Dimensionally Reduced Gravity, Hermitian Symmetric Spaces and the Ashtekar Variables

Abstract

Dimensional reductions of various higher dimensional (super)gravity theories lead to effectively two-dimensional field theories described by gravity coupled G/H nonlinear sigma-models. We show that a new set of complexified variables can be introduced when G/H is a Hermitian symmetric space. This generalizes an earlier construction that grew out of the Ashtekar formulation of two Killing vector reduced pure 4d general relativity. Apart from giving some new insights into dimensional reductions of higher dimensional (super)gravity theories, these Ashtekar-type variables offer several technical advantages in the context of the exact quantization of these models. As an application, an infinite set of conserved charges is constructed. Our results might serve as a starting point for probing the quantum equivalence of the Ashtekar and the metric formalism within a non-trivial midi-superspace model of quantum gravity.

20 pages, no figures, Latex2e

References in corpus (3)

• On Nonlinear $\sigma$-Models arizing in (Super-)Gravity
• Yangian Symmetry in Integrable Quantum Gravity
• Metric Versus Ashtekar Variables in Two Killing Field Reduced Gravity

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• Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces
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