On the existence of the global conformal gauge in string theory

TL;DR

This paper proves the global existence of the conformal gauge in string theory for various two-dimensional metrics, including Lorentzian and Riemannian cases, on different geometries such as the plane and infinite strips.

Contribution

It establishes the global existence of the conformal gauge on the entire plane and on infinite strips, extending previous local results in string theory.

Findings

Global conformal gauge exists on the entire plane for Lorentzian metrics.

Conformal gauge exists globally on infinite strips with straight boundaries.

Existence of conformal gauge is proven for positive definite Riemannian metrics.

Abstract

The global conformal gauge is playing the crucial role in string theory providing the basis for quantization. Its existence for two-dimensional Lorentzian metric is known locally for a long time. We prove that if a Lorentzian metric is given on a plain then the conformal gauge exists globally on the whole ${\mathbb R}^2$. Moreover, we prove the existence of the conformal gauge globally on the whole worldsheets represented by infinite strips with straight boundaries for open and closed bosonic strings. The global existence of the conformal gauge on the whole plane is also proved for the positive definite Riemannian metric.