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OU-HET 457 October 2003 A Note on Hamilton-Jacobi Formalism and D-brane Effective Actions 4 0 0 Matsuo Sato∗ and Asato Tsuchiya† 2 n a J Department of Physics, Graduate School of Science 5 Osaka University, Toyonaka, Osaka 560-0043, Japan 2 v 5 2 1 0 Abstract 1 3 0 Wefirstreviewthecanonicalformalismwithgeneralspace-likehypersurfacesdeveloped / by Dirac by rederiving the Hamilton-Jacobi equations which are satisfied by on-shell h t actions defined on such hypersurfaces. We compare the case of gravitational systems - p withthatoftheflatspace. Next,weremarkasasupplementtoourpreviousresultsthat e the effective actions of D-brane and M-brane given by arbitrary embedding functions h : are on-shell actions of supergravities. v i X r a ∗e-mail address : [email protected] †e-mail address : [email protected] 1 Introduction We showed in [1, 2] that the effective actions of D-brane and M-brane are on-shell actions of supergravities. We derived the Hamilton-Jacobi (H-J) equations of supergravities, which are satisfied by on-shell actions, regarding a radial direction as time, and solved those equations. We also found that these solutions to the H-J equations are the on-shell actions around the supergravity solutions that are conjectured to be dual to various gauge theories, which in particular include noncommutative super Yang Mills. In the gauge/gravity correspondence, the on-shell actions in gravities are considered to be generating functionals of correlation functions in gauge theories. Therefore, our results in [1, 2] should be useful for going beyond the AdS/CFT correspondence and studying the more general gauge/gravity correspondence. However, it is not clear why the D-brane effective actions which includes the all-order contributions in the α′ expansion are obtained within supergravities, which are just the low- est orderapproximationforstringtheoriesintheα′ expansion. Inordertoclarifythisreason, we must establish the exact correspondence between our calculations and the derivations of the D-brane effective action in string theory. On one hand, the worldvolumes of the D-brane effective actions in our solutions are fixed-time hypersurfaces, since the ordinary H-J formalism gives on-shell actions defined on the boundary hypersurfaces specified by the final time. On the other hand, the worldvolumes of D-branes in string theories are defined by embedding functions which can specify arbitrary hypersurfaces in the target space [3]. Therefore, before trying to establish the above correspondence, we should first investigate whether the D-brane effective action whose worldvolume is defined by such embedding functions is an on-shell action of supergravity or not. Hence, thisbriefnoteisconcernedwiththeon-shellactionthatisobtainedbysubstituting intotheactiontheclassicalsolutionwhichsatisfiesaboundaryconditiongivenonnotafixed- time