Time-Dependent Supergravity Solutions in Null Dilaton Background

Physics Letters B 640 (2006) 214–218 www.elsevier.com/locate/physletb

Time-dependent supergravity solutions in null dilaton background

Rashmi R. Nayak a, Kamal L. Panigrahi a,b,∗, Sanjay Siwach c

a Dipartimento di Fisica, Università and INFN, Sezione di Genova, “Genova”, Via Dodecaneso 33, 16146 Genova, Italy b Physics Department, Indian Institute of Technology, Guwahati 781 039, India c Centre for Theoretical Physics, Seoul National University, Seoul 151-747, South Korea Received 3 July 2006; received in revised form 26 July 2006; accepted 27 July 2006 Available online 7 August 2006 Editor: M. Cveticˇ

Abstract A class of time dependent pp-waves with NS–NS ﬂux in type IIA string theory is considered. The background preserves 1/4 supersymmetry and may provide a toy model of Big Bang cosmology with nontrivial ﬂux. At the Big Bang singularity in early past, the string theory is strongly coupled and matrix string model can be used to describe the dynamics. We also construct some time dependent supergravity solutions for D-branes and analyze their supersymmetry properties. © 2006 Elsevier B.V. All rights reserved.

1. Introduction tions of the string theory with a null dilaton are considered in the literature [11] and the matrix string theory of a class of pp- The study of time dependent backgrounds in string theory is wave background has been analyzed [9,18] (see also [29,30] for a challenging problem. Until now quite a few time dependent some early studies of time dependent pp-waves). The AdS/CFT solutions in string theory/supergravity are known. Recently a correspondence has also been considered for a time dependent simple cosmological background with a null dilaton has been type IIB background [17,19–21] and the dual gauge theory has proposed [1] as a toy model of Big Bang cosmology. The back- been realized as a time dependent supersymmetric Yang–Mills ground preserves half of the thirty-two supersymmetries and theory living on the boundary. In AdS/CFT correspondence, su- one expects that there is a good control over the singularity at persymmetry plays a very important role and the existence of early times. The matrix degrees of freedom, rather than point supersymmetry gives us a better control over the nonperturba- particles or the perturbative string states describe the correct tive behavior of the dual quantum field theory. In this sense, the physics near the singularity. In discrete light cone type of quan- time dependent backgrounds preserving some supersymmetry tization, a matrix string [2–4] with a time dependent coupling are useful tools for the study of the quantum gravity and dual constant is used to study the dynamics at the Big Bang. The suc- field theories. The study of open strings in time dependent back- cess of the model has to be tested by the further investigation grounds is also an interesting problem. In this context, some of the relevant matrix model and its cosmological predictions. D-brane solutions in a light-like linear dilaton background has Some steps have already been taken in this direction [5–27] and also been discussed in [31] by taking a suitable Penrose limit of the future studies may seed the revolution for the string cosmol- an intersecting brane solution in supergravity. ogy. See [28] for a recent review of the subject. In this Letter, we study the type IIA pp-waves with NS–NS A nontrivial extension of the background is to include the three form flux in a light-like linear dilaton background. This RR and NS–NS fluxes and some time dependent pp-wave solu- can be considered as the simplest extension of the model of [1] with nontrivial flux. We study the supersymmetry of the background and show that it preserves 1/4 of type IIA supersym- * Corresponding author. metry. At early times, the string theory is strongly coupled and E-mail addresses: [email protected] (R.R. Nayak), [email protected], [email protected] (K.L. Panigrahi), the matrix string description may be used. We also construct [email protected] (S. Siwach). some time dependent supergravity p-brane solutions preserving

