A Deformation of Sasakian Structure in the Presence of Torsion and Supergravity Solutions

TL;DR

This paper introduces a deformation of Sasakian geometry incorporating torsion, constructs explicit metrics with hidden symmetries, and applies these to find exact supergravity solutions with global structures extending known Einstein manifolds.

Contribution

It develops a new class of deformed Sasakian structures with torsion, provides explicit metric examples, and constructs novel supergravity solutions based on these geometries.

Findings

Explicit local metrics for deformed Sasakian structures with torsion.

Existence of hidden symmetries via generalized conformal Killing-Yano tensors.

New supergravity solutions with global structures extending Sasaki–Einstein manifolds.

Abstract

We discuss a deformation of Sasakian structure in the presence of totally skew-symmetric torsion by introducing odd dimensional manifolds whose metric cones are K\"ahler with torsion. It is shown that such a geometry inherits similar properties to those of Sasakian geometry. As an example of them, we present an explicit expression of local metrics and see how Sasakian structure is deformed by the presence of torsion. We also demonstrate that our example of the metrics admits the existence of hidden symmetries described by non-trivial odd-rank generalized closed conformal Killing-Yano tensors. Furthermore, using these metrics as an {\it ansatz}, we construct exact solutions in five dimensional minimal (un-)gauged supergravity and eleven dimensional supergravity. Finally, we discuss the global structures of the solutions and obtain regular metrics on compact manifolds in five dimensions, which give natural generalizations of Sasaki--Einstein manifolds $Y^{p,q}$ and $L^{a,b,c}$. We also discuss regular metrics on non-compact manifolds in eleven dimensions.