Quantization commutes with singular reduction: Cotangent bundles of compact Lie groups
SourceReviews in Mathematical Physics, 31, 6, (2019), pp. 1-40, article 1950016
Article / Letter to editor
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Reviews in Mathematical Physics
SubjectMathematical Physics; Mathematics
We analyze the 'quantization commutes with reduction' problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin-Sternberg Conjecture) for the conjugate action of a compact connected Lie group G on its own cotangent bundle T*G. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden-Weinstein quotient) T*G//AdG is typically singular. In the spirit of (modern) geometric quantization, our quantization of T*G (with its standard Kahler structure) is defined as the kernel of the Dolbeault-Dirac operator (or, equivalently, the spin(C)-Dirac operator) twisted by the pre-quantum line bundle. We show that this quantization of T*G reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarization. We then define the quantization of the singular quotient T*G//AdG as the kernel of the twisted Dolbeault-Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space L-2(T)(W(G,T)).
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