String Quantization and the Shuffle Hopf Algebra

TL;DR

This paper explores the quantization of the Poisson algebra of string invariants using the Shuffle Hopf algebra, providing a new algebraic framework to compare different quantization methods.

Contribution

It introduces a reformulation of the auxiliary Lie algebra within the Shuffle Hopf algebra, enabling better comparison of quantization approaches for the string invariants.

Findings

Reformulation of the auxiliary Lie algebra in the Shuffle Hopf algebra.

Facilitation of comparison between different quantization methods.

Enhanced algebraic understanding of string invariants.

Abstract

The Poisson algebra $\mathfrak h$ of invariants of the Nambu-Goto string, which was first introduced by K. Pohlmeyer in 1982, is described using the Shuffle Hopf algebra. In particular, an underlying auxiliary Lie algebra is reformulated in terms of the image of the first Eulerian idempotent of the Shuffle Hopf algebra. This facilitates the comparison of different approaches to the quantization of $\mathfrak h$.