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 E2-Hecke category HG,P=IndCoh(LSG,P(D2,S1)), where LSG,P(D2,S1) 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 Y→Z satisfying some mild conditions and any map between topological spaces N→M, we define (Y,Z)N,M=YN×ZNZM to be the space of maps from M to Z along with a lift to Y of its restriction to N. Using the pair of pants construction, we define an En-category Hn(Y,Z)=IndCoh0(((Y,Z)Sn−1,Dn)Y∧) and compute its factorization homology on any d-dimensional manifold M with d≤n, ∫MHn(Y,Z)≃IndCoh0(((Y,Z)∂(M×Dn−d),M)YM∧), where IndCoh0 is the sheaf theory introduced by Arinkin–Gaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi–Francis–Nadler and Beraldo.