OU-HET 496 October 2004 M5-brane Effective Action as an On-shell Action in Supergravity 5 0 0 2 Matsuo Sato1)∗ and Asato Tsuchiya2)† n a J 5 1) Department of Physics and Astronomy 3 University of Rochester, Rochester, NY 14627-0171, USA v 1 6 2) Department of Physics, Graduate School of Science 2 0 Osaka University, Toyonaka, Osaka 560-0043, Japan 1 4 0 / h t - Abstract p e h We show that the covariant effective action for M5-braneis a solution to theHamilton- : v Jacobi (H-J) equations of 11-dimensional supergravity. The solution to the H-J equa- i tions reproduces the supergravity solution that represents the M2-M5 bound states. X r a ∗e-mail address : [email protected] †e-mail address : [email protected] 1 Introduction D-branesandM-braneshave beenplaying acrucialroleinanalyzing nonperturbative aspects of string theory (M-theory). D-branes can be regarded in string perturbation theory as a boundary state with the Dirichlet boundary condition imposed while they also emerge as classical solutions of supergravity. In a series of publications [1, 2, 3], we showed that the D-brane effective action is a so- lution to the Hamilton-Jacobi (H-J) equations of type IIA (IIB) supergravity and that the M2-braneandM5-braneeffective actionsaresolutionstotheH-Jequationsof11-dimensional supergravity. We also showed that these solutions to the H-J equations reproduce the super- gravity solutions which represent a stack of D-branes in a B field, a stack of M2-branes and 2 a stack of the M2-M5 bound states, respectively. This fact means that those effective ac- tions of branes are on-shell actions around the corresponding brane solutions in supergravity. The near-horizon limit of these supergravity solutions are conjectured to be dual to various gauge theories [4, 5]. For instance, the near-horizon limit of the supergravity solution that represents D3-branes in a B field is conjectured to be dual to noncommutative super Yang 2 Mills in four dimensions [6, 7]. In gauge/gravity correspondence, the on-shell action around the background dual to a gauge theory is in general a generating functional of correlation functions in the gauge theory and the zero-mode part of the on-shell action should reproduce the holographic renormalization flow [8]. Hence, our findings should be useful for the study of the gauge/gravity (string) correspondence. (For other applications of our results, see the introduction in [2].) In this paper, we revisit the M5-brane case. The near-horizon limit of the supergravity solution representing the M2-M5 bound states is conjectured to be dual to (a noncommu- tative version of) 6-dimensional = (2,0) superconformal gauge field theory [4, 7]. There N are two versions [11, 12] of the M5-brane effective action, which are equivalent in the sense that both give the same equations of motion for M5-brane [9]. One version of the action was suggested in [10] and explicitly constructed in [11]. In this version, in order to obtain a complete set of the equations of motion for M5-brane, one needs to add the self-duality condition to the equations obtained by varying the action. In fact, we showed in [2] that this action is a solution to the H-J equations up to the self-duality condition. This implies that this