Abstract
We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus \( {\mathbbm{T}}^4 \) with ’t Hooft twisted boundary conditions. These instantons possess topological charge \( Q=\frac{r}{N} \), 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, kL1L2 = rℓL3L4, on the lengths of the periods of \( {\mathbbm{T}}^4 \) and yields solutions with abelian field strengths. However, by introducing a detuning parameter ∆ ≡ (rℓL3L4 – kL1L2)/\( \sqrt{L_1{L}_2{L}_3{L}_4} \), we generate self-dual nonabelian solutions on a general \( {\mathbbm{T}}^4 \) 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 \( \frac{1}{N} \) and k ≠ r possess non-compact moduli spaces, along which the \( \mathcal{O}\left(\Delta \right) \) gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with \( Q=\frac{r}{N} \) 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.
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Acknowledgments
We would like to thank F. David Wandler for comments on the manuscript. M.A. acknowledges the hospitality of the University of Toronto, where this work was completed. M.A. is supported by STFC through grant ST/T000708/1. E.P. is supported by a Discovery Grant from NSERC.
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ArXiv ePrint: 2307.04795
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Anber, M.M., Poppitz, E. Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted \( {\mathbbm{T}}^4 \). J. High Energ. Phys. 2023, 95 (2023). https://doi.org/10.1007/JHEP09(2023)095
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DOI: https://doi.org/10.1007/JHEP09(2023)095