Lectures on Supergravity p-branes

TL;DR

This work surveys classical $p$-brane solutions in supergravity, detailing the electric/electric and magnetic/magnetic branches, their horizons and singularity structures, and how mass/charge densities relate to preserved supersymmetry. It develops the general $p$-brane ansatz in $D$ dimensions, presents explicit $D=11$ membranes and five-branes, and extends to black-brane generalizations. The authors then explain how diagonal and vertical Kaluza–Klein reductions preserve the key quantity $\Delta$, enable multi-center and Scherk–Schwarz domain-wall constructions, and organize the full spectrum of branes via duality and Weyl-group symmetries, including charge quantization. These results illuminate the nonperturbative landscape of string/M-theory, showing how classical supergravity solutions encode the structure of brane spectra across dimensions and connect to domain walls and moduli spaces through discretized dualities.

Abstract

We review the properties of classical p-brane solutions to supergravity theories, i.e. solutions that may be interpreted as Poincare-invariant hyperplanes in spacetime. Topics covered include the distinction between elementary/electric and solitonic/magnetic solutions, examples of singularity and global structure, relations between mass densities, charge densities and the preservation of unbroken supersymmetry, diagonal and vertical Kaluza-Klein reduction families, Scherk-Schwarz reduction and domain walls, and the classification of multiplicities using duality symmetries.

Figures (4)

• Figure 1: Carter-Penrose diagram for the $D=11$ elementary/electric 2-brane solution.
• Figure 2: The $D=11$ elementary/electric 2-brane solution interpolates between flat space at ${\cal J}^\pm$ and $(\hbox{AdS})_4\times{\cal S}^7$ at the horizon.
• Figure 3: Carter-Penrose diagram for the solitonic/magnetic 5-brane solution.
• Figure 4: Brane-scan of supergravity $p$-brane solutions ($p\le (D-3)$)