States and Curves of Five-Dimensional Gauged Supergravity

TL;DR

The paper addresses supersymmetric solutions in ${\cal N}=8$, $D=5$ gauged supergravity with scalar fields in ${SL(6,\mathbb{R})/SO(6)}$, showing the general BPS state depends on six real moduli and can be analyzed through algebraic curves. The authors develop a Schwarz–Christoffel framework that maps the problem to Riemann surfaces, revealing a genus ${g=7}$ curve in the generic case and lower-genus curves when symmetries are enhanced, with explicit uniformizations in terms of elliptic functions for low-genus sectors. They provide a ten-dimensional type-IIB interpretation as D3-brane distributions on hypersurfaces in the transverse space, clarifying the geometric meaning of the moduli and the corresponding brane profiles. The spectra of massless scalar and graviton fluctuations reduce to one-dimensional Schrödinger problems with Calogero-type potentials (rational or elliptic), reflecting the underlying integrable structures and their dependence on the background symmetry. Overall, the work connects algebro-geometric classifications, higher-dimensional brane pictures, and holographic fluctuation analyses to map out a rich family of supersymmetric states on and away from the AdS$_5$ vacuum, with several open questions in elliptic cases and potential extensions to other gauged supergravities.

Abstract

We consider the sector of N=8 five-dimensional gauged supergravity with non-trivial scalar fields in the coset space SL(6,R)/SO(6), plus the metric. We find that the most general supersymmetric solution is parametrized by six real moduli and analyze its properties using the theory of algebraic curves. In the generic case, where no continuous subgroup of the original SO(6) symmetry remains unbroken, the algebraic curve of the corresponding solution is a Riemann surface of genus seven. When some cycles shrink to zero size the symmetry group is enhanced, whereas the genus of the Riemann surface is lowered accordingly. The uniformization of the curves is carried out explicitly and yields various supersymmetric configurations in terms of elliptic functions. We also analyze the ten-dimensional type-IIB supergravity origin of our solutions and show that they represent the gravitational field of a large number of D3-branes continuously distributed on hyper-surfaces embedded in the six-dimensional space transverse to the branes. The spectra of massless scalar and graviton excitations are also studied on these backgrounds by casting the associated differential equations into Schrodinger equations with non-trivial potentials. The potentials are found to be of Calogero type, rational or elliptic, depending on the background configuration that is used.