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

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.)