Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted $$ {\mathbbm{T}}^4 $$

A bstract We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus $$ {\mathbbm{T}}^4 $$ T 4 with ’t Hooft twisted boundary conditions. These instantons possess topological charge $$ Q=\frac{r}{N} $$ Q = 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, kL 1 L 2 = rℓL 3 L 4 , on the lengths of the periods of $$ {\mathbbm{T}}^4 $$ T 4 and yields solutions with abelian field strengths. However, by introducing a detuning parameter ∆ ≡ ( rℓL 3 L 4 – kL 1 L 2 )/ $$ \sqrt{L_1{L}_2{L}_3{L}_4} $$ L 1 L 2 L 3 L 4 , we generate self-dual nonabelian solutions on a general $$ {\mathbbm{T}}^4 $$ 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 $$ \frac{1}{N} $$ 1 N and k ≠ r possess non-compact moduli spaces, along which the $$ \mathcal{O}\left(\Delta \right) $$ O ∆ gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with $$ Q=\frac{r}{N} $$ Q = 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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