Topological aspects of spin and statistics in nonlinear sigma models

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 π d (M ). Any topological soliton α ∈ π d (M ) determines a quotient Stat n (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 Stat n (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 ∈ π 2 (S 2 ) is a generator, we compute that Stat n (R 2 , α) = Z, while Stat n (S 2 , α) = Z 2n . 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.

Topological Terms and Phases of Sigma Models

“…The bulk meaning of this term is that when the string wraps the target circle an odd number of times, the ground state has odd fermion parity. When the string has boundary, the winding number becomes ill-defined and so does the fermion parity [35]. However, it is possible to make the winding number defined mod 2 if we lift the boundary map to the double cover circle S 1 2 − → S 1 .…”

Quantum Particles as Conceptual Entities: A Possible Explanatory Framework for Quantum Theory

“…The spin-statistics theorem proves that the 'changing sign' and 'remaining unchanged' in both cases correspond, for the 'sign changing situation for rotation' manifested with particles with half integer spin coincides with the 'sign changing situation for permutation' for fermions, and the 'remaining unchanged situation for rotation' manifested with integer spin coincides with the 'remaining unchanged situation for permutation' for bosons. There have been attempts to prove the spin statistics theorem by relying less on technical aspects of relativistic field theory and more on topological aspects of the situation [88,89,90], without however being able to make the full circle. It has also been shown that the fact that space has at least three dimensions is crucial, and that for a two or one-dimensional space, para-statistics, i.e.…”

Higher-dimensional algebra and topological quantum field theory

“…Lest the reader think we have drifted hopelessly far from physics by now, we should note that elements of π 2 (X) correspond to 'topological solitons' in a nonlinear sigma model with target space X, for a spacetime of dimension 3. In this context, Figures 26 and 27 show the worldlines of such topological solitons, and the fact that the two pictures cannot be deformed into each other is why the statistics of such solitons is described using representations of the braid group [8]. In a spacetime of dimension 4 or more, the analogous pictures can be deformed into each other, since there is enough room to pass the two strands across each other.…”