×

Symplectic surgery and the Spin\(^c\)-Dirac operator. (English) Zbl 0929.53045

Let \(G\) be a compact Lie group acting on a compact symplectic manifold \((M,\omega)\) in a Hamiltonian fashion and with an equivariant moment map \(J:M\to{\mathfrak g}^*\). If these data are pre-quantizable, one can construct an associated \(\text{Spin}^c\)-Dirac operator \(\eth^c\), whose equivariant index yields a virtual representation of \(G\). The author proves a conjecture of V. Guillemin and S. Sternberg [Invent. Math. 67, 515-538 (1982; Zbl 0503.58018)] that if \(0\) is a regular value of \(J\), the multiplicity \(N(0)\) of the trivial representation in the index space \(\text{ind}(\eth^c)\), is equal to the index of the \(\text{Spin}^c\)-Dirac operator for the symplectic quotient \(J^{-1}(0)/G\). The result generalizes some previous articles where the conjecture could only be proven in special cases [loc. cit., see also Zbl 0849.53027, Zbl 0814.58020, Zbl 0851.53020, Zbl 0870.53022].
The present article uses the symplectic cutting technique of E. Lerman [Math. Res. Lett. 2, 247-258 (1995; Zbl 0835.53034)] in order to get the full result. Readers having a certain familiarity with the \(\text{Spin}^c\)-Dirac operator on symplectic manifolds will enjoy the very well-written introduction and the clear presentation of the details in the following sections. An extension to the case that \(0\) is no longer a regular value of \(J\) is discussed in a separate paper [E. Meinrenken and R. Sjamaar, Topology 38, 699-762 (1999)].

MSC:

53D05 Symplectic manifolds (general theory)
58J20 Index theory and related fixed-point theorems on manifolds
53D50 Geometric quantization
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Atiyah, M. F., Elliptic Operators and Compact Groups. Elliptic Operators and Compact Groups, Springer Lecture Notes in Mathematics, 401 (1974), Springer-Verlag: Springer-Verlag Berlin
[2] Atiyah, M. F., Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14, 1-15 (1982) · Zbl 0482.58013
[3] Atiyah, M. F.; Singer, I., The index of elliptic operators, I, Ann. Math., 87, 484-530 (1968) · Zbl 0164.24001
[4] Atiyah, M. F.; Segal, G., The index of elliptic operators, II, Ann. Math., 87, 531-545 (1968) · Zbl 0164.24201
[5] Atiyah, M. F.; Singer, I., The index of elliptic operators, III, Ann. Math., 87, 546-604 (1968) · Zbl 0164.24301
[6] Berline, N.; Vergne, M., Classes charactéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris, 295, 539-541 (1982) · Zbl 0521.57020
[7] Berline, N.; Getzler, E.; Vergne, M., Heat Kernels and Dirac Operators (1992), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0744.58001
[8] Delzant, T., Hamiltoniens périodique et images convexes de l’application moment, Bull. Soc. Math. France, 116, 315-339 (1988) · Zbl 0676.58029
[9] Duistermaat, J. J., The Heat Kernel Lefschetz Fixed Point Formula for the \(Spin^c\)-Dirac Operator (1995), Birkhäuser: Birkhäuser Basel · Zbl 0858.58045
[10] Duistermaat, J. J.; Guillemin, V.; Meinrenken, E.; Wu, S., Symplectic reduction and Riemann-Roch for circle actions, Math. Res. Lett., 2, 259-266 (1995) · Zbl 0839.58026
[11] Duistermaat, J. J.; Heckman, G., On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., 69, 259-268 (1982) · Zbl 0503.58015
[12] Fulton, W., Introduction to Toric Varieties (1994), Princeton Univ. Press: Princeton Univ. Press Princeton
[13] Gompf, R., A new construction of symplectic manifolds, Ann. of Math., 142, 527-595 (1995) · Zbl 0849.53027
[14] Gotay, M. J., On coisotropic embeddings of presymplectic manifolds, Proc. Am. Math. Soc., 84, 111-114 (1982) · Zbl 0476.53020
[15] Guillemin, V., Reduction and Riemann-Roch, Lie Groups and Geometry in Honour of B. Kostant. Lie Groups and Geometry in Honour of B. Kostant, Progress in Mathematics (1994), Birkhäuser: Birkhäuser Boston
[16] Guillemin, V.; Sternberg, S., Geometric quantization and multiplicities of group representations, Invent. Math., 67, 515-538 (1982) · Zbl 0503.58018
[17] Guillemin, V.; Sternberg, S., Homogeneous quantization and multiplicities of group representations, J. Funct. Anal., 47, 344-380 (1982) · Zbl 0733.58021
[18] Guillemin, V.; Sternberg, S., Convexity properties of the moment map, Invent. Math., 67, 491-513 (1982) · Zbl 0503.58017
[19] Guillemin, V.; Sternberg, S., Symplectic Techniques in Physics (1984), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0576.58012
[20] Guillemin, V.; Sternberg, S., Birational equivalence in the symplectic category, Invent. Math., 97, 485-522 (1989) · Zbl 0683.53033
[21] Jeffrey, L.; Kirwan, F., On localization and Riemann-Roch numbers for symplectic quotients, Quart. J. Math. Oxford Ser. (2), 47, 165-185 (1996) · Zbl 0870.53022
[22] Kalkman, J., Cohomology rings of symplectic quotients, J. Reine Angew. Math., 485, 37-52 (1995) · Zbl 0851.57029
[23] Kawasaki, T., The signature theorem for V-manifolds, Topology, 17, 75-83 (1978) · Zbl 0392.58009
[24] Kawasaki, T., The Riemann-Roch theorem for complex V-manifolds, Osaka J. Math., 16, 151-157 (1979) · Zbl 0405.32010
[25] Kirwan, F., Convexity properties of the moment map III, Invent. Math., 77, 547-552 (1984) · Zbl 0561.58016
[26] Lerman, E., Symplectic cuts, Math. Res. Lett., 2, 247-258 (1995) · Zbl 0835.53034
[27] Lerman, E.; Tolman, S., Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Am. Math. Soc., 349, 4201-4230 (1997) · Zbl 0897.58016
[28] Lerman, E.; Meinrenken, E.; Tolman, S.; Woodward, C., Non-abelian convexity by symplectic cuts, Topology, 37, 245-260 (1998) · Zbl 0913.58023
[29] Meinrenken, E., On Riemann-Roch formulas for multiplicities, Amer. Math. Soc., 9, 373-389 (1996) · Zbl 0851.53020
[30] Meinrenken, E., Vielfachheitsformeln für die Quantisierung von Phasenräumen (September 1994), University of Freiburg
[31] Satake, I., The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan, 9, 464-492 (1957) · Zbl 0080.37403
[32] Sternberg, S., On minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field, Proc. Natl. Acad. Sci., 74, 5253-5254 (1977) · Zbl 0765.58010
[33] Vergne, M., Quantification géométrique et multiplicités, C. R. Acad. Sci. Paris, 319, 327-332 (1994) · Zbl 0814.58020
[34] Vergne, M., Multiplicity formula for geometric quantization, I and II, Duke Math. J., 82, 143-194 (1996)
[35] Vergne, M., Equivariant index formula for orbifolds, Duke Math. J., 82, 637-652 (1996) · Zbl 0874.57029
[36] Woodward, C., The classification of transversal multiplicity-free group actions, Ann. Global Anal. Geom., 14, 3-42 (1996) · Zbl 0877.58022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.