Topological aspects of spin and statistics in nonlinear sigma models

Abstract

Spin and statistics for topological solitons in nonlinear sigma models are studied using topological methods based on the classical configuration space. Taking as space the connected d-manifold X, and considering nonlinear sigma models with the connected manifold M as target space, topological solitons are given by elements of πd(M). Any topological soliton α∈πd(M) determines a quotient Statn(X,α) of the group of framed braids with n strands on X, such that a framed braid determines a contractible path in the n-soliton sector of the configuration space if and only if its image in Statn(X,α) is the identity. In particular, when M=S2, as in the O(3) nonlinear sigma model with Hopf term, and α∈π2(S2) is a generator, we compute that Statn(R2,α)=Z, while Statn(S2,α)=Z2n. Thus one expects the phase exp(iθ) for interchanging two solitons of type α on S2 to satisfy the constraint θ=kπ/n, k∈Z, when n such solitons are present.