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

TL;DR

This paper constructs and analyzes fractional self-dual instanton solutions in $SU(N)$ Yang-Mills theory on a four-torus with twisted boundary conditions, revealing complex moduli spaces and zero modes.

Contribution

It introduces explicit analytical fractional instanton solutions with topological charge $r/N$, explores their moduli spaces, and examines their zero modes, advancing understanding of nonabelian gauge configurations on $ ext{T}^4$.

Findings

Solutions with topological charge $Q=r/N$ have compact moduli spaces.

Solutions with higher charge and $k eq r$ exhibit non-compact moduli spaces with runaway behavior.

Each instanton lump supports two adjoint fermion zero modes.

Abstract

We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus $\mathbb{T}^4$ with 't Hooft twisted boundary conditions. These instantons possess topological charge $Q=\frac{r}{N}$, where $1\leq r< N$. To implement the twist, we employ $SU(N)$ transition functions that satisfy periodicity conditions up to center elements and are embedded into $SU(k)\times SU(\ell)\times U(1)\subset SU(N)$, where $\ell+k=N$. The self-duality requirement imposes a condition, $k L_1L_2=r\ell L_3L_4$, on the lengths of the periods of $\mathbb{T}^4$ and yields solutions with abelian field strengths. However, by introducing a detuning parameter $\Delta\equiv (r\ell L_3L_4-k L_1 L_2)/\sqrt{L_1 L_2L_3L_4}$, we generate self-dual nonabelian solutions on a general $\mathbb{T}^4$ as an expansion in powers of $\Delta$. We explore the moduli spaces associated with these solutions and find that they exhibit intricate structures. Solutions with topological charges greater than $\frac{1}{N}$ and $k\neq r $ possess non-compact moduli spaces, along which the $O(\Delta)$ gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with $Q=\frac{r}{N}$ 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.