Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 419

On Family Rigidity Theorems II

Kefeng Liu and Xiaonan Ma

In [LM], we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems [LM] in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop group representation. Second we prove a family rigidity theorem for $spin^c$-manifolds. Several vanishing theorems on both cases are also obtained.

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