Abstract
An analytic proof of the Atiyah-Singer index, theorem is given with the help of the tools of supermathematics. The index formula for the Dirac operator on a spinor manifold is obtained here by direct calculation. A large portion of the paper is devoted to questions of quantization on supermanifolds, using spinors as example.
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References
V. I. Arnol'd and A. B. Givental', “Symplectic geometry,” Itogi Nauki Tekhn., Ser. Sovrem. Probl. Mat., Fundam. Napravl.,4, 7–139 (1985).
A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics [in Russian], Nauka, Moscow (1981).
F. A. Berezin, Method of Secondary Quantization [in Russian], 2nd edition, supplemented, Nauka, Moscow (1986).
F. A. Berezin, Introduction to Algebra and Analysis with Anticommutative Variables [in Russian], Mosk. Gos. Univ. (MGU), Moscow (1983).
F. A. Berezin and M. A. Shubin, Schrödinger's Equation [in Russian], MGU, Moscow (1983).
A. M. Vershik, “Metagonal and metaplectic infinite-dimensional groups. 1. General concepts and the metagonal group,” Zapiski Nauchn. Semin. Leningr. Otdel. Mat. Inst.,123, 3–35 (1983).
V. A. Ginzburg, “Quantization and the method of orbits,” Itogi Nauki Tekh., Ser. Matem. Analiz,22, 37–58 (1984).
B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable systems,” Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravl.,4, 179–284 (1985).
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979).
B. A. Dudrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods of Homology Theory [in Russian], Nauka, Moscow (1984).
A. A. Kirillov, Elements of the Theory of Representations [in Russian], Nauka, Moscow (1972).
A. A. Kirillov, “Geometric quantization,” Itogi Nauki Tekh., Ser. Sovrem. Problem. Matem., Fundam. Napravl.,4, 141–178 (1985).
D. A. Leites, Theory of Supermanifolds [in Russian], Petrozavodsk (1983).
A. S. Mishchenko, Vector Bundles and Their Applications [in Russian], Nauka, Moscow (1984).
L. A. Takhtadzhyan, “Noncommutative homology of quantum tori,” Funkts. Analiz Prilozhen.,23, No. 2, 75–76 (1989).
A. S. Shvarts, “Definition of superspace,” Teor. Mat. Fiz.,60, No. 1, 37–42 (1984).
A. S. Shvarts, Quantum Field Theory and Topology [in Russian], Nauka, Moscow (1989).
M. A. Shubin, Psuedodifferential Operators and Spectral Theory [in Russian], Nauka, Moscow (1978).
L. Alvarez-Gaumé, “Supersymmetry and the Atiyah-Singer index theorem,” Commun. Math. Phys.,90, 161–174 (1983).
M. F. Atiyah, R. Bott, and V. K. Patodi, “On the heat equation and the index theorem,” Invent. Math.,19, 279–230 (1973); “Errata,” Invent. Math.,28, 277–280 (1975).
M. F. Atiyah and I. M. Singer, “The index of elliptic operators on compact manifolds,” Bull. Amer. Math. Soc.,69, 422–433 (1963).
M. F. Atiyah and I. M. Singer, “The index of elliptic operators. I,” Ann. Math.,87, 484–530 (1968).
M. F. Atiyah and I. M. Singer, “The index of elliptic operators. III,” Ann. Math.,87, 546–604 (1968).
B. S. de Witt, Dynamical Theory of Groups and Fields, Gordon and Breach, New York (1965).
E. Getzler, “Psuedodifferential operators on supermanifolds and the Atiyah-Singer index theorem,” Commun. Math. Phys.,92, 163–178 (1983).
J. Glimm and A. Jaffe, Quantum Physics, Springer, New York (1981).
V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc. Surv. No. 14, Providence, R. I. (1977).
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, New York e. a. (1966).
N. Hurt, Geometric Quantization in Action, Reidel, Dordrecht e. a. (1983).
V. G. Kac, “Lie superalgebras,” Adv. Math.,26, 8–96 (1976).
M. Karoubi, K Theory, Springer, Berlin e. a. (1978).
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience, New York e. a. (1963, Vol. I; 1969, Vol. II).
B. Kostant, “Graded manifolds, graded Lie theory and prequantization,” Lect. Notes Math.,570, 177–306 (1977).
J. Martin, “Generalized classical dynamics and the classical analogue of a Fermi oscillator,” Proc. Roy. Soc. A,251, No. 1267, 536–542 (1959).
J. Martin, “The Feynman principle for a Fermi system,” Proc. Roy. Soc. A,251, No. 1267, 543–549 (1959).
R. Palais, Seminar on the Atiyah-Singer Index Theorem, Ann. Math. Stud. No. 57, Princeton Univ. Press, Princeton, N. J. (1965).
V. K. Patodi, “An analytic proof of the Riemann-Roch-Hirzebruch theorem for Kaehler manifolds,” J. Different. Geom.,5, 251–283 (1971).
M. Rothstein, “Deformations of complex supermanifolds,” Proc. Amer. Math. Soc.,95, No. 2, 255–260 (1985).
M. Rothstein, “Theaxioms of supermanifolds and a new structure arising from them,” Trans. Amer. Math. Soc.,297, No. 1, 159–180 (1986).
Th. Voronov, “Geometric integration theory on supermanifolds,” in: Soviet Scientific Reviews, Math. Phys., Gordon and Breach (1991).
F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, New York e. a. (1983).
H. Weyl, The Theory of Groups and Quantum Mechanics, Dover (1931).
H. Widom, “A complete symbolic calculus for psuedodifferential operators,” Bull. Sci. Math., 2e série,104, 19–63 (1980).
B. V. Fedosov, “Analytic formulas for the index of elliptic operators,” Tr. Mosk. Mat. Obshch.,30, 159–242 (1974).
B. V. Fedosov, “Index theorem in the algebra of quantum observables,” Dokl. Akad. Nauk SSSR,305, No. 4, 835–838 (1989).
J. Bokobza-Haggiang, “Operateurs pseudo-differentiel sur une varieté différentiable,” Ann. Inst. Fourier,19, 125–177 (1969).
E. Getzler, “A short proof of the local Atiyah-Singer index theorem,” Topology,25, No. 1, 111–117 (1986).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 38, pp. 3–118, 1991.
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Voronov, F.F. Quantization of supermanifolds and an analytic proof of the Atiyah-Singer index theorem. J Math Sci 64, 993–1069 (1993). https://doi.org/10.1007/BF01097407
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DOI: https://doi.org/10.1007/BF01097407