The emergence of geometry: a two-dimensional toy model

TL;DR

This paper introduces a two-dimensional toy model demonstrating how a metric and gravity-like dynamics can emerge from a topological theory through symmetry breaking, suggesting a novel approach to quantum gravity.

Contribution

It proposes a mechanism where gravitons emerge as Goldstone bosons in a 2D model with no initial metric, providing insights into emergent geometry and gravity.

Findings

The model is renormalizable and describes 2D gravity at long distances.

A metric appears only after symmetry breaking, making distance an induced concept.

The induced cosmological constant relates to fermion mass and acts as an infrared regulator.

Abstract

We review the similarities between the effective chiral lagrangrian, relevant for low-energy strong interactions, and the Einstein-Hilbert action. We use these analogies to suggest a specific mechanism whereby gravitons would emerge as Goldstone bosons of a global SO(D) X GL(D) symmetry broken down to SO(D) by fermion condensation. We propose a two-dimensional toy model where a dynamical zwei-bein is generated from a topological theory without any pre-existing metric structure, the space being endowed only with an affine connection. A metric appears only after the symmetry breaking; thus the notion of distance is an induced effective one. In spite of several non-standard features this simple toy model appears to be renormalizable and at long distances is described by an effective lagrangian that corresponds to that of two-dimensional gravity (Liouville theory). The induced cosmological constant is related to the dynamical mass M acquired by the fermion fields in the breaking, which also acts as an infrared regulator. The low-energy expansion is valid for momenta k >M, i.e. for supra-horizon scales. We briefly discuss a possible implementation of a similar mechanism in four dimensions.