On indefinite Einstein solvmanifolds admitting a Killing spinor

TL;DR

This paper explores indefinite Einstein solvmanifolds, showing that only hyperbolic half-spaces admit Killing spinors, and extends the nilsoliton concept to higher-dimensional indefinite metrics.

Contribution

It demonstrates that the only pseudo-Iwasawa Einstein solvmanifolds with Killing spinors are hyperbolic half-spaces, extending the understanding of spinor fields in indefinite Einstein geometries.

Findings

Only hyperbolic half-spaces admit Killing spinors among pseudo-Iwasawa solvmanifolds.

Nilsolitons can be used to construct higher-dimensional indefinite Einstein solvmanifolds.

The pseudo-Iwasawa condition involves a Lie algebra splitting with symmetric derivations.

Abstract

Riemannian Einstein solvmanifolds can be described in terms of nilsolitons, namely nilpotent Lie groups endowed with a left-invariant Ricci soliton metric. This characterization does not extend to indefinite metrics; nonetheless, nilsolitons can be defined and used to construct Einstein solvmanifolds of a higher dimension in any signature. An Einstein solvmanifold obtained by this construction turns out to satisfy the pseudo-Iwasawa condition, meaning that its Lie algebra splits as the orthogonal sum of a nilpotent ideal and an abelian subalgebra, the latter acting by symmetric derivations. We prove that the only pseudo-Iwasawa solvmanifolds that admit a Killing spinor, invariant or not, are the hyperbolic half-spaces.