Extending the Theory of Random Surfaces

TL;DR

This paper reviews and extends the theory of embedded random surfaces, connecting two-dimensional quantum gravity to four dimensions, revealing new terms in the Liouville action, and applying 2D gravity methods to 4D Euclidean gravity.

Contribution

It introduces previously unnoticed terms in the Liouville action and generalizes the theory to four dimensions, linking random surfaces to quantum gravity models.

Findings

New terms in Liouville field theory action.

Explanation of the Sine-Gordon model phase diagram.

Predictions of critical exponents and the $c=1$ barrier analog.

Abstract

The theory of embedded random surfaces, equivalent to two--dimensional quantum gravity coupled to matter, is reviewed, further developed and partly generalized to four dimensions. It is shown that the action of the Liouville field theory that describes random surfaces contains terms that have not been noticed previously. These terms are used to explain the phase diagram of the Sine--Gordon model coupled to gravity, in agreement with recent results from lattice computations. It is also demonstrated how the methods of two--dimensional quantum gravity can be applied to four--dimensional Euclidean gravity in the limit of infinite Weyl coupling. Critical exponents are predicted and an analog of the ``$c=1$ barrier'' of two--dimensional gravity is derived.