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Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. (English) Zbl 0543.58026
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

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
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[1] Alvarez-Gaumé, L.: Supersymmetry and the Atiyah-Singer index theorem. Commun. Math. Phys.90, 161-173 (1983) · Zbl 0528.58034
[2] Atiyah, M. F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. II. Ann. Math.88, 451-491 (1968) · Zbl 0167.21703
[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
[4] Atiyah, M. F., Singer, I. M.: The index of elliptic operators, III. Ann. Math.87, 546-604 (1968) · Zbl 0164.24301
[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
[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
[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
[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
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