HOCHSCHILD HOMOLOGY OF STRUCTURED

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. 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 [57, 27, 29, 31]. 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 [11, 38, 27, 29]. Most of these results use a very appealing recipe for constructing such operations introduced by Kontsevich in [39]. Starting from a model for the moduli of curves in terms of the combinatorial data of fatgraphs, the graphs can be used to guide the local-to-global construction of an operation on the Hochschild complex of an A∞-Frobenius algebra A – at every vertex of valence n, an n-ary trace is performed. 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. In constrast to the above, 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; by fiat these operations preserve composition, something that can be hard to verify in the setting of [39]. After constructing the operations globally, we then show how to read-off the action explicitly, so that formulas for individual operations can also be obtained. Doing this we recover the same formuli as in the local-to-global approach. Our construction can be seen as a formalization and extension of the method of [11] which considered the case of A∞–Frobenius algebras. Our main result, which we will explain now in more details, gave rise to new computations, including a complete description of the operations on the Hochschild complex of commutative algebras [36], a description of a large complex of operations on the Hochschild complex of commutative Frobenius algebras [35] and a description of the universal operations given any type of algebra [61]. 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 Gerstenhaber-Voronov [18], we call such a pair E = (E , i) a prop with A∞-multiplication. An E–algebra is a symmetric Date: October 21, 2015.