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

We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus T4 with ’t Hooft twisted boundary conditions. These instantons possess topological charge 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, kL1L2 = rℓL3L4, on the lengths of the periods of T4 and yields solutions with abelian field strengths. However, by introducing a detuning parameter ∆ ≡ (rℓL3L4 – kL1L2)//L1L2L3L4, we generate self-dual nonabelian solutions on a general T4 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 and k ≠ r possess non-compact moduli spaces, along which the O(^) gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with 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