ScalingStacks

Let $2{\rm Cob}$ denote the $\infty$-category whose objects are compact oriented $1$-manifolds, and whose morphisms $2{\rm Cob}(C_{1},C_{2})$ consist of the classifying spaces of oriented topological surfaces with fixed incoming and outgoing components $C_{1},C_{2}$. This $\infty$-category has a symmetric monoidal structure given by disjoint union of 1-manifolds. We will abuse notation by denoting the object consisting of the disjoint union of $m$ copies of $S^{1}$ by $m$.

Fix a base commutative derived ring $k$.

Note that ${\rm Cat}_{\infty}^{\rm ex}$ has a symmetric monoidal structure, the tensor product of presentable stable categories as studied in [L4, 4.2], and the unit of the monoidal structure is the $\infty$-category of $k$-modules. Since the empty $1$-manifold is the unit of $2{\rm Cob}$, a topological field theory $\mathcal{Z}$ sends its endomorphisms, which are closed surfaces $\Sigma$, to endomorphisms of the unit of the target $\infty$-category $\mathcal{C}$. When $\mathcal{C}={\rm Cat}_{\infty}^{\rm ex}$, the unit is $\Mod_{k}$ and $\mathcal{Z}(\Sigma)$ is a $k$-module.

As an example, take $X=BG$ (in characteristic zero). Then $\mathcal{Z}_{BG}(S^{1})$ is the $\infty$-category of sheaves on the adjoint quotient $G/G={\rm Loc}_{G}(S^{1})$, while $\mathcal{Z}_{BG}(\Sigma)$ is the cohomology of the structure sheaf of the moduli stack $BG^{\Sigma}={\rm Loc}_{G}(\Sigma)$ of $G$-local systems on $\Sigma$.

More generally, the particular topological field theory operations we are considering are not sensitive to the structure of manifolds, and the same construction provides a field theory living over “bordisms” of spaces. In what follows, by a space we will mean a space homotopy equivalent to a finite simplicial set.

The proof of the previous proposition immediately extends to give the following:

The 2d topological field theory $\mathcal{Z}_{X}$ above is a categorified analogue of the 2d TFTs defined by string topology on a compact oriented manifold, or the topological B-model defined by a Calabi-Yau variety [C]. Namely, we assign to the circle the $\infty$-category of quasi-coherent sheaves, rather than the complex of chains, on the loop space. This has the advantage that we may construct maps for arbitrary correspondences without the assumption that $X$ is a Calabi-Yau or satisfies Poincaré duality. On the other hand, we have gone up a level of categoricity, pushing off the problems of duality and orientation to the definition of operations between $\Gamma(X^{\Sigma},\mathcal{O}_{X^{\Sigma}})$ for cobordisms between surfaces or invariants for 3-manifolds. Recall [BK] that an extended 3-dimensional topological field theory assigns a braided (in fact ribbon) category to the circle. This category must however satisfy a strong finiteness and nondegeneracy condition (modularity) coming from its extension to three-manifolds. It would be interesting to see what conditions need to be imposed on the sheaves we consider and on the stack $X$ in order to extend $\mathcal{Z}_{X}$ in such a fashion. Note that Corollary 4.8 shows that $\qc(X)$ for $X$ perfect satisfies an analogue of the Calabi-Yau or Frobenius property: namely it is self-dual as a $\Mod_{k}$-module.

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.