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Elliptic genera, involutions, and homogeneous spin manifolds

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We study the normalized elliptic genera Φ(X)=ϕ(X)/εk/2 for 4k-dimensional homogeneous spin manifolds X and show that they are constant as modular functions. The basic tool is a reduction formula relating Φ(X) to that of the self-intersection of the fixed point set of an involution γ on X. When Φ(X) is a constant it equals the signature of X. We derive a general formula for sign(G/H), G⊃H compact Lie groups, and determine its value in some cases by making use of the theory of involutions in compact Lie groups.

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Authors and Affiliations

  1. Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, 5300, Bonn 3, Germany

    F. Hirzebruch

  2. Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, 7000, Stuttgart 80, Germany

    P. Slodowy

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  1. F. Hirzebruch
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  2. P. Slodowy
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Dedicated to Jacques Tits on the occasion of his sixtieth birthday

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Hirzebruch, F., Slodowy, P. Elliptic genera, involutions, and homogeneous spin manifolds. Geom Dedicata 35, 309–343 (1990). https://doi.org/10.1007/BF00147351

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