Dimensionally Reduced Gravity, Hermitian Symmetric Spaces and the Ashtekar Variables

TL;DR

This paper introduces a new set of complex variables for two-dimensional gravity theories derived from higher-dimensional supergravity, offering advantages for quantization and revealing conserved charges, with potential implications for quantum gravity models.

Contribution

It generalizes Ashtekar variables to Hermitian symmetric spaces in dimensionally reduced gravity, facilitating exact quantization and revealing conserved charges.

Findings

Introduction of complexified variables for Hermitian symmetric spaces

Construction of an infinite set of conserved charges

Potential for probing quantum equivalence of gravity formalisms

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.