ScalingStacks

An exciting recent development (postdating the submission of this paper) is Jacob Lurie’s announced proof of the Cobordism Hypothesis [L6] which characterizes extended TFTs in arbitrary dimensions. The results of this paper may be used to prove that the monoidal $\infty$-category $\qc(X)$, for a perfect stack $X$, defines an extended two-dimensional TFT. We briefly summarize this here (see [BN2] for more details regarding an analogous extended two-dimensional TFT).

The extended two-dimensional TFT $\mathcal{Z}_{X}$ associated to a perfect stack $X$ is a symmetric monoidal functor

$$\mathcal{Z}_{X}:2Bord\to 2Alg$$

from the $(\infty,2)$-category of unoriented bordisms:

1. $\bullet$

   objects: $0$-manifolds,

2. $\bullet$

   1-morphisms: $1$-dimensional bordisms between $0$-manifolds,

3. $\bullet$

   2-morphisms: classiying spaces of $2$-dimensional bordisms between $1$-bordisms,

to the Morita $(\infty,2)$-category of algebras $2Alg$:

1. $\bullet$

   objects: algebra objects in stable presentable $\infty$-categories,

2. $\bullet$

   1-morphisms: bimodule objects in stable presentable $\infty$-categories,

3. $\bullet$

   2-morphisms: classifying spaces of morphisms of bimodules.

Note that given $A\in 2Alg$, we can pass to the $(\infty,2)$-category of modules $\Mod_{A}$, and bimodules correspond to functors between $(\infty,2)$-categories of modules. Thus if a field theory assigns $A$ to a point, we can also think of it as assigning $\Mod_{A}$ to a point.

The field theory $\mathcal{Z}_{X}$ assigns the following to closed $0$, $1$ and $2$-manifolds:

1. $\bullet$

   To a point, $\mathcal{Z}_{X}$ assigns the monoidal $\infty$-category $\qc(X)$, or equivalently, the $(\infty,2)$-category of $\qc(X)$-linear $\infty$-categories.

2. $\bullet$

   To a circle, $\mathcal{Z}_{X}$ assigns the Hochschild homology category of $\qc(X)$, which by our results can be identified with $\qc(\mathcal{L}X)$.

3. $\bullet$

   To a closed surface $\Sigma$, $\mathcal{Z}_{X}$ assigns the $k$-module of derived global sections $\Gamma(X^{\Sigma},\mathcal{O}_{X^{\Sigma}})$ of the structure sheaf of the mapping space $X^{\Sigma}$.

The proof that $\qc(X)$ satisfies the dualizability conditions of [L6] to define an extended TFT follows closely from the identification of the Hochschild homology and cohomology categories of $\qc(X)$ (which is a necessary consequence of the TFT structure). One could consult [BN2] for more details in an analogous setting.