0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.07.057 R.R. Nayak et al. / Physics Letters B 640 (2006) 214–218 215 some fraction of supersymmetry in this background. The solu- Solving the above equation, one gets the affine parameter tions have singularities at early times similar to the Big Bang = + model but perhaps the matrix degrees of freedom may be used σ exp Qx /2 , (7) to resolve the singularity. upto a reparameterization. Therefore the singularity at x+ → The rest of the Letter is organized as follows. In Section 2 we −∞ correspond to σ = 0, and it has finite affine distance to all study the pp-waves with three form NS–NS flux as a toy model points interior of the spacetime and the spacetime is geodesi- of Big Bang cosmology. We analyze the geodesic equations in cally incomplete. One can try to compute the Riemann tensors this background and also give the matrix string description of and can show that there is a curvature singularity at σ = 0 and this class of solutions. We find that the background preserves gives a divergent tidal force felt by the inertial observer. At 1/4 of the type II supersymmetry. In Section 3, we present late times, x+ →∞the affine parameter diverges and it cor- classical solutions of D-branes and analyze the supersymme- responds to the asymptotic region of spacetime. try properties of these solutions. We show that they preserve 1/8 supersymmetry. Finally in Section 4, we present our con- 2.1. Supersymmetry clusions. The supersymmetry variation of dilatino and gravitino in string frame is given by 2. The type II pp-waves with null dilaton 1 1 δλ = Γ μ∂ φ − Γ μνρH Γ +···, (8) We begin with the following ansatz for the background met- 2 μ 12 μνρ 11 ric and other fields, 1 1 aˆbˆ + − + + + δΨμ = ∂μ + w ˆ − H ˆΓ11 Γ +···, (9) ds2 =−2 dx dx − μ2 x xi 2 dx dx 4 μaˆb 2 μaˆb i where the dots stand for the terms coming for the RR charges, + dxi dxi + dya dya, and we have used (μ,ν,ρ) to describe the ten-dimensional + + spacetime indices, and hated indices represent the flat tangent Φ =−Qx ,H+12 = H+34 = 2f x . (1) space indices. Note that the dilaton is linear in light cone time and the indices Vielbeins and spin connection for the above metric are, run as i = 1,...,4,a= 5,...,8. Only nonzero component of +ˆ −ˆ −ˆ 1 2 2 iˆ iˆ Ricci curvature computed from the above metric is e+ = 1,e− = 1,e+ = μ x ,e= δ , i j j + 2 = 2 ˆ R++ 4μ x (2) aˆ = aˆ −ˆ i = 2 i eb δb ,ω+ μ x . (10) and to be a solution of type IIA supergravity equations of mo- For our metric and linear null dilaton background, putting the tion, one should satisfy supersymmetry variations of dilatino and gravitino equal zero 1 to gives, μ2 =− Φ¨ + f 2 = f 2, (3) 2 ˆ 1 ˆ ˆ ˆ + ≡ − + − +ij = where dots denotes derivative with respect to x .AttheBig δλ QΓ Γ H+ij Γ11 0, + 12 Bang, x →−∞ the dilaton diverges and string theory is 1 2 i +ˆ iˆ 1 + 1ˆ2ˆ 3ˆ4ˆ strongly coupled and a DLCQ type of description can be given δΨ+ ≡ ∂+ − μ x Γ − μ x Γ + Γ Γ11 = 0, (see below). To study the nature of singularity at early times let 2 2 ≡ = = = us consider the geodesic motion of a point particle in our back- δΨ− ∂− 0,δΨa ∂a 0, ground. For this we change our metric to the Einstein frame 1 +ˆ jˆ δΨ ≡ ∂ − H+ Γ Γ = 0. (11) metric, i i 8 ij 11 + + − + + + ds2 = eQx /2 −2 dx dx − μ2 x xi 2 dx dx The dilatino variation is solved by imposing E +ˆ + dxi dxi + dya dya . (4) Γ = 0, (12) + At x →−∞, the metric components shrink to zero, which which breaks half of the supersymmetry. The δΨ+ = 0 equation corresponds to Big Bang singularity. We would like to study the further requires the condition geodesic equation for a test particle near the singularity. Null ˆ ˆ ˆ ˆ geodesic in the spacetime at constant X−,xi and at Xa = 0is Γ 12 + Γ 34 = 0 (13) given by which breaks another half of the supersymmetry. Thus the back- 2 + α β d X + dx dx ground preserve 1/4 supersymmetry. + Γ = 2 αβ 0 (5) dσ dσ dσ 2.2. Matrix string description which, for our metric, can be written as Near the Big Bang singularity, the dilaton diverges and the d2X+ Q dx+ 2 + = 0. (6) string theory is strongly coupled. However we can do a discrete dσ2 2 dσ light cone type of quantization (DLCQ) at the early times. For 216 R.R. Nayak et al. / Physics Letters B 640 (2006) 214–218 the DLCQ of the background, we pick up the direction y8 and 3. D-branes solutions make the following identifications, + − + − In general it is a difficult task to find p-brane solutions of x ,x ,y8 ∼ x ,x + R,y8 + R . (14) supergravity in time dependent backgrounds. However in the Making a Lorentz transformation as in [1] linear dilaton background the following factorized ansatz solve + − 5 the type II supergravity equations of motion. + + − X X Y x = X ,x= + + , f − 1 + − + + 2 2 2 = 2 2 − − 2 2 + 2 + ds e H 2 dx dx μ x x dx dyα y8 = X + Y 8,x1,2,3,4 = X1,2,3,4,y5,6,7 = Y 5,6,7 (15) i − f 1 2 + e 2 H 2 dx , and applying the T duality along Y 8 followed by an S duality a − − 2Φ − (7 p) f (3 p) + we get the IIB background, e = e 2 H 2 ,H+12 = H+34 = 