Spin-quantization commutes with reduction. (English) Zbl 1272.81091

The author states and proves a statement of the form that “quantization commutes with reduction” in the sense of Guillemin-Sternberg in the case of non-abelian group actions and under very general assumptions. Herewith, he extends previous results by L. C. Jeffrey and F. C. Kirwan [Topology 36, No. 3, 647–693 (1997; Zbl 0876.55007)] and P.-É. Paradan [Ann. Sci. Éc. Norm. Supér. (4) 36, No. 5, 805–845 (2003; Zbl 1091.53059)]. Let us briefly indicate the setting of the paper: Let \(K\) be a compact connected Lie group. Then a Hamiltonian \(K\)-manifold \((M,\omega,\Phi)\) is a compact \(K\)-manifold \(M\) with symplectic form \(\omega\) and momentum map \(\Phi\) from \(M\) into the adjoint of the Lie algebra of \(K\). Such a \(K\)-manifold is Spin-prequantizied if \(M\) carries an equivariant \(\mathrm{Spin}^c\)-structure \(P\) with determinant bundle being a Kostant-Souriau line bundle over \((M,2\omega,2\Phi)\). Now, consider the \(\mathrm{Spin}^c\) Dirac operator \(\mathcal{D}_P\) attached to \(P\). Then the spin of \((M,\omega,\Phi)\) quantization corresponds to the equivariant index of the elliptic operator \(\mathcal{D}_P\), and is denoted by \(\mathcal{Q}^K_{\mathrm{spin}}(M)\). The main result computes the multiplicities of \(\mathcal{Q}^K_{\mathrm{spin}}(M)\) as in the work of V. Guillemin and S. Sternberg [Invent. Math. 67, 515–538 (1982; Zbl 0503.58018)].


81R25 Spinor and twistor methods applied to problems in quantum theory
53D20 Momentum maps; symplectic reduction
53D50 Geometric quantization
81S10 Geometry and quantization, symplectic methods
22E70 Applications of Lie groups to the sciences; explicit representations
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
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