On the moduli space of multi-fractional instantons on the twisted $\mathbb T^4$

TL;DR

The work analyzes the moduli space of self-dual $SU(N)$ fractional instantons on a twisted ${\mathbb T}^4$, distinguishing the gcd$(k,r)$-dependent structure of constant-$F$ backgrounds from non-abelian deformations. It combines an analytic $\lambda$-expansion around constant-$F$ solutions with a $\Delta$-expansion for detuned tori and corroborates findings via lattice simulations in $SU(3)$, showing that gcd$(k,r)=r$ yields a fully abelian, constant-$F$ moduli space while gcd$(k,r)\neq r$ introduces additional moduli that render the field strength non-constant and non-abelian. The paper demonstrates strong agreement between analytic predictions and numerical results for several cases, notably $N=3$, $r=k=2$, and provides a quantitative test of the $\Delta$-expansion for $Q=\frac{2}{3}$ through lattice data. These results illuminate the semiclassical landscape of fractional instantons on twisted backgrounds and set the stage for deeper understanding of zero modes, moduli-space global structure, and the interplay between $\lambda$- and $\Delta$-expansions in nonperturbative gauge dynamics.

Abstract

The moduli space of self-dual $SU(N)$ Yang-Mills instantons on $\mathbb T^4$ of topological charge $Q = r/N$, $1 \leq r \leq N-1$, is of current interest, yet is not fully understood. In this paper, starting from 't Hooft's constant field strength ($F$) instantons, the only known exact solutions on $\mathbb T^4$, we explore the moduli space via analytical and lattice tools. These solutions are characterized by two positive integers $k, \ell$, $k+\ell=N$, and are self-dual for $\mathbb T^4$ sides $L_μ$ tuned to $k L_1 L_2 = r \ell L_3 L_4$. For gcd$(k,r) = r$, we show, analytically and numerically (for $N = 3$) that the constant-$F$ solutions are the only self-dual solutions on the tuned $\mathbb T^4$, with $4r$ holonomy moduli. In contrast, when gcd$(k,r) \ne r$, we argue that the self-dual constant-$F$ solutions acquire, in addition to the $4\text{gcd}(k,r)$ holonomies, $4r - 4\text{gcd}(k,r)$ extra moduli, whose turning on makes the field strength nonabelian and non-constant. Thus, for gcd($k,r) \ne r$, 't Hooft's constant-$F$ solutions are a measure-zero subset of the moduli space on the tuned $\mathbb T^4$, a fact explaining a puzzle encountered in arXiv:2307.04795. We also show that, for $r = k = 2$, $N = 3$, the agreement between the approximate analytic solutions on the slightly detuned $\mathbb T^4$ and the $Q=2/3$ self-dual configurations obtained by minimizing the lattice action is remarkable.

Figures (12)

• Figure 1: Action density of the $|Q|=1/3$ self-dual solution with $r=k=1$. On the four plots, we show the action density, in each case integrated over all but one $\mathbb{T}^4$ coordinate. The lattice size is $L_1, L_2, L_3, L_4 = 18, 4, 6, 6$, corresponding to a tuned $\mathbb{T}^4$ with $\Delta(1,1,2)=0$. Thus, multiplying the integrated action density by the appropriate $L_\mu$ in each case yields $8 \pi^2 \times {1 \over 3} \simeq 26.3$ (this number can be verified simply by a look at the above plot). We generate $104$ minimum-action configurations starting from random initial conditions. We display the action density of all the $104$ configurations plotted on top of each other. The action densities of all these configurations coincide within numerical error.
• Figure 2: The imaginary vs. the real parts of the fundamental representation Wilson loops winding once in $x_\mu=1,2,3,4$, evaluated at fixed values of the coordinates, for the $|Q|=1/3$ self-dual solution on the tuned-$\mathbb{T}^4$ with $r=k=1$ (whose action density is shown on Figure \ref{['fig:r=k=1action']}). The solid lines show the analytical curves of eqn. (\ref{['wilsoneqns1']}) that the real and imaginary part of the winding Wilson loop $W_\mu$ trace as the translational modulus $\phi_\mu^{[C']_1}$ is varied. The $104$ configurations generated randomly, each denoted by a dot, appear to cover the moduli space of the constant-$F$ solutions.
• Figure 3: Action density of the $Q=2/3$ self-dual solution with $r=k=2$. The lattice size is $L_1, L_2, L_3, L_4 = 16, 4, 8, 8$, corresponding to a tuned $\mathbb{T}^4$ with $\Delta(2,2,1)=0$, eqn. (\ref{['deltadef']}). In each case, we show the action density integrated over all coordinates but the one shown. We show the action densities for all $85$ configurations generated, plotted on top of each other.
• Figure 4: The imaginary vs. the real parts of the fundamental representation Wilson loops winding once in $x_\mu=1,2,3,4$, evaluated at fixed values of the coordinates, for the $Q=2/3$ self-dual solution with $\Delta(2,2,1)=0$, eqn. (\ref{['deltadef']}), with $r=k=2$ (with action density of Figure \ref{['fig:r=k=2action']}). The analytical results of eqn. (\ref{['wilsonr=k=2']}) as the $8$ moduli are varied fill the inside of the smooth curves shown (the boundary of the region corresponds to varying the translational moduli only). As in Figure \ref{['fig:r=k=1wilson']}, the $85$ configurations appear to cover the moduli space of the constant-$F$ solutions.
• Figure 5: The action density, integrated over all but one direction of the torus, for $SU(3)$ with $r=2, k=1$, on the tuned torus with $\Delta(2,1,2)=0$ and sides $(16, 4, 4,4)$. All $175$ self dual configurations generated starting from a random one are found to have non-constant action density.