Three dimensional strings. I. Classical theory

TL;DR

This paper explores a three-dimensional string theory with an additional volume term, linking its solutions to constant mean curvature surfaces, SU(2) models, and spinorial descriptions, revealing deep geometric and algebraic structures.

Contribution

It introduces a novel three-dimensional string model with a volume term and establishes its equivalence to various integrable models and geometric representations.

Findings

Solutions correspond to constant mean curvature surfaces

Equivalence to SU(2) principal chiral model coupled to gravity

Mapping to CP^1 nonlinear sigma model and Gross-Neveu spinorial model

Abstract

I consider a three-dimensional string theory whose action, besides the standard area term, contains one of the form $\int_{\Sigma} \epsilon_{\mu\nu\sigma} X^{\mu} d X^{\nu} \wedge d X^{\sigma}$. In the case of closed strings this extra term has a simple geometrical interpretation as the volume enclosed by the surface. The associated variational problem yields as solutions constant mean curvature surfaces. One may then show the equivalence of this equation of motion to that of an SU(2) principal chiral model coupled to gravity. It is also possible by means of the Kemmotsu representation theorem, restricted to constant curvature surfaces, to map the solution space of the string model into the one of the $CP^1$ nonlinear sigma model. I also show how a description of the Gauss map of the surface in terms of SU(2) spinors allows for yet a different description of this result by means of a Gross-Neveu spinorial model coupled to 2-D gravity. The standard three-dimensional string equations can also be recovered by setting the current-current coupling to zero.