Four-Dimensional Random Surfaces and One-Dimensional String Theory

TL;DR

This paper introduces a novel two-dimensional field theory action representing the area of a surface in four-dimensional Minkowski space, revealing new symmetries and connections to string theory and matrix models.

Contribution

It proposes a new geometric action with an additional symmetry, explores gauge fixing to a free scalar theory, and discusses quantum anomalies and links to matrix quantum mechanics.

Findings

The action describes a surface in four-dimensional Minkowski space.

A new infinite-dimensional local symmetry is identified.

Gauge fixing leads to a free scalar field theory.

Abstract

We consider a new action of a two-dimensional field theory interacting with gravitational field. The action is interpreted as the area of a surface imbedded into four-dimensional Mincowski target space. In addition to reparametrization invariance the new action has one extra infinite-dimensional local symmetry with a clear geometrical meaning. The special gauge choice, which includes the gauge condition of tracelessness of the energy-momentum tensor, leads to an effective free scalar field theory. The problem of anomalies in quantum theory and possible connection with matrix quantum mechanics are also discussed.