On Witten's global anomaly for higher SU(2) representations

TL;DR

This paper numerically investigates Witten's global anomaly in higher SU(2) representations by analyzing spectral flow along a specific gauge field path, confirming the predicted even-odd eigenvalue pairing pattern.

Contribution

It provides the first numerical computation of spectral flow for higher SU(2) representations, verifying Witten's anomaly predictions beyond the fundamental case.

Findings

Eigenvalue pairs change along the path for all representations studied

The even-odd pattern predicted by Witten is confirmed

Spectral flow behavior varies with representation

Abstract

The spectral flow of the overlap operator is computed numerically along a particular path in gauge field space. The path connects two gauge equivalent configurations which differ by a gauge transformation in the non-trivial class of pi_4(SU(2)). The computation is done with the SU(2) gauge field in the fundamental, the 3/2, and the 5/2 representation. The number of eigenvalue pairs that change places along this path is established for these three representations and an even-odd pattern predicted by Witten is verified.