TCDMATH 08–02 Negative-Tension Branes and Tensionless 1Brane 2 8 in 0 0 Boundary Conformal Field Theory 2 n a J Akira Ishida(1), Chanju Kim(2), Yoonbai Kim(1), O-Kab Kwon(3) 2 2 (1)Department of Physics and BK21 Physics Research Division, ] h Sungkyunkwan University, Suwon 440-746, Korea t - p [email protected], [email protected] e h [ (2)Department of Physics, Ewha Womans University, Seoul 120-750, Korea 1 [email protected] v 5 5 (3)School of Mathematics, Trinity College Dublin, Ireland 4 [email protected] 3 . 1 0 8 0 : v Abstract i X r In the framework of boundary conformal field theory we consider a flat unstable Dp- a brane in the presence of a large constant electromagnetic field. Specifically, we study the case that the electromagnetic field satisfy the following three conditions: (i) a constant electric field is turned on along the x1 direction (E = 0); (ii) the determinant of the matrix 1 6 (η + F) is negative so that it lies in the physical region ( det(η + F) > 0); (iii) the 11- − componentofitscofactorispositivetothelargeelectromagneticfield. Inthiscase,weidentify exactly marginal deformations depending on the spatial coordinate x1. They correspond to tachyon profiles of hyperbolic sine, exponential, and hyperbolic cosine types. Boundary states are constructed for these deformations by utilizing T-duality approach and also by directly solving the overlap conditions in BCFT. The exponential type deformation gives a tensionless half braneconnecting theperturbativestringvacuum andone of thetruetachyon vacua,whiletheothershavenegative tensions. Thisisinagreementwiththeresultsobtained in other approaches. 1 1 Introduction Type IIA (IIB) string theory supports even-dimensional (odd-dimensional) stable BPS D-branes and odd-dimensional (even-dimensional) unstable nonBPS D-branes [1]. The physics of unstable D-branesinvolves variousnonperturbative aspects of string theory. Specifically, two representative examples are the tachyon solitons interpreted as lower-dimensional D-branes [2, 3] and the rolling tachyon describing homogeneous real-time decay process of the D-brane [4, 5]. The instability of a nonBPS Dp-brane in superstring theory or a Dp-brane in bosonic string theory results in the appearance of a tachyonic degree. In the context of boundary conformal field theory (BCFT), the tachyon vertex operator with a single spatial dependence which represents the exactly marginal deformation is given by a sinusoidal function with a single multiplicative parameter. When the parameter has the value 1/2, the deformation is interpreted as an array of D(p 1)-branes in bosonic string theory or as a periodic array of a pair of a D(p 1)-brane and a − − D¯(p 1)-brane in superstring theory [2, 3, 6]. The homogeneous rolling tachyon in BCFT is de- − scribed by introducing a marginal deformation corresponding to the tachyon profiles of hyperbolic sine, exponential, or hyperbolic cosine type. The physical quantities like the energy-momentum tensor suggest real-time decay of an unstable D-brane when the tachyon is displaced from the maximum of the tachyon potential and rolls down towards its minimum. Both homogeneous rolling tachyons and lower-dimensional D-branes from an unstable D-brane have also been studied in the context of effective field theories (EFTs) such as Dirac-Born-Infeld (DBI) type EFT [7,8, 9,10] andboundary string field theory (BSFT) EFT [11, 12, 13]. Compared with the BCFT approach, physical quantities such as energy-momentum tensor obtained from these EFTs are qualitatively the same as, but slightly different from that in BCFT [7]–[13]. For the case of the half S-brane with the exponential type of the tachyon profile [14] which is a special case of homogeneous rolling tachyons, the energy-momentum tensor based on the formula in Ref. [5] coincides