Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T 4 $$ {\mathbbm{T}}^4 $$
Abstract We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus T 4 $$ {\mathbbm{T}}^4 $$ with ’t Hooft twisted boundary conditions. These instantons possess topological charge Q = r N $$ Q=\frac{r}{N} $$ , where 1 ≤ r < N. To implement the twist, we employ SU...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-09-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP09(2023)095 |
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| author | Mohamed M. Anber Erich Poppitz |
| author_facet | Mohamed M. Anber Erich Poppitz |
| author_sort | Mohamed M. Anber |
| collection | DOAJ |
| description | Abstract We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus T 4 $$ {\mathbbm{T}}^4 $$ with ’t Hooft twisted boundary conditions. These instantons possess topological charge Q = r N $$ Q=\frac{r}{N} $$ , where 1 ≤ r < N. To implement the twist, we employ SU(N) transition functions that satisfy periodicity conditions up to center elements and are embedded into SU(k) × SU(ℓ) × U(1) ⊂ SU(N), where ℓ + k = N. The self-duality requirement imposes a condition, kL 1 L 2 = rℓL 3 L 4, on the lengths of the periods of T 4 $$ {\mathbbm{T}}^4 $$ and yields solutions with abelian field strengths. However, by introducing a detuning parameter ∆ ≡ (rℓL 3 L 4 – kL 1 L 2)/ L 1 L 2 L 3 L 4 $$ \sqrt{L_1{L}_2{L}_3{L}_4} $$ , we generate self-dual nonabelian solutions on a general T 4 $$ {\mathbbm{T}}^4 $$ as an expansion in powers of ∆. We explore the moduli spaces associated with these solutions and find that they exhibit intricate structures. Solutions with topological charges greater than 1 N $$ \frac{1}{N} $$ and k ≠ r possess non-compact moduli spaces, along which the O ∆ $$ \mathcal{O}\left(\Delta \right) $$ gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with Q = r N $$ Q=\frac{r}{N} $$ and k = r have compact moduli spaces, whose coordinates correspond to the allowed holonomies in the SU(r) color space. These solutions can be represented as a sum over r lumps centered around the r distinct holonomies, thus resembling a liquid of instantons. In addition, we show that each lump supports 2 adjoint fermion zero modes. |
| first_indexed | 2024-03-08T18:16:54Z |
| format | Article |
| id | doaj.art-b95b78928e834d1f99f36d8e34eb33fa |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-03-08T18:16:54Z |
| publishDate | 2023-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-b95b78928e834d1f99f36d8e34eb33fa2023-12-31T12:08:14ZengSpringerOpenJournal of High Energy Physics1029-84792023-09-012023914910.1007/JHEP09(2023)095Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T 4 $$ {\mathbbm{T}}^4 $$Mohamed M. Anber0Erich Poppitz1Centre for Particle Theory, Department of Mathematical Sciences, Durham UniversityDepartment of Physics, University of TorontoAbstract We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus T 4 $$ {\mathbbm{T}}^4 $$ with ’t Hooft twisted boundary conditions. These instantons possess topological charge Q = r N $$ Q=\frac{r}{N} $$ , where 1 ≤ r < N. To implement the twist, we employ SU(N) transition functions that satisfy periodicity conditions up to center elements and are embedded into SU(k) × SU(ℓ) × U(1) ⊂ SU(N), where ℓ + k = N. The self-duality requirement imposes a condition, kL 1 L 2 = rℓL 3 L 4, on the lengths of the periods of T 4 $$ {\mathbbm{T}}^4 $$ and yields solutions with abelian field strengths. However, by introducing a detuning parameter ∆ ≡ (rℓL 3 L 4 – kL 1 L 2)/ L 1 L 2 L 3 L 4 $$ \sqrt{L_1{L}_2{L}_3{L}_4} $$ , we generate self-dual nonabelian solutions on a general T 4 $$ {\mathbbm{T}}^4 $$ as an expansion in powers of ∆. We explore the moduli spaces associated with these solutions and find that they exhibit intricate structures. Solutions with topological charges greater than 1 N $$ \frac{1}{N} $$ and k ≠ r possess non-compact moduli spaces, along which the O ∆ $$ \mathcal{O}\left(\Delta \right) $$ gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with Q = r N $$ Q=\frac{r}{N} $$ and k = r have compact moduli spaces, whose coordinates correspond to the allowed holonomies in the SU(r) color space. These solutions can be represented as a sum over r lumps centered around the r distinct holonomies, thus resembling a liquid of instantons. In addition, we show that each lump supports 2 adjoint fermion zero modes.https://doi.org/10.1007/JHEP09(2023)095Nonperturbative EffectsSolitons Monopoles and InstantonsAnomalies in Field and String Theories |
| spellingShingle | Mohamed M. Anber Erich Poppitz Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T 4 $$ {\mathbbm{T}}^4 $$ Journal of High Energy Physics Nonperturbative Effects Solitons Monopoles and Instantons Anomalies in Field and String Theories |
| title | Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T 4 $$ {\mathbbm{T}}^4 $$ |
| title_full | Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T 4 $$ {\mathbbm{T}}^4 $$ |
| title_fullStr | Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T 4 $$ {\mathbbm{T}}^4 $$ |
| title_full_unstemmed | Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T 4 $$ {\mathbbm{T}}^4 $$ |
| title_short | Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T 4 $$ {\mathbbm{T}}^4 $$ |
| title_sort | multi fractional instantons in su n yang mills theory on the twisted t 4 mathbbm t 4 |
| topic | Nonperturbative Effects Solitons Monopoles and Instantons Anomalies in Field and String Theories |
| url | https://doi.org/10.1007/JHEP09(2023)095 |
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