On family rigidity theorems, I

Abstract

Note that Ind(P ) is a virtual G-representation. Let chg(Ind(P )) with g ∈G be the equivariant Chern character of Ind(P ) evaluated at g. In this paper, we first prove a family fixed-point formula that expresses chg(Ind(P )) in terms of the geometric data on the fixed points X of the fiber of π . Then by applying this formula, we generalize the Witten rigidity theorems and several vanishing theorems proved in [Liu3] for elliptic genera to the family case. Let G = S1. A family elliptic operator P is called rigid on the equivariant Chern character level with respect to this S1-action, if chg(Ind(P )) ∈H ∗(B) is independent of g ∈ S1. When the base B is a point, we recover the classical rigidity and vanishing theorems. When B is a manifold, we get many nontrivial higher-order rigidity and vanishing theorems by taking the coefficients of certain expansion of chg . For the history of the Witten rigidity theorems, we refer the reader to [T], [BT], [K], [L2], [H], [Liu1], and [Liu4]. The family vanishing theorems that generalize those vanishing theorems in [Liu3], which in turn give us many higher-order vanishing theorems in the family case. In a forthcoming paper, we extend our results to general loop group representations and prove much more general family vanishing theorems that generalize the results in [Liu3]. We believe there should be some applications of our results to topology and geometry, which we hope to report on a later occasion. This paper is organized as follows. In Section 1, we prove the equivariant family index theorem. In Section 2, we prove the family rigidity theorem. In the last part of Section 2, motivated by the family rigidity theorem, we state a conjecture. In Section 3, we generalize the family rigidity theorem to the nonzero anomaly case. As corollaries, we derive several vanishing theorems.