Eisenstein series via factorization homology of Hecke categories

Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group $G$, a parabolic subgroup $P$, and a topological surface $M$, the (enhanced) spectral Eisenstein series category of $M$ is the factorization homology over $M$ of the ${\mathrm{E}}_2$-Hecke category ${\mathrm{H}}_{G,P}=\mathrm{I}\mathrm{n}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{h}({\mathrm{L}\mathrm{S}}_{G,P}(D^2,S^1))$, where ${\mathrm{L}\mathrm{S}}_{G,P}(D^2,S^1)$ denotes the moduli stack of $G$-local systems on a disk together with a $P$-reduction on the boundary circle. More generally, for any pair of stacks $\mathcal{Y}\to \mathcal{Z}$ satisfying some mild conditions and any map between topological spaces $N\to M$,wedefine$($\mathcal{Y}$, $\mathcal{Z}$)^{N, M} = $\mathcal{Y}$^N \times_{$\mathcal{Z}$^N} $\mathcal{Z}$^M$tobethespaceofmapsfrom$M$to$$\mathcal{Z}$$alongwithaliftto$$\mathcal{Y}$$ofitsrestrictionto$N$. Using the pair of pants construction, we define an ${\mathrm{E}}_n$-category $$\mathrm{H}$_n($\mathcal{Y}$, $\mathcal{Z}$) = $\mathrm{I}\mathrm{n}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{h}$0\left(\left(($\mathcal{Y}$, $\mathcal{Z}$)^{S^{n-1}, D^n}\right)^\wedge{$\mathcal{Y}$}\right)$ and compute its factorization homology on any $d$-dimensional manifold $M$ with $d\le n$, $${\int }_M{\mathrm{H}}_n(\mathcal{Y},\mathcal{Z})\simeq {\mathrm{I}\mathrm{n}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{h}}_0\left({((\mathcal{Y},\mathcal{Z}{)}^{\mathrm{\partial }(M\times D^{n-d}),M})}_{{\mathcal{Y}}^M}^{\wedge }\right),$$ where ${\mathrm{I}\mathrm{n}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{h}}_0$ is the sheaf theory introduced by Arinkin–Gaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi–Francis–Nadler and Beraldo.