KU-TP 029 Supersymmetric Intersecting Branes in Time-dependent Backgrounds 0 1 Kei-ichi Maeda,a,b,1 Nobuyoshi Ohta,c,2 Makoto Tanabea,3 and Ryo Wakebea,4 0 2 n aDepartment of Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan a J bWaseda Research Institute for Science and Engineering, Shinjuku, Tokyo 169-8555, Japan 9 cDepartment of Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan 1 ] h t - p Abstract e h We construct a fairy general family of supersymmetric solutions in time- and space- [ dependent backgrounds in general supergravity theories. One class of the solutions are in- 2 tersecting brane solutions with factorized form of time- and space-dependent metrics, the v secondclassarebranesolutionsinpp-wavebackgroundscarryingspacetime-dependence,and 8 the final class are the intersecting branes with more nontrivial spacetime-dependence, and 9 theirintersectionrulesaregiven. Physicalpropertiesofthesesolutionsarediscussed,andthe 2 3 relation to existing literature is also briefly mentioned. The number of remaining supersym- . metriesareidentifiedforvariousconfigurationsincludingsinglebranes,D1-D5,D2-D6-branes 3 with nontrivial dilaton, and their possible dual theories are briefly discussed. 0 9 0 : v i X r a 1E-mail address: maeda“at”waseda.jp 2E-mail address: ohtan“at”phys.kindai.ac.jp 3E-mail address: tanabe“at”gravity.phys.waseda.ac.jp 4E-mail address: wakebe“at”gravity.phys.waseda.ac.jp Contents 1 Introduction 1 2 Time-dependent brane system in supergravity 2 3 Solutions with time-independent harmonic functions 7 3.1 Branes with factorized metrics in time and space . . . . . . . . . . . . . . . . . . 7 3.2 Branes in pp-wave backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Solutions with time-dependent harmonic functions 9 4.1 Single brane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Intersecting two branes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Some Properties of the Solutions 12 5.1 Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2 Spacetime structure near branes and horizons . . . . . . . . . . . . . . . . . . . . 14 5.3 Asymptotic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6 Supersymmetry 16 6.1 D3-brane system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2 Intersecting D1-D5-brane system . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.3 Intersecting D2-D6-brane system . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7 Concluding Remarks 20 1 1 Introduction The understandingof the fundamental nature and quantum properties of spacetime is one of the most important questions in theoretical physics. An example of such problems is the spacetime singularities that general relativity predicts. A well-known one is the big-bang singularity of a time-dependent spacetime, where general relativity breaks down. One needs a quantum the- ory of gravity to understand physics close to the singularity. String theory is one of the most promising candidate for such a theory. Although we know some static solutions in string theory, e.g. products of Minkowski space and compact manifolds, these static spacetimes are not so much useful in clarifying the dynamics of string theory in the strong curvature regime or near the singularities. Therefore it is necessary to understand string theory on time-dependent back- grounds. Unfortunately time-dependent backgrounds are difficult to work with in string theories in general, though some special cases are analyzed [1]-[5]. Recently a model of big-bang cosmology has been proposed in matrix string theory based on AdS/CFT correspondence which is a powerful nonperturbative formulation of string theory [6]. This corresponds to a simple time-dependent solution of supergravities which