TOPOLOGY OF GAUGE FIELDS AND RELATED PROBLEMS
87
the so-called duality equation. If (6.7) is satisfied then, owing to (6.6),
the action has the absolute minimum value equal to 4nq/el.
Solution of (6.7) is easy. We introduce a complex field w by a
stereographic projection;
«1 -h i«2 = 2w/(l -f lw|^)
n^ = l
(6.8)
The substitution of (6.8) into (6.7) reduces this equation to:
d^w = (di -h i^2)^ =
(6.9)
Therefore (6.7) are just Cauchy-Riemann equations for the function
w. This function must be not only analytic, but also meromorphic, since
otherwise n would have branch cuts. Hence the most general solution
has the form:
w(z) =
/= 1 2 - h
(6.10)
where we have normalized w by w(oo) = 1 , corresponding to n(oo)
pointing in the x-direction. The integer q in (6.10) is just the topological
number. This can be seen without explicit computation since it is clear
from (6.10) that the inverse function z = z(w) is ^-valued. That implies
that the w-sphere is covered q times by the z-sphere. Of course, it is also
not difficult to substitute (6.10) into (6.4) and to compute q explicitly.
The remarkable thing about the instantons (6.10) is that they do not
interact classically. There are, however, anti-instantons
= n z — b,
(6.11)
having negative topological charge. The mixed configuration of instantons and anti-instantons is not a strict classical solution (as it was for
kinks and anti-kinks in Chapter 4) and a dipole-dipole like interaction
is present. Notice also, that the instantons (6.10) have a natural
structure of dipoles with poles placed at Uj and bj.
As we turn from this nice, clean classical mathematics to functional
integrals we encounter a difficulty. Namely, while in the zeroth order
approximation the one instanton contribution is proportional to
6 “^*= ^ =
it is quite obvious that quantum fluctuations should
renormalize the bare coupling
The resulting contribution can be
found without explicit computation by the following argument. As we
see from (6.10), the effective size of the instanton with parameters a and
87
the so-called duality equation. If (6.7) is satisfied then, owing to (6.6),
the action has the absolute minimum value equal to 4nq/el.
Solution of (6.7) is easy. We introduce a complex field w by a
stereographic projection;
«1 -h i«2 = 2w/(l -f lw|^)
n^ = l
(6.8)
The substitution of (6.8) into (6.7) reduces this equation to:
d^w = (di -h i^2)^ =
(6.9)
Therefore (6.7) are just Cauchy-Riemann equations for the function
w. This function must be not only analytic, but also meromorphic, since
otherwise n would have branch cuts. Hence the most general solution
has the form:
w(z) =
/= 1 2 - h
(6.10)
where we have normalized w by w(oo) = 1 , corresponding to n(oo)
pointing in the x-direction. The integer q in (6.10) is just the topological
number. This can be seen without explicit computation since it is clear
from (6.10) that the inverse function z = z(w) is ^-valued. That implies
that the w-sphere is covered q times by the z-sphere. Of course, it is also
not difficult to substitute (6.10) into (6.4) and to compute q explicitly.
The remarkable thing about the instantons (6.10) is that they do not
interact classically. There are, however, anti-instantons
= n z — b,
(6.11)
having negative topological charge. The mixed configuration of instantons and anti-instantons is not a strict classical solution (as it was for
kinks and anti-kinks in Chapter 4) and a dipole-dipole like interaction
is present. Notice also, that the instantons (6.10) have a natural
structure of dipoles with poles placed at Uj and bj.
As we turn from this nice, clean classical mathematics to functional
integrals we encounter a difficulty. Namely, while in the zeroth order
approximation the one instanton contribution is proportional to
6 “^*= ^ =
it is quite obvious that quantum fluctuations should
renormalize the bare coupling
The resulting contribution can be
found without explicit computation by the following argument. As we
see from (6.10), the effective size of the instanton with parameters a and