TOPOLOGY OF GAUGE FIELDS AND RELATED PROBLEMS
97
will also be satisfied (the converse is not true). To check this, let us
differentiate (6.46):
^ 0
(6.48)
(where the last equality is a consequence of the Bianichi identity:
+ ^P^yoL F ^y^ap “ ^
if ^ap ~ ^a^p ~ ^p^a F
^^])
(6.49)
Notice by the way that for constant
this reduces to the Jacobi
identity:
lAp, Ay']'] + lAp, [Ay, AJ] + lAy,
Ap]] = 0
(6.50)
We see that the “duality” equations (6.46) are in some sense a
four dimensional analogue of the Cauchy-Riemann equations (6.9).
Their most surprising property is that they possess multi-instanton
solutions. Before discussing them let us present a solution with q = i.
The ansatz for this solution can be found by the following trick.
Let us consider instead of the gauge group S'1/(2), a group St/(2) ®
SC/(2) ^ 0(4). Then equations (6.46) will have the symmetry group
0(4) (X) 0(4), where the first factor is space rotations, and the second
isotopic rotations. We shall be looking for a solution which breaks
0(4)® 0(4) but preserves the single 0(4) formed by simultaneous
rotations in jc-space and isotopic space. After that we shall return to
S0(2).
Generators of 0(4) are described by matrices
skew-symmetric in
(a, j8), which represent a rotation in the (a, jS)-plane. Therefore, gauge
fields for this group also have these indices:
A , =
(X)
The most general 0(4)-symmetric ansatz is given by:
Afix) =
(r^ =
(6.51)
(6.52)
On symmetry grounds it must be compatible with the Yang-Mills and
duality equations. The six fields Al^ can be split into two sets, each of
three fields, and
each corresponding to an SU(2). This splitting is
described by:
a; = ^(Af +
= kri^fAl^
(6.53)