The odd side of torsion geometry

TL;DR

This paper introduces Sasaki with torsion (ST) structures as an odd-dimensional analogue of KT geometry, exploring their properties, examples, and relations to other geometries, along with reduction techniques.

Contribution

It defines ST structures, shows their existence on compact Lie groups, relates them to KT geometry, and develops a moment map framework for their reduction.

Findings

Any compact Lie group admits a left-invariant ST structure with closed torsion.

The cylinder and cone over an ST manifold are KT, with the cylinder maintaining torsion closedness.

A G-moment map concept is introduced, with criteria for existence and applications to reduction.

Abstract

We introduce and study a notion of `Sasaki with torsion structure' (ST) as an odd-dimensional analogue of K\"ahler with torsion geometry (KT). These are normal almost contact metric manifolds that admit a unique compatible connection with 3-form torsion. Any odd-dimensional compact Lie group is shown to admit such a structure; in this case the structure is left-invariant and has closed torsion form. We illustrate the relation between ST structures and other generalizations of Sasaki geometry, and explain how some standard constructions in Sasaki geometry can be adapted to this setting. In particular, we relate the ST structure to a KT structure on the space of leaves, and show that both the cylinder and the cone over an ST manifold are KT, although only the cylinder behaves well with respect to closedness of the torsion form. Finally, we introduce a notion of `G-moment map'. We provide criteria based on equivariant cohomology ensuring the existence of these maps, and then apply them as a tool for reducing ST structures.