The Polynomial Formulation of the U(1) Non-Linear Sigma-Model in 2   Dimensions
  We investigate some properties of a first-order polynomial formulation of the U(1) non-linear sigma-model in two Euclidean dimensions. The variables in this description are a 1-form field plus a 0-form Lagrange multiplier field. The usual spin variables are non-local functions of the new fields. As this construction incorporates O(2) invariance ab initio, only O(2)-invariant correlation functions (the only non-vanishing ones in the model) can be constructed. We show that the vortices play a dual role to the spin variables in the partition function. The equivalent Sine-Gordon description is obtained in a natural way, when one integrates out the 1-form field to get an effective partition function for the Lagrange multiplier. We also show how to introduce strings of vortices within this formulation. 
C. D. Fosco, T. Matsuyama