories with Calabi-Yau structures, extending previous results to include framed $E_2$-algebras.

Findings

Hochschild cochains of Calabi-Yau categories form a framed $E_2$-algebra.

The approach applies to smooth, proper, and dualizable $$-categories.

Constructs chain-level string topology operations for manifolds with boundary.

Abstract

The Deligne conjecture (many times a theorem) endows Hochschild cochains of a linear category with the structure of an $E_2$-algebra, that is, of an algebra over the little 2-disks operad. In this paper, we prove the cyclic Deligne conjecture, stating that for a linear category equipped with a Calabi-Yau structure (a kind of non-commutative orientation), the Hochschild cochains is endowed with the finer structure of a framed $E_2$-algebra, that is, of a circle-equivariant algebra over the little 2-disks operad. Our approach applies simultaneously to both smooth and proper linear categories, as well as to linear functors equipped with a relative Calabi-Yau structure, and works for a very general notion of linear category, including any dualizable presentable $\infty$-category. As a particular application, given a compact oriented manifold with boundary $\partial M \subset M$, our construction gives chain-level genus zero string topology operations on the relative loop homology $H_{*}(LM,L\partial M)$.