Gauging and symplectic blowing up in nonlinear sigma-models: I. point singularities

TL;DR

This paper explores how symplectic blowing up of point singularities in nonlinear sigma-models can be understood through gauging symmetries, providing a method to smooth out singularities in the target space.

Contribution

It establishes a connection between symplectic reduction and gauging in sigma-models, demonstrating how blowing up singular points can be achieved via gauging techniques.

Findings

Gauging symmetries mimics symplectic blowing up at singular points.

Constructs symplectic diffeomorphisms to smoothly transition between singular and regular models.

Provides a method to eliminate singularities in the target space of sigma-models.

Abstract

In this paper a two dimensional non-linear sigma model with a general symplectic manifold with isometry as target space is used to study symplectic blowing up of a point singularity on the zero level set of the moment map associated with a quasi-free Hamiltonian action. We discuss in general the relation between symplectic reduction and gauging of the symplectic isometries of the sigma model action. In the case of singular reduction, gauging has the same effect as blowing up the singular point by a small amount. Using the exponential mapping of the underlying metric, we are able to construct symplectic diffeomorphisms needed to glue the blow-up to the global reduced space which is regular, thus providing a transition from one symplectic sigma model to another one free of singularities.