Can fermions save large N dimensional reduction?

TL;DR

The paper probes whether large-N dimensional reduction via orbifold projections can relate a 4D Yang-Mills theory with adjoint fermions to a lower-dimensional theory. By constructing a 3D SU(NΓ) parent theory and performing a volume-expanding orbifold projection, it derives a 4D candidate and computes the one-loop effective potential for the eigenvalues of the link field φ to test Z_Gamma symmetry. The main finding is that, for most parameter choices, Z_Gamma in the 3D theory is spontaneously broken, causing the expected large-N equivalence to fail; adjoint fermions do not generically stabilize the necessary symmetry in the 3D setup. The authors argue that stabilizing center symmetry likely requires a full tower of double-trace deformations, and the viability of dimensional reduction depends critically on regulator-dependent UV effects and the order of limits, indicating that fermions alone do not rescue large-N dimensional reduction. These results highlight subtle limitations of orbifold-based dimensional reduction and guide future work toward more complete symmetry-preserving deformations and careful treatment of continuum vs lattice regularizations.

Abstract

This paper explores whether Eguchi-Kawai reduction for gauge theories with adjoint fermions is valid. The Eguchi-Kawai reduction relates gauge theories in different numbers of dimensions in the large $N$ limit provided that certain conditions are met. In principle, this relation opens up the possibility of learning about the dynamics of 4D gauge theories through techniques only available in lower dimensions. Dimensional reduction can be understood as a special case of large $N$ equivalence between theories related by an orbifold projection. In this work, we focus on the simplest case of dimensional reduction, relating a 4D gauge theory to a 3D gauge theory via an orbifold projection. A necessary condition for the large N equivalence between the 4D and 3D theories to hold is that certain discrete symmetries in the two theories must not be broken spontaneously. In pure 4D Yang-Mills theory, these symmetries break spontaneously as the size of one of the spacetime dimensions shrinks. An analysis of the effect of adjoint fermions on the relevant symmetries of the 4D theory shows that the fermions help stabilize the symmetries. We consider the same problem from the point of view of the lower dimensional 3D theory and find that, surprisingly, adjoint fermions are not generally enough to stabilize the necessary symmetries of the 3D theory. In fact, a rich phase diagram arises, with a complicated pattern of symmetry breaking. We discuss the possible causes and consequences of this finding.

Figures (2)

• Figure 1: Phase plot of one-loop effective potential with axes $r$ vs. $am$ in the large $N\Gamma$ limit. Configurations of the $N\Gamma$ eigenvalues distributed at $1, \ldots,i\ldots, N \Gamma$ points evenly spaced around the circle were tested against each other, and the configuration with the lowest energy was determined at each point in the plot above. We refer to the configuration with the eigenvalues distributed on $i$ points on the circle as the $Z_{i}$ configuration in the plot legend above. The purple region is the region where the full $Z_{N \Gamma}$ symmetry is unbroken.
• Figure 2: The figure on the left shows the $\mathbb{Z}_{N\Gamma}$ phase diagram following from numerical minimization of the one-loop effective potential with only the first two double-trace terms included with coefficients $d_1, d_2$. The figure on the right shows the phase diagram with the full tower of double trace terms, with all double trace terms having the same coefficient $d$, with varying bare fermion mass $m$ and Wilson term coefficient $r=1$. In both figures $N\Gamma=25$ and $N_f=1$.