exactly with that of DBI type effective action with 1/cosh potential [15]. When the fundamental strings exist in the worldvolume of unstable Dp-brane, they couple to the second-rank antisymmetric tensor field (or equivalently to the electromagnetic field strength tensor on the D-brane [16]) and the string current density is given by the Lorentz-covariant conjugate momentum of U(1) gauge field [17, 18, 19]. Then one may study the effect of the electromagnetic field. For rolling tachyons, the three types of deformations mentioned above are not changed by the presence of constant electric [20, 21, 22], or both electric and magnetic [23, 24] fields as far as they satisfy the physical condition, det(η + F) > 0 where F denotes the − electromagnetic field strength tensor. The situation is more interesting for the case of tachyon kinks which are identified as lower- dimensional D-branes of codimension one. The spectrum of the tachyon kink is not changed when the constant electric field is turned on with keeping det(η + F) > 0; only the period of − D(p 1)D¯(p 1) in the array becomes large as the electric field increases. However, when the − − electric field eventually reaches the critical value for which the determinant vanishes, the period 2 becomes infinite and we obtain a single regular BPS tachyon kink with constant electric flux. It is interpreted as a thick BPS D(p 1)-brane in the background of fundamental string charge − density. This has been checked in various languages including DBI EFT [9, 25], BCFT [26], noncommutative field theory (NCFT) [27], and BSFT [13]. In the presence of both constant electric and constant magnetic fields in an unstable Dp-brane with p 2, it turns out that other types of deformations are possible. This is because the 11- ≥ component of the cofactor C11 of (η+F) can have either negative or positive value while keeping µν det(η + F) > 0. (Here, x1 denotes the coordinate on which the tachyon depends.) For small − electromagnetic fields the cofactor C11 is negative. In this case the species of tachyon kinks are essentially the same as those without electromagnetic field. On the other hand, when p 2, ≥ electromagnetic fields can take large values for which C11 becomes positive while maintaining the condition det(η+F) > 0. In this case, three new codimension-one objects are supported, which − correspond to tachyon profiles of hyperbolic sine, exponential, and hyperbolic cosine types. These objects have been obtained in aforementioned EFTs including DBI EFT [9, 25], NCFT [28], and BSFT [13]. They, however, have not yet been reproduced in the context of BCFT for type II superstring theory. The purpose of this paper is to analyze these three kinks in the context of BCFT. In section 2, we describe new tachyon vertices of hyperbolic sine, exponential, and hyperbolic cosine types with the dependence on a single spatial coordinate, and show that they give marginal deformations in the context of BCFT. In addition, these tachyon profiles are obtained as static solutions of the linearized tachyon equation in the background of a nontrivial open string metric and a noncommutativity parameter of open string field theory (OSFT). In section 3 we construct the boundary states corresponding to the marginal deformations given in section 2. We utilize Lorentz transformation and T-duality in subsection 3.1 while the overlap condition is directly solved in subsection 3.2 to construct the boundary states. In section 4 we read the corresponding physical quantities, specifically the energy-momentum tensor T and the fundamental string µν current density Π . Forthecaseofhyperbolic sineandhyperbolic cosinetypesoftachyon profiles, µν they are slightly different from those in EFTs as expected [9, 25, 28], but, for the exponential type, they coincide exactly with the results of EFTs. They are interpreted as negative tension branes for hyperbolic sine and cosine profiles and tensionless half brane for exponential profile in the huge constant backgroundofpositiveenergydensity. Weconclude insection5. Intheappendix, wegive an alternative derivation of the energy-momentum tensor for the exponential type deformation by calculating the partition function of the worldsheet action following Ref. [29]. 