Topological aspects of spin and statistics in nonlinear sigma models

Baez, John C.; Ody, Michael S.; Richter, Wiliam

doi:10.1063/1.531304

Topological aspects of spin and statistics in nonlinear sigma models

May, 1992

14 pages

Published in:

J.Math.Phys. 36 (1995) 247-257

e-Print:

cond-mat/9208005 [cond-mat]

DOI:

10.1063/1.531304

Report number:

PRINT-92-0332 (WELLESLEY)

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Abstract: (arXiv)

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

pi_d(M)

. Any topological soliton

\alpha \in \pi_d(M)

determines a quotient \Stat_n(X,\alpha) of the group of framed braids on

X

, such that choices of allowed statistics for solitons of type

\alpha

are given by unitary representations of \Stat_n(X,\alpha) when

n

solitons are present. In particular, when

M = S~2

, as in the

O(3)

nonlinear sigma model with Hopf term, and

\alpha \in \pi_2(S~2)

is a generator, we compute that \Stat_n(\R~2,\alpha) = \Z, while \Stat_n(S~2,\alpha) = \Z_{2n}. It follows that phase

\exp(i\theta)

for interchanging two solitons of type

\alpha

on

S~2

must satisfy the constraint

\theta = k\pi/n

,

k \in \Z

, when

n

such solitons are present.

sigma model: nonlinear
topology
spin: statistics
statistics: spin
field equations: soliton
soliton: topological
Hopf term
braid group
bibliography

References(31)

Figures(0)

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