SU(N) fractional instantons and the Fibonacci sequence

Abstract

We study, by means of numerical methods, new $SU(N)$ self-dual instanton solutions on $\mathbf{R}\times \mathbf{T}^3$ with fractional topological charge $Q=1/N$. They are obtained on a box with twisted boundary conditions with a very particular choice of twist: both the number of colours and the 't Hooft $\mathbf{Z}_N$ fluxes piercing the box are taken within the Fibonacci sequence, i.e. $N=F_n$ (the $nth$ number in the series) and $|\vec m| = |\vec{k}|=F_{n-2}$. Various arguments based on previous works and in particular on ref. \cite{Chamizo:2016msz}, indicate that this choice of twist avoids the breakdown of volume independence in the large $N$ limit. These solutions become relevant on a Hamiltonian formulation of the gauge theory, where they represent vacuum-to-vacuum tunneling events lifting the degeneracy between electric flux sectors present in perturbation theory. We discuss the large $N$ scaling properties of the solutions and evaluate various gauge invariant quantities like the action density or Wilson and Polyakov loop operators.