Asymptotic spectral flow

In this paper we study the asymptotic behavior of the spectral flow of a one-parameter family {Ds} of Dirac operators acting on the spinor bundle S twisted by a vector bundle E of rank k, with the parameter s is an element of [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 (eta) over bar (D-r) which is established via local index theory technique and heat kernel estimate.

Automated summary

In this paper, we investigate the behavior of the spectral flow of a one-parameter family {Ds} of Dirac operators acting on a twisted spinor bundle S with a vector bundle E. The study focuses on the asymptotic behavior of the spectral flow when the parameter s belongs to the interval [0,r] and r is sufficiently large. Our method involves the variation of eta invariant and local index theory technique, with a key step being the establishment of a uniform estimate of the eta invariant (eta) over bar (D-r) using local index theory technique and heat kernel estimate.