  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.
