Flow Equations In Arbitrary Signature

TL;DR

This paper derives unified flow equations for extremal solutions in four-dimensional N=2 supergravity with arbitrary signatures, using para-special Kähler geometry and analyzing supersymmetry constraints.

Contribution

It introduces a unified approach to derive flow equations for all signatures in supergravity using para-complex geometry and stabilisation equations.

Findings

Flow equations are unified for Euclidean, neutral, and Lorentzian signatures.

Solutions with different signatures are related via field redefinitions.

Stabilisation equations are expressed in terms of para-complex sections.

Abstract

We discuss general bosonic configurations of four-dimensional N=2 supergravity coupled to vector multiplets in (t,s) space-time. The supergravity theories with Euclidean and neutral signature are described by the so-called para-special K\"ahler geometry. For extremal solutions, we derive in a unified fashion, using the equations of motion, the flow equations for all space-time signatures. Demanding that the solutions with neutral and Euclidean signatures admit unbroken supersymmetry, we derive the constraints, known as the stabilisation equations, on the para-covariantly holomorphic sections expressed in terms of the adapted coordinates. The stabilisation equations expressed in terms of the para-complex sections imply generalised flow equations in terms of para-complex central charge. For Euclidean and neutral signature, it is demonstrated that solutions for either signs of gauge kinetic terms are mapped into each other via field redefinitions.