Random Geometry, Quantum Gravity and the K\"ahler Potential

TL;DR

This paper introduces a novel approach to modeling random geometries via a map between metrics and hermitian matrices, enabling background-independent measures and new quantum gravity models.

Contribution

It presents a new formalism connecting metrics to matrices, leading to background-independent measures and generalized quantum gravity models beyond Liouville theory.

Findings

A new method for defining random geometries using matrix mappings.

Construction of background-independent measures on metric spaces.

Identification of a new gravitational effective action in 2D gravity.

Abstract

We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a background-independent measure on the space of metrics can be easily constructed from first principles. Our framework suggests the relevance of a new gravitational effective action and we show that it occurs when coupling the massive scalar field to two-dimensional gravity. This yields new types of quantum gravity models generalizing the standard Liouville case.