THE LARGE N EXPANSION
8.3 The
-Model
1 39
Being unable to proceed further with principal chiral fields, we consider
in this section another interesting model, namely the field which
belongs to the coset space CP^~^ = SU(N)/SU(N - 1)® U(l). It
resembles the n-field in the respect that the large N expansion in this
case is easy, with no planar graphs arising. On the other hand the model
is topologically nontrivial and contains instantons. This gives the
possibility to analyse effects of topological charge quantitatively. Another remarkable feature of this model, as we shall see, is the dynamical
generation of gauge fields.
Complex projective space CP^~^ is defined by taking A-dimensional
complex space and identifying in it the point
..., z^) with the point
(Azj,...,
where A 7^ 0 is an arbitrary complex number. By choice of
k we can parametrize the points of CP^~^ by a unit sphere:
with the identification
(8.53)
(8.54)
The complex dimensionality of the resulting space is N — 1.
The Lagrangian of the z(x)-field must be invariant under the gauge
group:
z{x)
(8.55)
Only in this case does it describe a field belonging to CP^~^ and not to
the A-dimensional complex sphere (8.53) (which would be the same as
the 2A-dimensional «-field). Such a Lagrangian can be written as
5 = i
^0 J
with
being a new independent field which transforms as
(8.56)
(8.57)
It is quite obvious that (8.56) is invariant under simultaneous transformations (8.55) and (8.57). Also, since (8.56) does not contain a kinetic
term for the A^-fidd this field can be eliminated (at least classically) by
minimizing S. We have:
ss
s
- (d^z*)z)Ai, + Al) = 0
(8.58)