3 2 New Tachyon Vertices as Exactly Marginal Deforma- tions In this section we show that there exist three new tachyon vertices as marginal deformations in the presence of the constant electromagnetic field. They are hyperbolic sine, hyperbolic cosine, and exponential types which depend on a single spatial coordinate. We shall show this first in the scheme of BCFT and then in the context of linearized OSFT. In the BCFT description of bosonic string theory, the worldsheet action of a Dp-brane in the presence of a background U(1) gauge field A is given by1 µ 1 i S = d2w∂Xµ∂¯X dtA (X)∂ Xµ, (2.1) BCFT µ µ t 2π − 2π ZΣ Z∂Σ where Σ denotes the worldsheet and t parametrizes the boundary of the worldsheet. It satisfies the boundary condition (∂ Xµ +iFµ ∂ Xν) = 0, (2.2) n ν t ∂Σ | where∂ and∂ denoterespectivelythenormalandtangentialderivativesattheboundary∂Σ. The n t dynamics of an unstable Dp-brane is described by introducing a conformally invariant boundary interaction to the worldsheet action, S = dtT(X). (2.3) T Z∂Σ When there is no background gauge field (A = 0), it is well-known that for any spatial µ direction X the operator T(X) = λcosX (2.4) is exactly marginal and has been used to describe lower-dimensional D-branes [2, 3]. The relevant boundary state is given by [30, 31, 32] j = Dj (R) j;m,m , (2.5) |BiX m,−m | ii j=0,1/2,1,...m=−j X X where R is the SU(2) rotation matrix cosπλ isinπλ R = , (2.6) isinπλ cosπλ (cid:18) (cid:19) Dj (R) is the spin j representation matrix of R in the J eigenbasis, and j;m,m is the m,−m z | ii Virasoro-Ishibashi state [33] built over the primary state j;m,m = j,m j,m . | i | i| i 1 Throughout this paper we use the α′ =1 unit. 4 The temporal version of (2.4) T(X0) = λcoshX0 (2.7) describes the dynamics of the rolling tachyon [4, 5]. For example the relevant boundary state for a Dp-brane with the deformation (2.7) is given by = p N 25 D ghost , (2.8) |Bi |BiX0 ⊗µ=1 | iXµ ⊗i=p+1 | iXi ⊗| i where ∞ 1 N = exp αµ α¯µ 0 , | iXµ − n −n −n | i " # n=1 X ∞ 1 D = exp αi α¯i 0 , | iXi n −n −n | i " # n=1 X ∞ ¯ ghost = exp (b c +b c¯ ) (c +c¯ )c c¯ 0 , (2.9) −n −n −n −n 0 0 1 1 | i − | i " # n=1 X and is the boundary state (2.5) in the Wick-rotated variable X iX0. X0 |Bi → − In the presence of a constant background electric field the operator (2.7) is modified by a Lorentz factor [20], T(X0) = λcosh(√1 E2X0), (2.10) − where E is the magnitude of the electric field. Rolling tachyons have further been generalized to the case that both electric and magnetic fields are turned on [23, 24]. In this section we would like to discuss the exactly marginal deformation S (2.3) in the T background of a general constant electromagnetic field for which we take the symmetric gauge, 1 A = F Xν. (2.11) µ µν −2 As usual [4], we first consider the Wick-rotated theory obtained by the replacement X0 iX0 → and make the inverse Wick-rotation back to the Minkowski time later. Under the deformed boundary condition (2.2), the correlation function on the upper-half plane is obtained as [34], Xµ(w)Xν(w′) = δµν ln w w′ +δµν ln w w¯′ h i − | − | | − | w w¯′ G¯µν ln w w¯′ 2 θ¯µν ln − . (2.12) − | − | − w¯ w′ (cid:18) − (cid:19) Here G¯ and θ¯µν are the Wick-rotated version of the open string metric Gµν and the noncommu- µν tativity parameter θµν given as 1 µν Cνµ 1 µν Cνµ Gµν = = S , θµν = = A , (2.13) η +F η +F (cid:18) (cid:19)S Yp (cid:18) (cid:19)A Yp 5 where Cµν and Cµν are respectively the symmetric and antisymmetric parts of Cµν, the cofactor S A matrix of (η +F) , and det(η +F). When w′ is on the boundary, (2.12) reduces to µν p Y ≡ w t Xµ(w)Xν(t) G¯µν ln w t 2 θ¯µν ln − , (2.14) ∼ − | − | − w¯ t (cid:18) − (cid:19) where t represents the boundary. Since the OPE of the energy-momentum tensor T = ∂Xµ∂X µ − with the tachyon boundary vertex operator eik·X(t) is G¯µνk k 1 T(w)eik·X(t) µ ν eik·X(t) + ∂ (eik·X(t)), (2.15) ∼ (w t)2 w t t − − this operator becomes marginal when G¯µνk k = 1. (2.16) µ ν In this paper we are interested in the operators which depend on a single spatial coordinate. For definiteness we take k δ and denote X = X1 for the sake of simplicity. In this case (2.16) µ µ1 ∝ reduces to k2 = 1/G¯11. Then the boundary operator is not only marginal but actually exactly 1 marginal [32] since the second term containing the noncommutative parameter in (2.14) plays no role. After the inverse Wick rotation, the marginality condition then leads to 1 k2 = = Yp . (2.17) 1 G11 C11 As an example, let us consider an unstable D2-brane (p = 2) with a constant electromagnetic field F = E (E2 = E2) and F = B. In this case, = (1 E2 +B2) and C11 = 1+E2 so that 0i i i 12 Y2 − − − 2 the marginality condition (2.17) becomes (1 E2 +B2) k2 = − − . (2.18) 1 (1 E2) − − 2 Before discussing the physical meaning of these marginal deformations, we extend our analysis to the superstring case with worldsheet fermions, ψµ and ψ¯µ, µ = 0,1,...,9. The worldsheet action with a constant electromagnetic field is 1 1 1 i S = d2z ∂Xµ∂¯X + ψµ∂¯ψ + ψ¯µ∂ψ¯ dt(A ∂ Xµ F ΨµΨν), (2.19) w µ µ µ µ t µν 2π 2 2 − 2π − ZΣ (cid:18) (cid:19) Z∂Σ where the fermions in the boundary interaction always appear as the following combination, 1 Ψµ = (ψµ +ψ¯µ). (2.20) 2 In addition to the boundary condition for bosonic degrees (2.2), we impose that for fermionic degrees, (η F )ψν = ǫ(η +F )ψ¯ν , (2.21) µν − µν ∂Σ µν µν ∂Σ (cid:12) (cid:12) (cid:12) 6 (cid:12) where ǫ = . Without the background gauge field, the following operator which represents the ± tachyon field T(x) = √2λcos(x/√2), i√2λψsin(X/√2) σ (2.22) 1 − ⊗ is known to be exactly marginal. Here we have assigned the Chan-Paton factorσ and the relevant 1 boundary state is j ,ǫ = Dj (R) j;m,m,ǫ , (2.23) |B iX,ψ m,−m | ii j=0,1,...m=−j X X where j;m,m,ǫ is the super-Virasoro-Ishibashi state built over the primary state j;m,m,ǫ | ii | i and ǫ = correspond to the two different boundary conditions for the fermions in (2.21). ± Taking the inverse Wick rotation, the deformation (2.22) describes the rolling tachyon in superstring theory [5]. The boundary state for the Dp-brane with this interaction is given by = ,+ , , (2.24) |Bi |B i−|B −i where ,ǫ = ,ǫ p N,ǫ 9 D,ǫ ghost,ǫ . (2.25) |B i |B iX0,ψ0 ⊗µ=1 | iXµ,ψµ ⊗i=p+1 | iXi,ψi ⊗| i Here ,ǫ is the boundary state (2.23) in the Wick-rotated variables X = iX0, ψ = iψ0, X0,ψ0 |B i − − and ψ¯ = iψ¯0, and the spatial and ghost parts are usual ones, which are respectively given by − ∞ ∞ 1 N,ǫ = exp αµ α¯µ iǫ ψµ ψ¯µ 0 , | iXµ,ψµ − n −n −n − −r −r| i n=1 r=1/2 X X   ∞ ∞ 1 D,ǫ = exp αi α¯i +iǫ ψi ψ¯i 0 , | iXi,ψi  n −n −n −r −r| i n=1 r=1/2 X X   ∞ ∞ ¯ ¯ ghost,ǫ = exp (b c +b c¯ )+iǫ (β γ¯ β γ ) Ω , −n −n −n −n −r −r −r −r | i − − | i n=1 r=1/2 X X Ω = (c +c¯ )c c¯ e−φ(0)−φ¯(0) 0 .  (2.26) 0 0 1 1 | i | i Now we consider the marginality condition of the tachyon vertex operator with momentum k in the presence of the constant electromagnetic field. The vertex operator in the zero-picture is given by √2k Ψeik·X(t). (2.27) − · Since the fermion Ψ has conformal weight 1/2, the marginality condition becomes 1 Gµνk k = . (2.28) µ ν 2 7 If we consider the operators which depend only on X1, this condition reduces to 1 k2 = = Yp . (2.29) 1 2G11 2C11 In the absence of the electromagnetic field, C11 = = 1 and hence k2 = 1 and 1/2, Yp − 1 respectively, for the bosonic and superstring case, as it should be. As the electromagnetic field is turned on C11 and change and so does k2. It has a positive value as long as both and C11 Yp 1 Yp are negative. In (2.18), this is the case when E2 < 1 with an appropriate B to keep negative. 