hypersurface but an general hypersurface. (See Fig.1.) We need a generalization of the H-J formalism that gives such on-shell actions. Indeed, this generalization was studied by Dirac around fifty years ago [4]. The generalization in gravitational systems is much simpler. In fact, by performing a general coordinate transformation that transforms the general hypersurface defined by the embedding functions to a fixed-time hypersurface, one can show that the answer to the above question is positive. However, it is interesting and useful for further developments to compare the case of gravitational system with that of the 1 flat space, so that we will give a heuristic argument below. We will first derive Dirac’s result in the flat space in a different way, and next make the comparison. Finally, we remark that the effective actions of D-brane and M-brane given by arbitrary embedding functions are on-shell actions of supergravities x0 x0 XM(σi) x1, x2, x3 x1, x2, x3 Figure 1: Generalization of a boundary surface 2 Introduction of dynamical coordinates As we mentioned, we pay attention to on-shell actions with the boundary values of fields given on general space-like hypersurfaces. Let us consider a four-dimensional scalar field theory in the flat space as an example. The action is given by 1 I = d4x ηMN∂ φ(x)∂ φ(x)+V(φ(x)) , (2.1) M N − 2 Z (cid:18) (cid:19) where η = diag( 1,1,1,1). The equation of motion is MN − ηMN∂ ∂ φ = V′(φ). (2.2) M N The space-like hypersurface which specifies the boundary is parametrized by the XM(σi), where i = 1,2,3. (See Fig.1.) In principle, we can obtain the solution φ¯(x) to the equation of motion (2.2) which satisfies a boundary condition φ¯(XM(σi)) = Φ(XM(σi)). (2.3) By substituting this solution into I, we obtain the on-shell action in which we are interested, 1 S(X,Φ(X)) = d4x ηMN∂ φ¯(x)∂ φ¯(x)+V(φ¯(x)) . (2.4) M N − 2 Z (cid:18) (cid:19) 2 It is in general difficult to solve the equation of motion in the above situation and to obtain the on-shell action directly. Instead, we seek for the Hamilton-Jacobi equations, which are the differential equations satisfied by the on-shell action of this kind. In order to develop the generalized Hamilton-Jacobi formalism, we consider an action, ∂x(σ) 1 ∂σα ∂σβ I˜= d4σdet ηMN ∂ φ˜(σ)∂ φ˜(σ)+V(φ˜(σ)) , (2.5) − ∂σ 2 ∂xM(σ)∂xN(σ) α β Z (cid:18) (cid:19)(cid:18) (cid:19) where σα = (τ,σi) (α = 0,1,2,3, i = 1,2,3, σ0 = τ). In this action, let both the induced scalar φ˜(σ) and the coordinates xM(σ) be dynamical variables as in [4]. We regard τ as time in this system. Moreover, let the boundary values of the xM(σ) parametrize the same space-like hypersurface, xM(T,σi) = XM(σi), (2.6) whereT istheboundaryvalueofτ thatisthefinaltime. I˜isinvariantunderthereparametriza- tionofσ bywhichbothφ˜(σ)andthexM(σ)aretransformedasscalars. Ifthisdiffeomorphism