Towards a semi-classical approximation of the non-linear σ-model

We consider the contribution of instantons of arbitrary topological charge to the expectation value of any observable in the CP¹ model defined on the sphere S².We replace the integration over one of the parameters of the instantons by an integration over the group of isospin transformations and carry it out explicitly by using the isospin invariance. This allows us to express the result as the expectation value of an observable of a neutral gas of 2 species of particles, computed in the grand canonical ensemble. The interactions are complicated and involve multi-body forces (2 × k, k being the topological charge).