Unitarity in “Quantization Commutes with Reduction”

Abstract

Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M // G. Guillemin and Sternberg [Invent. Math. 67, 515–538 (1982)] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M //G and the G-invariant subspace of the quantum Hilbert space over M.

Reproducing other recent results in the literature, we prove that in general the natural map of Guillemin and Sternberg is not unitary, even to leading order in Planck’s constant. We then modify the quantization procedure by the “metaplectic correction” and show that in this setting there is still a natural invertible map between the Hilbert space over M // G and the G-invariant subspace of the Hilbert space over M. We then prove that this modified Guillemin–Sternberg map is asymptotically unitary to leading order in Planck’s constant. The analysis also shows a good asymptotic relationship between Toeplitz operators on M and on M // G.

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Correspondence to Brian C. Hall.

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Supported in part by NSF Grant DMS-02000649.

Communicated by J.Z. Imbrie

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Hall, B.C., Kirwin, W.D. Unitarity in “Quantization Commutes with Reduction”. Commun. Math. Phys. 275, 401–442 (2007). https://doi.org/10.1007/s00220-007-0303-6

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Keywords

  • Line Bundle
  • Toeplitz Operator
  • Holomorphic Section
  • Bergman Kernel
  • Geometric Quantization