2μ x , 2 QX+ + − 2 + 2 i 2 + + 2f −1 ds = re −2 dX dX − μ X X dX dX F+− = e ∂ H (for p 3), α...a a + dXi dXi + dYa dYa , F = ∂ H(for p 3), (21) a1...an a1...ana a + + Φ = Q X + log r, F+12 = F+34 = 2f X , where i = x1,...,x4, and xα and xa correspond to the di- i = 1,...,4,a= 5,...,8, (16) rections parallel and transverse to the Dp-brane, respectively. ˜ = + + = + r0 d = R 8 ∼ 8 + 2πls f Qx , is a function of x only and H 1 ( r ) (where where r 2πl and Y Y r . s d˜ = 7 − p in ten dimensions), is the harmonic function in trans- Let us focus on the DBI action of a single D1-brane in this + background, verse space and is independent of x . However, asymptotically the above solutions do not go to 1 − S =− dτ dσ e Φ the pp-wave background with null linear dilaton discussed in D1 2 2πls the previous section. To remedy the situation one has to give × − μ ν + 2 up the factorized ansatz and replace the harmonic function in det ∂αX ∂β X Gμν 2πls Fαβ . (17) + ˜ the transverse space by H → 1 + e−Qx ( r0 )d and modify and = 8 = σ + − r Let us take Fαβ 0, Y r and X and X depending on rewrite the field strengths accordingly. Explicitly one can check τ only. The equations of motion derived from the above action that the following solutions obey the supergravity equations of + − require, ∂τ X = ∂τ X and a classical solution is given by, motion, 1 τ 1 τ 1 + − 8 2 ˜ − 1 + − 2 + 2 + 2 2 X = √ ,X= √ ,Y= σ, ds = H 2 −2dx dx − μ x x dx + dy r 2 r 2 r i α 1 5,6,7 i + ˜ 2 2 Y = 0,X= 0. (18) H dxa , − + 1 τ 8 σ 2Φ −2f ˜ (3 p) + Choosing the gauge, X = √ ,Y = and defining the new e = e H 2 ,H+ = H+ = 2μ x , r √2 r 12 34 − 1 √τ f −1 variable Z as, X = + 2Z, and expanding the action +− = ˜ r 2 F α...a e ∂aH (for p 3), around the classical solution upto quadratic terms in the fluctu- f ˜ Fa ...a = a ...a ae ∂aH(for p 3), (22) ations, we get, 1 n 1 n + ˜ −Qx+ r d˜ 2 where f = Qx as before and H = 1 + e ( 0 ) . 1 1 1 (μ ) 2 r S = dτ dσ − + Xi We would like to study the geodesic equations of a point D1 2πl2 r2 2 r2 s particle in this background. To see the nature of the trajectory, 1 2 2 i 2 i 2 we pass on to the Einstein frame and the metric for the Dp- + (∂τ Z) − (∂σ Z) + ∂τ X − ∂σ X 2 brane is given by, 5,6,7 2 5,6,7 2 + ∂τ Y − ∂σ Y − 2 f/2 ˜ − 7 p + − 2 + 2 + 2 2 √ ds = e H 8 −2 dx dx − μ x x dx + dy 2 Qτ E i α + 2π2l4 exp − F 2 . (19) p+1 s τσ + f/2 ˜ 8 2 r e H dxa . (23) This is the action for a single D1 branes that follows from the We would like to examine the trajectories near the singularity matrix string action [2,3] for N D1 branes, − α a = at constant X ,y and at X 0. Namely we check the trajectories moving along the X+, which is given by 1 1 i 2 1 a 2 S = tr DμX + DμY D1 2πl2 2 2 ˙ s d2X+ f˙ p − 7 H˜ dx+ dx+ 1 (μ )2 1 + + = 0. (24) + Xi 2 + g2l4π 2F 2 − Xi,Xj 2 dσ2 2 8 H˜ dσ dσ 2 s s μν 2 2 4 2 r 4π gs ls Near the Big Bang singularity (at x+ →−∞), the above equa- 1 1 − Y a,Yb 2 − Xi,Ya 2 +··· , 2 2 4 2 2 4 tion can be written as 4π gs ls 4π gs ls d2X+ dx+ dx+ (20) + c = 0, (25) where the dots stand for fermionic terms. dσ2 dσ dσ R.R. Nayak et al. / Physics Letters B 640 (2006) 214–218 217 where f = Qx+ and c = ((11 − p)/8)Q. Hence the affine pa- we are left with the following equations to be solved rameter is ˜ ˜ 1 ∂+H 1 ∂ H + − = − a = = cx ∂+ 0, ∂a 0. (33) σ e (26) 8 H˜ 8 H˜ upto an affine reparameterization invariance. The singularity at The above equations are solved by the spinor of the form = x+ →−∞correspond to σ = 0, and can be approached in a fi- ˜ 1 H 8 0, with 0 being a constant spinor. Now putting together the nite affine time and the spacetime is geodesically incomplete. conditions, (29), (30), and (32), we conclude that our solutions + →∞ On the other hand, at x , the string coupling gs goes to (22) preserve 1/8 supersymmetry. zero, and the spacetime theory is free at late times. So the geodesic behavior is qualitatively similar to that of our background 4. Conclusions in the previous section. In this Letter, we discussed a class of time dependent pp- 3.1. Supersymmetry waves with NS–NS three form flux in null linear dilaton background. It may serve as a toy model of Big Bang cosmology Next we would like to analyze the supersymmetry of the Dp- with nontrivial flux. At the beginning of the time, the dilaton brane solutions presented above. The supersymmetry variation diverges and the singularity may be interpreted as Big Bang of the gravitino and dilatino fields in type IIA supergravity in singularity at the origin of the time. The background preserves string frame is given by [32,33], 1/4 supersymmetry as opposed to 1/2 supersymmetry of the 1 1 background in the absence of the flux. The geodesic equation δλ = Γ λ∂ Φ − Γ μνρH Γ 2 λ 12 μνρ 11 near the singularity has been analyzed. Near the singularity, the perturbative string theory fails as dilaton is divergent, however 1 3 1 + eΦ F 0 − Γ μνF (2)Γ + Γ μνρσ F (4) , one can use a discrete light cone type of quantization and the 5 ! μν 11 ! μνρσ 8 2 4 corresponding matrix string description is given.

