98
GAUGE FIELDS AND STRINGS
(here
rj^i^Q =
etc.). The origin of this decomposition is
based on a check that the commutation relations:
^jaP jyd-^ _ ^ctyjPS _|_ ^pSjay
_ ^a^jpy _ ^Pyjoi6
get decomposed as:
=
[ x ; , x ; ] = o
for
2^2^abc^be i ^ao)
We conclude from (6.52) and (6.53) that the ansatz:
(6.54)
(6.55)
is bound to be compatible with the duality equation. The substitution
of (6.55) into (6.46) gives after some simple calculations:
Alix) =
2y?a^v(^v - ^v)
(jc - ay +
K M =
(6.56)
{{x-af + p y
with arbitrary scale parameter p and position parameter a^.
This Non-Abelian instanton can be viewed as a magnetic dipole of
size p. If we consider now the contribution of one instanton to the
partition function Z we find, just as in case of n-field, several factors.
First of all we have a factor e “^*= ^ =
which gets replaced, after
taking account of the one loop correction by
where, for
SU(N)
is an effective coupling for the size p. The contribution has to be
integrated over p and a. The measure must be both scale and translationally invariant. The only combination with these properties is
dp p~^. We find from this consideration:
7(1) ,
-^INST
y
g - 8«2/g2(p)
lN/3
(6.58)
(F is the 4-volume).
As happened in the case of the /i-field, the instanton contribution has
an infrared divergence. This implies that in the multi-instanton picture.