are the low-energy effective theories of string theories that preserves 1/2 supersymmetry in ten dimensions, with the light-like linear dilaton background, and various extensions have been considered [7]-[30]. As is usual in AdS/CFT correspondence, supersymmetry is expected to play an important role. The existence of supersymmetry allows us to better control the behaviors of the solutions in string/supergravity backgrounds and the quantum and nonperturbative properties of the field theories. Therefore there has been much interest in time-dependent supersymmetric solutions of string/supergravity theories. For a detailed review of the big-bang models in string theory, see [31]. On the other hand, D-branes can probe the nonperturbative dynamics of the string theory and they have been used to study various duality aspects of string theory. It is thus interesting to find if we can have such brane solutions in time-dependent backgrounds with time-dependent dilaton. In fact, D3-brane solutions have been found and discussed in [17, 19] and other single brane solutions in [24, 26]. A systematic derivation of the general brane solutions in the pp-wave backgrounds has been given in [32]. It is also interesting to study intersecting brane systems because it is known that some such configurations can describe the standard model of particle physics. More recently, gauge theories on D-branes are examined to gain into the dynamical supersymmetry breaking [33]. These solutions are interesting since the metrics of these solutions depend on both space and time, but the dependence is restricted to the product form of these functions. Thequestion thennaturally arises iftherearesolutions withmoregeneraldependence on space and time and if such solutions can give more physical insight. In this paper, we investigate more general time-dependent supersymmetric solutions in su- pergravity theories in ten and eleven dimensions in order to understand the nature of spacetime. In sect. 2, we derive brane solutions in general supergravities with dilaton and forms of arbi- trary ranks in spacetime-dependent backgrounds. In sect. 3, we give time-dependent solutions restricted to those with time-independent harmonic functions. All known solutions belong to this class of solutions, butour solutions are more general. We clarify the relation of our solutions and the known ones. In sect. 4, we give more general solutions with time-dependent harmonic functions for one brane and two intersecting branes. These are new solutions and the physi- cal properties of these solutions including spacetime and asymptotic structures are discussed in sect. 5. In sect. 6, we show that these solutions have unbroken supersymmetry, and identify the amount of remaining supersymmetries. Sect. 7 is devoted to conclusions and discussions. 1 2 Time-dependent brane system in supergravity Thelow-energy effective action for the supergravity system coupled to dilaton and n - form field A strength is given by m 1 1 1 I = dDx −g R− (∂Φ)2− eaAΦF2 , (2.1) 16πG 2 2n ! nA D " A # Z A=1 p X where G is the Newton constant in D dimensions and g is the determinant of the metric. The D last term includes both RR and NS-NS field strengths, and a = 1(5−n ) for RR field strength A 2 A and a = −1 for NS-NS 3-form. In the eleven-dimensional supergravity, there is a four-form and A no dilaton. We put fermions and other background fields to be zero. From the action (2.1), one can derive the field equations 1 1 n −1 R = ∂ Φ∂ Φ+ eaAΦ n F2 − A F2 g , (2.2) µν 2 µ ν 2n ! A nA µν D−2 nA µν A " # A X (cid:0) (cid:1) a 2Φ = A eaAΦF2 , (2.3) 2n ! nA A A X ∂µ1 −geaAΦFµ1···µnA = 0, (2.4) (cid:16)p (cid:17) where Fn2A denotes Fµ1···µnAFµ1···µnA and (Fn2A)µν denotes