On the Bogomol'nyi bound in Einstein-Maxwell-dilaton gravity

TL;DR

This paper examines the Bogomol'nyi bound in Einstein-Maxwell-dilaton gravity, revealing conditions under which supersymmetric solutions exist and clarifying the role of the dilaton coupling constant in ensuring the solutions satisfy all field equations.

Contribution

It demonstrates that the dilatino equation alone does not guarantee the dilaton field equation unless specific conditions on the solution or coupling constant are met, and classifies supersymmetric solutions accordingly.

Findings

Dilatino equation fails to ensure dilaton field equation unless conditions are met.

Rotating solutions occur only for dyonic cases or specific coupling constants.

The theory with arbitrary coupling constant cannot generally be embedded into supergravity.

Abstract

It has been shown that the 4-dimensional Einstein-Maxwell-dilaton theory allows a Bogomol'nyi-type inequality for an arbitrary dilaton coupling constant $\alpha $, and that the bound is saturated if and only if the (asymptotically flat) spacetime admits a nontrivial spinor satisfying the gravitino and the dilatino Killing spinor equations. The present paper revisits this issue and argues that the dilatino equation fails to ensure the dilaton field equation unless the solution is purely electric/magnetic, or the dilaton coupling constant is given by $\alpha=0, \sqrt 3$, corresponding to the Brans-Dicke-Maxwell theory and the Kaluza-Klein reduction of 5-dimensional vacuum gravity, respectively. A systematic classification of the supersymmetric solutions reveals that the solution can be rotating if and only if the solution is dyonic or the coupling constant is given by $\alpha=0, \sqrt 3$. This implies that the theory with $\alpha \ne 0, \sqrt 3$ cannot be embedded into supergravity except for the static truncation. Physical properties of supersymmetric solutions are explored from various points of view.