×

zbMATH — the first resource for mathematics

A computation of the equivariant index of the Dirac operator. (English) Zbl 0592.58044
Authors’ abstract: ”We give a proof of the Lefschetz fixed-point formulas of Atiyah-Singer for the twisted Dirac operator on a compact spin manifold M by using the heat equation on the frame bundle P. In particular, the \(\hat A\) class makes its entrance as the Jacobian of the exponential map on the manifold P.”
Reviewer: G.Warnecke

MSC:
58J20 Index theory and related fixed-point theorems on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
81T20 Quantum field theory on curved space or space-time backgrounds
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] ALVAREZ-GAUMÉ (L.) , Supersymmetry and the Atiyah-Singer Index theorem, Commun. Math. Phys. Vol. 90, No. 101, 1983 , pp. 161-173. Article | MR 85d:58078 | Zbl 0528.58034 · Zbl 0528.58034 · doi:10.1007/BF01205500 · minidml.mathdoc.fr
[2] ATIYAH (M.F.) , Circular symmetry and stationary phase approximation. In Proceedings of the Conference in honor of L. Schwartz, Asterisque , Paris, Vol. 131, 1985 , pp. 43-60. MR 87h:58206 | Zbl 0578.58039 · Zbl 0578.58039
[3] ATIYAH (M.F.) and BOTT (R.) , A Lefschetz fixed point formula for elliptic complexes, I. Ann. Math. , Vol. 86, 1967 , pp. 374-407 ; II, 88, 1968 , pp. 451-481. MR 35 #3701 | Zbl 0161.43201 · Zbl 0161.43201 · doi:10.2307/1970694
[4] ATIYAH (M.F.) and SEGAL (G.B.) , The index of elliptic operators II, Ann. of Math. , Vol. 87, 1968 , pp. 531-545. MR 38 #5244 | Zbl 0164.24201 · Zbl 0164.24201 · doi:10.2307/1970716
[5] ATIYAH (M.F.) and SINGER (I.M.) , The index of elliptic operators, I Ann. of Math. , Vol. 87, 1968 , pp. 484-530 ; III Ann. of Math., Vol. 87, 1968 , pp. 546-604. Zbl 0164.24301 · Zbl 0164.24301 · doi:10.2307/1970717
[6] BERGER (M.) , GAUDUCHON (P.) and MAZET (E.) , Le spectre d’une variété riemannienne, Lecture notes in Mathematics , 194, Springer-Verlag, Berlin-Heidelberg-New York. Zbl 0223.53034 · Zbl 0223.53034
[7] BERLINE (N.) and VERGNE (M.) , The equivariant index and Kirillov’s character formula, Amer. Math. Soc, Amer. J. of Math. , Vol. 107, n^\circ 5, 1985 , pp. 1159-1190. MR 87a:58143 | Zbl 0604.58046 · Zbl 0604.58046 · doi:10.2307/2374350
[8] BERLINE (N.) and VERGNE (M.) , Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante , Comptes rendus Acad. Sc., Vol. 295, 1982 , pp. 539-541. MR 83m:58002 | Zbl 0521.57020 · Zbl 0521.57020
[9] BISMUT (J.M.) , The Atiyah-Singer theorems. A Probabilistic approach. I The index theorem , J. Funct. Anal., Vol. 57, 1984 , pp. 56-99. II The Lefschetz fixed point formulas , J. Funct. Anal., Vol. 57, 1984 , pp. 329-348. Zbl 0556.58027 · Zbl 0556.58027 · doi:10.1016/0022-1236(84)90106-X
[10] DONNELLY (H.) and PATODI (V.K.) , Spectrum and the fixed point set of Isometries II , Topology, Vol. 16, 1977 , pp. 1-11. MR 55 #6489 | Zbl 0341.53023 · Zbl 0341.53023 · doi:10.1016/0040-9383(77)90027-1
[11] FRIEDAN (D.) and WINDEY (H.) , Supersymmetric Derivation of the Atiyah-Singer Index and the Chiral Anomaly , Nuclear Physics B., Vol. B., No. 235 (FS 11), 1984 , pp. 395-416. MR 89b:58198
[12] GETZLER (E.) , Pseudo differential operators on super manifolds and the Atiyah-Singer index theorem , Commun. Math. Phys., Vol. 92, 1983 , pp. 163-178. Article | MR 86a:58104 | Zbl 0543.58026 · Zbl 0543.58026 · doi:10.1007/BF01210843 · minidml.mathdoc.fr
[13] GETZLER (E.) , A short proof of the Atiyah-Singer Index Theorem (To appear). · Zbl 0607.58040 · doi:10.1016/0040-9383(86)90008-X
[14] GILKEY (P.) , Curvature and eigenvalues of the Laplacian for elliptic complexes . Advances in Math., Vol. 10, 1973 , pp. 344-382. MR 48 #3081 | Zbl 0259.58010 · Zbl 0259.58010 · doi:10.1016/0001-8708(73)90119-9
[15] GILKEY (P.) , Lefschetz fixed point formulas and the heat equation, in Partial Differential Equations and Geometry, Proceedings, Park-City Conference , 1977 , Lecture Notes in Pure Appl. Math., No. 48, pp. 91-147, M. Dekker, New York, 1979 . Zbl 0405.58044 · Zbl 0405.58044
[16] KIRILLOV (A.A.) , Characters of unitary representations of Lie groups , Funct. Anal. App., Vol. 2-2, 1967 , pp. 40-55. MR 38 #4615 | Zbl 0174.45001 · Zbl 0174.45001 · doi:10.1007/BF01075947
[17] LICHNEROWICZ (A.) , Spineurs Harmoniques , C.R. Acad. Sc., Vol. 257, 1963 , pp. 7-9. MR 27 #6218 | Zbl 0136.18401 · Zbl 0136.18401
[18] MCKEAN (H.P.) , SINGER (I.M.) , Curvature and the eigenvalues of the Laplacian , J. Differential Geometry, Vol. 1, 1967 , pp. 43-69. MR 36 #828 | Zbl 0198.44301 · Zbl 0198.44301
[19] MINAKSHISUNDARAM (S.) and PLEUEL (A.) , Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds , Canad. J. Math., Vol. 1, 1949 , pp. 242-256. MR 11,108b | Zbl 0041.42701 · Zbl 0041.42701 · doi:10.4153/CJM-1949-021-5
[20] PATODI (V.K.) , Curvature and the Eigenforms of the Laplace Operators , J. Diff. Geometry, Vol. 5, 1971 , pp. 233-249. MR 45 #1201 | Zbl 0211.53901 · Zbl 0211.53901
[21] PATODI (V.K.) , Analytic proof of the Riemann-Roch theorem for Kaehler manifolds , J. Diff. Geometry, Vol. 5, 1971 , pp. 251-283. MR 44 #7502 | Zbl 0219.53054 · Zbl 0219.53054
[22] VERGNE (M.) , Formule de Kirillov et indice de l’opérateur de Dirac, Proceedings of the International Congress of Mathematicians , 1983 , Warszawa. Zbl 0597.58033 · Zbl 0597.58033
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.