84
Topological defects
continuously defonned to the trivial configuration. Thus, magnetic monopole
configurations are stabilized in a topological way.
The topological quantum number N is related to the magnetic charge. This
can be demonstrated as follows. Denote the elements of surface and the unit
nonnal on the sphere in coordinate space by dS and ii. Then
-_ (a, a,)
dS n = - x - du dv
(3.103)
au
av
from which it follows that
a('Jt rk) du dv.
€;jk dS
a(lI, v)
Noting that
a. x a. = ~ (a_ x a_) a(rj"k)
(3.105)
au
av
2 a'j
a'k
a(u, v)
and using (3.103), the topological quantum number N may be recast in tenns of
coordinate-space derivatives of _ as
3
= I ~ _ (
41rrl N
-2
€jjknj. . -a a. x - a. ) ­ dS. (3.106)
E
'l
ark
The integral (3.106) may be fonnulated as a surface integral of.· F jk on a sphere
1; of large radius, where the Fjk are the spatial components of the gauge field
strength defined in (3.70). For this purpose, we need a solution for F jk in tenns
of _ valid for large,. As discussed in section 3.7, for a finite-energy solution
there is a cancellation between the two tenns in the covariant derivative
Dj. = aj. - g(A; x _)
(3.107)
such that the covariant derivative is of order ,-2 for, -+ 00, whereas, separately,
the two tenns are of order' -I. Thus, for large r,
aj. ~ g(Aj x _)
(3.108)
from which it follows that
1
Aj = - 2 (. x a;.) +
1
laj.
(3.109)
g"
"
where
aj =.·Aj.
(3.110)
The corresponding expression
-2.·
for _ . F lA: obtained from (3.70) is
1
•. FjA: =
(aj. x at.) + (ajak - akaj).
(3.111)
g"
nj =
(3.104)