States and Curves of Five-Dimensional Gauged Supergravity

TL;DR

This paper explores supersymmetric solutions in five-dimensional gauged supergravity with scalar fields, analyzing their geometric properties via algebraic curves, and connects these solutions to D3-brane distributions in string theory.

Contribution

It introduces a classification of supersymmetric solutions using algebraic curves and explicitly constructs solutions with elliptic functions, linking supergravity backgrounds to D3-brane configurations.

Findings

Algebraic curves of solutions are genus seven Riemann surfaces in the generic case.

Symmetry enhancement occurs when cycles of the algebraic curve shrink, lowering the genus.

Massless spectra are described by Schrödinger equations with Calogero-type potentials.

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.