Equivariant higher Hochschild homology and topological field theories

Abstract

We present a version of higher Hochschild homology for spaces equipped with principal bundles for a structure group G. As coefficients, we allow E∞-algebras with G-action. For this homology theory, we establish an equivariant version of excision and prove that it extends to an equivariant topological field theory with values in the (∞, 1)-category of cospans of E∞-algebras.

The Little Bundles Operad

“…The operadic approach is particularly well-suited for the study of (∞, 1) equivariant topological field theories [MW20]. As a main application, we prove that in dimension two the value of such a theory on the circle produces a homotopy little bundles algebra (Theorem 5.2), i.e.…”

“…For two such collections of decorated circles, the morphism space between them is the space of twodimensional compact oriented bordisms between these two collections of circles equipped with a map to BG extending the ones prescribed on the boundary components. We refer to [MW20] for the technical details and an example of such a field theory constructed using an equivariant version of higher derived Hochschild chains.…”

Finite symmetries of quantum character stacks

“…Our work can be understood as exploring this (expected) equivariant field theory in the oriented setting and dimension n = 2 with values in Pr c . As a complementary example it was shown in [MW20a] that equivariant higher Hochschild homology, that is factorisation homology for E ∞ -algebras with D-action in chain complexes, gives examples of equivariant field theories in any dimension n. D-equivariant field theories can also be studied through the cobordism hypothesis, which implies that 2-dimensional framed fully extended D-equivariant field theories with values in BrTens are described by functors BD −→ BrTens. Such a functor is described by picking out an object A ∈ BrTens, together with a central algebra M d for every d ∈ D, a central M d2 •M d1 -M d2d1 -bimodule for every pair d 1 , d 2 ∈ D and furthermore 3-and 4-morphisms for all triples and quadruples of group elements, respectively, satisfying a coherence condition involving five group elements.…”