The topological chiral homology of the spherical category

We consider the spherical DG category $Sph_G$ attached to an affine algebraic
group $G$. By definition, $Sph_G := IndCoh(LS_G(S^2))$ 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 $Sph_G$ with an
$E_3$-monoidal structure. More generally, for an algebraic stack $Y$
(satisfying some mild conditions) and $n \geq -1$, we can look at the
$E_{n+1}$-monoidal DG category $Sph(Y,n) := IndCoh_0((Y^{S^n})^\wedge_Y)$,
where $IndCoh_0$ is the sheaf theory introduced in [AG2] and [centerH]. % The
case of $Sph_G$ 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 $d \leq n+1$ (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 $$ \int_{M^d} Sph(Y,n) \simeq IndCoh_0 ( (Y^{\partial(M^d
\times D^{n+1-d})})^\wedge_{Y^M} ) , $$ 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 $Sph_G \simeq
Sph(BG,2)$, the original spherical category, and $X$ a Riemann surface. In this
case, we obtain a monoidal equivalence $\int_X Sph_G \simeq
H(LS_G^{Betti}(X))$, where $LS_G^{Betti}(X)$ is the stack of $G$-local systems
on the topological space underlying $X$ and $H$ is the sheaf theory introduced
in [centerH].