The topological chiral homology of the spherical category

Abstract

We consider the spherical DG category $Sp{h}_G$ attached to an affine algebraic group $G$. By definition, $Sp{h}_G:=IndCoh(L{S}_G(S^2))$ consists of ind-coherent sheaves of the stack of $G$-local systems on the $2$-sphere $S^2$. The $3$-dimensional version of the pair of pants endows $Sp{h}_G$ with an $E_3$-monoidal structure. More generally, for an algebraic stack $Y$ (satisfying some mild conditions) and $n\ge -1$, we can look at the $E_{n+1}$-monoidal DG category $Sph(Y,n):=IndCo{h}_0((Y^{S^n}{)}_Y^{\wedge })$, where $IndCo{h}_0$ is the sheaf theory introduced in [AG2] and [centerH]. % The case of $Sp{h}_G$ is recovered by setting $Y=BG$ and $n=2$. The cobordism hypothesis associates to $Sph(Y,n)$ an $(n+1)$-dimensional TQFT, whose value of a manifold $M^d$ of dimension $d\le n+1$ (possibly with boundary) is given by the {topological chiral homology} ${\int }_{M^d}Sph(Y,n)$. % In this paper, we compute such homology (in virtually all cases): we have the Stokes style formula $${\int }_{M^d}Sph(Y,n)\simeq IndCo{h}_0((Y^{\partial (M^d\times D^{n+1-d})}{)}_{Y^M}^{\wedge }),$$ where the formal completion is constructed using the obvious projection $\partial (M^d\times D^{n+1-d})\to M^d$. The most interesting instance of this formula is for $Sp{h}_G\simeq Sph(BG,2)$, the original spherical category, and $X$ a Riemann surface. In this case, we obtain a monoidal equivalence ${\int }_{X}Sp{h}_G\simeq H(L{S}_G^{Betti}(X))$, where $L{S}_G^{Betti}(X)$ is the stack of $G$-local systems on the topological space underlying $X$ and $H$ is the sheaf theory introduced in [centerH].

Accepted for publication by Journal of Topology