Equivariant higher Hochschild homology and topological field theories

We present a version of higher Hochschild homology for spaces equipped with principal bundles for a structure group $G$. As coefficients, we allow $E_{\infty}$-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 $(\infty , 1)$-category of cospans of $E_{\infty}$-algebras.

Keywords

Hochschild homology, topological field theory, bordism category, principal bundle, $E_{\infty}$-algebra

2010 Mathematics Subject Classification

13D03, 81T45

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L.M. is supported by the Doctoral Training Grant ST/N509099/1 from the UK Science and Technology Facilities Council (STFC). L.W. is supported by the RTG 1670 “Mathematics inspired by String theory and Quantum Field Theory” and thanks the Heriot-Watt University in Edinburgh and, in particular, Richard Szabo for their hospitality during the time when part of this project was completed.

Received 9 October 2018

Received revised 6 May 2019

Accepted 7 May 2019

Published 18 September 2019