92
GAUGE FIELDS AND STRINGS
where
and
is a constant field, invariant under H:
=
for heH
(6.24)
The matrix Qati^) in (6.23) need not be continuous. Let us consider a set
of matrices g^^\x) defined in the northern hemisphere and ^^^^(x)
defined in the southern one. Suppose on the equator we have the
relation:
g^^\x) = g^^\x) • h(x) x e equator =
heH
(6.25)
We see from (6.23) and (6.24) that in spite of the discontinuity in ^(x)
the field cPaix) is continuous and defines a map
G/H. From (6.25)
we deduce that these maps can be classified according to the maps of
the equator to H:
H. If H = U{\) this is just a 5^
map,
classified by the winding number. If H = 1/(1) x something, then we
can map
onto the first factor in H. In mathematical notation the
statement we have proved is written as:
7i2(G/H)^n^(H) if 712(G) = 0
(6.26)
(where 7r^(M) is the kth homotopy group, elements of which are classes
of nontrivial maps of -► M).
The most familiar example of a chiral theory with instanton structure
is the so-called
-model where the field belongs to the complex
projective space:
(/>6 CP^ N -I _
SU(N)
SU{N - I )® U (l)
(6.27)
(The case N = 2 is the 0(3) /i-field.) There are some interesting
dynamics in this model. We shall discuss it in Chapter 8.
6.2 Instantons in Non-Abeiian Gauge Theories
Non-Abelian gauge theories with any symmetry group G possess
topologically nontrivial fields. This can be seen from the following
consideration. In order that the Yang-Mills action be finite one must
require that:
P ,,(x ) - 0(1/x^)
(6.28)