Domain walls of gauged supergravity, M-branes, and algebraic curves

TL;DR

This work provides an algebraic classification of all supersymmetric domain-wall solutions in maximal gauged supergravity in D=4 and D=7 with nontrivial scalars in the cosets $SL(8,\mathbb{R})/SO(8)$ and $SL(5,\mathbb{R})/SO(5)$, using Bogomol\'nyi first-order equations. The authors show how these solutions correspond to continuous distributions of M2- and M5-branes in M-theory and develop a framework based on the Christoffel–Schwarz transformation and algebraic-curve uniformization to compute the Schrödinger potentials for scalar and graviton fluctuations, often solvable by supersymmetric quantum mechanics. They provide a comprehensive algebraic classification, yielding 22 M2-brane models, 11 D3-brane models, and 7 M5-brane models, with explicit uniformizations and spectra in many low-genus cases, and they connect these results to previous D=5 constructions and to Wilson-surface observables in the six-dimensional $(0,2)$ theories. The analysis highlights how domain-wall physics across dimensions can be captured by a unified algebro-geometric approach, offering insights into AdS/CFT on the Coulomb branch and suggesting intriguing duality-driven relationships among brane configurations. The work also discusses Lamé-type fluctuation problems and outlines directions for future exploration of higher-genus cases and duality structures.

Abstract

We provide an algebraic classification of all supersymmetric domain wall solutions of maximal gauged supergravity in four and seven dimensions, in the presence of non-trivial scalar fields in the coset SL(8,R)/SO(8) and SL(5,R)/SO(5) respectively. These solutions satisfy first-order equations, which can be obtained using the method of Bogomol'nyi. From an eleven-dimensional point of view they correspond to various continuous distributions of M2- and M5-branes. The Christoffel-Schwarz transformation and the uniformization of the associated algebraic curves are used in order to determine the Schrodinger potential for the scalar and graviton fluctuations on the corresponding backgrounds. In many cases we explicitly solve the Schrodinger problem by employing techniques of supersymmetric quantum mechanics. The analysis is parallel to the construction of domain walls of five-dimensional gauged supergravity, with scalar fields in the coset SL(6,R)/SO(6), using algebraic curves or continuous distributions of D3-branes in ten dimensions. In seven dimensions, in particular, our classification of domain walls is complete for the full scalar sector of gauged supergravity. We also discuss some general aspects of D-dimensional gravity coupled to scalar fields in the coset SL(N,R)/SO(N).