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.