Subsection 1.5. (0NZ5) · ScalingStacks

1.5. Topological field theory

As we discuss in Section 6, our results may be viewed from the perspective of topological field theory.

The $\infty$ -category $\qc(\mathcal{L}X)$ (the Drinfeld center $\mathcal{Z}(\qc(X))$ ) carries a rich collection of operations generalizing the braided tensor structure on modules for the classical Drinfeld double. Namely, for any cobordism $\Sigma$ between disjoint unions of circles $(S^{1})^{\coprod m}$ and $(S^{1})^{\coprod n}$ , we obtain restriction maps between the corresponding mapping stacks

(\mathcal{L}X)^{\times m}\leftarrow X^{\Sigma}\rightarrow(\mathcal{L}X)^{\times n}.

Pullback and pushforward along this correspondence defines a functor

\qc(\mathcal{L}X)^{\otimes m}\to\qc(\mathcal{L}X)^{\otimes n}

which is compatible with composition of cobordisms. In particular, we obtain a map from the configuration space of $m$ small disks in the standard disk to functors

\qc(\mathcal{L}X)^{\otimes m}\to\qc(\mathcal{L}X).

This equips $\qc(\mathcal{L}X)$ , and hence the Drinfeld center $\mathcal{Z}(\qc(X))$ , with the structure of a (framed) $\mathcal{E}_{2}$ -category. This establishes the categorified (cyclic) Deligne conjecture in the current geometric setting.

In particular for $X=BG$ , $X^{\Sigma}$ is the derived moduli stack of $G$ -local systems on $\Sigma$ and we obtain a topological gauge theory.

More generally, we have the following corollary of our main results:

0NW9

Corollary 1.13. Let $X$ be a perfect stack, and let $\Sigma$ be a finite simplicial set. Then there is a canonical equivalence $\qc(X^{\Sigma})\simeq\qc(X)\otimes\Sigma$ , where $X^{\Sigma}=\Map(\Sigma,X)$ denotes the derived mapping stack, and $-\otimes\Sigma$ the tensor of stable $\infty$ -categories over simplicial sets.

One can view this corollary as providing for a TFT over finite simplicial sets. We assign to such a simplicial set $U$ the $\infty$ -category $\qc(X^{U})$ , and for any diagram of finite simplicial sets

U\rightarrow\Sigma\leftarrow V,

(for example a cobordism of manifolds) we obtain a correspondence of mapping stacks

X^{U}\leftarrow X^{\Sigma}\rightarrow X^{V},

and hence by pullback and pushforward a functor

\qc(X^{U})\to\qc(X^{V}),

satisfying composition laws with respect to gluings.

In particular, since $\qc(X^{S^{n}})$ is equivalent to the $\mathcal{E}_{n}$ -center $\mathcal{Z}_{\mathcal{E}_{n}}(\qc(X))$ , we obtain an action of the (framed) $\mathcal{E}_{n+1}$ -operad on $\mathcal{Z}_{\mathcal{E}_{n}}(\qc(X))$ . This establishes the categorified (cyclic) Kontsevich conjecture in the current geometric setting.

1.5.1. Extended TFT

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:

$\bullet$

objects: $0$ -manifolds,
$\bullet$

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

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

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

$\bullet$

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

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

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:

$\bullet$

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.
$\bullet$

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

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.

1.5.2. Rozansky-Witten theory

The topological field theory $\mathcal{Z}_{X}$ introduced above is closely related to well-known three-dimensional topological field theories. When the target $X$ is the classifying stack $BG$ of a finite group, $\mathcal{Z}_{X}$ is the untwisted version of Dijkgraaf-Witten theory. More generally, if $X$ is a 2-gerbe ( $BB{\mathbb{G}}_{m}$ -torsor) over $BG$ (classified by a class in $H^{3}(G,{\mathbb{G}}_{m})$ ), we obtain the twisted version (as studied in [Fr]).

When $X$ is a smooth complex projective variety, $\mathcal{Z}_{X}$ is closely related to the $\mathbb{Z}/2$ -graded three-dimensional topological field theory associated to the holomorphic symplectic manifold $T^{*}X$ by Rozansky-Witten [RW], Kontsevich [K2] and Kapranov [Ka]. In particular see [RobW] and [KRS] for work on Rozansky-Witten theory as an extended topological field theory.

We confine ourselves here to a brief comparison of the two theories on the circle. On the one hand, $\mathcal{Z}_{X}$ assigns to $S^{1}$ the stable $\infty$ -category $\qc(\mathcal{L}X)$ . Since $X$ is a smooth scheme, $\mathcal{L}X$ is the total space $T_{X}[-1]=\Spec\operatorname{Sym}^{\bullet}\Omega_{X}[1]$ of the shifted tangent bundle of $X$ . Thus we can identify $\qc(\mathcal{L}X)$ with module objects in the $\infty$ -category $\qc(X)$ for the commutative algebra object $\operatorname{Sym}^{\bullet}\Omega_{X}[1]$ . Via Koszul duality, this category is closely related to modules for $\operatorname{Sym}^{\bullet}T_{X}[-2]$ , or in other words, sheaves on $T^{*}X$ with an unusual grading. On the other hand, Rozansky-Witten theory assigns to $S^{1}$ the $\infty$ -category $\operatorname{Perf}(T^{*}X)$ of perfect complexes. It would be very interesting to develop Rozansky-Witten theory as a fully extended TFT using the results of [L6], and explore its relation with the derived algebraic geometry of $\qc(X)$ .

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5