Strings with Discrete Target Space

TL;DR

This paper explores the field theory of strings with discrete target spaces, establishing conditions for continuum limits and mapping to known models like the SOS model, with implications for string amplitudes and matrix models.

Contribution

It introduces a framework for string theories with discrete target spaces related to Lie algebra Dynkin diagrams, connecting them to Coulomb gas and matrix model techniques.

Findings

Continuum limit exists for target spaces as Dynkin diagrams of simply laced Lie algebras.

String amplitudes can be computed via perturbative expansion around saddle points.

Partition functions relate to Gaussian fluctuations and random matrix models.

Abstract

We investigate the field theory of strings having as a target space an arbitrary discrete one-dimensional manifold. The existence of the continuum limit is guaranteed if the target space is a Dynkin diagram of a simply laced Lie algebra or its affine extension. In this case the theory can be mapped onto the theory of strings embedded in the infinite discrete line $\Z$ which is the target space of the SOS model. On the regular lattice this mapping is known as Coulomb gas picture. ... Once the classical background is known, the amplitudes involving propagation of strings can be evaluated by perturbative expansion around the saddle point of the functional integral. For example, the partition function of the noninteracting closed string (toroidal world sheet) is the contribution of the gaussian fluctuations of the string field. The vertices in the corresponding Feynman diagram technique are constructed as the loop amplitudes in a random matrix model with suitably chosen potential.