O ct 2 00 5 PROJECTIVELY INVARIANT STAR PRODUCTS
نویسنده
چکیده
It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth functions on the cotangent bundle, fiberwise-polynomial of bounded degree, a one-parameter family of graded star products. For a particular value of the parameter (corresponding to half-densities) the star product is symmetric, and specializes in the projectively flat case to the one constructed previously by C. Duval, P. Lecomte and V. Ovsienko. These star products are built from a projectively invariant quantization map associating to a symmetric polyvector a formally self-adjoint operator on densities. A limiting form of this family of star products yields a commutative deformation of the symmetric tensor algebra of the manifold which is closely related to the limiting commutative multiplication of modular forms defined by P. Cohen, Y. Manin, and D. Zagier. A basic ingredient of the proof is the construction of projectively invariant multilinear differential operators on bundles of weighted symmetric k-vectors. The construction works except for a discrete set of excluded weights and generalizes the Rankin-Cohen brackets of modular forms.
منابع مشابه
00 5 Projectively Invariant Star Products
It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth, fiberwise-polynomial functions on the cotangent bundle a one-parameter family of graded star products. For a particular value of the parameter (corresponding to half-densities) the star product is symmetric, and specializes in the projectively flat case to the one constructed previ...
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