On Classification of Geometries with SO(2,2) Symmetry

TL;DR

This paper classifies solutions with SO(2,2) symmetry in various Einstein-Maxwell-Dilaton and supergravity theories, revealing geometric structures relevant to EVH black holes and proving a rigidity theorem for certain solutions.

Contribution

It constructs and classifies geometries with SO(2,2) symmetry in multiple theories, including a new rigidity theorem for the structure of these solutions.

Findings

Classified 4D and 5D solutions with SO(2,2) symmetry.

Proved that the 2D surface in 4D solutions must have a U(1) isometry.

Constructed all Einstein vacuum solutions with specific isometries.

Abstract

Motivated by the Extremal Vanishing Horizon (EVH) black holes, their near horizon geometry and the EVH/CFT proposal, we construct and classify solutions with (local) SO(2,2) symmetry to four and five dimensional Einstein-Maxwell-Dilaton (EMD) theory with positive, zero or negative cosmological constant Lambda, the EMD-$\Lambda$ theory, and also $U(1)^4$ gauged supergravity in four dimensions and $U(1)^3$ gauged supergravity in five dimensions. In four dimensions the geometries are warped product of AdS3 with an interval or a circle. In five dimensions the geometries are of the form of warped product of AdS3 and a 2d surface $\Sigma_2$. For the Einsten-Maxwell-$\Lambda$ theory we prove that $\Sigma_2$ should have a U(1) isometry, a rigidity theorem in this class of solutions. We also construct all d dimensional Einstein vacuum solutions with $SO(2,2) \times U(1)^{d-4}$ or $SO(2,2) \times SO(d-3)$ isometry.