TY - JOUR

T1 - The differential graded Verlinde formula and the Deligne Conjecture

AU - Schweigert, Christoph

AU - Woike, Lukas

N1 - Publisher Copyright: © 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

PY - 2023

Y1 - 2023

N2 - A modular category (Formula presented.) gives rise to a differential graded modular functor, that is, a system of projective mapping class group representations on chain complexes. This differential graded modular functor assigns to the torus the Hochschild chain complex and, in the dual description, the Hochschild cochain complex of (Formula presented.). On both complexes, the monoidal product of (Formula presented.) induces the structure of an (Formula presented.) -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 (Formula presented.) a Calabi–Yau structure so that the cyclic Deligne Conjecture endows the Hochschild cochain and chain complex of (Formula presented.) with a second (Formula presented.) -structure. Our main result is that the action of a specific element (Formula presented.) in the mapping class group of the torus transforms the differential graded Verlinde algebra into this second (Formula presented.) -structure afforded by the Deligne Conjecture. This result is established for both the Hochschild chain and the Hochschild cochain complex of (Formula presented.). 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 (Formula presented.). 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.

AB - A modular category (Formula presented.) gives rise to a differential graded modular functor, that is, a system of projective mapping class group representations on chain complexes. This differential graded modular functor assigns to the torus the Hochschild chain complex and, in the dual description, the Hochschild cochain complex of (Formula presented.). On both complexes, the monoidal product of (Formula presented.) induces the structure of an (Formula presented.) -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 (Formula presented.) a Calabi–Yau structure so that the cyclic Deligne Conjecture endows the Hochschild cochain and chain complex of (Formula presented.) with a second (Formula presented.) -structure. Our main result is that the action of a specific element (Formula presented.) in the mapping class group of the torus transforms the differential graded Verlinde algebra into this second (Formula presented.) -structure afforded by the Deligne Conjecture. This result is established for both the Hochschild chain and the Hochschild cochain complex of (Formula presented.). 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 (Formula presented.). 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.

U2 - 10.1112/plms.12518

DO - 10.1112/plms.12518

M3 - Journal article

AN - SCOPUS:85152137374

VL - 126

SP - 1811

EP - 1841

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 6

ER -