Magnetic monopoles in grand unified theories
85
Combining (3.106) with (3.111), the topological quantum number N may be
written in terms of the field strength as
3
g'1
4H'1 N = T 21 1: fijkni.· Fjk dS.
(3.112)
Finally. N may be recast as a surface integral of the magnetic field on a sphere
1: of large radius. As discussed in section 3.7, the magnetic field in the model
with SO(3) gauge group should be identified with the component Br in (3.96)
along the direction about which the surviving U (I) gauge symmetry is the group
of rotations. In general. this is the direction 1.1- 1 •• Consequently, the magnetic
field B is given by
I
Bi = -€ijk • . F jk.
(3.113)
2'1
It follows from (3.112) that
N = - g 1 B ·iidS.
4H 1:
-
(3.114)
Thus, the topological quantum number N measures the magnetic charge in units
of 4H/g.
3.9 Magnetic mono poles in grand unified theories
In general, if we start with a grand unified group G and the symmetry is
spontaneously broken to H (at a phase transition), then the action of any element
of H leaves a vacuum state invariant. Consequently. distinct vacuum states
correspond to the coset manifold G / H. The logic behind this is that G invariance
of the effective potential means that starting from any vacuum state. we can
generate further vacuum states (degenerate in energy) by acting with elements
of G. However. when the element of G in question is an element of the subgroup
H, it does not produce a new vacuum state. The topological entity underlying
the existence of stable magnetic monopoles is the second homotopy group for
G / H denoted by H2 (G / H), whose elements are inequivalent mappings from the
surface of a two-sphere S2 to G / H, i.e. mappings which cannot be continuously
deformed into each other. There is a theorem that H2 (G / H) can be identified with
HI (H)/H)(G). Here, HI (G) is the first homotopy group of G whose elements are
inequivalent mappings from a circle S I to G and similarly for H.
In the example just considered. G = SO(3) and H =
(1»
V(I). Also
1rt (U
= Z, the integers, with the value of the integer being the winding
number, i.e. the number of times we wind around the circle defined by U (I) as we
wind once around the circle in coordinate space. Less obviously, HI (SO(3» =
Z2, the integers modulo 2. In this case, therefore. H2(G/ H) = Z/Z2 or the even
integers. This is why we found magnetic charges in multiples of 4H / g which is
twice the Dirac magnetic monopole charge.
85
Combining (3.106) with (3.111), the topological quantum number N may be
written in terms of the field strength as
3
g'1
4H'1 N = T 21 1: fijkni.· Fjk dS.
(3.112)
Finally. N may be recast as a surface integral of the magnetic field on a sphere
1: of large radius. As discussed in section 3.7, the magnetic field in the model
with SO(3) gauge group should be identified with the component Br in (3.96)
along the direction about which the surviving U (I) gauge symmetry is the group
of rotations. In general. this is the direction 1.1- 1 •• Consequently, the magnetic
field B is given by
I
Bi = -€ijk • . F jk.
(3.113)
2'1
It follows from (3.112) that
N = - g 1 B ·iidS.
4H 1:
-
(3.114)
Thus, the topological quantum number N measures the magnetic charge in units
of 4H/g.
3.9 Magnetic mono poles in grand unified theories
In general, if we start with a grand unified group G and the symmetry is
spontaneously broken to H (at a phase transition), then the action of any element
of H leaves a vacuum state invariant. Consequently. distinct vacuum states
correspond to the coset manifold G / H. The logic behind this is that G invariance
of the effective potential means that starting from any vacuum state. we can
generate further vacuum states (degenerate in energy) by acting with elements
of G. However. when the element of G in question is an element of the subgroup
H, it does not produce a new vacuum state. The topological entity underlying
the existence of stable magnetic monopoles is the second homotopy group for
G / H denoted by H2 (G / H), whose elements are inequivalent mappings from the
surface of a two-sphere S2 to G / H, i.e. mappings which cannot be continuously
deformed into each other. There is a theorem that H2 (G / H) can be identified with
HI (H)/H)(G). Here, HI (G) is the first homotopy group of G whose elements are
inequivalent mappings from a circle S I to G and similarly for H.
In the example just considered. G = SO(3) and H =
(1»
V(I). Also
1rt (U
= Z, the integers, with the value of the integer being the winding
number, i.e. the number of times we wind around the circle defined by U (I) as we
wind once around the circle in coordinate space. Less obviously, HI (SO(3» =
Z2, the integers modulo 2. In this case, therefore. H2(G/ H) = Z/Z2 or the even
integers. This is why we found magnetic charges in multiples of 4H / g which is
twice the Dirac magnetic monopole charge.