is fixed by the gauge fixing condition xM(σ) = σM, I˜ reduces to I. Therefore I˜ and I are equivalent. We will actually see below that the on-shell action of I˜is equal to that of I. The equations of motions of Iãre ∂x ∂σγ 1 ∂σα ∂σβ ∂σα ∂σγ ∂σβ ∂ det ηMN ∂ φ˜∂ φ˜+V(φ˜) ηMN ∂ φ˜∂ φ˜ = 0, γ ∂σ ∂xL 2 ∂xM ∂xN α β − ∂xL ∂xM ∂xN α β (cid:20) (cid:18) (cid:19)(cid:18) (cid:18) (cid:19) (cid:19)(cid:21) ∂x ∂σγ ∂σβ ∂x ∂ det ηMN ∂ φ˜ det V′(φ˜) = 0. (2.7) γ ∂σ ∂xM ∂xN β − ∂σ (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:19) Let us consider the classical solution x¯M(σ) and φ¯˜(σ) which satisfies the boundary condition, x¯M(T,σi) = XM(σi), φ¯˜(T,σi) = Φ˜(σi) Φ(X(σi)). (2.8) ≡ By substituting this classical solution into I˜, we obtain the on-shell action, S˜(T,Φ˜,X) = T dτ d3σidet ∂x¯(σ) 1ηMN ∂σα ∂σβ ∂ φ¯˜(σ)∂ φ¯˜(σ)+V(φ¯˜(σ)) , − ∂σ 2 ∂x¯M(σ)∂x¯N(σ) α β ZT0 Z (cid:18) (cid:19)(cid:18) (cid:19) (2.9) where T is the initial time. If we define φ¯(x) by φ¯˜(σ) = φ¯(x¯(σ)) and x = x¯(σ), φ¯(x) satisfies 0 (2.2) and (2.3). Therefore the coordinate transformation x = x¯(σ) gives the desired relation S(X,Φ(X)) = S˜(T,Φ˜,X). (2.10) 3 Let us derive the Hamilton-Jacobi equations satisfied by S˜. By solving these equations we obtain S˜ and hence S as well. S˜ is a functional of the final time T and the boundary values, Φ˜ and X. The variation of S˜ with respect to T, Φ˜, and X is given by δS˜ = d3σi det ∂x¯ 1ηMN ∂σα ∂σβ ∂ φ¯˜∂ φ¯˜+V(φ¯˜) δT − ∂σ 2 ∂x¯M ∂x¯N α β Z (cid:18) (cid:19)(cid:18) (cid:19) (cid:12)τ=T − d3σidet ∂∂σx¯ ηMN∂∂x¯τM ∂∂x¯σNβ ∂βφ¯˜δφ¯˜ (cid:12)(cid:12)(cid:12) Z (cid:18) (cid:19) (cid:12)τ=T − d3σidet ∂∂σx¯ ∂∂x¯τL 21ηMN∂∂x¯σMα ∂∂(cid:12)(cid:12)(cid:12)x¯σNβ ∂αφ¯˜∂βφ¯˜+V(φ¯˜) Z (cid:18) (cid:19)(cid:18) (cid:18) (cid:19) ∂σαηMN ∂τ ∂σβ ∂ φ¯˜∂ φ¯˜ δx¯L . −∂x¯L ∂x¯M ∂x¯N α β (cid:19) (cid:12)τ=T (cid:12) We have used the equations of motion (2.7) and δφ¯˜(T0,σi) = δx¯(T0,σi) =(cid:12)(cid:12) 0. Moreover, the condition (2.8) implies δx¯M(T,σi) = δXM(σi) ∂ x¯M(T,σi)δT, τ − δφ¯˜(T,σi) = δΦ˜(σi) ∂ φ¯˜(T,σi)δT. (2.11) τ − Then, ∂S˜ = 0, ∂T δS˜ = det ∂x¯ ∂τ 1ηMN ∂σα ∂σβ ∂ φ¯˜∂ φ¯˜+V(φ¯˜) δXL − ∂σ ∂x¯L 2 ∂x¯M ∂x¯N α β (cid:18) (cid:19)(cid:18) (cid:18) (cid:19) ∂σαηMN ∂τ ∂σβ ∂ φ¯˜∂ φ¯˜ , −∂x¯L ∂x¯M ∂x¯N α β (cid:19)(cid:12)τ=T δδΦS˜˜ = −det ∂∂σx¯ ηMN∂∂x¯τM ∂∂x¯σNβ ∂βφ¯˜ . (cid:12)(cid:12)(cid:12) (2.12) (cid:18) (cid:19) (cid:12)τ=T (cid:12) Here we define L and γ for convenience: (cid:12) M ij (cid:12) ∂x¯ ∂τ L ǫ ∂ XL1∂ XL2∂ XL3 = det (ǫ = 1), M ≡ ML1L2L3 1 2 3 ∂σ ∂x¯M 0123 (cid:18) (cid:19) (cid:12)τ=T γ ∂ XM∂ XNη , γ det(γ ). (cid:12) (2.13) ij i j MN ij (cid:12) ≡ ≡ (cid:12) From (2.12) and (2.13), noting that ∂ XML = 0, ηMNL L = γ, i M M N − ∂σα ∂ φ¯˜ = 1 δS˜L +γij∂ Φ˜η ∂ XN, (2.14) ∂x¯M α γ δΦ˜ M i MN j (cid:12)τ=T (cid:12) (cid:12) (cid:12) 4 we obtain the Hamilton-Jacobi equations 2 δS˜ 1 1 δS˜ 1 δS˜ = + γij∂ Φ˜∂ Φ˜ +V(Φ˜) L γij∂ Φ˜η ∂ XN ,(2.15) δXM −2 √γ δΦ˜! 