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

Abstract

The moduli space of self-dual $SU(N)$ Yang-Mills instantons on ${\mathbb{T}}^4$ of topological charge $Q=r/N$, $1\le r\le N-1$, 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 $L_{\mu }$ 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.

Comments: 38 pages+appendices; typos fixed and font in figures increased, matches the published version