**1.5. Conditions for their existence: topological criteria**

Let us now explore the conditions for the existence of topological
defects. It is widely accepted that the final goal of particle physics
is to provide a unified gauge theory comprising strong, weak and
electromagnetic interactions (and some day also gravitation). This
unified theory is to describe the physics at very high temperatures,
when the age of the universe was slightly bigger than the Planck time.
At this stage, the universe was in a state with the highest possible
symmetry, described by a symmetry group **G**, and the Lagrangian
modeling the system of all possible particles and interactions present
should be invariant under the action of the elements of **G**.

As we explained before, the form of the finite temperature effective
potential of the system is subject to variations during the cooling
down evolution of the universe. This leads to a chain of phase
transitions whereby some of the symmetries present in the beginning
are not present anymore at lower temperatures. The first of these
transitions may be described as **G**
**H**,
where now **H**
stands for the new (smaller) unbroken symmetry group ruling the
system. This chain of symmetry breakdowns eventually ends up with
SU(3) × SU(2) × U(1), the symmetry group underlying the
`standard model' of particle physics.

A broken symmetry system (with a Mexican-hat potential for the Higgs
field) may have many different minima (with the same energy), all related
by the underlying symmetry. Passing from one minimum to another is
included as one of the symmetries of the original group **G**, and
the system will not change due to one such transformation. If a
certain field configuration yields the lowest energy state of the
system, transformations of this configuration by the elements of the
symmetry group will also give the lowest energy state. For example,
if a spherically symmetric system has a certain lowest energy value,
this value will not change if the system is rotated.

The system will try to minimize its energy and will spontaneously
choose one amongst the available minima. Once this is done and the
phase transition achieved, the system is no longer ruled by **G**
but by the symmetries of the smaller group **H**. So, if **G**
**H** and the system is in one of the lowest energy states
(call it *S*_{1}), transformations of *S*_{1}
to *S*_{2} by elements of **G** will leave the energy
unchanged. However, transformations of *S*_{1}
by elements of **H** will leave *S*_{1} *itself*
(and not just the
energy) unchanged. The many distinct ground states of the system
*S*_{1}, *S*_{2} , ... are given by all
transformations of **G** that are
*not* related by elements in **H**. This space of distinct
ground states is called the *vacuum manifold* and denoted
.

is the space of all elements ofGin which elements related by transformations inHhave been identified. Mathematicians call it thecoset spaceand denote itG/H. We then have =G/H.

The importance of the study of the vacuum manifold lies in the fact
that it is precisely the *topology* of
what determines the
type of defect that will arise. Homotopy theory tells us how to map
into physical space
in a non-trivial way, and what ensuing
defect will be produced. For instance, the existence of non
contractible loops in
is the requisite for
the formation of
cosmic strings. In formal language this comes about whenever we have
the first homotopy group
_{1}
()
**1**, where **
1** corresponds to the trivial group. If the vacuum manifold is
disconnected we then have
_{0}
()
**1**, and domain
walls are predicted to form in the boundary of these regions where the
field is away from the
minimum of the potential. Analogously,
if _{2}
()
**1** it follows that
the vacuum manifold
contains non contractible two-spheres, and the ensuing defect is a
monopole. Textures arise when
contains non contractible
three-spheres and in this case it is the third homotopy group,
_{3}(), the one that is non
trivial. We summarize this in Table 1.1 .

_{0}
()
1 |
disconnected
| Domain Walls |

_{1}
()
1 |
non contractible loops in
| Cosmic Strings |

_{2}
()
1 |
non contractible 2-spheres in
| Monopoles |

_{3}
()
1 |
non contractible 3-spheres in
| Textures |