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