Fµρ···σFνρ···σ. The Bianchi identity for the form field is given by ∂ F = 0. (2.5) [µ µ1···µnA] In this paper we assume the following metric form: d−2 ds2 = e2Ξ(u,r) −2dudv+K(u,yα,r)du2 + e2Zα(u,r)(dyα)2+e2B(u,r) dr2+r2dΩ2 ,(2.6) D d˜+1 (cid:2) (cid:3) αX=1 (cid:16) (cid:17) whereD = d+d˜+2,thecoordinatesu,v andyα,(α =1,...,d−2)parameterizethed-dimensional worldvolume where the branes belong, and the remaining d˜+ 2 coordinates r and angles are transverse to the braneworldvolume, dΩ2 is theline element of the (d˜+1)-dimensional sphere. d˜+1 Note that u and v are null coordinates. The metric components Ξ,Z ,B and the dilaton Φ are α assumed to be functions of u and r, whereas K depends on u,yα and r. Our ansatz includes more general solutions than those in [17, 19], which consider only single D3-brane solutions with themetrics of productformof time- and space-dependentfactors; ours allows intersecting branes as well as more general spacetime dependence. For the field strength backgrounds, we take FnA = EA′ (u,r)du∧dv∧dyα1 ∧···∧dyαqA−1 ∧dr, (2.7) where n = q +2. Throughout this paper, the dot and prime denote derivatives with respect A A to u and r, respectively. The ansatz (2.7) means that we have an electric background. We 2 could, however, also include magnetic background in the same form as the electric one with the replacement g → g , F → eaΦ ∗F , Φ → −Φ. (2.8) µν µν n n This is dueto the S-duality symmetry of the original system (2.1). Sowe donot have to consider it separately. With our ansatz, the Einstein equations (2.2) reduce to d˜+1 D−q −3 Ξ′′+ U′+ Ξ′ = A S (E′ )2, (2.9) r 2(D−2) A A (cid:16) (cid:17) XA d−2 d−2 d−2 Z¨ +(d˜+2)B¨ + Z˙2 +(d˜+2)B˙2−2Ξ˙ Z˙ +(d˜+2)B˙ α α α " # α=1 α=1 α=1 X X X 1 d−2 1 1 d˜+1 + e2(Ξ−Zα)∂2K +e2(Ξ−B) KΞ′′+ K′′+ Ξ′K + K′ U′+ 2 α 2 2 r " # αX=1 (cid:16) (cid:17)(cid:16) (cid:17) D−q −3 1 = A e2(Ξ−B)KS (E′ )2− (Φ˙)2, (2.10) 2(D−2) A A 2 A X d−2 d−2 d−2 d−2 1 Ξ˙′+ Z˙′ +(d˜+1)B˙′− Z˙ +(d˜+2)B˙ Ξ′−B˙ Z′ + Z˙ Z′ =− Φ˙Φ′, (2.11) α α α α α 2 " # α=1 α=1 α=1 α=1 X X X X d˜+1 δ(α) Z′′+ U′+ Z′ = A S (E′ )2, (2.12) α r α 2(D−2) A A (cid:16) (cid:17) XA d−2 d˜+1 d−2 U′′+B′′− 2Ξ′+ Z′ − B′+2(Ξ′)2+ (Z′)2 α r α (cid:16) αX=1 (cid:17) αX=1 1 D−q −3 = − (Φ′)2+ A S (E′ )2, (2.13) 2 2(D−2) A A A X d˜+1 U′ q +1 B′′+ U′+ B′+ = − A S (E′ )2, (2.14) r r 2(D−2) A A (cid:16) (cid:17) XA (α) where U, S and δ are defined by A A d−2 U ≡ 2Ξ+ Z +d˜B, (2.15) α α=1 X S ≡ exp ǫ a Φ−2 2Ξ+ Z , (2.16) A A A α " !# αX∈qA and D−q −3 yα belonging to q -brane δ(α) = A for A , (2.17) A −(q +1) otherwise A (cid:26) (cid:26) respectively, and ǫ = +1(−1) is for electric (magnetic) backgrounds. The sum of α in Eq. A (2.16) runs over the q -brane components in the (d−2)-dimensional yα-space, for example A qA−1 Z = Z . (2.18) α αA αX∈qA αXA=1 3 Eqs. (2.9), ···, (2.13) and (2.14) are the uv,uu,ur,αβ,rr and ab components of the Einstein equations(2.2), respectively. Thedilatonequation (2.3)andtheequations fortheformfield(2.4) and (2.5) yield e−Ur−(d˜+1)(eUrd˜+1Φ′)′ = −1 ǫ a S (E′ )2, (2.19) 2 A A A A A X ′ rd˜+1eUS E′ = 0, (2.20) A A (cid:16) (cid:17)(cid:5) rd˜+1eUS E′ = 0. (2.21) A A (cid:16) (cid:17) We assumethat U is independentof r butdependsonly on u. In the case of static spacetime, it is known that under this condition (U is constant in case of no dependence on u), all the supersymmetric intersecting brane solutions have been derived [34]. If this condition is relaxed, onemaygetmoregeneralnon-BPSsolutions[35],buthereweareinterestedintheBPSsolutions. We extend them to the time-dependent case. From Eqs. (2.20) and (2.21), we learn that rd˜+1eUS E′ = c , (2.22) A A A is a constant. Combined with Eq. (2.19), we then get 1 c E˜ Φ′ =− ǫ a A A, (2.23) 2 A A rd˜+1 A X where we have defined E˜ = e−UE . (2.24) A A Similarly from Eqs. (2.9), (2.12), (2.14), we find D−q −3c E˜ Ξ′ = A A A, 2(D−2) rd˜+1 A X δ(α) c E˜ Z′ = A A A, α 2(D−2) rd˜+1 A X q +1 c E˜ B′ = − A A A. (2.25) 2(D−2) rd˜+1 A X Note thatthereisnointegral constantintherighthandsidesof(2.23) and(2.25). Thisisrelated to the BPS condition. Substituting these into (2.13), we get cAM +rd˜+1 1 ′δ cB E˜AE˜B = 0, (2.26) 2 AB E˜ AB 2 r2d˜+2 XA,Bh (cid:18) A(cid:19) i where d−2 (α) (α) 2(D−q −3)(D−q −3) δ δ (q +1)(q +1) 1 M = A B + A B +d˜ A B + ǫ a ǫ a .