# Why are some solitons formed from bosonic fields fermionic?

Some topological solitons formed from bosonic fields have fermionic statistics. Why?

-

The fermionic statistics has to do with a minus sign acquired when two objects are interchanged. Usually this is unambiguously attributed to statistics, as opposed to interaction between the objects, since any interaction falls off rapidly when they are far apart, but the relative phase acquired due to interchange is the same whether they are close or far.

However, in low dimensional systems the interaction strength between well-separated objects does not always fall off very fast (or not at all). So, one can get fermionic (or more generally anyonic) behaviour from interaction. For some solitons the interactions is such that they acquire a phase when interchanged, no matter how far apart they are.

This arguments only shows it can happen in low dimensional systems, more detailed arguments will show you how it happens (for example google sine-Gordon and Thirring models, or bosonization). The why questions I am never sure how to answer.

-
Additional comment is that even in 4d, the WZW term can make a soliton constructed from scalar fields alone fermionic; the skyrmion becoming a baryon when the number of color is odd is the typical example. In a sense the Lagrangian in this case has a term which produces long-range phase. –  Yuji Feb 25 '11 at 6:39
Great point, thanks! –  user566 Feb 25 '11 at 6:43

There are many examples. Moshe focused on lower-dimensional theories; for example, in 1+1 dimensions, bosonic and fermionic conformal field theories are typically totally equivalent to each other (and one may get fermions as kinks of the bosons, and bosons as bilinear currents of the fermions). I will focus on higher-dimensional theories.

I don't really know how to get fermions in the $D=26$ bosonic string theory. But in other, $D=10$ string theories that normally contain bosons only, fermions may be obtained - see e.g. this paper by Justin David, Shiraz Minwalla, and Carlos Nunez (whose writing I remember pretty well, back from Santa Barbara 2001 etc.)

http://arxiv.org/abs/hep-th/0107165

The basic trick, known already in 3+1-dimensional bosonic gauge theory (by Jackiw-Rebbi-Hasenfratz-’t Hooft), is "spin from isospin".

The soliton is only invariant under the "diagonal group" of the rotations and some internal isospin group. Noether's theorem dictates that the right definition of the spin of the excitations is associated with whatever symmetry we have. In this case we have the diagonal symmetry, so the new conserved angular momentum will be the diagonal one. Because half-integer-isospin excitations are possible, they will look like half-integer-spin excitations of the soliton.

-