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On modular invariance and rigidity theorems. (English) Zbl 0836.57024

We study the elliptic operators naturally derived from loop spaces and show that their modularity implies their rigidity. As consequences we first prove the rigidity of the Dirac operator on loop space twisted by loop group representations of positive energy of any level, while the rigidity theorems conjectured by Witten are the special cases of level 1. Then we generalize these rigidity theorems to non-zero anomaly case from which we obtain holomorphic Jacobi forms and many vanishing theorems, especially an \(\widehat {\mathfrak A}\)-vanishing theorem for loop spaces. We also discuss elliptic genera of level 1, mod 2 elliptic genera and the relationships between elliptic genera and the geometry of elliptic modular surfaces, and the classical elliptic modular functions.

MSC:

57S15 Compact Lie groups of differentiable transformations
11F27 Theta series; Weil representation; theta correspondences
58J20 Index theory and related fixed-point theorems on manifolds
22E67 Loop groups and related constructions, group-theoretic treatment
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
81T20 Quantum field theory on curved space or space-time backgrounds
11F03 Modular and automorphic functions
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