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On the rigidity theorems of Witten. (English) Zbl 0667.57009
This paper proves that the elliptic genus is rigid, as was predicted by Edward Witten in 1968. Specifically, the paper describes a sequence of genera having the property that for an \(S^ 1\) action on a Spin manifold the corresponding equivariant extension is a trivial representation. The proof given here is based on the standard methods of index theory for elliptic complexes. Taubes also gave a proof, based upon Witten’s outline, using the idea of elliptic complexes on the loop space of a manifold.
Reviewer: R.E.Stong

MSC:
57R20 Characteristic classes and numbers in differential topology
58J10 Differential complexes
58J20 Index theory and related fixed-point theorems on manifolds
55N91 Equivariant homology and cohomology in algebraic topology
55N99 Homology and cohomology theories in algebraic topology
55M20 Fixed points and coincidences in algebraic topology
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[1] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. · Zbl 0395.30001
[2] Michael Atiyah and Friedrich Hirzebruch, Spin-manifolds and group actions, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 18 – 28. · Zbl 0193.52401
[3] M. F. Atiyah and R. Bott, The Lefschetz fixed point theorem for elliptic complexes I and II, Ann. of Math. 86 (1967), 374 and 88 (1968), 451. · Zbl 0167.21703
[4] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1 – 28. · Zbl 0521.58025 · doi:10.1016/0040-9383(84)90021-1 · doi.org
[5] M. F. Atiyah and G. B. Segal, The index of elliptic operators. II, Ann. of Math. (2) 87 (1968), 531 – 545. · Zbl 0164.24201 · doi:10.2307/1970716 · doi.org
[6] Raoul Bott, On the fixed point formula and the rigidity theorems of Witten lectures at Cargèse 1987, Nonperturbative quantum field theory (Cargèse, 1987) NATO Adv. Sci. Inst. Ser. B Phys., vol. 185, Plenum, New York, 1988, pp. 13 – 32.
[7] Raoul Bott and Hans Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964 – 1029. · Zbl 0101.39702 · doi:10.2307/2372843 · doi.org
[8] J.-L. Brylinski, Remark on Witten’s modular forms, 1987, preprint.
[9] D. V. Chudnovsky and G. V. Chudnovsky, Elliptic modular functions and elliptic genera, Topology 27 (1988), no. 2, 163 – 170. · Zbl 0653.57015 · doi:10.1016/0040-9383(88)90035-3 · doi.org
[10] Allan L. Edmonds, Orientability of fixed point sets, Proc. Amer. Math. Soc. 82 (1981), no. 1, 120 – 124. · Zbl 0466.57016
[11] Friedrich Hirzebruch, Elliptic genera of level \? for complex manifolds, Differential geometrical methods in theoretical physics (Como, 1987) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 250, Kluwer Acad. Publ., Dordrecht, 1988, pp. 37 – 63. · Zbl 0667.32009
[12] Peter S. Landweber, Elliptic genera: an introductory overview, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986) Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 1 – 10. · doi:10.1007/BFb0078036 · doi.org
[13] Peter S. Landweber, Elliptic cohomology and modular forms, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986) Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 55 – 68. · doi:10.1007/BFb0078038 · doi.org
[14] Peter S. Landweber and Robert E. Stong, Circle actions on Spin manifolds and characteristic numbers, Topology 27 (1988), no. 2, 145 – 161. · Zbl 0647.57013 · doi:10.1016/0040-9383(88)90034-1 · doi.org
[15] Haynes Miller, The elliptic character and the Witten genus, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 281 – 289. · doi:10.1090/conm/096/1022688 · doi.org
[16] Serge Ochanine, Genres elliptiques équivariants, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986) Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 107 – 122 (French). · Zbl 0649.57023 · doi:10.1007/BFb0078041 · doi.org
[17] G. Segal, Elliptic cohomology, Séminaire Bourbaki, 40e année, 1987-88, no. 695 (to appear).
[18] C. H. Taubes, \( {S^1}\)-actions and elliptic genera, 1987, preprint.
[19] Edward Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987), no. 4, 525 – 536. · Zbl 0625.57008
[20] Don Zagier, Note on the Landweber-Stong elliptic genus, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986) Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 216 – 224. · Zbl 0653.57016 · doi:10.1007/BFb0078047 · doi.org
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