Tachyonic instabilities in 2  +  1 dimensional Yang–Mills theory and its connection to number theory

We consider the 2  +  1 dimensional Yang–Mills theory with gauge group $SU\left(N\right)$ 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. (paper)