Chaos from Equivariant Fields on Fuzzy $S^4$

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.