2 i j  M − i MN j δΦ˜ ∂S˜   = 0. (2.16) ∂T It is instructive to rederive the above Hamilton-Jacobi equations in the canonical formalism following Dirac [4]. I˜is rewritten in the canonical formalism as follows: I˜= dτd3σi P ∂ xM +P ∂ φ˜ CMΣ (x,φ˜,P ,P ) , (2.17) M τ φ˜ τ − M L φ˜ Z (cid:16) (cid:17) where 2 1 1 1 Σ (x,φ˜,P ,P ) = P + P + γij∂ φ˜∂ φ˜+V(φ˜) L +P ∂ φ˜γijη ∂ xN, M L φ˜ M 2 √γ φ˜ 2 i j ! M φ˜ i MN j (cid:18) (cid:19) (2.18) and CM are Lagrange multipliers. The constraints Σ = 0 are first-class ones 1 and come M from the invariance under the reparametrization of σ. By applying the relation between the on-shell action and the canonical momenta, δS˜ δS˜ P = , P = , (2.19) M δXM φ˜ δφ˜ to this constraint, we obtain (2.15). (See section 2 in [2].) On the other hand, the ordinary Hamilton-Jacobi equation δS˜ + H = 0 reduces to (2.16), since the Hamiltonian δT H = d3σiCMΣ vanishes on shell. M R 3 Application to gravitational systems Inthissection, weconsider thesameproblemastheprevioussectioningravitationalsystems. Let us consider as an example a four-dimensional gravity given by 1 I = d4x g(x) R(x) gMN(x)∂ φ(x)∂ φ(x)+V(φ(x)) . (3.1) M N − − 2 Z (cid:18) (cid:19) p As before we consider the classical solution g¯ (x) and φ¯(x) which satisfies the boundary MN condition ∂XM(σk)∂XN(σk) ∂XM(σk)∂XN(σk) g¯ (X(σk)) = G (X(σk)), ∂σi ∂σj MN ∂σi ∂σj MN φ¯(X(σk)) = Φ(X(σk)), (3.2) 1Indeed, the Poisson brackets between the ΣM are ΣM(τ,σi),ΣN(τ,σ′i) =0. { } 5 where the XM(σ) parametrize an space-like hypersurface.2 Note that this boundary condition is invariant under general coordinate transformations of x. We also consider the on-shell action 1 S(X,G(X),Φ(X)) = d4x g¯(x) R¯(x) g¯MN(x)∂ φ¯(x)∂ φ¯(x)+V(φ¯(x)) , (3.3) M N − − 2 Z (cid:18) (cid:19) p which is an analogue of (2.4). Note that we can obtain (2.5) from (2.1) by performing a general coordinate transformation x = x(σ). Similarly, we obtain from (3.1) 1 I˜= d4σ g˜(σ) R˜(σ) g˜αβ(σ)∂ φ˜(σ)∂ φ˜(σ)+V(φ˜(σ)) . (3.4) α β − − 2 Z (cid:18) (cid:19) p NotethatI˜doesnotdependonthexM(σ)andtakesthesameformasI,becauseI isinvariant under the general coordinate transformations. Now we consider the classical solution g¯˜ (σ) αβ and φ¯˜(σ) which satisfies the boundary condition on the fixed-time hypersurface, ∂x¯M(T,σk)∂x¯N(T,σk) g¯˜ (T,σk) = G˜ (σk) G (X(σk)), ij ij ≡ ∂σi ∂σj MN φ¯˜(T,σk) = Φ˜(σk) Φ(X(σk)), ≡ x¯M(T,σk) = XM(σk), (3.5) Then the on-shell action of I˜is S˜(T,G˜,Φ˜) = T dτ d3σi g¯˜(σ) R¯˜(σ) 1g¯˜αβ(σ)∂ φ¯˜(σ)∂ φ¯˜(σ)+V(φ¯˜(σ)) . (3.6) α β − − 2 ZT0 Z q (cid:18) (cid:19) As in the previous section, if we define g¯ (x) and φ¯(x) by g¯ (x¯(σ)) = ∂σα ∂σβg¯˜ (σ), MN MN ∂x¯M ∂x¯N αβ φ¯(x¯(σ)) = φ¯˜(σ) and x = x¯(σ), g¯ (x) and φ¯(x) satisfy the equation of motion of I and the MN boundary condition (3.2). It follows again that the