# Domain wall seeds in CSO-gauged supergravity

## TL;DR

This paper studies gravitational domain wall solutions in CSO-gauged supergravity, revealing how the restriction to solutions with zero Noether charge influences their symmetry orbit structure and the diagonalization of the coset matrix.

## Contribution

It demonstrates that seed solutions in CSO-gauged supergravity have diagonal coset matrices, and that boundary transformations can diagonalize the entire flow due to charge restrictions.

## Key findings

- Seed solutions have diagonal coset matrices.
- Boundary transformations diagonalize the entire flow.
- Charge restrictions influence solution orbit structure.

## Abstract

Gravitational domain wall solutions in gauged supergravity are often constructed within truncations that do not include vectors. As a consequence the gauge group is only a global symmetry of this truncation. The consistency of the truncation requires the restriction to solutions with vanishing Noether charge under this global symmetry, since otherwise vector fields are sourced. We show that this has interesting consequences for the orbit structure of the solutions under the global symmetries. We investigate this for $\text{CSO}(p,q,r)$-gaugings in various dimensions with scalar fields truncated to the $\text{SL}(n,\mathbb{R})/\text{SO}(n)$ subcoset. We prove that the seed solution $-$ which generates all other solutions using only global transformations $-$ has a diagonal coset matrix. This means that there exists a transformation at the boundary of the geometry that diagonalises the coset matrix and that this same transformation also diagonalises the whole flow as a consequence of the vanishing charge.