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The Manifold of Compatible Almost Complex Structures and Geometric Quantization

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Abstract

Let (M, ω) be an integral symplectic manifold. We study a family of hermitian vector bundles on the space \({\mathcal{J}}\) of almost complex structures on M compatible with ω, whose fibers consist of nearly holomorphic sections of powers of a prequantum line bundle. We obtain asymptotics of the curvature of a natural connection in these bundles. These results, together with Toeplitz operator theory, provide another proof of Donaldson’s result that the action of HAM(M) on \({\mathcal{J}}\) is hamiltonian with moment map the scalar curvature. We also give an example involving Teichmüller space and discuss a relationship between parallel transport and the Schrödinger equation.

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Authors and Affiliations

  1. Department of Mathematics, University of Western Ontario, London, Ontario, Canada, N6A 5B7

    T. Foth

  2. Mathematics Department, University of Michigan, Ann Arbor, Michigan, 48109, USA

    A. Uribe

Authors
  1. T. Foth
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  2. A. Uribe
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Correspondence to A. Uribe.

Additional information

Communicated by P. Sarnak

T.F. supported in part by NSF grant DMS-0204154 and by NSERC.

A.U. supported in part by NSF grant DMS-0070690 and DMS-0401064.

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Foth, T., Uribe, A. The Manifold of Compatible Almost Complex Structures and Geometric Quantization. Commun. Math. Phys. 274, 357–379 (2007). https://doi.org/10.1007/s00220-007-0280-9

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  • DOI: https://doi.org/10.1007/s00220-007-0280-9

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