Universal operations in Hochschild homology

TL;DR

This paper develops a universal method to identify all natural operations on Hochschild complexes of E-algebras, linking them to moduli spaces of Riemann surfaces and Sullivan diagrams, and producing new higher string topology operations.

Contribution

It introduces a general approach to classify natural operations on Hochschild complexes for various algebraic structures, unifying different known operations and identifying them with geometric moduli spaces.

Findings

Chain complex of natural operations approximated by formal operations

Identifies operations with moduli space of Riemann surfaces for topological conformal field theories

Relates operations to Sullivan diagrams for topological quantum field theories

Abstract

We provide a general method for finding all natural operations on the Hochschild complex of E-algebras, where E is any algebraic structure encoded in a prop with multiplication, as for example the prop of Frobenius, commutative or A_infty-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations, for which we provide an explicit model that we can calculate in a number of cases. When E encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich-Soibelman via different methods identify with all formal operations. When E encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by Tradler-Zeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.