Physical Foundations of Cosmology

220 The very early universe

far away from it. Strings are topologically stable, classical solutions of the field
equations.
In the language of homotopy groups, strings correspond to the mappings of a
circleS^1 to the vacuum manifoldM=S^1. The corresponding group is nontrivial:
π 1

(

S^1

)

=Z,whereZis the group of integers under addition. Taking a circleγ(τ)
inxspace, let us consider its map toM:φ(τ)=σexp(iθ(τ)). Because the complex
fieldφis an unambiguous function of the spatial coordinatesx,the phaseθchanges
by 2πmaround the circleγ,wheremis an integer. Ifm= 0 ,the mapping is trivial
and the contourγcan be continuously deformed to a point without passing through
the region of false vacuum. Hence it does not contain any topological defects. The
map withm=0 wraps the circleγaround the vacuum manifoldmtimes. Form= 1
there is a string inside the contourγ. In fact, considering two points withθ= 0
andθ=π, whereφis equal toσand−σrespectively, and shrinking the contour,
one necessarily arrives at a place where the fieldφvanishes because otherwise it
would have infinite derivative. This is where the string lives.
Let us take for simplicity a straight global string (no gauge fields are present)
and consider its energy per unit length. At a large distancerfrom the core of the
string, the derivative∂iφcan be estimated on dimensional grounds asσ/r.Hence
the gradient term gives a logarithmically divergent contribution to the energy per
unit length:

μs∼σ^2

∫(

1 /r^2

)

d^2 x. (4.234)

In this case the natural regularization factor is the distance to the nearest string.
As we have seen, axions assume a globalUPQ( 1 )symmetry and therefore global
axionic strings can exist.
In a theory with local gauge invariance the derivative∂iφis replaced byDiφ=
∂iφ+ieAiφ; the properties of thelocalstrings are very different from those of
global strings. Solving the coupled system of equations forφandAi,one can show
that the gauge field compensates the leading term in∂iφand that the covariant
derivativeDiφdecays faster than 1/rasr→∞.As a result the energy per unit
length converges. Writingφ=χexp(iθ),and assuming that∂iχdecays faster than
1 /r,we find that the compensation takes place only if

Ai→( 1 /e)∂iθ

asr→∞.Taking a contourγ,far away from the core of the string, and calculating
the flux of the magnetic field(B=∇×A)we immediately find
∫
Bd^2 σ=

∮

Aidxi=

2 π

e

m. (4.235)