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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2504.06344 |
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| _version_ | 1866912867542892544 |
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| author | Anber, Mohamed M. Cox, Andrew A. Poppitz, Erich |
| author_facet | Anber, Mohamed M. Cox, Andrew A. Poppitz, Erich |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the moduli space of multi-fractional instantons on the twisted $\mathbb T^4$ Anber, Mohamed M. Cox, Andrew A. Poppitz, Erich High Energy Physics - Theory High Energy Physics - Lattice Mathematical Physics 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. |
| title | On the moduli space of multi-fractional instantons on the twisted $\mathbb T^4$ |
| topic | High Energy Physics - Theory High Energy Physics - Lattice Mathematical Physics |
| url | https://arxiv.org/abs/2504.06344 |