(27) We have also constructed a class of D-brane solutions in this 1 aˆbˆ 1 aˆbˆ δΨ = ∂ + ω ˆΓ − H ˆΓ Γ μ μ 8 μaˆb 8 μaˆb 11 background, by solving the type II ﬁeld equations explicitly.

Φ Though we restrict ourselves to the type IIA brane solutions + e (0) − 1 μν (2) + 1 μνρσ (4) in this Letter, the general expression is also valid for p-branes F Γ Fμν Γ11 Γ Fμνρσ Γμ , 8 2! 4! in type IIB theory as well. We have also worked out the geo- (28) desic equations, and the nature of singularity at early times. The wherewehaveused(μ,ν,ρ) to describe the ten-dimensional supersymmetry variation revealed that this class of branes pre- spacetime indices, and the hated indices are the correspond- serve 1/8 of the full types IIA spacetime supersymmetry. ing tangent space indices. Solving the dilatino variations for the There are several directions for future work. The lift of the Dp-brane solutions presented in (22) gives the following con- background to eleven dimensional M-theory is straightforward. ditions on the spinor, One can extend the discussion with RR three form ﬂux and cor- +ˆ responding type IIB description can be written down. The list Γ = 0, (29) of D-brane solutions given by us, is based on a clever ansatz and and perhaps not exhaustive. One may try to ﬁnd more time dependent solutions and particularly some intersecting branes in ˆ +ˆ −ˆ ˆ ˆ ˆ Γ a − Γ α2...αpa = 0 (for p 3), this background. A matrix string description can also be given

aˆ 1 aˆ aˆ ...aˆ aˆ to brane solutions presented in this Letter. Γ + ˆ ˆ ˆ Γ 1 2 n = 0 (for p 3). (30) (n + 1)! a1...ana Acknowledgements We would like to emphasize that we need to impose both the conditions, (29) and (30), for the dilatino variation to be satis- We would like to thank D. Bak, S.F. Hassan, and A. Kumar ﬁed. Next we would like analyze the gravitino variation, for various useful discussions. The work of R.R.N. was sup- ˜ 1 ∂+H μ − 1 ˆ ˆ ˆ ˆ ported by INFN. The work K.L.P. was supported partially by ≡ − − ˜ 2 12 + 34 = δΨ+ ∂+ H Γ Γ Γ11 0, PRIN 2004, “Studi perturbativi e nonperturbativi in teorie quan- 8 H˜ 2 tistiche dei campi per le interazioni fondamentali”. The work of δΨ− ≡ ∂− = 0,δΨα ≡ ∂α = 0, ˜ S.S. was supported by Korea Science Foundation ABRL pro- 1 ∂aH gram (R14-2003-012-01001-0). δΨa ≡ ∂a − = 0, (31) 8 H˜ where in writing the above variations we have used the con- References dition (29) and the brane supersymmetry condition (30).Now using the condition [1] B. Craps, S. Sethi, E.P. Verlinde, JHEP 0510 (2005) 005, hep-th/0506180. [2] T. Banks, N. Seiberg, Nucl. Phys. B 497 (1997) 41, hep-th/9702187. 1ˆ2ˆ + 3ˆ4ˆ = [3] R. Dijkgraaf, E.P. Verlinde, H.L. Verlinde, Nucl. Phys. B 500 (1997) 43, Γ Γ 0, (32) hep-th/9703030. 218 R.R. Nayak et al. / Physics Letters B 640 (2006) 214–218

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