标题： 关于在扭曲的$\mathbb T^4$上多分数瞬子的模空间 显示英文标题

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

摘要： 自对偶$SU(N)$ Yang-Mills 瞬子在$\mathbb T^4$上的模空间，其拓扑电荷为$Q = r/N$,$1 \leq r \leq N-1$，目前受到关注，但尚未完全理解。 在本文中，从 't Hooft 的常场强（$F$）瞬子出发，这是在$\mathbb T^4$上已知的唯一精确解，我们通过分析和格点工具来探索模空间。 这些解由两个正整数$k, \ell$,$k+\ell=N$表征，并且对于$\mathbb T^4$边$L_\mu$调整为$k L_1 L_2 = r \ell L_3 L_4$时是自对偶的。 对于 gcd$(k,r) = r$，我们通过分析和数值方法（对于$N = 3$）证明，在调谐的$\mathbb T^4$上，常数-$F$解是唯一的自对偶解，具有$4r$全纯模。 相反，当 gcd$(k,r) \ne r$时，我们认为自对偶常量$F$解除了除$4\text{gcd}(k,r)$拓扑外，还获得了$4r - 4\text{gcd}(k,r)$额外模数，其开启会使场强非阿贝尔且非常数。 因此，当 gcd($k,r) \ne r$, 't Hooft 的常量$F$解是调谐$\mathbb T^4$上模空间的一个测度为零的子集，这一事实解释了在 arXiv:2307.04795 中遇到的一个谜题。 我们还表明，对于$r = k = 2$，$N = 3$，近似解析解在略微失谐的$\mathbb T^4$上与通过最小化晶格作用得到的$Q=2/3$自对偶配置之间的吻合是显著的。 显示英文摘要

摘要： 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 $L_\mu$ 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.