Some Features of Blown-Up Nonlinear $\sigma$-Models

TL;DR

This paper explores the effects of symplectic blowing up in gauged nonlinear sigma-models, analyzing classical and quantum implications with examples from toric and Kähler manifolds, and discussing connections to Mirror symmetry.

Contribution

It introduces a method to perform symplectic blow-ups in gauged sigma-models and identifies the resulting modes, linking geometric modifications to quantum field theory.

Findings

Classical and quantum aspects of blow-up modes are characterized.

Examples from toric projective space and Kähler manifolds illustrate the theory.

Implications for Mirror symmetry and quantum cohomology are discussed.

Abstract

In terms of the gauged nonlinear $\sigma$-models, we describe some results and implications of solving the following problem: Given a smooth symplectic manifold as target space with a quasi-free Hamiltonian group action, perform the symplectic blowing up of the point singularity and identify the blow-up modes in the corresponding (gauged) $\sigma$-model. Both classical and quantum aspects of the construction are explained, along with illustrating examples from the toric projective space and the K\"ahler manifold. We also discuss related problems such as the origin of Mirror symmetry and the quantum cohomologies.(Talk to be given at ICHEP94, Glasgow, July 20-27.)