58
N. Crampé and A. M. Grundland
1 The CP 2s Sigma Model
The dynamical fields in the CP 2s sigma models are maps from the Riemann sphere
S 2 to the complex projective space CP 2s S 4s(s+1) /U (1)
S
2
ξ ± = ξ
1
± iξ
2
→ z = (z 0 , z 1 , . . . , z 2s ) ∈ C
2s+1
\ {∅},
(where the value of the index s is either an integer or half-integer) which are
stationary points of the action functional [1]
A =
1
4
S 2
(D μ z)
†
· (D μ z)dξ + dξ − ,
(1)
and hence are solutions of the Euler–Lagrange (EL) equations
D μ D μ z + (D μ z)
†
· (D μ z)z = 0,
(2)
subjected to z † z = 1, where D μ are the covariant derivatives defined by
D μ z = ∂ μ z − (z
† ∂ μ z)z,
∂ μ =
∂
∂ξ μ ,
μ= 1, 2.
We require that the action (1) over the whole Riemann sphere S 2 be finite.
2 Projective Formalism
Equivalently, representing the z’s by their homogeneous representatives, i.e. maps
into C 2s+1 \ {∅}
z =
f
(f † · f ) 1/2 ,
we may use (fields of) rank-1 Hermitian projectors
P =
f ⊗ f †
f † · f
,
P
2
= P ,
P
†
= P .
(3)
This places the EL equations in the form of the conservation law (CL)
∂[ ¯
∂P , P ] + ¯
∂[∂P , P ] = 0,
(4)