Bosonization and Current Algebra of Spinning Strings

TL;DR

This paper develops a geometric action principle for spinning strings in Minkowski space, analyzing their current algebra and anomalies, and explores special cases including the Nambu string and strings with spin currents.

Contribution

It introduces a Grassmann-free geometric formulation of spinning strings, derives the Virasoro generators, and analyzes the associated current algebra and anomalies.

Findings

Virasoro generators are first class constraints.

Anomaly terms in current algebra obstruct quantization.

Anomalies vanish for the Nambu string, recovering the Poincaré loop group algebra.

Abstract

We write down a general geometric action principle for spinning strings in $d$-dimensional Minkowski space, which is formulated without the use of Grassmann coordinates. Instead, it is constructed in terms of the pull-back of a left invariant Maurer-Cartan form on the $d$-dimensional Poincar\'e group to the world sheet. The system contains some interesting special cases. Among them are the Nambu string (as well as, null and tachyionic strings) where the spin vanishes, and also the case of a string with a spin current - but no momentum current. We find the general form for the Virasoro generators, and show that they are first class constraints in the Hamiltonian formulation of the theory. The current algebra associated with the momentum and angular momentum densities are shown, in general, to contain rather complicated anomaly terms which obstruct quantization. As expected, the anomalies vanish when one specializes to the case of the Nambu string, and there one simply recovers the algebra associated with the Poincar\'e loop group. We speculate that there exist other cases where the anomalies vanish, and that these cases give the bosonization of the known pseudoclassical formulations of spinning strings.