ScalingStacks

The notion of Drinfeld center for monoidal stable categories is a categorical analogue of Hochschild cohomology of associative (or $A_{\infty}$) algebras (or more precisely of Hochschild cochains, or its spectral analogue, topological Hochschild cohomology). In the case of algebras, Deligne’s conjecture states that the Hochschild cochain complex has the structure of an $\mathcal{E}_{2}$-algebra (lifting the Gerstenhaber algebra, or $H_{*}(\mathcal{E}_{2})$-algebra, structure on Hochschild cohomology). There is also a cyclic version of the conjecture which states that the Hochschild cochains for a Frobenius algebra possesses the further structure of a framed $\mathcal{E}_{2}$, or ribbon, algebra. See [T1, K1, KS1, MS, C, KS2] for various proofs of the Deligne conjecture and [Ka, TZ] for its cyclic version. The Kontsevich conjecture (see [T2, HKV]) generalizes this picture to higher algebras, asserting that the Hochschild cochains on an $\mathcal{E}_{n}$-algebra have a natural $\mathcal{E}_{n+1}$-structure.

The work of Costello [C] and Kontsevich-Soibelman [KS2] explains how a strong form of the Deligne conjecture follows from the structure of topological field theory – specifically, by constructing a topological field theory $\mathcal{Z}_{A}$ from an algebra $A$ such that $\mathcal{Z}_{A}(S^{1})$ is the Hochschild cochain complex of $A$. Observe that the configuration space of $m$-tuples of circles in the disk (the $m$-th space of the $\mathcal{E}_{2}$-operad) maps to the space of functors $\mathcal{Z}(S^{1})^{\otimes m}\to\mathcal{Z}(S^{1})$ compatibly with

compositions.

A natural categorified analogue of the Deligne conjecture states that the Drinfeld center of a monoidal $\infty$-category is an $\mathcal{E}_{2}$-category. For dualizable and self-dual $k$-linear $\infty$-categories, one can ask if this structure comes from a natural framed $\mathcal{E}_{2}$, or ribbon, structure. We observe that our identification of the Drinfeld center of $\qc(X)$ with sheaves on the loop space, combined with the construction of the topological field theory $\mathcal{Z}_{X}$, automatically solves this conjecture in the applicable cases:

In fact, the same arguments prove a case of the categorified form of the Kontsevich conjecture. We consider $\qc(X^{S^{n}})$ as part of an $n+1$-dimensional topological field theory, with operations given by cobordisms of $n$-manifolds.