Constituent monopoles through the eyes of fermion zero-modes

TL;DR

This work shows that fermion zero-modes in caloron backgrounds with non-trivial holonomy can probe constituent monopoles and reveal their localization properties. Using the ADHM/Nahm framework, the authors relate zero-mode densities to conserved Nahm data and derive exact charge-2 results with Jacobi-elliptic parametrizations, highlighting both extended disk-like cores and the point-like limit for well-separated constituents. In the high-temperature limit the zero-modes localize to monopole cores, yielding a simple abelian description at large distances, while general Nahm data produce richer, disk-bound structures that encode the full moduli of the caloron. These findings bridge non-abelian caloron solutions to an effective monopole picture and point toward lattice investigations of caloron monopole content and their role in confinement mechanisms.

Abstract

We use the fermion zero-modes in the background of multi-caloron solutions with non-trivial holonomy as a probe for constituent monopoles. We find in general indication for an extended structure. However, for well separated constituents these become point-like. We analyse this in detail for the SU(2) charge 2 case, where one is able to solve the relevant Nahm equation exactly, beyond the piecewize constant solutions studied previously. Remarkably the zero-mode density can be expressed in the high temperature limit as a function of the conserved quantities that classify the solutions of the Nahm equation.

Figures (7)

• Figure 1: The logarithm of the properly normalized zero-mode density for a typical $SU(3)$ caloron of charge 1, cycling through $z$. Shown are $z=\mu_j$ (for linear plots see Fig. \ref{['fig:zmrcycle']}) and three values of $z$ roughly in the middle of each interval $z\in[\mu_j,\mu_{j+1}]$. All plots are on the same scale, cutoff for values of the logarithm below -5. The zero-mode with anti-periodic boundary conditions is found at $z=30/60$. For the action density of the associated gauge field, see Ref. MTPWWW.
• Figure 3: Zero-mode densities for a typical charge 2, $SU(2)$ axially symmetric solution. For comparison the action density (cmp. Fig. 2 of Ref. BrvB) is shown in the middle. All are on a logarithmic scale, cutoff below $e^{-3}$. On the left is shown the two periodic zero-modes ($z=0$) and on the right the two anti-periodic zero-modes ($z=1/2$).
• Figure 4: The two zero-mode densities at $z=\mu_2=1/4$ (same configuration as Fig. \ref{['fig:zmlocal']}). The bipole zero-mode (right) is at 5 times the vertical scale of the second zero-mode (left). Top for the high temperature limit and bottom for finite temperature ($\beta=1$).
• Figure 6: Plotting $|\Delta\vec{a}|$ versus $D$, Eq. (\ref{['eq:bcI']}), for ${\bf k}=0,$ 0.9, 0.99, 0.999 and 0.9999 (left to right).
• Figure 7: On the left we plot ${\bf k}$ versus $\rho$ for $\alpha=0$ (${\bf k}\equiv1$), $\alpha=-\pi/100$ and $\alpha=-\pi/2$ (giving the lower bound for ${\bf k}$ at fixed $\rho$). On the right are shown the locations, Eq. (\ref{['eq:conloc']}), of monopoles and antimonopoles (fat vs. thin curves) in the $1,3$-plane for $\rho=1/4$, by varying $\alpha$ from $-\pi$ (indicated by the arrows) to 0.