On the moduli space of multi-fractional instantons on the twisted $$ {\mathbbm{T}}^4 $$

A bstract The moduli space of self-dual SU( N ) Yang-Mills instantons on $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> 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 $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> , 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 $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> sides L μ tuned to kL 1 L 2 = rℓL 3 L 4 . 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 $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> , with 4 r 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, 4 r − 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 $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> , 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 $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> and the Q = 2 / 3 self-dual configurations obtained by minimizing the lattice action is remarkable.