on-shell actions are equivalent: S(X,G(X),Φ(X)) = S˜(T,G˜,Φ˜). (3.7) Contrary to the case of the flat space, the Hamilton-Jacobi equations of I˜ clearly take the same forms as the ordinary ones of I, which are satisfied by the on-shell action with the boundary values of the fields given on the fixed-time hypersurface. Therefore if one knows 2In fact, we must add the Gibbons-Hawking term to the action in order to impose a boundary condition such as (3.2) consistently. However, the following argument is still valid after adding this term. 6 theon-shellactionwiththeboundaryvaluesofthefieldsgivenonthefixed-timehypersurface, one can obtain the on-shell action with those given on an arbitrary space-like hypersurface. Note that this consequence directly comes from the facts that I is invariant under general coordinate transformations andthat anarbitrary space-like hypersurface can be transformed to a fixed-time hypersurface by a general coordinate transformation. Finally we apply the above argument to our results in [1, 2]. We reported in [1, 2] that the D-brane and M-brane effective actions are on-shell actions of supergravities, which are defined on fixed-radial-time hypersurfaces. Obviously, the above argument holds even when we replace the time with the radial time that is a space-like direction. It follows that the D-brane and M-brane effective actions given by arbitrary embedding functions are also on- shell actions in supergravities. We take the D3-brane case as an example below and write down the explicit form of the on-shell action. The on-shell actions corresponding to the other branes can be written down explicitly in the same way. We start with the five-dimensional gravity which we obtained in [1, 2] by reducing type IIB supergravity on S5, 5 1 I5 = d5x√ h e−2φ+54ρ R+4∂αφ∂αφ+ ∂αρ∂αρ 5∂αφ∂αρ H3 2 − 4 − − 2| | Z (cid:20) (cid:18) (cid:19) 1 e45ρ F1 2 + F3 +c0H3 2 + F5 +c2H3 2 +e−2φ+43ρR(S5) ,(3.8) −2 | | | | | | (cid:21) (cid:0) (cid:1) where H = db , F = dc (p = 1,1,3). The xM (M = 0, ,4) are five-dimensional 3 2 p+2 p+1 − ··· coordinates. h , φ, ρ, b and c are the five-dimensional metric, the dilaton, the warp MN 2 p+1 factor of S5, the Kalb-Ramond field and the R-R (p + 1)-form, respectively. R(S5) is the constant curvature of S5. In this case, we are interested in on-shell actions with the boundary values of fields given on general time-like hypersurfaces, since we interpret these hypersurfaces as the worldvolumes of D3-brane. Let XM(σµ) (µ = 0, ,3) be embedding functions satisfying such a hy- ··· persurface. We consider the classical solution h¯ (x), φ¯(x), ρ¯(x), ¯b (x) and c¯ (x) MN MN M1···Mp+1 which satisfies the boundary condition ∂XM(σλ)∂XN(σλ) ∂XM(σλ)∂XN(σλ) h¯ (X(σλ)) = G (X(σλ)), ∂σµ ∂σν MN ∂σµ ∂σν MN φ¯(X(σµ)) = Φ(X(σµ)), ρ¯(X(σµ)) = Υ(X(σµ)), 7 ∂XM(σλ)∂XN(σλ) ∂XM(σλ)∂XN(σλ) ¯b (X(σλ)) = B (X(σλ)), ∂σµ ∂σν MN ∂σµ ∂σν MN ∂XM1(σν) ∂XMp+1(σν) c¯ (X(σν)) ∂σµ1 ··· ∂σµp+1 M1···Mp+1 ∂XM1(σν) ∂XMp+1(σν) = C (X(σν)). (3.9) ∂σµ1 ··· ∂σµp+1 M1···Mp+1 As before I˜ is obtained by replacing the fields in I with those with tilde, h˜ (σα), φ˜(σα), 5 5 αβ ρ˜(σα), ˜b (σα) and c˜ (σα), where σα = (σµ,σ4) (µ = 0, ,3). We regard σ4 as time, 2 p+1 ··· and denote the boundary value of σ4 by U. The classical solution of I˜, x¯M(σα), h˜¯ (σγ), 5 αβ φ¯˜(σα), ρ¯˜(σα), ¯˜b (σγ) and ¯c˜ (σβ), corresponding to the above solution of I satisfies αβ α1···αp+1 5 the boundary condition x¯M(σµ,U) = XM(σµ), h˜¯ (σλ,U) = G˜ (σλ) ∂XM(σλ)∂XN(σλ)G (X(σλ)), µν µν ≡ ∂σµ ∂σν MN φ¯˜(σµ,U) = Φ˜(σµ) Φ(X(σµ)), ≡ ρ¯˜(σµ,U) = Υ˜(σµ) Υ(X(σµ)), ≡ ¯˜b (σλ,U) = B˜ (σλ) ∂XM(σλ)∂XN(σλ)B (X(σλ)), µν µν ≡ ∂σµ ∂σν MN ∂XM1(σν) ∂XMp+1(σν) ¯c˜ (σν,U) = C˜ (σν) C (X(σν)). µ1···µp+1 µ1···µp+1 ≡ ∂σµ1 ··· ∂σµp+1 M1···Mp+1 (3.10) In order to derive the H-J equations, we perform the ADM decomposition as follows: ds2 = h dσαdσβ 5 αβ = (n2 +nµn )(dσ4)2 +2n dσµdσ4 +h dσµdσν, (3.11) µ µ µν where n and n are the lapse function and the shift functions, respectively. By adding the µ Gibbons-Hawking term, I˜ can be rewritten in the canonical form as 5 I˜ = d5σ√ h(πµν∂ h˜ +π ∂ φ˜+π ∂ ρ˜+πµν∂ ˜b + πµ1···µp+1∂ c˜ 5 − σ4 µν φ˜ σ4 ρ˜ σ4 ˜b2 σ4 µν c˜p+1 σ4 µ1···µp+1 Z p X nH n Hµ ˜b Zµ c˜ Zµ c˜ Zµνλ), (3.12) − − µ − 4µ ˜b2 − 4µ c˜2 − 4µνλ c˜4 with 1 1 4 2 H = −e2φ˜−45ρ˜ (πµν)2 + 2πφ˜2 + 2πµµπφ˜+ 5πρ˜2 +πφ˜πρ˜+ π˜bµ2ν −c˜πc˜µ2ν −6c˜λρπc˜µ4νλρ (cid:18) (cid:19) (cid:16) (cid:17) 8 1 −e−54ρ˜ 2πc˜02 +(πc˜µ2ν)2 +12(πc˜µ4νλρ)2 −L, (3.13) (cid:18) (cid:19) Hµ = 2 πµν +π ∂µφ˜+π ∂µρ˜+π H˜µνλ − ∇ν φ˜ ρ˜ ˜b2νλ +π ∂µc˜+π F˜µνλ +π (F˜µνλρσ +4c˜µνH˜λρσ), (3.14) c˜0 c˜2νλ c˜4νλρσ Zµ = 2 πµν, (3.15) ˜b2 ∇ν ˜b2 Zµ = 2 πµν 4πµνλρH˜ , (3.16) c˜2 ∇ν c˜2 − c˜4 νλρ Zµνλ = 4 πµνλρ, (3.17) c˜4 ∇ρ c˜4 where h = deth , µν H˜ = 3∂ ˜b , F˜ = (p+1)∂ c˜ , µνλ [µ νλ] µ1···µp+1 [µ1 µ2···µp+2] 5 15 1 = e−2φ˜+45˜ρ˜ R(4) +4 µ µφ˜ µ µρ˜ 4∂µφ˜∂µφ˜ ∂µρ˜∂µρ˜+5∂µφ˜∂µρ˜ H˜µνλH˜µνλ L ∇ ∇ − 2∇ ∇ − − 8 − 12 (cid:18) (cid:19) 1 1 +e54ρ˜ ∂µc˜∂µc˜ (F˜µνλ +c˜H˜µνλ)(F˜µνλ +c˜H˜µνλ) +e−2φ˜+43ρ˜R(S5). (3.18) −2 − 12 (cid:18) (cid:19) The H-J equations of this system are ∂S˜ = 0, (3.19) ∂U and H = 0, Hµ = 0, Z = 0, Z = 0, Z = 0 (3.20) ˜b2 c˜2 c˜4 with the following replacements: h˜ G˜ , φ˜ Φ˜, ρ˜ Υ˜, ˜b B˜ , c˜ C˜ , µν → µν → → µν → µν µ1···µp+1 → µ1···µp+1 1 δS˜ 1 δS˜ 1 δS˜ πµν , π , π , → G˜ δG˜µν φ˜ → G˜ δΦ˜ ρ˜ → G˜ δΥ˜ − − − 1 δS˜ 1 δS˜ πµν p , πµ1···µp+p1 p . (3.21) ˜b2 → G˜ δB˜µν c˜p+1 → G˜ δc˜µ1···µp+1 − − The H-J equation (3.p19) implies that S˜ does nopt depend on the final time explicitly while the last four equations in (3.20) imply that S˜ is invariant under the diffeomorphism in four dimensions and the U(1) gauge transformations for B˜ and C˜ . The first equation µν µ1···µp+1 H = 0 in (3.20) is only a nontrivial one that can be determine the form of S˜. 9