Tachyonic instabilities in 2+1 dimensional Yang-Mills theory and its connection to Number Theory

TL;DR

This paper explores the connection between tachyonic instabilities in 2+1D Yang-Mills theory on a torus and number theory, showing that stability conditions relate to Diophantine approximation problems.

Contribution

It establishes a novel link between gauge theory stability and number theory, specifically relating phase transitions to Diophantine approximation of fractions.

Findings

Absence of tachyonic instabilities correlates with number-theoretic problems.

Proves that stability conditions depend on Diophantine approximation.

Connects gauge theory phase transitions to mathematical number theory concepts.

Abstract

We consider the $2+1$ dimensional Yang-Mills theory with gauge group $\text{SU}(N)$ on a flat 2-torus under twisted boundary conditions. We study the possibility of phase transitions (tachyonic instabilities) when $N$ and the volume vary and certain chromomagnetic flux associated to the topology of the bundle can be adjusted. Under natural assumptions about how to match the perturbative regime and the expected confinement, we prove that the absence of tachyonic instabilities is related to some problems in number theory, namely the Diophantine approximation of irreducible fractions by other fractions of smaller denominator.