Physics Letters B 599 (2004) 313–318 www.elsevier.com/locate/physletb Supergravity approach to tachyon condensation on the brane–anti-brane system J.X. Lu a,b,c, Shibaji Roy d a Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, PR China b Interdisciplinary Center of Theoretical Studies, Chinese Academy of Sciences, Beijing 100080, PR China c Michigan Center for Theoretical Physics, Randall Laboratory, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA d Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India Received 25 May 2004; accepted 23 August 2004 Available online 7 September 2004 Editor: L. Alvarez-Gaumé Abstract We study the tachyon condensation on the D-brane–anti-D-brane system from the supergravity point of view. The non- supersymmetric supergravity solutions with symmetry ISO(p, 1) × SO(9 − p) are known to be characterized by three pa- rameters. By interpreting this solution as coincident N Dp-branes and N¯ D¯ p-branes we give, for the first time, an explicit representation of the three parameters of supergravity solutions in terms of N, N¯ and the tachyon vev. We demonstrate that the solution and the corresponding ADM mass capture all the required properties and give a correct description of the tachyon condensation advocated by Sen on the D-brane–anti-D-brane system. 2004 Elsevier B.V. All rights reserved. It is well known that a coincident D-brane–anti-D- scription using either the string field theory approach brane pair (or a non-BPS D-brane) in type II string [3,4] or the tachyon effective action approach [5] on theories is unstable due to the presence of tachyonic the brane. However, a closed string (or supergravity) mode on the D-brane worldvolume [1]. As a result, understanding of this process is far from complete these systems decay and the decay occurs by a process and the purpose of this Letter is precisely to have a known as tachyon condensation [2]. Tachyon con- closed string understanding of the tachyon conden- densation is well understood in the open string de- sation. An earlier attempt in this direction has been made in [6] by giving an interesting interpretation to the previously known [7,8] non-supersymmetric, E-mail addresses: [email protected] (J.X. Lu), three parameter supergravity solutions with a symme- [email protected] (S. Roy). try ISO(p, 1) × SO(9 − p) in ten space–time dimen- 0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.08.047 314 J.X. Lu, S. Roy / Physics Letters B 599 (2004) 313–318 sions as the coincident Dp–D¯ p system. The three pa- examining how it reduces to a supersymmetric con- rameters in this solution were argued [6] to be related figuration which either corresponds to a BPS N Dp- (although the exact relations were not given) to the branes (for N¯ = 0) or BPS N¯ D¯ p-branes (for N = 0) physically meaningful parameters, namely, the num- or the final supersymmetric state at the end of tachyon ber of Dp-branes (N), number of D¯ p-branes (N¯ )and condensation. We also expect in taking the BPS limit the tachyon vev1 of the Dp–D¯ p system. that only one parameter corresponding to the num- In this Letter we use the static counterpart of the as- ber of branes remains and the other parameters of ymptotically flat time dependent supergravity solution the solution get automatically removed. For general obtained in [9]. This is also a non-BPS, three parame- case when both N and N¯ are non-zero, the solution ter solution with the symmetry ISO(p, 1) × SO(9 − p) is not supersymmetric and there must be a tachyon and can be identified with the solution given in Ref. [6] on the worldvolume of Dp–D¯ p system belonging to once the proper parameter relations are given. This the complex (N, N)¯ representation of the gauge group solution can also be naturally interpreted as the coin- U(N) × U(N)¯ . The end of the tachyon condensation cident N Dp-branes and N¯ D¯ p-branes system given should give BPS (N − N)¯ D-branes if N>N¯ or BPS its aforementioned symmetry ISO(p, 1) × SO(9 − p) (N¯ − N) anti-D-branes if N>N¯ . We will give a gen- (rather than R × SO(p) × SO(9 − p) as in a black eral description for arbitrary N and N¯ where N = N¯ p-brane case) and its non-BPS nature. Given this, we appears as a special case. We will show how the in- should be able to gain some information about the