Getzler, Ezra Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. (English) Zbl 0543.58026 Commun. Math. Phys. 92, No. 2, 163-178 (1983). Some results on Clifford algebras and the spinor representation, as well as an extension of a symbol calculus on spin manifolds to pseudodifferential operators are combined to simplify the proof of the Atiyah-Singer index theorem for the Dirac operator on a spin manifold. Reviewer: N. Papaghiuc Cited in 7 ReviewsCited in 71 Documents MSC: 58J40 Pseudodifferential and Fourier integral operators on manifolds 35S05 Pseudodifferential operators as generalizations of partial differential operators 58J20 Index theory and related fixed-point theorems on manifolds 53C27 Spin and Spin\({}^c\) geometry 53C20 Global Riemannian geometry, including pinching 58A50 Supermanifolds and graded manifolds Keywords:Clifford algebras; spinor representation; spin manifolds; pseudodifferential operators; Atiyah-Singer index theorem; Dirac operator × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Alvarez-Gaumé, L.: Supersymmetry and the Atiyah-Singer index theorem. Commun. Math. Phys.90, 161-173 (1983) · Zbl 0528.58034 · doi:10.1007/BF01205500 [2] Atiyah, M. F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. II. Ann. Math.88, 451-491 (1968) · Zbl 0167.21703 · doi:10.2307/1970721 [3] Atiyah, M. F., Bott, R., Patodi, V. K.: On the heat equation and the index theorem. Inv. Math.19, 279-330 (1973) · Zbl 0257.58008 · doi:10.1007/BF01425417 [4] Atiyah, M. F., Singer, I. M.: The index of elliptic operators, III. Ann. Math.87, 546-604 (1968) · Zbl 0164.24301 · doi:10.2307/1970717 [5] Berezin, F. A.: The method of second quantization, New York: Academic Press 1966 · Zbl 0151.44001 [6] Berezin, F. A., Marinov, M. S.: Particle spin dynamics as the Grassman variant of classical mechanics. Ann. Phys.104, 336-362 (1977) · Zbl 0354.70003 · doi:10.1016/0003-4916(77)90335-9 [7] Bokobza-Haggiag, J.: Opérateurs pseudo-différentiel sur une variété différentiable. Ann. Inst. Four. (Grenoble)19, 125-177, (1969) · Zbl 0176.08702 [8] Boutet de Mouvel, L.: Hypoelliptic operators with double characteristics and related pseudo-differential operators. Commun. Pure Appl. Math.27, 585-639 (1976) · Zbl 0294.35020 [9] Bourbaki, N.: Eléments de Mathematique, Algebre, ch. 9, Formes sesquilinéaires et formes quadratiques. Paris: Herman 1959 [10] Brauer, R., Weyl, H.: Spinors inn dimensions. Am. J. Math.57, 425 (1935) · Zbl 0011.24401 · doi:10.2307/2371218 [11] Chevalley, C.: The algebraic theory of spinors. New York: Columbia University, 1954 · Zbl 0057.25901 [12] Glimm, J., Jaffe, A.: Quantum physics. New York: Springer 1981 · Zbl 0461.46051 [13] Hörmander, L.: Pseudodifferential operators. Commun. Pure Appl. Math.17, 501-517 (1965) · Zbl 0125.33401 · doi:10.1002/cpa.3160180307 [14] Kostant, B.: Symplectic spinors. Symp. Math,16, 139-152 (1973) [15] Leites, D. A.: Introduction to the theory of supermanifolds. Usp. Mat. Nauk.35, 3-57 (1980) (Russ. Math. Surv.35, 1-64 (1980)) · Zbl 0439.58007 [16] Patodi, V. K.: An analytic proof of the Riemann-Roch-Hirzebruch theorem for Kaehler manifolds. J. Diff. Geom.5, 251-283 (1971) · Zbl 0219.53054 [17] Widom, H.: Families of pseudodifferential operators. In: Topics in Functional analysis. Gohberg, I., and Kac, M. (eds.) New York: Academic Press 1978 · Zbl 0477.58035 [18] Widom, H.: A complete symbolic calculus for pseudodifferential operators. Bull. Soc. Math.104, 19-63 (1980) · Zbl 0434.35092 [19] Witten, E.: Supersymmetry and Morse theory. J. Diff. Geom.17, 661 (1982) · Zbl 0499.53056 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.