We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any prop with A∞-multiplication-we think of such algebras as A∞-algebras "with extra structure". As applications, we obtain an integral version of the Costello-Kontsevich-Soibelman moduli space action on the Hochschild complex of open TCFTs, the Tradler-Zeinalian and Kaufmann actions of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex.
Bibliographical noteFunding Information:
We would like to thank Alexander Berglund, Kevin Costello, Daniela Egas, Richard Hepworth, Ralph Kaufmann, Anssi Lahtinen and Bob Penner for helpful algebraic, geometric and twisted conversations. The first author was supported by the Danish National Sciences Research Council (DNSRC) and the European Research Council (ERC) under the European Union's Seventh Framework Programme (FP/2007–2013), ERC Grant Agreement n. 239807 , as well as by the Danish National Research Foundation (DNRF92) through the Centre for Symmetry and Deformation. The second author was supported by the Australian Research Council (ARC) under the Future Fellowship ( FT100100307 ) and Discovery Project ( DP1095831 ) schemes, as well as a US National Science Foundation ( NSF ) standard grant ( DMS-1406162 ). He thanks the University of Copenhagen for its hospitality and support.
- Frobenius algebras
- Hochschild homology
- Moduli space of Riemann surfaces