Loop Variables with Chan-Paton Factors

TL;DR

This paper extends the Loop Variable method to non-Abelian gauge invariance using Chan-Paton factors, showing how the continuum limit yields Yang-Mills equations from string theory.

Contribution

It introduces a generalized Loop Variable approach incorporating Chan-Paton factors, connecting string symmetries to low-energy Yang-Mills theory.

Findings

Symmetries involve massive modes and include scale and rotation transformations.

The continuum limit of the equations reproduces Yang-Mills equations.

An infinite sum of terms in the cutoff theory converges to a smooth low-energy limit.

Abstract

The Loop Variable method that has been developed for the U(1) bosonic open string is generalized to include non-Abelian gauge invariance by incorporating "Chan-Paton" gauge group indices. The scale transformation symmetry $k(s) \to k(s) \lambda (s)$ that was responsible for gauge invariance in the U(1) case continues to be a symmetry. In addition there is a "rotation" symmetry. Both symmetries crucially involve the massive modes. However it is plausible that only a linear combination, which is the usual Yang-Mills transformation on massless fields, has a smooth (world sheet) continuum limit. We also illustrate how an infinite number of terms in the equation of motion in the cutoff theory add up to give a term that has a smooth continuum limit, and thus contributes to the low energy Yang-Mills equation of motion.