M5-brane effective action is not an on-shell in the ordinary sense. So, it seems unclear 1 what role this effective action as a solutions to the H-J equations plays in the gauge/gravity correspondence. On the other hand, the other version of the action which was constructed in [12] and is called the ’covariant action’ directly gives a complete set of the equations of motion for M5-brane. Although it contains an auxiliary scalar field that causes a problem in defining the partition function [10], it is well-defined at least at classical level. Thus, for the sake of the study of the gauge/gravity correspondence, it is worth investigating whether this covariant action satisfies the H-J equations of 11-dimensional supergravity and repro- duces the supergravity solution representing the M2-M5 bound states so that the action is an on-shell action around the M2-M5 bound state solution in the ordinary sense. In this paper, we show that this is indeed the case. The present paper is organized as follows. In section 2, we perform a reduction of 11- dimensional supergravity on S4 and obtain a 7-dimensional gravity. In section 3, we develop the canonical formalism for the 7-dimensional gravity to derive the H-J equations. In section 4, we write down the covariant action for M5-brane explicitly. In section 5, we show that the covariant effective action is a solution to the H-J equations obtained in section 3. In section 6, we show that the solution to the H-J equations in the previous section reproduces the supergravity solution representing a stack of the M2-M5 bound states. In section 7, by using the relation of 11-dimensional supergravity with type IIA supergravity, we obtain an effective action for NS 5-branes that is a solution to the H-J equations of type IIA supergravity and reproduces the supergravity solution representing a stack of NS 5-branes, which is relevant for the duality between gravity and little string theory [15]. Section 8 is devoted to discussion. We give an argument which is expected to intuitively explain why the D-brane effective action satisfies the H-J equations of supergravity. In appendix, we present some equations which are useful for the calculations in sections 5 and 6. 2 Reduction of 11-dimensional supergravity on S4 In this section, we perform a reduction of 11-dimensional supergravity on S4 and obtain a 7-dimensional gravity. Wewillregarda radial-time-fixed surfacein the7-dimensional gravity as a worldvolume of M5-brane. 2 The bosonic part of 11-dimensional supergravity is given by 1 1 1 I = d11X√ G R F 2 A F F , (2.1) 11 2κ2 Z − (cid:18) G − 2| 4| (cid:19)− 12κ2 Z 3 ∧ 4 ∧ 4 11 11 where F = dA (2.2) 4 3 and 1 F 2 = F FM1M2M3M4. (2.3) | 4| 4! M1M2M3M4 We drop the fermionic degrees of freedom consistently. The equations of motion derived from (2.1) are 1 1 1 RG F F L1L2L3 +G R + F 2 = 0, MN − 12 ML1L2L3 N MN (cid:18)−2 G 4| 4| (cid:19) 1 D FLM1M2M3 εM1M2M3L1···L8F F = 0, (2.4) L − 2(4!)2 L1L2L3L4 L5L6L7L8 where D stands forthecovariant derivative ineleven dimensions, whiletheBianchi identity M which follows from (2.2) is dF = 0. (2.5) 4 We split the 11-dimensional coordinates XM (M = 0,1, ,10) into two parts as XM = ··· (ξα,θ ) (α = 0, ,6, i = 1, ,4), where the ξα are 7-dimensional