Weyl-Dirac zero-mode for calorons

TL;DR

The paper analyzes the Weyl-Dirac fermion zero-mode in SU(2) calorons with non-trivial holonomy and unit charge. Using the ADHM construction in conjunction with the Nahm-Fourier transform, it derives an analytic zero-mode and shows the mode is localized on a single constituent monopole, with the anti-periodic boundary condition selecting the monopole that carries Taubes-winding. Explicit densities are obtained as functions of the holonomy parameter $omega$ and inter-monopole separation $rho$, demonstrating sharp localization in the static limit. The work clarifies the role of monopole constituents and Taubes-winding in finite-temperature chiral dynamics and supports a monopole-based interpretation of calorons in QCD.

Abstract

We give the analytic result for the fermion zero-mode of the SU(2) calorons with non-trivial holonomy. It is shown that the zero-mode is supported on ONLY ONE of the constituent monopoles. We discuss some of its implications.

Figures (1)

• Figure 1: For the two figures on the sides we plot on the same scale the logarithm of the zero-mode densities (cutoff below $1/e^5$) for $\omega=1/8$ (left $\Psi^-$ / right $\Psi^+$) and $\omega=3/8$ (right $\Psi^-$ / left $\Psi^+$), with $\beta=1$ and $\rho=1.2$. In the middle figure we show for the same parameters (both choices of $\omega$ give the same action density) the logarithm of the action density (cutoff below $1/2e^2$).