2 Y2 The marginal tachyon vertex operator is then of the trigonometric type: T(X) = λcos(k X) or 1 λsin(k X). If we examine the corresponding energy-momentum tensor and the R-R coupling, the 1 resulting configuration with λ = 1/2 for pure tachyon case ( = C11 = 1) is interpreted as an p Y − array of D-branes for the bosonic case or an array of D(p 1)D¯(p 1) for superstrings [2, 3]. It − − was also discussed in the presence of electric field ( = 1+E2, C11 = 1) [26, 9, 25]. 1 Y − − On the other hand, if the electromagnetic field is sufficiently strong, and/or C11 can be p Y flipped to be positive. From the physical ground, the determinant should be negative and then p Y the question is whether C11 can become positive while keeping negative. It turns out that this p Y is possible when p 2 as we see in (2.18). Note that when E2 > 1, C11 becomes positive while ≥ 2 remains negative as long as the magnetic field B is sufficiently strong. p Y Then the tachyon profile becomes of the hyperbolic type T(X) = e±κX where −Yp for bosonic string, κ ik = C11 (2.30) 1 ≡  q−Yp for superstrings.  2C11 q This is in contrast with the rolling tachyons in which k02 = Yp/C00 is always negative. Actually C00 is positive irrespective of the values of constant electromagnetic field and the dimension p of D-brane. Therefore turning on the electromagnetic field does not give rise to a new type of deformation for the case of rolling tachyons. Without loss of the generality, the tachyon profile may be classified into the following three cases depending on the asymptotic behaviors, (i) λsinh(κX), T(X) = (ii) λexp( κX), (2.31)  α ± (iii) λcosh(κX), where α = 1 for bosonic string and α = √2 for superstrings. Note that the coordinate X is a spatial direction along the D-brane. Nevertheless the form of the operator looks like that of rolling tachyons thanks to the strong electromagnetic field. This deformation is however entirely physical and can be obtained through a chain of maps involving T-duality, Lorentz boost, and rotation [23] as discussed in subsection 3.1 where the corresponding boundary state is constructed. One comment is in order. For bosonic string, the boundary term is the same as the tachyon profile(2.31). Forsuperstrings, tachyon vertex operatorscorresponding to(2.31) inthe 1-picture − 8 are of the form e−φT(X). Since the picture number of the boundary term should be zero, each boundary term corresponding to the tachyon vertex (2.31) is obtained by picture-changing from 1 to 0-picture, − (i) 2iλκΨcosh(κX) σ , 1 ⊗ (ii) 2iλκΨexp( κX) σ , (2.32)  1 ± ± ⊗ (iii) 2iλκΨsinh(κX) σ , 1 ⊗ where the Chan-Paton factor σ is necessary to describe GSO-odd states. 1 The tachyon profiles (2.31) canalso beobtained in the framework of OSFT. Let us consider the linearized equations of motion in OSFT ignoring the interaction among various fields except the coupling to constant electromagnetic field. Near the perturbative string vacuum, the tachyonic degree due to the instability of an unstable D-brane can be described by a real scalar field T and its action in the presence of the constant electromagnetic field is expressed in terms of an open string metric Gµν and a noncommutativity parameter θµν in (2.13), Gµν m2 S = dp+1x√ G ∂ T ∂ T T T , (2.33) L µ ν − − 2 ∗ − 2 ∗ Z (cid:18) (cid:19) where G = detG and denotes star product between the tachyon fields. m2 < 0 is the square of µν ∗ the tachyon mass