Asymptotic Spectral Flow

TL;DR

This paper investigates the asymptotic behavior of spectral flow for families of Dirac operators on spinor bundles, employing eta invariant variation and local index theory to derive uniform estimates as the parameter grows large.

Contribution

It introduces a novel approach combining eta invariant variation and local index theory to analyze spectral flow asymptotics for Dirac operators.

Findings

Established uniform eta invariant estimates for large parameters

Derived asymptotic formulas for spectral flow behavior

Applied heat kernel techniques for local index analysis

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 bunldle $S$ 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.