Supersymmetric solitons and a degeneracy of solutions in AdS/CFT

TL;DR

This work uncovers a novel degeneracy of supersymmetric saddles in AdS/CFT: in both $D=4$ and $D=5$ gauged $$ supergravity, there exist Lorentzian 1/2-BPS planar solitons with AdS asymptotics and a Wilson line on the shrinking circle, as well as a separate, pure-AdS solution with a constant gauge field. At a special Wilson line value $\Phi=\Phi_S$ these two configurations become degenerate in energy, and a phase transition occurs as $\Phi$ is varied; the degeneracy extends to higher-genus Euclidean boundaries and to the five-dimensional minimal gauged supergravity with similar structure. The authors provide explicit Killing spinors, analyze the phase diagram and Euclidean partition function, and discuss the intriguing possibility that the CFT ground state is a quantum superposition of the two geometries. The results have potential implications for holographic indices and twisted partition functions, and suggest that SUSY degeneracies of bulk saddles may be a generic feature in AdS/CFT with $U(1)$ gauge fields and planar boundaries.

Abstract

We study Lorentzian supersymmetric configurations in $D=4$ and $D=5$ gauged $\mathcal{N}=2$ supergravity. We show that there are smooth $1/2$ BPS solutions which are asymptotically AdS$_{4}$ and AdS$_{5}$ with a planar boundary, a compact spacelike direction and with a Wilson line on that circle. There are solitons where the $S^{1}$ shrinks smoothly to zero in the interior, with a magnetic flux through the circle determined by the Wilson line, which are AdS analogues of the Melvin fluxtube. There is also a solution with a constant gauge field, which is pure AdS. Both solutions preserve half of the supersymmetries at a special value of the Wilson line. There is a phase transition between these two saddle-points as a function of the Wilson line precisely at the supersymmetric point. Thus, the supersymmetric solutions are degenerate, at least at the supergravity level. We extend this discussion to one of the Romans solutions in four dimensions when the Euclidean boundary is $S^{1}\timesΣ_{g}$ where $Σ_{g}$ is a Riemann surface with genus $g > 0$. We speculate that the supersymmetric state of the CFT on the boundary is dual to a superposition of the two degenerate geometries.

Figures (2)

• Figure 1: The rescaled energy density, $Y= \frac{27 \Delta\phi^{3}}{8\pi^{3}\ell^{2}}\langle T_{tt} \rangle$, vs the rescaled magnetic flux, $X=\Phi/\Phi_{max}$, for the two branches of soliton solutions. The lower branch crosses the axis at $X_{S} = \Phi_{S}/\Phi_{max} = \sqrt{3}/2$.
• Figure 2: The action $\epsilon =\frac{\ell}{2\pi\left\vert Q\right\vert }S^{\prime}/V$ vs $\Delta=\frac{\sqrt{\left\vert Q\right\vert \ell}}{4\pi\ell^{2}}\Delta\phi$, at fixed $Q$. There are two solitons for each boundary condition for $\Delta\phi< \Delta \phi_{max}$. The supersymmetric solution is at the maximum value $\Delta\phi= \Delta\phi_{max}$.