Additivity of spin^{𝑐}-quantization under cutting

Fuchs, Shay

doi:10.1090/S0002-9947-09-04863-6

Additivity of spin$^c$-quantization under cutting
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Trans. Amer. Math. Soc. 361 (2009), 5345-5376 Request permission

Abstract:

A $G$-equivariant spin$^c$-structure on a manifold gives rise to a virtual representation of the group $G$, called the spin$^c$-quantization of the manifold. We present a cutting construction for $S^1$-equivariant spin$^c$-manifolds and show that the quantization of the original manifold is isomorphic to the direct sum of the quantizations of the cut spaces. Our proof uses Kostant-type formulas, which express the quantization in terms of local data around the fixed point set of the $S^1$-action.

References

Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol. 25, American Mathematical Society, Providence, RI, 2000. Translated from the 1997 German original by Andreas Nestke. MR 1777332, DOI 10.1090/gsm/025
Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs, vol. 98, American Mathematical Society, Providence, RI, 2002. Appendix J by Maxim Braverman. MR 1929136, DOI 10.1090/surv/098
H. Lawson and M. Michelson, Spin Geometry, Princeton, 1989.
Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247–258. MR 1338784, DOI 10.4310/MRL.1995.v2.n3.a2
Victor Guillemin, Shlomo Sternberg, and Jonathan Weitsman, Signature quantization, J. Differential Geom. 66 (2004), no. 1, 139–168. MR 2128715
Ana Cannas da Silva, Yael Karshon, and Susan Tolman, Quantization of presymplectic manifolds and circle actions, Trans. Amer. Math. Soc. 352 (2000), no. 2, 525–552. MR 1714519, DOI 10.1090/S0002-9947-99-02260-6
Michael D. Grossberg and Yael Karshon, Equivariant index and the moment map for completely integrable torus actions, Adv. Math. 133 (1998), no. 2, 185–223. MR 1604738, DOI 10.1006/aima.1997.1686
Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992. MR 1215720, DOI 10.1007/978-3-642-58088-8
Shrawan Kumar and Michèle Vergne, Equivariant cohomology with generalized coefficients, Astérisque 215 (1993), 109–204. Sur la cohomologie équivariante des variétés différentiables. MR 1247061
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
S. Fuchs, spin$^c$-prequantization and symplectic cutting (in preparation).
Eckhard Meinrenken, Symplectic surgery and the $\textrm {Spin}^c$-Dirac operator, Adv. Math. 134 (1998), no. 2, 240–277. MR 1617809, DOI 10.1006/aima.1997.1701
Victor W. Guillemin and Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999. With an appendix containing two reprints by Henri Cartan [ MR0042426 (13,107e); MR0042427 (13,107f)]. MR 1689252, DOI 10.1007/978-3-662-03992-2