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Baryogenesis
as Ix I ..... 00. Then U effectively maps the S3 of real space. obtained by
identifying points at infinity. into the S3 which is the SU (2) group space [36].
Such mappings are characterized by their homotopy class. Since
1r3(S3) = Z.
(4.147)
where Z is the group of integers. any such mapping is associated with an integer
which counts the number of times the spacetime S3 is wrapped around the (unit)
S3 of the internal SU(2) space.
Now. the field strength W;~c constructed from w;ac is zero but the current
K JI. (4.137) is no longer zero:
KJl.vac = _ 4ig2 fJl.\lfXI tr[WvacWvacWvac]
3
\J
(4.148)
P
a .
So
KOvac = g2 fOij/cfabc(W~vacW~vacW/ccvac)
3
I
J
= - 2g2 det W vac •
(4.149)
Also
det w vac = J det G
(4.150)
where
Gij = Wrac. Wj8C = ~Yij
(4.151)
and Yij is the metric (4.146) on S3 for the spatial coordinates. Thus. if we define
K == / d 3 x KOvac
(4.152)
we have that
K = - 2' 16/ d 3 xJdety
g2
= - ~ { dV
g~ ls3
321r 2
= ---n
(4.153)
g~
where n is an integer which counts the number of wrappings of the internal S3
provided by the mapping U; the volume of the unit S3 is 21r2. For a general
(non-vacuum) field configuration.
321r 2
K=-2- Ncs
(4.154)
g2