Spontaneous generation of geometry in four dimensions

TL;DR

This paper extends a 2D model to four dimensions where a metric spontaneously emerges, with gravitons as Goldstone bosons, suggesting a quasi-topological, renormalizable microscopic theory that reproduces Einstein gravity at long distances.

Contribution

It introduces a 4D model where geometry arises spontaneously without assuming a predefined metric, with gravitons emerging as Goldstone bosons from symmetry breaking.

Findings

Emergence of a metric as a condensate of fermions

Effective Einstein-Hilbert action with cosmological constant

Renormalizable microscopic theory in four dimensions

Abstract

We present the extension to 4 dimensions of an euclidean 2-dimensional model that exhibits spontaneous generation of a metric. In this model gravitons emerge as Goldstone bosons of a global SO(D) X GL(D) symmetry broken down to SO(D). The microscopic theory can be formulated without having to appeal to any particular space-time metric and only assumes the pre-existence of a manifold endowed with an affine connection. We emphasize that not even a flat metric needs to be assumed; in this sense the microscopic theory is quasi-topological. The vierbein appears as a condensate of the fundamental fermions. In spite of having non-standard characteristics, the microscopic theory appears to be renormalizable. The effective long-distance theory is obtained perturbatively around a vacuum that, if the background affine connection is set to zero, is (euclidean) de Sitter space-time. If perturbatively small connections are introduced on this background, fluctuations of the metric (i.e. gravitons) appear; they are described by an effective theory at long distances whose more relevant operators correspond to the Einstein-Hilbert action with a cosmological constant. This effective action is derived in the large N limit, N being the number of fermion species in the fundamental theory. The counterterms required by the microscopic theory are directly related to the cosmological constant and Newton constant and their couplings could eventually be adjusted to the physical values of Mp and \Lambda.