Finite Volume Phases of Large N Gauge Theories with Massive Adjoint Fermions

TL;DR

This work studies SU($N$) gauge theories with $N_f$ massive adjoint fermions on $S^3\times S^1$ with periodic boundary conditions in the weak-coupling, small-volume regime. Using a one-loop effective action for the Polyakov loop, it shows that the phase structure at large $N$ comprises a confining phase (all $f_n>0$) and a sequence of $Z_p$ partially-confined phases with $p$ gaps in the eigenvalue density, separated by first-order transitions; for small $mR$ confinement persists, with critical values increasing with $N_f$. Finite-$N$ analyses and comparisons with ${\mathbb{R}}^3\times S^1$ and lattice results confirm qualitative agreement and support volume independence in appropriate limits. The results illuminate how volume and mass control confinement in adjoint QCD, providing a bridge between perturbative finite-volume insights and nonperturbative lattice studies, with potential implications for AdS/CFT and planar equivalences.

Abstract

The phase structure of QCD-like gauge theories with fermions in various representations is an interesting but generally analytically intractable problem. One way to ensure weak coupling is to define the theory in a small finite volume, in this case S^3 x S^1. Genuine phase transitions can then occur in the large N theory. Here, we use this technique to investigate SU(N) gauge theory with a number N_f of massive adjoint-valued Majorana fermions having non-thermal boundary conditions around S^1. For N_f =1 we find a line of transitions that separate the weak-coupling analogues of the confined and de-confined phases for which the density of eigenvalues of the Wilson line transform from the uniform distribution to a gapped distribution. However, the situation for N_f >1 is much richer and a series of weak-coupling analogues of partially-confined phases appear which leave unbroken a Z_p subgroup of the centre symmetry. In these Z_p phases the eigenvalue density has p gaps and they are separated from the confining phase and from one-another by first order phase transitions. We show that for small enough mR (the mass of the fermions times the radius of the S^3) only the confined phase exists. The large N phase diagram is consistent with the finite N result and with other approaches based on R^3 x S^1 calculations and lattice simulations.

Figures (12)

• Figure 1: The lines where $f=0$ indicating the regions where $f>0$ and $f<0$ in the $(L/R,m L)$ plane for $N_f =1,\ldots,4$.
• Figure 2: The structure of configuration space showing $|\rho_p|$ and one additional direction. The boundary is indicated by the dotted line and it is important that the allowed region is convex. For $f_p<0$, the lines of vanishing action define a cone and the lines of constant negative action being hyperbolae therein. The density (i) is the uniform distribution characteristic of the confining phase; (ii) is the density with $|\rho_p|=\tfrac{1}{2}$ which lies at the boundary of configuration space; and (iii) is the density with minimal action as lying at the boundary of configuration space where the lines of constant $S$ lie tangent to boundary.
• Figure 3: The behaviour of the density across the transition at $f_p=0$: (i) the uniform density in the confining phase and (iii) the ${\mathbb Z}_p$ phase (for $p=3$). At the transition point (ii) the mode $\rho_p$ becomes massless and the density develops $p$ zeros as shown in the middle.
• Figure 4: The phase diagrams in $(L/R,m L)$ coordinates for (a) $N_f =1$ and (b) $N_f = 2$. We have shown the transitions between the ${\mathbb Z}_p$ and ${\mathbb Z}_{p+1}$ phases along the continuation of the $f_p=0$ line for simplicity since this seems to be a good approximation and matches the value calculated in the ${\mathbb{R}}^3 \times S^1$ from Myers:2009df indicated by the arrow pointing to the $m L$-axis.
• Figure 5: The phase diagram in $(L/R,m R)$ coordinates for (a) $N_f =1$ and (b) $N_f = 2$. As above, we have shown the transitions between the ${\mathbb Z}_p$ and ${\mathbb Z}_{p+1}$ phases along the continuation of the $f_p=0$ which seems to be a good approximation.