Abstract

We study the purely topological restrictions on allowed spin and statistics of topological solitons in nonlinear sigma models. 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 on X, such that choices of allowed statistics for solitons of type α are given by unitary representations of Statn(X,α) when n solitons are present. In particular, when M = S 2, as in the O(3) nonlinear sigma model with Hopf term, and α ∈ π2(S 2) is a generator, we compute that Statn(R 2,α) = Z, while Statn(S 2,α) = Z2n. It follows that phase exp(iθ) for interchanging two solitons of type α on S 2 must satisfy the constraint θ = kπ/n, k ∈ Z, when n such solitons are present. 1

Citations