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

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.

latex 20 pages, 1 figure

References in corpus (4)

• A precise calculation of the fundamental string tension in SU(N) gauge theories in 2+1 dimensions
• Volume independence for Yang-Mills fields on the twisted torus
• The spectrum of 2+1 dimensional Yang-Mills theory on a twisted spatial torus
• Renormalization on noncommutative torus

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• Perturbative contributions to Wilson loops in twisted lattice boxes and reduced models
• Perturbative study of large $N$ principal chiral model with twisted reduction
• Scale setting of SU($N$) Yang--Mills theory, topology and large-$N$ volume independence
• Memory efficient finite volume schemes with twisted boundary conditions