coordinates and the θ i i ··· ··· parametrize S4. We make an ansatz for the fields as follows: ds11 = hαβ(ξ)dξαdξβ +eρ(2ξ)dΩ4, 1 F = F (ξ)dξα1 dξα4 4 4! α1···α4 ∧···∧ 1 + eρ(ξ)εα1···α7F˜ (ξ)ε dθ dθ . (2.6) 4!7! α1···α7 θi1···θi4 i1 ∧···∧ i4 where h is a 7-dimensional metric. Note that this ansatz preserves the 7-dimensional αβ general covariance. By substituting (2.6) into the equations of motion (2.4) and the Bianchi identity (2.5), we obtain the following equations in seven dimensions. 1 1 R ρ ∂ ρ∂ ρ F F γ1γ2γ3 αβ −∇α∇β − 4 α β − 12 αγ1γ2γ3 β 3 1 5 1 1 hαβ R+e−ρ2R(S4) 2 γ γρ ∂γρ∂γρ F4 2 + F˜7 2 = 0, −2 (cid:18) − ∇ ∇ − 4 − 2| | 2| | (cid:19) 1 3 3 1 1 R+ e−21ρR(S4) γ γρ ∂γρ∂γρ F4 2 F˜7 2 = 0, 2 − 2∇ ∇ − 4 − 2| | − 2| | 1 (eρFγα1α2α3)+ eρF F˜α1α2α3γ1γ2γ3γ4 = 0, ∇γ 4! γ1γ2γ3γ4 εαβγ1γ2γ3γ4γ5∂ F = 0, γ1 γ2γ3γ4γ5 (eρF˜γα1···α6) = 0, (2.7) γ ∇ where stands for the covariant derivative in seven dimensions, and R(S4) = 12. By using α ∇ a relation in seven dimensions, 1 F˜ F˜ γ1···γ6 = h F˜ F˜γ1···γ7, αγ1···γ6 β 7 αβ γ1···γ7 we can check that these equations are derived from the 7-dimensional gravity given by 3 1 1 I7 = d7ξ√ heρ Rh +e−ρ2R(S4) + ∂αρ∂αρ F4 2 F˜7 2 , (2.8) Z − (cid:18) 4 − 2| | − 2| | (cid:19) where F = dA , 4 3 1 F˜ = dA A F . (2.9) 7 6 3 4 − 2 ∧ This reduction is a consistent truncation in the sense that every solution of I can be lifted 7 to a solution of 11-dimensional supergravity. 3 Canonical formalism and the H-J equations In this section, we develop the canonical formalism for I obtained in the previous section 7 and derive the H-J equations. First, we rename the 7-dimensional coordinates: ξµ = xµ (µ = 0, ,5), ξ6 = r. ··· Adopting r as time, we make the ADM decomposition for the 7-dimensional metric. ds2 = h dξαdξβ 7 αβ = (n2 +gµνn n )dr2 +2n dr dxµ +g dxµdxν, (3.1) µ ν µ µν 4 where n and n are the lapse function and the shift function, respectively. Henceforce µ, ν µ run from 0 to 5. In what follows, we consider a boundary surface specified by r = const. and impose the Dirichlet condition for the fields on the boundary. Here we need to add the Gibbons- Hawking term [16] to the actions, which is defined on the boundary and ensures that the Dirichlet condition can be imposed consistently. Then, the 7-dimensional action I with the 7 Gibbons-Hawking term on the boundary can be expressed in the canonical form as follows: I = drd6x√ g (πµν∂ g +π ∂ ρ+πµ1µ2µ3∂ A +πµ1···µ6∂ A 7 Z − r µν ρ r r µ1µ2µ3 r µ1···µ6 nH n Hµ A Gµν A Gµ1···µ5) (3.2) − − µ − rµν 2 − rµ1···µ5 5 with 1 5 4 H = e−ρ (πµν)2 + (πµ )2 π2 + πµ π 3(πµ1µ2µ3 +10πµ1µ2µ3ν1ν2ν3A )2 (cid:18)− 9 µ − 9 ρ 9 µ ρ − ν1ν2ν3 6! (πµ1···µ6)2 , −2 (cid:19)−L 15 Hµ = 2 gπµν +∂µρπ +Fµ πν1ν2ν3 + Fµ Aµ F πν1···ν6, − ∇ν ρ ν1ν2ν3 (cid:18) ν1···ν6 − 2 ν1ν2 ν3···ν6(cid:19) Gµν = 3 gπλµν, 2 − ∇λ Gµ1···µ5 = 6 gπρµ1···µ5, (3.3) 5 − ∇ρ where 5 1 1 L = eρ(cid:18)Rg −2∇gµ∇µgρ− 4∂µρ∂µρ− 2|F4|2 − 2|F˜7|2(cid:19)+eρ2R(S4), (3.4) πµν is the canonical momentum conjugate to g , and so on. The relations between the µν canonical momenta and the r derivatives of the fields are given by 1 π = eρ K +g K + g (∂ ρ nλ∂ ρ) , µν µν µν µν r λ (cid:18)− n − (cid:19) 3 1 π = eρ 2K + (∂ ρ nµ∂ ρ) , ρ r µ (cid:18) 2n − (cid:19) 1 1 1 1 π = eρ (F nνF )+ (F˜ nνF˜ )Aν1ν2ν3 , µ1µ2µ3 (cid:18)−6n rµ1µ2µ3 − νµ1µ2µ3 72n rµ1µ2µ3ν1ν2ν3 − νµ1µ2µ3ν1ν2ν3 (cid:19) 1 1 π = eρ (F˜ nνF˜ ), (3.5) µ1···µ6 −6! n rµ1···µ6 − νµ1···µ6 5 where K is the extrinsic curvature given by µν 1 K = (∂ g gn gn ), K = gµνK . (3.6) µν 2n r µν −∇µ ν −∇ν µ µν n, n and A and A behave like Lagrange multipliers and give the constraints: µ rµν rµ1···µ5 H = 0, Hµ = 0, Gµν = 0 and Gµ1···µ5 = 0. (3.7) 2 5 The first one and the second one are called the Hamiltonian constraint and the momentum constraint respectively, while the third one and the last one are called the Gauss law con- straints. Note that the hamiltonian density, = nH +n Hµ +A Gµν +A Gµ1···µ5, H µ rµν 2 rµ1···µ5 5 vanishes on shell due to these constraints. As usual, the H-J equation is obtained by performing the following replacements in the hamiltonian. 1 δS 1 δS 1 δS πµν(x) = , π (x) = , πµ1µ2µ3(x) = , ρ −g(x)δgµν(x) −g(x)δρ(x) −g(x)δAµ1µ2µ3(x) p 1 δS p p πµ1···µ6(x) = , (3.8) −g(x)δAµ1···µ6(x) p where S isanon-shellaction, andg (x), ρ(x), A (x)andA (x) represent thevalues µν µ1µ2µ3 µ1···µ6 of the fields on the boundary r = const.. The fact that the hamiltonian vanishes on shell simplifies the ordinary H-J equation: ∂S ∂S + d6x = 0 = 0. (3.9) ∂r Z H → ∂r This implies that S does not depend on the boundary ’time’ r explicitly but depend only on the boundary values of the fields. In addition to the ordinary H-J equation (3.9), there are a set of equations for S which is obtained by applying the replacements (3.8) to the constraints (3.7). These equations should also be called the H-J equations. For instance, the H-J equation coming from Hµ = 0 takes the form 1 δS 1 δS 1 δS 2 g +∂µρ +Fµ − ∇ν (cid:18)√ gδg (cid:19) √ g δρ ν1ν2ν3√ gδA − µν − − ν1ν2ν3 15 1 δS + Fµ Aµ F = 0. (3.10) (cid:18) ν1···ν6 − 2 ν1ν2 ν3···ν6(cid:19) √ gδA − ν1···ν6 The H-J equations coming from Hµ = 0, Gµν = 0 and Gµ1···µ5 = 0 gives a condition 2 5 that S must be invariant under the diffeomorphism in six dimensions and the U(1) gauge 6 transformations A A +dΣ , 3 3 2 → 1 A A +dΣ + Σ F . (3.11) 6 6 5 2 4 → 2 ∧ (See appendix C in Ref.[1]). The H-J equation coming from H = 0 is a nontrivial equation that can determine the form of S. 4 Covariant M5-brane action Before solving the H-J equations obtained in the previous section, we write down the covari- ant effective action for M5-brane, which was constructed in [12].1 It takes the form 1 S = d6σ det( + ˜ )+ √ ˇµν ˜ M5 µν µν µν −Z (cid:18)q− G H 4 −GH H (cid:19) 1 + d6σ + F , (4.1) 6 3 3 Z (cid:18)A 2A ∧ (cid:19) where the σµ (µ = 0, ,5) parametrize the worldvolume of the M5-brane and , and µν 3 ··· G A are the induced fields on the worldvolume of the corresponding fields in 11-dimensional 6 A supergravity. For instance, is given by µν G ∂YM(σ)∂YN(σ) (σ) = G (Y(σ)), (4.2) Gµν ∂σµ ∂σν MN where the YM(σ) (M = 0, ,10) are embedding functions of the worldvolume in eleven ··· dimensions. F is the gauge field strength on the worldvolume, which is a third-rank anti- 3 symmetric tensor. There also exists an auxiliary scalar field on the worldvolume, which is denoted by a. It is convenient to introduce a time-like unit vector field v : µ c ∂a v = µ , c = , v vµ = 1. (4.3) µ √ c cν µ ∂σµ µ − ν − Then ˇ and ˜ are defined in terms of , , F and v as follows: µν µν µν 3 3 µ H H G A = +F , µνλ µνλ µνλ H A 1 ∗µνλ = εµνλρστ , ρστ H 6 H ˇ = vλ, µν µνλ H H ˜ = ∗ vλ. (4.4) Hµν Hµνλ 1 For a canonical formulation and a gauge fixing for the covariant M5-brane action, see Ref.