which is equal to 1 for bosonic string theory and 1/2 for superstring theory − − in our convention. Since the background electromagnetic field is constant on the flat D-brane with the metric η , both Gµν and θµν in (2.13) are also constant. In addition, every star product in the µν action (2.33), quadratic in the tachyon, can be replaced by an ordinary product and the equation of motion for the tachyon field becomes Gµν∂ ∂ T = m2T. (2.34) µ ν For the static kink configurations of codimension-one objects, we assume T = T(x),(x = x1), and then the equation of motion (2.34) reduces to C11T′′ = m2T, (2.35) p − −Y where the prime ′ denotes differentiation of x. To keep the role of spacetime variables, the deter- minant should be nonpositive and, to obtain nontrivial configurations, C11 should be nonvan- p Y ishing. As discussed in the previous subsection, the types of the solution of (2.35) depend on k2 = 1 /C11 m2 . When C11 is positive, the solution is given by (2.31). In the absence of the elec- p Y | | tromagnetic field, C11 = 1 and although the obtained tachyon configurations (2.31) are static − solutions of the linearized tachyon equation with the coupling of constant electromagnetic field, the linear tachyon system is obtained through a consistent truncation of full open string field equations restricting the fields to a universal subspace and then the obtained solutions (2.31) are expected to be solutions of full open string equations [4, 5, 20]. 9 3 Construction of Boundary States for New Codimension- one Objects In this section we construct the boundary state for the hyperbolic type of marginal deformation (2.31) along a spatial direction in the presence of a strong constant electromagnetic field. For definiteness we concentrate on a D25-brane in bosonic string theory. Then the case of superstring theory will briefly be discussed. The generalization to lower-dimensional D-branes is straightfor- ward. 3.1 T-duality approach In the following we shall construct the corresponding boundary state through a chain of transfor- mations starting from a well-established configuration which turns out to be the rolling tachyon in the presence of the constant electric field. The order of the transformations is as follows. We begin with a D25-brane where a constant electric field is turned on along the y1-direction and all the other excitations are set to zero except the rolling tachyon. We compactify the y2-direction on a circle and T-dualize the D25-brane to a D24-brane. Then we boost it twice: first along the y1-direction and then along the y2-direction. Subsequently we rotate it in the y1y2-plane. Finally we T-dualize it back along the y2-direction. The resulting configuration will be a D25-brane with the deformation given in (2.31). Let us first consider a flat D25-brane (p = 25) with rolling tachyon in a constant electric field. We denote the worldsheet fields of the brane by Yµ and assume that the constant electric field denoted by E˜ is along the y1-direction. The rolling tachyon is then described by the exactly 1 marginal boundary operator2 [20] λ dtcosh 1 E˜2Y0(t) . (3.1) − 1 Z (cid:18)q (cid:19) We compactify the y2-direction on a circle of radius R and wrap the D25-brane on it. Under the T-dualization of the y2-direction, the right-moving part of Y2 changes its sign while the other fields remain unchanged, T dual: Y2(z¯) Y2(z¯). (3.2) R −→ − R The D25-brane is then turned into an array of D24-branes on the dual circle. Taking the decom- ˜ pactification limit R we get a localized D24-brane with a constant electric field E turned 1 → ∞ on along the y1-direction. The Y2 part of the boundary state is given by the Dirichlet state, ∞ 1 D = exp β2 β¯2 δ(yˆ2) 0 , (3.3) | iY2 n −n −n | i ! n=1 X 2We display only the cosh-type operator for simplicity. 10