CHAPTER 6
Topology of Gauge Fields
and Related Problems
We have seen in the previous chapters, that in Abelian systems the
problem of charge confinement is solved by instantons.
In Non-Abelian theories instanton solutions are also present. However, due to the large perturbative fluctuations, discussed in Chapter 2,
it is difficult to judge whether they play a decisive role in forming a mass
gap and a confining regime. In such theories we have a kind of
instanton liquid which is difficult to treat. It is possible that due to some
hidden symmetries, present in these systems, instantons may form a
useful set of variables for an exact description of the system, but this has
not yet been shown.
At the same time, due to the fact that instantons carry nontrivial
topology (they describe configurations of the fields which can not be
“disentangled”), some manifestations of instantons cannot be mixed up
with perturbative fluctuations.
In this chapter we shall analyse topological properties of (mainly)
nonabelian instantons and solitons and discuss some associated
peculiar effects.
6.1 Instantons for ^ = 2, = 3 /i-Fields
Let us find minima of the classical action for the /i-field in the case
N = 3. In order that this action be finite, we have to consider a
boundary condition:
if(x) -> /to
(6.1)
Therefore, since infinity can be viewed as one point, our x-space is
topologically a sphere. Each configuration n(x) defines a map of such a
sphere in x-space onto the sphere /i^ = 1, which in the case N = 3 gives
85
DOI: 10.1201/9780203755082-6
Topology of Gauge Fields
and Related Problems
We have seen in the previous chapters, that in Abelian systems the
problem of charge confinement is solved by instantons.
In Non-Abelian theories instanton solutions are also present. However, due to the large perturbative fluctuations, discussed in Chapter 2,
it is difficult to judge whether they play a decisive role in forming a mass
gap and a confining regime. In such theories we have a kind of
instanton liquid which is difficult to treat. It is possible that due to some
hidden symmetries, present in these systems, instantons may form a
useful set of variables for an exact description of the system, but this has
not yet been shown.
At the same time, due to the fact that instantons carry nontrivial
topology (they describe configurations of the fields which can not be
“disentangled”), some manifestations of instantons cannot be mixed up
with perturbative fluctuations.
In this chapter we shall analyse topological properties of (mainly)
nonabelian instantons and solitons and discuss some associated
peculiar effects.
6.1 Instantons for ^ = 2, = 3 /i-Fields
Let us find minima of the classical action for the /i-field in the case
N = 3. In order that this action be finite, we have to consider a
boundary condition:
if(x) -> /to
(6.1)
Therefore, since infinity can be viewed as one point, our x-space is
topologically a sphere. Each configuration n(x) defines a map of such a
sphere in x-space onto the sphere /i^ = 1, which in the case N = 3 gives
85
DOI: 10.1201/9780203755082-6