Chaos from Equivariant Fields on Fuzzy $S^4$

TL;DR

This paper investigates the chaotic dynamics and soliton solutions of a 5D Yang-Mills matrix model with fuzzy four-sphere configurations, revealing stability, chaos, and topological features through explicit equivariant parametrizations and numerical analysis.

Contribution

It provides explicit equivariant parametrizations of gauge fields on fuzzy S^4, derives stable reduced actions, and explores chaos and solitons in the resulting low-energy systems.

Findings

Reduced systems exhibit bounded potentials indicating stability.

Chaotic dynamics are confirmed via Lyapunov exponents.

Euclidean LEAs support instanton solutions related to topological fluxes.

Abstract

We examine the $5d$ Yang-Mills matrix model in $0+1$-dimensions with $U(4N)$ gauge symmetry and a mass deformation term. We determine the explicit $SU(4)\approx SO(6)$ equivariant parametrizations of the gauge field and the fluctuations about the classical four concentric fuzzy four sphere configuration and obtain the low energy reduced actions(LEAs) by tracing over the $S_F^4$s for the first five lowest matrix levels. The LEA's so obtained have potentials bounded from below indicating that the equivariant fluctuations about the $S_F^4$ do not lead to any instabilities. These reduced systems exhibit chaotic dynamics, which we reveal by computing their Lyapunov exponents.Using our numerical results, we explore various aspects of chaotic dynamics emerging from the LEAs. In particular, we model how the largest Lyapunov exponents change as a function of the energy. We also show that, in the Euclidean signature, the LEAs support the usual kink type soliton solutions, i.e. instantons in $1+0$-dimensions, which may be seen as the imprints of the topological fluxes penetrating the concentric $S_F^4$s due to the equivariance conditions, and preventing them to shrink to zero radius.