Note on orbit space of G membranes

TL;DR

This paper investigates how the motion of membranes with symmetry groups can be simplified by analyzing their behavior in quotient spaces, extending to cases with additional fields and different symmetry conditions.

Contribution

It demonstrates that Nambu-Goto membranes in symmetric spacetimes can be described by membranes in the quotient space under certain symmetry conditions, including coupling with scalar or form fields.

Findings

Membranes with Abelian symmetry groups reduce to membranes in quotient space.

Semisimple compact groups also allow reduction to quotient space.

The reduction applies when the orbit distribution is integrable, even with additional fields.

Abstract

The motion of test membranes on which the group $G$ of isometries of a spacetime $M$ acts has been considered in general settings. It has been shown that the configuration of Nambu-Goto membranes is described by the Nambu-Goto membranes in the quotient manifold $M/G$ with an appropriate projected metric if (i) $G$ is Abelian, (ii) $G$ is semisimple and compact, or (iii) the orthogonal distribution of the orbit of $G$ is integrable, but in general not. It has also been shown that a similar result holds when the membranes couple with scalar maps or differential form fields.