Hochschild homology of structured algebras

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 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. The Hochschild complex of an associative algebra A admits a degree 1 self-map, Connes-Rinehart’s boundary operator B. If A is Frobenius, the (proven) cyclic Deligne conjecture says that B is the ∆–operator of a BV-structure on the Hochschild complex of A. In fact B is part of much richer structure, namely an action by the chain complex of Sullivan diagrams on the Hochschild complex [45]. A weaker version of Frobenius algebras, called here A∞–Frobenius algebras, yields instead an action by the chains on the moduli space of Riemann surfaces [8, 27]. In this paper we develop a general method for constructing explicit operations on the Hochschild complex of A∞–algebras “with extra structure”, which contains these theorems as special cases. Our method is global to local: we give conditions on a composable collection of operations that ensures that it acts on the Hochschild complex of algebras of a given type. We then show how to read-off the action explicitly, so that formulas for individual operations can be also obtained. An A∞–algebra can be described as an enriched symmetric monoidal functor from a certain dg-category A∞ to Ch, the dg-category of chain complexes over Z. The category A∞ is what is called a dg-PROP, a symmetric monoidal dg-category with objects the natural numbers. We consider here more generally dg-PROPs E equipped with a dgfunctor i : A∞ → E . Expanding on the terminology of McClure–Smith [32], we call such a pair E = (E , i) a PROP with A∞-multiplication. An E–algebra is a symmetric monoidal dg-functors Φ : E → Ch. When E is a PROP with A∞–multiplication, any E–algebra comes with a specified A∞–structure by restriction along i, and hence we can talk about the Hochschild complex of E–algebras. We introduce in the present paper a generalization of the Hochschild complex which assigns to any dg-functor Φ : E → Ch a certain new functor C(Φ) : E → Ch. The assignment has the property that, for Φ symmetric monoidal, C(Φ) evaluated at 0 is the usual Hochschild complex of the underlying A∞–algebra. (The evaluation of C(Φ)(n) can more generally be interpreted in terms of higher Hochschild homology as in [39] associated to the union of a circle and n points.) This Hochschild complex construction can be iterated, and for Φ split monoidal, the iterated complex Cn(Φ) evaluated at 0 is the nth tensor power (C(Φ)(0))⊗n. Our main theorem, Theorem 5.11, says that if the iterated Hochschild complexes of the functors Φ = E(e,−) admit a natural action of a dg-PROP D of the form C(E(e,−))⊗D(n,m)→ C(E(e,−)) Date: June 16, 2014. 1i.e. such that the maps Φ(n)⊗Φ(m)→ Φ(n+m) are isomorphisms (also known as strong monoidal)