The topological chiral homology of the spherical category. Issue 3 (12th March 2019)

• Record Type: Journal Article
• Authors: Beraldo, Dario
• Is Part Of: Journal of topology. Volume 12:Issue 3(2019)
• Page Start: 685
• Page End: 704
• Publication Date: 2019-03-12
• Subjects: 14F05 -- 55N22 (primary) -- 14D24 (secondary); Topology -- Periodicals; 514.05
• DOI: 10.1112/topo.12098
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract: 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 on the (derived) stack of G ‐local systems on the 2‐sphere S 2 . The three ‐dimensional version of the pair of pants endows Sph G with an E 3 ‐monoidal structure. More generally, for an algebraic stack Y and n ⩾ − 1, we consider the E n + 1 ‐monoidal DG category Sph ( Y, n ) : = IndCoh 0 ( ( Y S n ) Y ∧ ), where IndCoh 0 is the sheaf theory introduced by Arinkin and Gaitsgory. The case of Sph G is recovered by setting Y = B G and n = 2 . The cobordism hypothesis associates to Sph ( Y, n ) an ( n + 1 ) ‐dimensional TFT, whose value on a manifold M d of dimension d ⩽ n + 1 (possibly with boundary) is given by the topological chiral homology ∫ M d Sph ( Y, n ) . In this paper, we compute such chiral homology, obtaining the Stokes style formula ∫ M d Sph ( Y, n ) ≃ IndCoh 0 Y ∂ ( M d × D n + 1 − d ) Y M d ∧, where the formal completion is constructed using the obvious projection ∂ ( M d × D n + 1 − d ) → M d . The most interesting instance of this formula is for Sph G ≃ Sph ( B G, 2 ), the original spherical category, and X a Riemann surface. In this case, we obtain a monoidal equivalence ∫ X Sph G ≃ 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 a sheaf theory related to Hochschild cochains.