In a 1993 letter, Deligne posed the following (paraphrased from a [paper of Gerstenhaber and Voronov's](http://www.math.umn.edu/~voronov/del.pdf)):



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> > **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 $E\_2$.
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("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](http://www.math.umn.edu/~voronov/del.pdf), [Tamarkin](http://arxiv.org/abs/math/9803025), [Voronov](http://lanl.arxiv.org/abs/math/9908040), [McClure-Smith](http://arxiv.org/abs/math/9910126) (see also [1](http://arxiv.org/abs/math/0106024), [2](http://arxiv.org/abs/math/0211368)), and [Kontsevich-Soibelman](http://arxiv.org/abs/math/0001151).


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](https://arxiv.org/abs/math/0210150) 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...



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> > **Question:** With suitable hypotheses on $A$, can $CC^\*(A)$ be given the structure of an algebra over a chain operad of $\tilde E\_2$?
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Can the Hochschild cochain complex be given the structure of a "homotopy BV algebra"?
Yes, this is the "cyclic Deligne conjecture", which also has several proofs by now. I believe the first one was Kaufmann's, using the cacti operad. 


