Special metric structures and closed forms

TL;DR

This thesis explores special metric structures induced by differential forms, focusing on PSU(3) and G_2×G_2 structures, their topological obstructions, integrability conditions, and connections to supergravity equations, with explicit examples and classifications.

Contribution

It provides new topological and geometric characterizations of PSU(3) and G_2×G_2 structures, linking critical points of Hitchin's functional to supergravity supersymmetry equations.

Findings

Derived obstructions to PSU(3)-structure reductions.

Characterized integrability via co-closed spinor fields.

Connected critical points to supergravity supersymmetry conditions.

Abstract

The primary aim of this thesis is to investigate metrics which are induced by a differential form and arise as a critical point of Hitchin's variational principle. Firstly, we investigate metrics associated with the structure group PSU(3) acting in its adjoint representation. We derive various obstructions to the existence of a topological reduction to PSU(3). For compact manifolds, we also find sufficient conditions if the PSU(3)-structure lifts to an SU(3)-structure. We give a Riemannian characterisation of topological PSU(3)-structures through an invariant spinor valued 1-form and show that the PSU(3)-structure is integrable if and only if the spinor valued 1-form defines a co-closed Rarita-Schwinger field. Moreover, we construct non-symmetric (compact) examples. Secondly, we consider even or odd forms which can be naturally interpreted as spinors for a spin structure on $T\oplus T^*$. As such, the forms we consider induce a reduction from $Spin(7,7)$ to $G_2\times G_2$. We give a topological classification of $G_2\times G_2$-structures. We prove that the condition for being a critical point is equivalent to the supersymmetry equations on spinors in supergravity theory of type IIA/B with NS-NS background fields. Examples are systematically constructed by the device of T-duality.