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

252 Infinite-dimensional symmetries in gravitywe get a new 2D LagrangianÄ MM =− 1 2 λ−1 ∂ µ λ∂ µ ρ + 1 8 ρ−2 ∂ µ ∂ µ + 1 8 ρ−1 2 ∂ µ ˜B∂ µ ˜B (14.52)for the fields ˜B and .The subscript MM stands for Matzner–Misner, who performed this analysisat the end of the 1960s. It is worth noting that the link between the fields B and˜B is given by three-dimensional duality. To see why it is so, let us consider theduality relatione 2 B mn = ɛ mnp ∂ p B (14.53)and then reduce the dimensionality using the properties of the Kaluza–Kleinvector B m . The duality relation becomes thenρ −1 2 ∂ µ ˜B = ɛ µν ∂ ν B. (14.54)The deep relation between these two distinct reducted Lagrangians will beexplained in the next section, treating nonlinear σ -models. In the language of thenonlinear sigma models, these two reducted actions correspond to two distinctSL(2, R)/SO(2) models.14.2 Nonlinear σ -modelsIn this section, we introduce nonlinear σ -models and discover that the reducedgravity is a certain nonlinear σ -model, connected to a certain symmetry group.The expression of the Lagrangian of the model depends on this symmetry group.Let us start from a non-compact Lie group G and consider the maximalcompact Lie subgroup H of G. The Lie algebra decomposition iswith the following commutation rulesG = H ⊕ K (14.55)[H, H] ⊂ H, [K , K ] ⊂ H, [ H, K ] ⊂ K . (14.56)This decomposition is invariant under the symmetric space automorphismτ(H) = H, τ(K ) =−K (14.57)which can alternatively be formulated in terms of Lie group elements g directlythroughτ(g) = η −1 (g T ) −1 η (14.58)where the matrix η depends on the group G (e.g. η = 1 for G = SL(n, R)).