Holography on the lattice: Evidence from 3D supersymmetric Yang--Mills theory

Abstract

We present new results from our lattice investigations of maximally supersymmetric Yang--Mills theory in three dimensions, focusing on its nonperturbative phase diagram. Using a lattice formulation that preserves part of the supersymmetry algebra at finite lattice spacing, we study the spatial deconfinement transition, which holography relates to the transition between localized and homogeneous black branes in the dual gravity theory. Our analysis employs $N_L^2 \times N_T$ lattices with $N = 8$ colors in the SU($N$) gauge group, considering $N_T = 8$, $10$ and $12$, in each case with aspect ratios $α= N_L/N_T \leq 3$. The resulting transition temperatures are consistent with the holographic low-temperature, large-$N$ prediction $T_c \propto α^3$, providing further evidence for the gauge--gravity correspondence in this setting.

Figures (4)

• Figure 1: Normalized violations of a ${\cal Q}$-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 $N_T = 8$ calculations considered here.
• Figure 2: The magnitude of the Polyakov loop vs. $T$. For all $N_T = 8$ 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 susceptibilities vs. $T$, considering both different aspect ratios for fixed $N_T = 8$ (left) and different $N_T = 8$, $10$, $12$ for fixed aspect ratio $\alpha = 2$. 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 in the form $r_T = 0.28 r_L^{3/2}$ (solid line). The constant comes from fitting $T = c \alpha^3$, which produces $c = 0.181(10)$. The dotted diagonal lines show the trajectories we scan with fixed aspect ratio $\alpha = r_L / r_T$.