Canonical Analysis of Algebraic String Actions

TL;DR

This paper analyzes the canonical structure of algebraic string actions, revealing a self-dual formulation with non-commutative coordinates and an extra degree of freedom in the non-self-dual case, impacting their equivalence to traditional string theory.

Contribution

It introduces a self-dual formulation with an Immirzi-like parameter and performs a comprehensive Hamiltonian analysis, uncovering new features like non-commutativity and additional degrees of freedom.

Findings

Self-dual formulation exhibits non-commutative coordinates.

The Hamiltonian analysis identifies the constraints and Dirac brackets.

Non-self-dual case has an extra propagating degree of freedom.

Abstract

We investigate the canonical aspects of the algebraic first order formulation of strings introduced two decades ago by Balachandran and collaborators. We slightly enlarge the Lagrangian framework and show the existence of a self-dual formulation and of an Immirzi-type parameter reminiscent of four-dimensional first order gravity. We perform a full Hamiltonian analysis of the self-dual case: we extract the first class constraints and construct the Dirac bracket associated to the second class constraints. The first class constraints contain the diffeomorphisms algebra on the world-sheet, and the coordinates are shown to be non-commutative with respect to the Dirac bracket. The Hamilton equations in a particular gauge are shown to reproduce the wave equation for the string coordinates. In the general, non-self-dual case, we also explicit the first class constraints of the system and show that, unlike the self-dual formulation, the theory admits an extra propagating degree of freedom than the two degrees of freedom of conventional string theory. This prevents the general algebraic string from being strictly equivalent to the Nambu-Goto string.