Supersymmetric solitons in gauged $\mathcal{N}=8$ supergravity

TL;DR

The work analyzes solitons in AdS$_4$ within a gauged ${\cal N}=8$ supergravity truncation (the STU/T${}^{3}$ model), focusing on two Wilson lines along a compact circle and the resulting phase structure under fixed flux or fixed charge boundary data. Hairy solitons are constructed as double Wick rotations of charged planar black holes, yielding explicit scalar and gauge-field profiles and enabling a detailed map between bulk data and boundary operators. The authors uncover a degeneracy of supersymmetric solutions and demonstrate that, under alternate boundary conditions, a non-supersymmetric soliton can have lower energy than the SUSY branches, a result reconciled with the positive energy theorem via asymptotic Killing spinor considerations. They also relate the new solutions to earlier Anabalon results, discuss flux quantization, and chart how the phase diagram evolves across fixed-flux and fixed-charge sectors, including domain-wall and Poincaré-AdS limits. The findings illuminate rich ground-state structure in holographic AdS/CFT with Wilson lines and provide a framework for further exploring solitons, phase transitions, and energy bounds in higher-dimensional supergravity truncations.

Abstract

We consider soliton solutions in AdS$_{4}$ with a flat slicing and Wilson loops around one cycle. We study the phase structure and find the ground state and identify supersymmetric solutions as a function of the Wilson loops. We work in the context of a scalar field truncation of gauged $\mathcal{N}=8$ supergravity, where all the dilatons are equal and all the axions vanish in the STU model. In this theory, we construct new soliton solutions parameterized by two Wilson lines. We find that there is a degeneracy of supersymmetric solutions. We also show that, for alternate boundary conditions, there exists a non-supersymmetric soliton solution with energy lower than the supersymmetric one.

Figures (7)

• Figure 1: The square $x_0^2$ of the roots of (\ref{['Pol']}) in the $y$-axis vs. the "rescaled charge" $q_1$ in the $x$-axis. The blue line shows the location of the supersymmetric solitons, where $q_2$ is determined as a function of $q_1$ by ${q_2=-\sqrt{3}\,q_1}$ (see Section \ref{['susy']}). The red and black lines are the roots of (\ref{['pol2']}) plotted for fixed ${q_2 = -0.1}$. The green line indicates the value of $q_1$ that that satisfies the susy condition $q_1=-\frac{1}{\sqrt{3}}\,q_2$ at $q_2=-0.1$. As expected, this intersects the red and black lines where they intersect the blue line: these are the supersymmetric solitons for fixed $q_2=-0.1$. There is also an intersection at $x_0=1$, where also the black and red roots intersect; we will see below that this corresponds to non-supersymmetric solutions with zero scalar.
• Figure 2: The square $x_0^2$ of the roots of (\ref{['Pol']}) in the $y$-axis vs. the "rescaled charge" $q_1$ in the $x$-axis. The blue line is the same as in the previous plot - it shows the location of the supersymmetric solitons, where $q_2$ is determined as a function of $q_1$ by ${q_2= - \sqrt{3}\,q_1}$ (see Section \ref{['susy']}). The red and black lines are the roots of (\ref{['pol2']}) plotted for fixed ${q_2 = -0.14}$. We see that for this value of $q_2$, there is both a lower and an upper bound on $q_1$ for the existence of solitons. The green line indicates the value of $q_1$ that satisfies the susy condition $q_1= - \frac{1}{\sqrt{3}}\, q_2$ at $q_2=-0.14$. As expected, this intersects the red and black lines where they intersect the blue line: these are the supersymmetric solitons for fixed $q_2=-0.14$. There is also an intersection at $x_0=1$, where also the black and red roots intersect; we will see below that this corresponds to non-supersymmetric solutions with zero scalar.
• Figure 3: Rescaled free energy $\frac{G_{\phi}}{\left\vert G_{0}\right\vert }$ (red, green) and rescaled vev $\left\langle \mathcal{O}\right\rangle \Delta/\pi$ (black, blue) for $\psi_{2}=\pm 0.6$. Different colours are used to represent different branches of the solution.
• Figure 4: Rescaled free energy $\frac{G_{\phi}}{\left\vert G_{0}\right\vert }$ at $\psi_{2}=\pm 0.4$. The different colours represent different branches of the solution.
• Figure 5: Rescaled free energy $\frac{G_{\phi}}{\left\vert G_{0}\right\vert}$ for $\psi_{2}=\pm 0.8$. The different colours represent different branches of the solution. We note that, for these values of $\psi_{2}$, only the $x>1$ solutions are necessary to describe the phase diagram.