On Nonlinear $\sigma$-Models arizing in (Super-)Gravity

TL;DR

This paper analyzes the group theoretical structure of nonlinear sigma-models derived from four-dimensional gravity theories with gauge fields, providing a unified framework and deriving a quadratic mass formula for black holes.

Contribution

It offers a detailed group-theoretical analysis and a uniform parametrization of sigma-models arising in (super-)gravity, extending previous work and simplifying derivations.

Findings

Unified treatment of sigma-models via triangular group elements

Derivation of a quadratic mass formula for stationary black holes

Enhanced understanding of the structure of gravity-related sigma-models

Abstract

In a previous paper with Gibbons [CMP 120 (1987) 295] we derived a list of three dimensional symmetric space $\sigma$-model obtained by dimensional reduction of a class of four dimensional gravity theories with abelian gauge fields and scalars. Here we give a detailed analysis of their group theoretical structure leading to an abstract parametrization in terms of `triangular' group elements. This allows for a uniform treatment of all these models. As an interesting application we give a simple derivation of a `Quadratic Mass Formula' for strictly stationary black holes.