[17]. 7 Note that the effective action (4.1) is the lowest order in derivative expansion so that the effective action is valid only when the fields are almost independent of σ. 5 Covariant M5-brane action as a solution to the H-J equations In solving the H-J equations of the 7-dimensional gravity, we drop the dependence of the fields on the 6-dimensional coordinates xµ. Correspondingly, any solution of supergravity obtained from such a solution to the H-J equations will depend only on the radial time r. In other words, we reduce the problem to a one-dimensional one. Let S be a solution to the 0 H-J equations under this simplification. It follows from (3.3), (3.4) and (3.8) that the H-J equations coming from the hamiltonian constraint H = 0 is simplified as 1 δS 2 1 1 δS 2 5 1 δS 2 0 g 0 + 0 µν (cid:18)√ gδg (cid:19) − 9 (cid:18) √ gδg (cid:19) 9 (cid:18)√ g δρ (cid:19) µν µν − − − 4 1 δS 1 δS 1 δS 1 δS 2 g 0 0 +3 0 +10A 0 −9 µν√ gδg √ g δρ (cid:18)√ gδA ρ1ρ2ρ3√ gδA (cid:19) − µν − − µνλ − µνλρ1ρ2ρ3 6! 1 δS 2 + 0 +e23ρR(S4) = 0. (5.1) 2 (cid:18)√ gδA (cid:19) − µ1···µ6 Let us consider the following form: S = S +S +S , (5.2) 0 c BI WZ with Sc = α d6x√ ge34ρ, Z − 1 S = β d6x det(g +H˜ )+ √ gHˇµνH˜ , BI µν µν µν Z (cid:18)q− 4 − (cid:19) 1 S = γ d6x A + A F , (5.3) WZ 6 3 3 Z (cid:18) 2 ∧ (cid:19) where Hˇ and H˜ are defined in terms of g , A , F and v in the same way as (4.4), µν µν µν 3 3 µ H = A +F , µνλ µνλ µνλ 1 H∗µνλ = εµνλρστH , ρστ 6 Hˇ = H vλ, µν µνλ H˜ = H∗ vλ, (5.4) µν µνλ 8 and v is defined in terms of c in the same way as (4.3). All of the fields in (5.3) are µ µ independent of xµ so that the integral over the 6-dimensional space-time is factored out. In what follows, we show that S (5.2) is a solution to the simplified H-J equations of the 0 7-dimensional gravity with c and F being arbitrary constants if µ 3 16 α2 = R(S4) = 64 and β = γ. (5.5) 3 − S trivially satisfies the simplified H-J equations of the 7-dimensional gravity except (5.1), 0 while S satisfies (5.1) quite nontrivially, as we will see below. 0 If (5.2) is substituted into (5.1), the lefthand side of (5.1) is decomposed into four parts: LHS of (5.1) = (1)+(2)+(3)+(4) (5.6) with 1 δS 2 1 1 δS 2 5 1 δS 2 4 1 δS 1 δS c c c c c (1) = g + g µν µν (cid:18)√ gδg (cid:19) − 9 (cid:18) √ gδg (cid:19) 9 (cid:18)√ g δρ (cid:19) − 9 √ gδg √ g δρ µν µν µν − − − − − +e23ρR(S4), 1 δS 1 δS 2 1 δS 1 δS 4 1 δS 1 δS c BI c BI c BI (2) = 2g g g g g , µλ νρ µν λρ µν √ gδg √ g δg − 9 √ gδg √ g δg − 9 √ g δρ √ g δg µν λρ µν λρ µν − − − − − − 1 δS 2 1 1 δS 2 BI BI (3) = g µν (cid:18)√ g δg (cid:19) − 9 (cid:18) √ g δg (cid:19) µν µν − − 1 δS 1 δS 1 δS 2 BI WZ WZ +3 + +10A , ρστ (cid:18)√ gδA √ gδA √ gδA (cid:19) µνλ µνλ µνλρστ − − − 6! 1 δS 2 WZ (4) = . (5.7) 2 (cid:18)√ gδA (cid:19) µνλρστ − By using the relations 1 δS 1 c = αe34ρgµν, √ gδg 2 µν − 1 δS 3 c = αe43ρ, (5.8) √ g δρ 4 − we can easily calculate (1), (2). The results are 3 (1) = α2e23ρ +e32ρR(S4), −16 2 1 1 δS BI (2) = 1 g = 0. (5.9) µν (cid:18) − 3 − 3(cid:19) √ g δg µν − 9