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

Appendix B. Duality symmetry 327By collecting the various contributions from φ, R and H 2 , the action (16.165) canbe rewritten as:S =− λ ∫sdt e −φ [ ˙φ 2 + 124 Tr(γ −1 ˙γ) 2 − Tr(γ −1 ¨γ)− 1 2 Tr( ˙γ −1 ˙γ)+ ˙φ Tr(γ −1 ˙γ)+ 1 4 Tr(γ −1 Ḃ) 2 ]. (16.171)We can now eliminate the second derivatives, and the mixed term (∼ ˙φ ˙γ ), bynoting thatddt [e−φ Tr(γ −1 ˙γ)] = e −φ [Tr(γ −1 ¨γ)+ Tr( ˙γ −1 ˙γ)− ˙φ Tr(γ −1 ˙γ)]. (16.172)Finally, by using the identity,(γ −1 )˙=−γ −1 ˙γγ −1 (16.173)(following from g −1 g = γ −1 γ = I), we can rewrite the action in quadratic form,modulo a total derivative, asS =− λ ∫ [sdt e −φ 1 ˙φ2 −24 Tr(γ −1 ˙γ) 2 + 1 ]4 Tr(γ −1 Ḃ) 2 . (16.174)This action can be set into a more compact form by using the (2d × 2d)matrix M, defined in terms of the spatial components of the metric and of theantisymmetric tensor,(G−1−GM =−1 )BBG −1 G − BG −1 ,BG = g ij ≡−γ ij , G −1 ≡ g ij , B ≡ B ij , (16.175)and using also the matrix η, representing the invariant metric of the O(d, d) groupin the off-diagonal representation,( )0 Iη =(16.176)I 0(I is the unit d-dimensional matrix). By computing Mη, Ṁη and (Ṁη) 2 we find,in fact,Tr(Ṁη) 2 = 2Tr[˙γ −1 ˙γ + (γ −1 Ḃ) 2 ] = 2Tr[−(γ −1 ˙γ) 2 + (γ −1 Ḃ) 2 ], (16.177)and the action becomesS =− λ s2∫dt e −φ [˙φ2 +18 Tr(Ṁη)2 ]. (16.178)