Nikita Markarian’s mathblog

The Hochschild cohomology and the Duflo isomorphism

Posted in Mathematics by nikitamarkarian on October 3, 2008

Here are slides of the talk given at the Algebra seminar of Paris 6,7. The subject strongly depends on the technique developed here and is inspired by the Alekseev-Meinrenken proof of the Duflo formula.

In more details, in the Riemann-Roch paper it is shown, that Hochschild cochains on a smooth complex manifold are isomorphic to the direct sum of polyvetor fields being placed in appropriate cohomlogical degree. If the manifold is a holomorphic symplectic, then some additional structures appear. Firstly, polyvector fields are isomorphic to differential forms. Secondly, on the Hochschild cochains a differential
operator of the first order appears. It is given by the Gerstenhaber bracket with the cochain that corresponds to the Poisson structure that corresponds to the symplectic structure. Or one may think about the non-commutative deformation of functions on the manifold along the symplectic structure and the mentioned differential operator is the deformation of the differential on the Hochschild complex. Anyway,
the natural question is: does this differential operator map to the de Rham differential under the isomorphism between Hochschild cochains and differential forms? The answer is ‘no’, if one uses naive (PBW) isomorphism between cochains and polyvector fields. But the situation may be improved if one corrects the isomorphism by a term that looks like the Duflo character.

This statement nearly immediately gives a formula for the Todd class in the Riemann-Roch theorem. It seems that this approach is very close the one developed in Tsygan papers e. g. this one.

The algebra of Hochschild cochains with the differential gives an example of ‘derived’ de Rham complex, that is a an object in the derived category of
\mathcal O-modules that is an algebra in the derived category and is equipped with a differential of degree 1. The constructed map gives an isomorphism between this object and the usual de Rham complex that respects the (de Rham) differential.If the object would be a usual algebra, rather than an algebra in the derived category, this map would be an isomorphism of algebras. As the case stands, there is only an isomorphism of subalgebras annihilated by the de Rham differential.

Note finally, that nowadays there are a lot of proofs of different kinds of the Duflo isomorphism between algebras in a derived category. We hope that our approach would give a sharper statement like an isomorphism of homotopy algebras.

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