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Deformation Quantization of Hermitian Vector Bundles

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Abstract

Motivated by deformation quantization, we consider in this paper *-algebras \(\mathcal{A}\) over rings \(C\)=\(R\)(i), where \(R\) is an ordered ring and I2=−1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) \(\mathcal{A}\)-valued inner product. For A=C ∞(M), M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of C ∞(M) and Γ∞(\(End\left( E \right)\)(E)) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of C *-algebras. We also discuss the semi-classical geometry arising from these deformations.

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Authors and Affiliations

  1. Department of Mathematics, UC Berkeley, Berkeley, CA, 94720, U.S.A.

    Henrique Bursztyn

  2. Département de Mathématique, Université Libre de Bruxelles, Campus Plaine, CP 218, Boulevard du Triomphe, B-1050, Bruxelles, Belgium

    Stefan Waldmann

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  1. Henrique Bursztyn
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  2. Stefan Waldmann
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Bursztyn, H., Waldmann, S. Deformation Quantization of Hermitian Vector Bundles. Letters in Mathematical Physics 53, 349–365 (2000). https://doi.org/10.1023/A:1007661703158

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