k-String Tension from Eguchi-Kawai Reduction

TL;DR

This work constructs an analytic route to the $k$-string tension in 3d SU(N) gauge theories with adjoint fermions by employing Eguchi-Kawai volume reduction to a 2d theory. Through a controlled reduction, bosonization, and analysis of the Cartan-subalgebra dynamics, the authors derive a sine-law for the string tension, $\sigma_k \sim N \sin\left(\frac{\pi k}{N}\right)$, up to model-dependent prefactors and under assumptions about confinement and center symmetry. They discuss the regime where the reduction is valid (large $\lambda_3 RN$) and emphasize potential breakdowns due to light W-bosons and KK modes, offering a careful caveat about the universality of the result. The paper also notes possible extensions to 4d theories and highlights a distinct Majorana case with a modified prefactor. Overall, the work connects 3d nonperturbative string tensions to a solvable 2d framework, illuminating conditions under which analytic sine-law behavior can emerge.

Abstract

We consider three-dimensional SU(N) gauge theories with massless Dirac or Majorana fermions in the adjoint representation. We use the Eguchi-Kawai volume reduction to two-dimensions to calculate the tension sigma_k of the k-string in such theories. Under some assumptions whose validity we discuss, we derive the previously conjectured sine formula, sigma_k ~ N sin pi k/N.