The moduli space of self-dual SU(N) Yang-Mills instantons on \( {\mathbbm{T}}^4 \) 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 \( {\mathbbm{T}}^4 \) , 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 \( {\mathbbm{T}}^4 \) 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 \( {\mathbbm{T}}^4 \) , 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 \( {\mathbbm{T}}^4 \) , 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 \( {\mathbbm{T}}^4 \) and the Q = 2/3 self-dual configurations obtained by minimizing the lattice action is remarkable.
