Maximally supersymmetric Yang--Mills in three dimensions

TL;DR

This paper investigates the non-perturbative phase structure of maximally supersymmetric Yang–Mills theory in three dimensions using a lattice formulation that preserves a subset of supersymmetry. By simulating a skewed 3-torus with a $\mathcal{Q}$-exact/$\mathcal{Q}$-closed action and soft deformations, the authors study the spatial deconfinement transition predicted by gauge/gravity duality, mapping it to a D2–D0 black brane transition in the holographic frame. They report preliminary critical temperatures for aspect ratios $\alpha=2,2.5,3$ that align with the holographic scaling $T_c\propto\alpha^3$, and they present a consistent, though noisier, picture for $\alpha=4$. The work lays out concrete paths for improving statistics, performing continuum extrapolations, and exploring the order and dynamics of the transition, contributing a non-perturbative test of holography in a maximally supersymmetric, lower-dimensional setting.

Abstract

We present the latest results from our ongoing lattice field theory investigations of maximally supersymmetric Yang--Mills theory in three space-time dimensions, focusing on its non-perturbative phase diagram. Exploiting a lattice formulation that preserves a subset of the supersymmetry algebra at non-zero lattice spacing, we study the 'spatial deconfinement' phase transition that holography relates to the transition between localized and homogeneous black branes in the dual quantum gravity. Fixing $N_L^2 \times 8$ lattice volumes and N=8 colors in the SU(N) gauge group, we consider four aspect ratios $α= N_L / N_T \leq 4$ corresponding to $N_L = 16$, $20$, $24$ and $32$. The transition temperatures we determine are in good agreement with the low-temperature, large-N holographic expectation $T_c \propto α^3$.

Figures (4)

• Figure 1: Normalized violations of a $\mathcal{Q}$-supersymmetry Ward identity involving the bosonic action, $s_B = 9N^2 / 2$, plotted vs. the dimensionless temperature from Eq. \ref{['eq:T']}. The violations increase for the stronger 't Hooft couplings at lower $T$, but remain under half a percent for all the calculations considered here.
• Figure 2: The magnitude of the Polyakov loop vs. $T$. For all calculations considered here, $|PL|$ is more than large enough to confirm the thermal deconfinement required for holographic duality to apply.
• Figure 3: Spatial Wilson line magnitudes (top) and the corresponding susceptibilities (bottom), both plotted vs. $T$. Each point combines the Wilson lines in both the $x$- and $y$-directions. The peaks in the susceptibilities for $\alpha = N_L / N_T \leq 3$ signal the spatial deconfinement transition.
• Figure 4: Numerical lattice field theory results for the three-dimensional SYM phase diagram in the $r_T$--$r_L$ plane, compared to the holographic expectation Eq. \ref{['eq:holo']} in the form $r_T = c r_L^{3/2}$ (solid line). We set $c = 0.3$ by hand (rather than fitting the points), which corresponds to $T = 0.21\alpha ^3$. The dotted diagonal lines show the trajectories with scan with fixed aspect ratio $\alpha = r_L / r_T$.