# Can the Hochschild cochain complex be given the structure of a "homotopy BV algebra"?

In a 1993 letter, Deligne posed the following (paraphrased from a paper of Gerstenhaber and Voronov's):

Conjecture (Deligne). The Hochschild cochain complex $CC^*(A)$ of an associative algebra $A$ has a natural structure of an algebra over a chain operad of the little 2-disks operad $E_2$.

("Chain operad of $E_2$" means any operad in $Ch$ whose homology is isomorphic to the homology of $E_2$, i.e. to the Gerstenhaber operad.)

Various versions of Deligne's conjecture have been proven, e.g. by Gerstenhaber-Voronov, Tamarkin, Voronov, McClure-Smith (see also 1, 2), and Kontsevich-Soibelman.

Now, in some situations, $HH^*(A)$ can be equipped with a BV operator $\Delta$ compatible with the Gerstenhaber algebra structure, so that it becomes a BV algebra. For instance, this was proven by Tradler in the case that $A$ is unital and equipped with a symmetric, invariant, and non-degenerate inner product. And BV algebras are the same thing as algebras over the framed little 2-disks operad $\tilde E_2$. So...

Question: With suitable hypotheses on $A$, can $CC^*(A)$ be given the structure of an algebra over a chain operad of $\tilde E_2$?

• Thanks! Do you know whether it is expected (proven or not) that $CC^*(A)$ has any more structure? Jun 23, 2016 at 16:51