Spontaneous generation of geometry in four dimensions

Abstract

We present the extension to four dimensions of a Euclidean two-dimensional model that exhibits spontaneous generation of a metric. In this model gravitons emerge as Goldstone bosons of a global S O ( 4 ) × G L ( 4 ) symmetry broken down to S O ( 4 ) . The microscopic theory can be formulated without having to appeal to any particular space-time metric and only assumes the preexistence of a manifold endowed with an affine connection. In this sense the microscopic theory is quasitopological. There are indications that suggest that the model may be renormalizable, a fact that seems to be due to the impossibility of constructing counterterms without a preexisting metric. The vierbein appears as a condensate of the fundamental fermions defining a vacuum that, if the background affine connection is set to zero, is the Euclidean continuation of de Sitter space-time. If perturbatively small affine connections are introduced on this background, fluctuations of the metric (i.e., gravitons) appear and 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 M p and Λ . The relevance of higher-dimensional operators is also briefly discussed.