(2.27) AB (D−2)2 (D−2)2 (D−2)2 2 A A B B α=1 X 4 We require that all the branes be independent, and so E are independent functions. We thus A learn from Eq. (2.26) that ′ c 1 AM +rd˜+1 δ = 0, (2.28) 2 AB E˜ AB (cid:18) A(cid:19) the off-diagonal part of which is M = 0 for A 6= B. As shown in Ref. [34, 27], this condition AB leadstotheintersection rulesfortwobranes. Ifq -braneandq -braneintersectoverq¯(≤ q ,q ) A B A B dimensions, this gives (q +1)(q +1) 1 A B q¯= −1− ǫ a ǫ a . (2.29) A A B B D−2 2 The rule (2.29) tells us that D1-branes with a = 1 can intersect with D3-brane with a = 0 1 3 on a point (q¯= 0) and with D5-brane with ǫ a = −1 over a string (q¯= 1), and D5-brane can 5 5 intersect with D5-brane over 3-brane (q¯= 3), in agreement with Refs. [36, 37, 38]. The second term in (2.28) must be constant. This, in particular, means 2(D−2) 1 H = , (2.30) A s ∆A E˜A is a harmonic function (rd˜+1H′ )′ = 0, (rd˜+1H′ )(cid:5) = 0, (2.31) A A where we have defined D−2 ∆ = (D−q −3)(q +1)+ a2. (2.32) A A A 2 A Note, however, that the condition (2.31) allows u-dependent term Q A H = h (u)+ , (2.33) A A rd˜ where h is an arbitrary function of u and Q is a constant. This class of solutions generalize A A those discussed in [27]. They are also similar to those discussed in [28] though time-dependence is taken differently. Using (2.30) in (2.25), we find D−q −3 A Ξ = − lnH +ξ(u), A ∆ A A X (α) δ Z = − A lnH +ζ (u), α A α ∆ A A X q +1 A B = lnH +β(u), A ∆ A A X D−2 Φ = ǫ a lnH +φ(u), (2.34) A A A ∆ A A X where ξ,ζ ,β,φ are functions of u only. It follows from the definition and the solutions (2.34) α that U reduces to d−2 U = 2ξ(u)+ ζ (u)+d˜β(u), (2.35) α α=1 X 5 consistent with our ansatz that U depends only on u. Thecondition(2.20)and(2.21)or(2.22),combinedwiththedefinition(2.16)andthesolution, gives ǫ a φ+2 ζ +2d˜β = 0, (2.36) A A α αX∈/qA 2(D−2) c = d˜Q . (2.37) A A s ∆A We then find that M = ∆A δ . It turns out that using the intersection rules, the condi- AB D−2 AB tion (2.11) is reduced to (H˙ )′ = 0. (2.38) A Namely we find that the harmonic function can be, at most, a sum of the functions of r and u. This is consistent with our previous result (2.33) and gives no further constraint. Note that separable forms for the metric of the type (2.34) was assumed from the beginning in [17, 19, 27, 29], but here we have naturally derived this property. Also the harmonic functions were taken to be independent of u, but they can be actually functions of u as well. We still have to take Eq. (2.10) into our account. This equation is rewritten as d−2 1 1 ′ W(u,r)+V(u)+ e2(Ξ−Zα)∂2K + e2(Ξ−B)r−(d˜+1) r(d˜+1)K′ = 0, (2.39) 2 α 2 αX=1 (cid:16) (cid:17) where (D−2)2 D−2 · · ·· W(u,r) ≡ (M +2)(lnH ) (lnH ) +2 (lnH ) AB A B A ∆ ∆ ∆ A B A A,B A X X · (lnH ) +4(D−2)(β˙ −ξ˙) A , (2.40) ∆ A A X d−2 d−2 1 V(u) ≡ ζ¨ +ζ˙2 +(d˜+2) β¨+β˙2 −2ξ˙ ζ˙ +(d˜+2)β˙ + (φ˙)2. (2.41) α α α 2 " # αX=1(cid:16) (cid:17) (cid:16) (cid:17) αX=1 Eq.