The moduli space of self-dual SU(N) Yang-Mills instantons on T4 of topological charge Q = r/N, 1 ≤ r ≤ 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 T4, we explore the moduli space via analytical and lattice tools. These solutions are characterized by two positive integers k, ℓ, k + ℓ = N, and are self-dual for T4 sides Lμ tuned to kL1L2 = rℓL3L4. 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 T4, with 4r holonomy moduli. In contrast, when gcd(k, r) ≠ r, we argue that the self-dual constant-F solutions acquire, in addition to the 4gcd(k, r) holonomies, 4r − 4gcd(k, r) extra moduli, whose turning on makes the field strength nonabelian and non-constant. Thus, for gcd(k, r) ≠ r, ’t Hooft’s constant-F solutions are a measure-zero subset of the moduli space on the tuned T4, a fact explaining a puzzle encountered in 1. We also show that, for r = k = 2, N = 3, the agreement between the approximate analytic solutions on the slightly detuned T4 and the Q = 2/3 self-dual configurations obtained by minimizing the lattice action is remarkable.
Anber et al. (Mon,) studied this question.