$G_2$ generating technique for minimal D=5 supergravity and black rings

TL;DR

This paper develops a $G_2$-based solution generating method for five-dimensional minimal supergravity, enabling the construction of new black ring solutions with multiple parameters by exploiting symmetries in a reduced three-dimensional sigma-model.

Contribution

It introduces a novel $G_2$ U-duality approach for generating solutions in D=5 supergravity, including a new charged black ring with two rotation parameters.

Findings

Derived a $7\times 7$ matrix representation of the coset $G_{2(2)}/(SL(2,R)\times SL(2,R))$.

Constructed a new charged black ring solution with two independent rotation parameters.

Demonstrated the use of $G_2$ symmetries to generate non-trivial supergravity solutions.

Abstract

A solution generating technique is developed for D=5 minimal supergravity with two commuting Killing vectors based on the $G_2$ U-duality arising in the reduction of the theory to three dimensions. The target space of the corresponding 3-dimensional sigma-model is the coset $G_{2(2)}/(SL(2,R)\times SL(2,R))$. Its isometries constitute the set of solution generating symmetries. These include two electric and two magnetic Harrison transformations with the corresponding two pairs of gauge transformations, three $SL(2,R) S$-duality transformations, and the three gravitational scale, gauge and Ehlers transformations (altogether 14). We construct a representation of the coset in terms of $7\times 7$ matrices realizing the automorphisms of split octonions. Generating a new solution amounts to transforming the coset matrices by one-parametric subgroups of $G_{2(2)}$ and subsequently solving the dualization equations. Using this formalism we derive a new charged black ring solution with two independent parameters of rotation.