(2.39) canberegardedformallyastheequationforK,whichisanelliptictypedifferential equation with respect to r and yα. However the source terms depend not only on r but also on u. Hence we have to solve the elliptic type differential equation at any time u. It may be very difficult to find the analytic solutions. Instead we may first assume K explicitly, and then solve Eq. (2.39). In this case, Eq. (2.39) must be regarded as a constraint equation for the formally solved variables Ξ,Z ,B and Φ. In this paper, we shall adopt the latter approach. α Here we assume m(u,yα) K = e−2ξ(u)k(u,yα)+ , (2.42) rd˜ with d−2 d−2 k(u,yα) = k (u)+ k (u)yα+ k (u)yαyβ + e2ζα(u)h (u)(yα)2,(2.43) 0 α αβ αα αX=1 α,β=X1(α6=β) αX∈∀qA d−2 m(u,yα) = m (u)+ m (u)yα, (2.44) 0 α α=1 X 6 where k (u),k (u),k (u),h (u),m (u) and m (u) are arbitrary functions of u. Here the sum 0 α αβ αα 0 α α ∈∀q is taken only over yα coordinates belonging to all the branes. A Given K(u,r,yα), we find that u-dependent terms (ξ,ζ ,β and φ) are constrained by two α conditions (2.36) and (2.39). The solution is then given by 2qA+1 −2D−2 ds2 = H ∆A e2ξ(u) H ∆A −2dudv+K(u,r,yα)du2 D A A " A A Y Y (cid:0) (cid:1) d−2 γ(α) −2 A + H ∆A e2ζα(u)(dyα)2+e2β(u) dr2+r2dΩ2 , A d˜+1 # αX=1YA (cid:16) (cid:17) 2(D−2) D−2 E˜ = H−1, Φ = ǫ a lnH +φ(u), (2.45) A s ∆A A A A ∆A A A X with two constraints (2.36) and (2.39), where H and K are given by Eqs. (2.33) and (2.42), A respectively, and D−2 yα belonging to q -brane γ(α) = for A . (2.46) A 0 otherwise (cid:26) (cid:26) Note that we still have one gauge freedom for the time coordinate u, by which we can choose any function for ξ(u). To give solutions explicitly, Eqs. (2.36) and (2.39) must still be solved. Let us now discuss explicit solutions. 3 Solutions with time-independent harmonic functions Toseetherelationofourresultswithearlierwork,letusfirstdiscusssolutionswithu-independent harmonic functions, i.e., Q (0) A H = h + , (3.1) A A rd˜ where h(0) and Q are constants. In this case, since H˙ = 0, we have W = 0. A A A We now discuss two examples. 3.1 Branes with factorized metrics in time and space If h (u) = 0(α ∈∀ q ), the conditions on u-dependent terms (ξ,ζ ,β and φ) should satisfy are αα A α Eq. (2.36) and Eq. (2.39) with W = 0, i.e. d−2 d−2 1 ζ¨ +ζ˙2 +(d˜+2) β¨+β˙2 −2ξ˙ ζ˙ +(d˜+2)β˙ + (φ˙)2 = 0. (3.2) α α α 2 " # αX=1(cid:16) (cid:17) (cid:16) (cid:17) αX=1 The solutions discussed in [19, 29] belong to this class. They consider a single D3-brane with d = 4,d˜= 4 and take R4 H = , e2ξ = e2ζ1 = e2ζ2 ≡ef(u), K = β = 0, a = 0, Φ= φ(u), (3.3) 3 r4 A 7 where R is a constant. The metrics here are of the factorized form in u- and r-dependent terms. Eq. (2.36) is trivially satisfied, and Eq. (3.2) gives 1 1 f¨− f˙2+ φ˙2 = 0, (3.4) 2 2 inagreementwiththeirresult. Herewehavemoregeneralintersectingsolutionswiththefunction K. 3.2 Branes in pp-wave backgrounds Here, we give an example with the pp-wave, K =e−2ξ(u)k(u,yα), (3.5) with (2.43). This is just the case with m = 0 in Eq. (2.42). The condition (2.39) reduces to V + h = 0, (3.6) αα αX∈∀qA where V is defined by Eq. (2.41). If this condition and Eq. (2.36) are satisfied, the solution is Eq. (2.45) with (3.5). Foracheck, letuscomparewiththeD3-branesolutionin[17],inwhichtheyhaved= 4,d˜= 4 and R4 H = , e2ξ ≡ k2 (u) , e2ξK = k(u,yα) ≡ h (u,r,yα), r4 CH CH ζ = ζ = β = 0, a = 0, Φ = φ(u) ≡ φ (u), (3.7) 1 2 A CH where k ,h ,φ are the variables adopted in [17]. Again Eq. (2.36) is trivial and Eq. (3.6) CH CH CH reduces to 1 φ˙2 = −h −h , (3.8) 11 22 2 in agreement with Eq. (12) in [17]. As a more interesting case, let us consider D1-D5-brane solution: 1 4 ds2 = H−43H−41e2ξ(u) −2dudv +K(u,yα)du2 + H1 4 eζα(u)dy2 1 5 H α (cid:18) 5(cid:19) α=1 (cid:2) (cid:3) X 1 3 + H4H4e2β(u) dr2+r2dΩ2 , 1 5 3 1 H1 2 (cid:0) (cid:1) Φ = ln +φ(u). (3.9) H (cid:18) 5(cid:19) In this case, K dependson yα linearly because only one spatial dimension in u-v coordinates can intersect, and so we take d−2 K = e−2ξ(u) k (u)+ k (u)yα . (3.10) 0 α ! α=1 X 8