terplay of the parameters describes the tachyon con- δ-function source to the bulk equations of motion, and densation in accordance with the conjecture made by therefore the worldvolume action of the Dp–D¯ p sys- Sen [1,2] for the Dp–D¯ p system. The recognition of tem in the presence of the tachyon. So long as the bulk having a supersymmetric background at the end of configuration is concerned, the worldvolume fields (in the tachyon condensation is crucial for us to find ex- particular the tachyon) do not need to satisfy their plicit representation of the parameters of the solution respective worldvolume equations of motion (for ex- in terms of N, N¯ and the tachyon2 vev T . ample, we can put worldvolume scalars and tachyon In order to understand the tachyon condensation, to constants and other fields to zero). In other words, we look at the expression of the total ADM mass of they can be off-shell. In this way, the tachyon vev will the solution representing the total energy of the sys- appear as a parameter labelling the solution. The static tem. We then express this total energy in terms of gravity configuration represents a given instant bulk the three physical parameters namely, N, N¯ and T configuration during the process of tachyon conden- of the Dp–D¯ p system using the aforementioned re- sation. Since there is no direct link between the pa- lations. The total energy can be seen to be equal or rameters characterizing the bulk configuration and the less than the sum of the masses of N Dp-branes and N, N¯ and the tachyon vev, our approach in this Letter N¯ D¯ p-branes, indicating the presence of tachyon con- is based on producing the required properties of the tributing the negative potential energy to the system. tachyon condensation from the bulk configuration. In We will see that the energy expression gives the right contrast to the attempt made in [6], we approach the picture of tachyon condensation conjectured by Sen problem by giving, for the first time, an explicit repre- [1,2]. We will reproduce all the expected results from sentation of the parameters of the solution in terms of this general mass formula under various special limits N, N¯ and the tachyon vev of the Dp–D¯ p system. for N = N¯ as well as N = N¯ at the top and at the We proceed as follows. Once the supergravity so- bottom of the tachyon potential. We will also show lution under consideration is realized to represent the how the various known BPS supergravity configura- N D-brane and N¯ anti-D-brane system, we can gain tions can be reproduced in these special limits. information about the parameters of the solution by ∗ 2 When N = N¯ = 1, tachyon is a complex field and |T |2 = TT . 1 By tachyon vev we mean the classical value of the tachyon for But for N,N>¯ 1, tachyon is a matrix and in that case we follow [6] | |2 = 1 ∗ which the total energy of the system takes a particular value which to define T N Tr(T T ). Here and in the rest of the Letter we includes the extremum as well as off-shell values [6]. denote |T | as T for simplicity. J.X. Lu, S. Roy / Physics Letters B 599 (2004) 313–318 315 The static, non-BPS supergravity p-brane solution for the NSNS branes. Also b is the magnetic charge analogous to the time dependent solution obtained in and the Vol(Ω8−p) is the volume-form of the (8 − p)- [9] has the form in d = 10, dimensional unit sphere. Without any loss of gener- ality, we take (α + β),b,θ 0 as we did already in − 7−p 2 = 8 − 2 + 2 +···+ 2 ds F dt dx1 dxp the above. We note from (2) that the solution has a p+1 2 curvature singularity at r = ω and the physically rel- + 8 ˜ 7−p 2 + 2 2 F (H H) dr r dΩ8−p , evant region is r>ω.Asr →∞, H,H˜ → 1and 2δ so, F → 1. The solution is therefore asymptotically − H e2φ = F a , flat. In order to obtain the electrically charged solu- H˜ tion we dualize F[8−p] in (1) and get the gauge field F[ −p] = b Vo l(Ω −p), (1) 8 8 A[p+1] as where we have written the metric in the Einstein C 0 p A[ + ] = sinh θ cosh θ dx ∧···∧dx , (6) frame. Note that the metric has the required sym- p 1 F metry and represents the magnetically charged p- brane solution. The corresponding electric solution where C is defined as can be obtained from (1) by dualizing the field strength H α H˜ β F[ + ] = eaφ ∗ F[ − ],where∗ denotes the Hodge C = − .