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We construct analytical self-dual Yang-Mills fractional instanton solutionson a four-torus $\mathbb{T}^4$ with 't Hooft twisted boundary conditions. Theseinstantons possess topological charge $Q=\frac{r}{N}$, where $1\leq r< N$. Toimplement the twist, we employ $SU(N)$ transition functions that satisfyperiodicity conditions up to center elements and are embedded into $SU(k)\timesSU(\ell)\times U(1)\subset SU(N)$, where $\ell+k=N$. The self-dualityrequirement imposes a condition, $k L_1L_2=r\ell L_3L_4$, on the lengths of theperiods of $\mathbb{T}^4$ and yields solutions with abelian field strengths.However, by introducing a detuning parameter $\Delta\equiv (r\ell L_3L_4-k L_1L_2)/\sqrt{L_1 L_2L_3L_4}$, we generate self-dual nonabelian solutions on ageneral $\mathbb{T}^4$ as an expansion in powers of $\Delta$. We explore themoduli spaces associated with these solutions and find that they exhibitintricate structures. Solutions with topological charges greater than$\frac{1}{N}$ and $k\neq r $ possess non-compact moduli spaces, along which the$O(\Delta)$ gauge-invariant densities exhibit runaway behavior. On the otherhand, solutions with $Q=\frac{r}{N}$ and $k=r$ have compact moduli spaces,whose coordinates correspond to the allowed holonomies in the $SU(r)$ colorspace. These solutions can be represented as a sum over $r$ lumps centeredaround the $r$ distinct holonomies, thus resembling a liquid of instantons. Inaddition, we show that each lump supports $2$ adjoint fermion zero modes.

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