The differential graded Verlinde Formula and the Deligne Conjecture

TL;DR

This paper explores the relationship between the differential graded Verlinde algebra and the Deligne Conjecture within the context of modular categories, revealing how specific mapping class group actions connect these structures and generalize the Verlinde formula.

Contribution

It establishes a link between the differential graded Verlinde algebra and the Deligne Conjecture via the mapping class group action, extending the Verlinde formula to non-semisimple cases.

Findings

Mapping class group element S transforms the Verlinde algebra into a second E2-structure.

In the semisimple case, results reduce to the classical Verlinde formula.

In the non-semisimple case, the work generalizes the Verlinde formula using Hochschild (co)homology.

Abstract

A modular category $\mathcal{C}$ gives rise to a differential graded modular functor, i.e. a system of projective mapping class group representations on chain complexes. This differential graded modular functor assigns to the torus the Hochschild complex and, in the dual description, the Hochschild cochain complex of $\mathcal{C}$. On both complexes, the monoidal product of $\mathcal{C}$ induces the structure of an $E_2$-algebra, to which we refer as the differential graded Verlinde algebra. At the same time, the modified trace induces on the tensor ideal of projective objects in $\mathcal{C}$ a Calabi-Yau structure so that the cyclic Deligne Conjecture endows the Hochschild cochain and chain complex of $\mathcal{C}$ with a second $E_2$-structure. Our main result is that the action of a specific element $S$ in the mapping class group of the torus transforms the differential graded Verlinde algebra into this second $E_2$-structure afforded by the Deligne Conjecture. This result is established for both the Hochschild chain and the Hochschild cochain complex of $\mathcal{C}$. In general, these two versions of the result are inequivalent. In the case of Hochschild chains, we obtain a block diagonalization of the Verlinde algebra through the action of the mapping class group element $S$. In the semisimple case, both results reduce to the Verlinde formula. In the non-semisimple case, we recover after restriction to zeroth (co)homology earlier proposals for non-semisimple generalizations of the Verlinde formula.