Physical Foundations of Cosmology

4.5 Instantons, sphalerons and the early universe 191

t

r

φ→φ 0 φ=const

φ→ 0

Fig. 4.15.

In particular, taking an arbitrary time-independent unitary matrixU(x),we find
that

A 0 = 0 ,Ai=

i

g

(∂iU)U−^1 (4.173)

also describes the vacuum. If an arbitraryU(x)could be continuously transformed
to the unit matrixeverywhere in spaceall vacua would be equivalent. However
this is not the case. Instead the set of all functionsU(x)can be decomposed into
homotopy classes. We say that two functions belong to the same homotopy class if
there exists a nonsingular continuous transformation relating them; otherwise the
functions belong to different homotopy classes.

Winding numberTo find and characterize the homotopy classes let us introduce
thewinding number, defined as

ν≡−

1

24 π^2

∫

tr

(

εijk(∂iU)U−^1

(

∂jU

)

U−^1 (∂kU)U−^1

)

d^3 x, (4.174)

whereεijkis a totally antisymmetric Levi–Civita symbol:ε^123 =1 and it changes
sign upon permutation of any two indices. We will show first that this number is a
topological invariant and second that it takes integer values characterizing different
homotopy classes. To prove the first statement let us consider a small nonsingular
variationU→U+δU, and show thatδν= 0 .Taking into account that

(δU)U−^1 =−U

(

δU−^1

)