Local cosmic strings
71
Like a domain wall, a (global) cosmic string is stabilized by topological
considerations. A cosmic string with asymptotic behaviour e in9 as p _ 00 is
said to have winding number n. The space of vacuum states (minima of V) is
characterized by e ifJ (as in (3.14» and so is just a circle SI. If (for fixed z) we
draw a circular path of large radius in real space encircling the core of the cosmic
string (where tP = 0), then, as we go once around this path in real space, the field
tP goes n times around the circle S I which is the space of vacuum states. Provided
that the cosmic string is either of infinite length or forms a closed loop. this is a
property of a cosmic string that cannot be changed by continuous deformations. It
is a topological quantum number which, at least at the classical level, guarantees
the continued existence of a vortex line once formed, unless it encounters other
vortex lines or divides into more vortex lines in such a way that n is conserved
(e.g. into n vortex lines with unit winding number.)
If we denote the space of true vacua (minima of V) by M. then the
topological entity involved is the homotopy group Jrl (M). In the present case,
the relevant homotopy group is Jrl (SI) which is known to be Z, i.e. isomorphic
to the integers. The winding number n E Z expresses this fact.
3.4 Local cosmic strings
If a complex scalar field is coupled to a gauge field, e.g. the electromagnetic field,
then the Lagrangian possesses a local symmetry, rather than a global symmetry
as in section 3.3, and a so-called 'local cosmic string' [41 or gauge string can
occur as a solution of the field equations. The simplest example is provided by
the Higgs model, which is the theory of a complex scalar field coupled to a U (I)
gauge field which we may take to be the electromagnetic field. The Lagrangian is
that of section 3.3 amended to incorporate the gauge coupling. Thus,
C = (D",tP)*(D"'tP) - !F",uF"'U - V(tP)
(3.23)
with V (tP) given by (3.12) and
D",tP == (a", + ieA",)tP
(3.24)
F",v = a",A v - avA",
(3.25)
where A", is the electromagnetic four-potential and e is the charge of the scalar
field. This is the same model as that studied in section 2.4. after adding a constant
to V and with a different definition of A.
As in the discussion of global cosmic strings, V has a minimum at tP = F1e ifJ •
However. the Lagrangian now possesses a local U ( I) symmetry under
tP _ eiA(x)tP
(3.26)
1
A", - A", - -a",A(x)
(3.27)
e
71
Like a domain wall, a (global) cosmic string is stabilized by topological
considerations. A cosmic string with asymptotic behaviour e in9 as p _ 00 is
said to have winding number n. The space of vacuum states (minima of V) is
characterized by e ifJ (as in (3.14» and so is just a circle SI. If (for fixed z) we
draw a circular path of large radius in real space encircling the core of the cosmic
string (where tP = 0), then, as we go once around this path in real space, the field
tP goes n times around the circle S I which is the space of vacuum states. Provided
that the cosmic string is either of infinite length or forms a closed loop. this is a
property of a cosmic string that cannot be changed by continuous deformations. It
is a topological quantum number which, at least at the classical level, guarantees
the continued existence of a vortex line once formed, unless it encounters other
vortex lines or divides into more vortex lines in such a way that n is conserved
(e.g. into n vortex lines with unit winding number.)
If we denote the space of true vacua (minima of V) by M. then the
topological entity involved is the homotopy group Jrl (M). In the present case,
the relevant homotopy group is Jrl (SI) which is known to be Z, i.e. isomorphic
to the integers. The winding number n E Z expresses this fact.
3.4 Local cosmic strings
If a complex scalar field is coupled to a gauge field, e.g. the electromagnetic field,
then the Lagrangian possesses a local symmetry, rather than a global symmetry
as in section 3.3, and a so-called 'local cosmic string' [41 or gauge string can
occur as a solution of the field equations. The simplest example is provided by
the Higgs model, which is the theory of a complex scalar field coupled to a U (I)
gauge field which we may take to be the electromagnetic field. The Lagrangian is
that of section 3.3 amended to incorporate the gauge coupling. Thus,
C = (D",tP)*(D"'tP) - !F",uF"'U - V(tP)
(3.23)
with V (tP) given by (3.12) and
D",tP == (a", + ieA",)tP
(3.24)
F",v = a",A v - avA",
(3.25)
where A", is the electromagnetic four-potential and e is the charge of the scalar
field. This is the same model as that studied in section 2.4. after adding a constant
to V and with a different definition of A.
As in the discussion of global cosmic strings, V has a minimum at tP = F1e ifJ •
However. the Lagrangian now possesses a local U ( I) symmetry under
tP _ eiA(x)tP
(3.26)
1
A", - A", - -a",A(x)
(3.27)
e