Hochschild homology of structured algebras

TL;DR

This paper develops a universal method for constructing natural operations on Hochschild complexes of structured algebras, with applications to topological field theories, string topology, and algebraic structures.

Contribution

It introduces a general approach to define operations on Hochschild complexes for algebras over PROPs with $A_$-multiplication, extending previous frameworks.

Findings

Provides an integral version of the moduli space action on Hochschild complexes.

Establishes the Tradler-Zeinalian action of Sullivan diagrams on Frobenius algebras.

Applies to string topology in characteristic zero.

Abstract

We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any PROP with $A_\infty$--multiplication---we think of such algebras as $A_\infty$--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 action 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.