Hochschild cohomology and characteristic classes for star-products

Details Hochschild cohomology and characteristic classes for star-products Hochschild cohomology and characteristic classes for star-products

1997-09-30

arxiv.org/abs/q-alg/9709043v2

Mathematics

Differential Geometry

LaTex2e, 22 pages, minor corrections in the last section and a reference added (Festschrift for V.I. Arnol'd's 60th birthday,

Scientific paper

We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class obtained by differentiating the star-product with respect to the deformation parameter is seen to be closely related to the characteristic class of the quantization. A fundamental role in the analysis is played by ``quantum Liouville operators,'' which rescale the deformation parameter in the same way in which Liouville vector fields scale the Poisson structure (or the units of action). Several examples are given.

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