Asymptotic spectral flow

Abstract

In this paper we study the asymptotic behavior of the spectral flow of a one-parameter family $$\{D_s\}$$ of Dirac operators acting on the spinor bundle S twisted by a vector bundle E of rank k, with the parameter $$s\in [0,r]$$ when r gets sufficiently large. Our method uses the variation of eta invariant and local index theory technique. The key is a uniform estimate of the eta invariant $$\bar{\eta }(D_r)$$ which is established via local index theory technique and heat kernel estimate.