Spinning the Fuzzy Sphere

TL;DR

This paper constructs exact solutions of the $SO(3)$ BMN matrix model representing rotating fuzzy spheres and tori, analyzing their phase structure, topology changes, and continuum limits.

Contribution

It introduces a method to find exact analytical solutions with $ ext{Z}_N$ symmetry, including topology transitions from spheres to tori, in the context of matrix models and field theories.

Findings

Discrete solutions exist for each angular momentum value.

Phase structure varies with matrix size $N$.

Topology change from sphere to torus is characterized.

Abstract

We construct various exact analytical solutions of the $SO(3)$ BMN matrix model that correspond to rotating fuzzy spheres and rotating fuzzy tori.These are also solutions of Yang Mills theory compactified on a sphere times time and they are also translationally invariant solutions of the $\mathcal{N} = 1^*$ field theory with a non-trivial charge density. The solutions we construct have a $\mathbb{Z}_N$ symmetry, where $N$ is the rank of the matrices. After an appropriate ansatz, we reduce the problem to solving a set of polynomial equations in $2N$ real variables. These equations have a discrete set of solutions for each value of the angular momentum. We study the phase structure of the solutions for various values of $N$. Also the continuum limit where $N\to \infty$, where the problem reduces to finding periodic solutions of a set of coupled differential equations. We also study the topology change transition from the sphere to the torus.