Equivalence of the path-integral theory of spinning particles and the topological nonlinear σ model in d=2 dimensions

We show that the path-integral theory of a spinning particle with spin S is equivalent to the topological O(3) nonlinear σ model (TNLSM) in 1+1 Euclidean dimensions. We prove this equivalence in two different ways. In particular, we use stochastic quantization of the theory of spin to show the identity of the generating functionals of correlation functions of both theories. Also, using canonical quantization of the TNLSM, we show that its ground-state wave function coincides with the Feynman weight for the path-integral theory of the spin. In addition, we give a full classification of the invariants of the TNLSM. The invariants are computed explicitly in the case of the sphere