ScalingStacks

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 $\qc(X)$.