Closed constraint algebras and path integrals for loop group actions

TL;DR

This paper develops a method for analyzing systems with closed second class constraints, especially those with loop group symmetry, by constructing reduced theories and computing symplectic forms via fiber integrals, accounting for anomalies.

Contribution

It introduces a novel approach to reduced theories with closed second class constraints, applying fiber integrals to compute symplectic forms in the presence of loop group symmetries and anomalies.

Findings

The symplectic form on the reduced space can be obtained via fiber integrals.

Loop group anomalies affect the closure of the 2-form but not the constraint algebra.

Examples include D-branes on group manifolds with WZW action.

Abstract

In this note we study systems with a closed algebra of second class constraints. We describe a construction of the reduced theory that resembles the conventional treatment of first class constraints. It suggests, in particular, to compute the symplectic form on the reduced space by a fiber integral of the symplectic form on the original space. This approach is then applied to a class of systems with loop group symmetry. The chiral anomaly of the loop group action spoils the first class character of the constraints but not their closure. Proceeding along the general lines described above, we obtain a 2-form from a fiber (path)integral. This form is not closed as a relict of the anomaly. Examples of such reduced spaces are provided by D-branes on group manifolds with WZW action.