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Multi-fractional instantons in SU( N ) Yang-Mills theory on the twisted T 4

Anber, Mohamed M.; Poppitz, Erich


Erich Poppitz


We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus T4 with ’t Hooft twisted boundary conditions. These instantons possess topological charge Q=rN, 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 T4 and yields solutions with abelian field strengths. However, by introducing a detuning parameter ∆ ≡ (rℓL3L4 – kL1L2)/L1L2L3L4, we generate self-dual nonabelian solutions on a general T4 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 1N and k ≠ r possess non-compact moduli spaces, along which the O∆ gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with Q=rN 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.


Anber, M. M., & Poppitz, E. (in press). Multi-fractional instantons in SU( N ) Yang-Mills theory on the twisted T 4. Journal of High Energy Physics, 2023(9), Article 95.

Journal Article Type Article
Acceptance Date Sep 7, 2023
Online Publication Date Sep 15, 2023
Deposit Date Nov 3, 2023
Publicly Available Date Nov 3, 2023
Journal Journal of High Energy Physics
Print ISSN 1126-6708
Publisher Scuola Internazionale Superiore di Studi Avanzati (SISSA)
Peer Reviewed Peer Reviewed
Volume 2023
Issue 9
Article Number 95
Keywords Anomalies in Field and String Theories, Solitons Monopoles and Instantons, Nonperturbative Effects
Public URL
Additional Information Received: 18 July 2023; Accepted: 7 September 2023; First Online: 15 September 2023


Published Journal Article (1.7 Mb)


Copyright Statement
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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