(ed.). Gravitational waves (IOP, 2001)(422s).

Nonlinear σ -models 253Example: G = SL(2, R), H= SO(2)The generators of the group are(Y 1 1 0=0 −1), Y 2 =( )0 1, Y 3 =1 0( )0 1. (14.59)−1 0We haveTr(Y 1 ) 2 = Tr(Y 2 ) 2 =−Tr(Y 3 ) 2 = 2 (14.60)so Y 1 and Y 2 are the non-compact generators, while Y 3 generates the SO(2)subgroup.The group can be decomposed on its generators, asH = RY 3 , K = RY 1 ⊕ RY 2 . (14.61)Let us introduce now an element of the group v(x) ∈ G with the propertyv(x) → v ′ (x) = g −1 v(x)h(x) (14.62)where g is a rigid G transformation and h(x) a local H transformation. This typeof transformation is needed for the necessity of preserving gauge choice. In fact,you can fix the gauge choosing a particular element of the group v. Then, whenyou act on v by an arbitrary g, that gauge choice will be lost. To restore the gaugeyou have to introduce the local transformation h(x) so that the rotation g can becompensed. It follows that h does not depend only on the coordinates x, but alsoon the vector v and the rotation g.Therefore, equation (14.62) is called a nonlinear realization of symmetries,because h depends nonlinearly on v.This is important for the following calculation, because we can fix a gauge,called triangular gauge, such thatv(x) = exp ϕ(x), ϕ(x) ∈ K → v ∈ G/H. (14.63)The next step is the construction of a Lagrangian with the required symmetry. Tothis aim, let us consider the Lie algebra valued expressionv −1 ∂ m v = Q m + P m , Q m ∈ H, P m ∈ K (14.64)or equivalentlyv −1 D m v = v −1 (∂ m v − vQ m ) = P m (14.65)which defines the H -covariant derivative D m . It is straightforward to verify thatQ m transforms like a gauge field with respect to the local group H , namelyQ ′ m = h−1 Q m h + h −1 ∂ m h and that P m ′ = h−1 P m h. The formula (14.62) impliesthe integrability relations∂ m Q m − ∂ n Q m + [Q m , Q n ] = −[P m , P n ]D m P n − D n P m = 0.