nLab Deligne conjecture

Idea

Statement

The original Deligne conjecture on the structure of Hochschild cohomology of an associative algebra and more generally an A-∞ algebra states that Hochschild cohomology naturally has the structure of a BV-algebra and that moreover the full Hochschild cohomology complex has the structure of an algebra over an operad over the (framed) little disk operad. (The second statement implies the first since (BV-)Gerstenhaber algebras are algebras over an operad over the (framed) little disk operad.)

Proofs of the conjecture have been given first by McClure-Smith and others, then also by Kontsevich-Soibelman, and Tamarkin.

A proof for general Ek-algebras in symmetric monoidal (∞,1)-categories is in (Lurie, section 2.5).

Geometric interpretation

A general geometric (higher geometric) interpretation has been indicated in Ben-ZviFrancisNadler. They observed (see Hochschild cohomology for details) that the Hochschild homology of $\mathcal{O}(X)$ is naturally interpreted as the algebra of functions on the derived loop space $\mathcal{L}X$. In the presence of good geometric ∞-function theory this naturally induces an action of the little disk operad on $\mathcal{O}(\mathcal{L}X)$. Since a Gerstenhaber algebra is an algebra over the homology of the little disk operad, this immediately explains the existence of this structure.

See also this prop.

For higher algebras and higher monoidal categories

More generally, analogs of the statement of the Deligne conjecture exist and work for $(\infty,n)$-algebras: k-tuply monoidal n-categories. This is closely related to the statement (and proofs) of the delooping hypothesis.

This case is discussed in (Francis) and (Lurie).

In (KockToen) it is shown that in a monoidal model category that is also a simplicial model category the derived hom-space $\mathbb{R} Hom(I,I)$ from the tensor unit to itself is a E-2 algebra.

History

Historically it was first found that there is the structure of a Gerstenhaber algebra on $HH^\bullet(A,A)$. By (Cohen) it was known that Gerstenhaber algebras arise as the homology of E2-algebras in chain complexes. In a letter in 1993 Deligne wondered whether the Gerstenhaber algebra structure on the Hochschild cohomology $HH^\bullet(A,A)$ lifts to an E2-algebra-structure on the cochain complex $C^\bullet(A,A)$.

In GerstenhaberVoronov (1994) a resolution of the Gerstenhaber algebra structure was given, but the relationship to $E_2$-algebras remained unclear.

In (Tamarkin (1998)) a genuine resolution in the model structure on operads of the Gerstenhaber operad was given and shown to act via the Gerstenhaber-Voronov construction on $C^\bullet(A,A)$. This proved Deligne’s conjecture.

Various authors later further refined this result. A summary of this history can be found in (Hess).

In Hu-Kriz-Voronov (2003) it was further shown that for $A$ an En-algebra, $C^\bullet(A,A)$ is an $E_{n+1}$-algebra.

References

Direct proofs of the Deligne conjecture have been given in.

• James McClure, Jeffrey Smith, A solution of Deligne’s conjecture (arXiv:math/9910126)

• Maxim Kontsevich, Yan Soibelman, Deformations of algebras over operads and Deligne’s conjecture (arXiv:math/0001151)

• Dmitry Tamarkin, Another proof of M. Kontsevich formality theorem (arXiv:math/9803025)

• Clemens Berger, Benoit Fresse, Combinatorial operad actions on cochains (arXiv:math/0109158)

A review is in

• Kathryn Hess, Deligne’s Hochschild cohomology conjecture (pdf)

• Cohen

• Gertstenhaber-Voronov

• Hu, Kriz, Voronov

A transparent higher geometric interpretation in a suitably dualizable context is indicated in

• David Ben-Zvi, John Francis, David Nadler,

  Integral transforms and Drinfeld centers in derived algebraic geometry (arXiv:0805.0157)

• John Francis, PhD thesis (web)

Section 2.5 of

• Jacob Lurie, Ek-Algebras

A construction in monoidal model categories is in

• Joachim Kock, Bertrand Toen, Simplicial localization of monoidal structures, and a non-linear version of Deligne’s conjecture Compositio Math. 141 (2005), 253-261 (arXiv)