AdS$_3$ solutions with exceptional supersymmetry

TL;DR

The paper constructs AdS$_3$ backgrounds in type IIA with exceptional supersymmetry by realizing the R-symmetries as isometries of an internal $S^6$ fibered over an interval, yielding two locally unique analytic solutions: one realizing the $F(4)$ algebra with ${\cal N}=8$ and another realizing $G(3)$ with ${\cal N}=7$ (the latter with $G_2$ flavor symmetry broken from $SO(7)$ by flux). It employs a metric ansatz and flux expansions based on $SU(3)$ structures on $S^6$ and uses pure spinor (generalized $G_2$) techniques to derive flow equations, obtaining a near-horizon analytic background for $F(4)$ and a corresponding SUSY analysis that proves local uniqueness; global properties reveal non-compact internal spaces that could be made finite by gluing with D8-branes, yielding a finite central charge scaling $c \,\sim\, p^{1/3} q^{5/3}$. Beyond the analytic solutions, the authors also present several ${\cal N}=1$ backgrounds with $G_2$ flavor symmetry, obtained numerically with varied localized sources, including O8- and O2-planes, and provide a systematic framework connecting exceptional algebras, internal geometry, and flux configurations in AdS$_3$ holography. These results illuminate how exceptional symmetries constrain AdS$_3$ vacua, reveal the role of fluxes in breaking R-symmetries, and offer tractable pathways to finite-volume dual CFTs with interesting scaling of central charges.

Abstract

Among the possible superalgebras that contain the AdS$_3$ isometries, two interesting possibilities are the exceptional $F(4)$ and $G(3)$. Their R-symmetry is respectively SO(7) and $G_2$, and the amount of supersymmetry ${\cal N}=8$ and ${\cal N}=7$. We find that there exist two (locally) unique solutions in type IIA supergravity that realize these superalgebras, and we provide their analytic expressions. In both cases, the internal space is obtained by a round six-sphere fibred over an interval, with an O8-plane at one end. The R-symmetry is the symmetry group of the sphere; in the $G(3)$ case, it is broken to $G_2$ by fluxes. We also find several numerical ${\cal N}=1$ solutions with $G_2$ flavor symmetry, with various localized sources, including O2-planes and O8-planes.

Figures (4)

• Figure 1: A solution with a single O8 (at the right extremum). In orange $e^{2A}$, in blue $e^{2Q}$ and in black the dilaton $e^{\phi}$.
• Figure 2: A solution with a single O2 (at the right extremum). In orange $e^{2A}$, in blue $e^{2Q}$ and in black the dilaton $e^{\phi}$.
• Figure 3: A solution with an O8 (left extremum) and an O2 (right extremum). In orange $e^{2A}$, in blue $e^{2Q}$ and in black the dilaton $e^{\phi}$.
• Figure 4: A solution with two O2-planes. In orange $e^{2A}$, in blue $e^{2Q}$ and in black the dilaton $e^{\phi}$.