Boundary String Field Theory at One-loop

Transcription

1 Boundary String Field Theory at One-loop Taejin Lee, K. S. Viswanathan and Yi Yang Department of Physics, Kangwon National University Chuncheon , Korea, Department of Physics, Simon Fraser University Burnaby, BC V5A 1S6 Canada We apply the boundary string field theory BSFT) to a unstable D-brane system to study the open-closed string duality at one loop in the presence of the tachyon condensation. The partition function at one-loop level is calculated by using both open and closed string channels. We find that the results from two different channels coincide, thus the open-closed string duality holds even off-shell. taejin@cc.kangwon.ac.kr kviswana@sfu.ca yyangc@sfu.ca

2 1 Introduction The boundary string field theory BSFT) [1, 2] has been proved to be a useful tool to study the dynamics of the D-branes in string theory. It is this framework where the tachyon condensation on unstable D-brane systems has been discussed extensively in recent works [3,4,5,6,7,8,9,10,11,12]. Givenaspecificformofthetachyonprofile,theBSFT yields the string partition function, thus, the effective action for the tachyon potential in the most efficient way. The BSFT also has been extended to the case of superstring theories in refs. [3, 11, 13, 14, 15, 16, 17, 18]. The results obtained in those studies confirm Sen s conjecture; as the tachyon condensation develops, the unstable open string vacuum spontaneously rolls down to the stable closed string vacuum. The extension of the BSFT to the one-loop level is also studied recently in refs. [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. One may employ two different schemes to calculate the one-loop partition function. One is to use the open string channel, which adopts the open string Green function on an annulus [19, 20, 21, 22, 23, 24, 25, 26], and the other is to use the closed string channel [24, 27, 28, 29, 30], which adopts the boundary state formulation [31]. In the second scheme the geometry of the string world sheet is given by a cylinder. A of figure 1 depicts the one-loop diagram in the open string channel. If one once calculates the Green function on the annulus, the partition function is readily obtained as d dy ln Z = 1 2π dσ X 2. 1) 8π 0 As depicted by B of figure 1), the same diagram can be viewed in the closed string channel as follows; a closed string appears from a D-brane and propagates to another or the same) D-brane. The initial and the final states of the closed string are described by boundary states B and the one-loop partition function is obtained as Z = B exp [ τl 0 + L 0 ) ] B. 2) As is well known, the the partition functions obtained in two different channels agree onshell thanks to the open-closed string duality. However, it remains to be checked whether 1

3 τ σ τ σ Α Β Figure 1: A. Open string channel. B. Closed string channel. the open-closed string duality is valid even in the presence of the tachyon condensation, off-shell. At the tree level, it already has been shown that the evaluation of the disk diagram in the open string channel agrees with that in the closed string channel in the tachyon background. Since the boundary state formulation entirely depends on the openclosed string duality, it is important to clarify this issue. In the next section, we discuss the bosonic string theory at one-loop in the background of the tachyon condensation and show that the open-closed string duality holds even offshell. In section 3, we extend it to the supersymmetric theory. Section 4 concludes the paper with some discussions. 2 Bosonic String Field Theory at One-loop We start with the action S = S bulk + S bndy = 1 d 2 x hh ij i X µ j X µ + 1 dσv. 3) 8π Σ 8π Σ 2

4 S bndy describes the interactions between the string and the background, which may be expanded as V = T X)+A µ X) σ X µ + C µν σ X µ σ X ν +. 4) In this paper, we only consider the tachyon field background, which take the following simple form p V = T X) u i X i ) 2. 5) i=1 Open string channel - Annulus At the tree level, the world sheet is a unit disk on which the boundary action is given S bndy = 1 2π dσ ux 2 ). 6) 8π 0 Throughout this paper we are only concerned with the string variables in the direction where the tachyon condensation takes place. So we omit the space-time indices hereafter. The geometry of the world sheet for the one loop diagram is an annulus with a r b, which has two boundaries. It leads us to introduce the following boundary action S bndy = 1 [ 2π 2π ] dσ u a X 2 ) a + dσ u b X 2 ) b. 7) 8π 0 0 One may consider an alternative form of the boundary action, which makes use of the induced metric on the world sheet S bndy = 1 [ 2π 2π ] adσu a X 2 ) a + bdσu b X 2 ) b. 8) 8π 0 0 However, if we take S bndy as the boundary action, we would encounter difficulties such as the breakdown of the Fischler-Susskind mechanism [32] as pointed out in ref. [25]. Hence, the theory may become singular. If we turn on the tachyon condensation, the system is no longer conformally invariant. But we expect that the system is invariant under the following transformation, z ab/ z, which maps the inner boundary of the annulus onto the outer boundary vice versa. It is obvious that S bndy does not respect this symmetry while S bndy does [24]. For these reasons we take S bndy Eq.7) as the boundary action. 3

5 It follows from the action 3) that the world sheet Green s function z Gz, w) = 4πδ 2) z w), 9) satisfies the following conditions a ) r u a G B z, w) r=a = 0, b ) r + u b G B z, w) r=b = 0, 10) which read in the complex coordinates z, z) =re iσ,re iσ ) [19, 20] as with the boundary conditions z z Gz, w) = 2πδ 2) z w), 11) z + z u a )G B z, w) r=a = 0, z + z + u b )G B z, w) r=b = 0. 12) An explicit form of the Green function is obtained by solving this boundary problem [19]. The partition function is then determined by Eq.1) up to a integration constant c B [ Z B u a,u b ; a, b) = c B Tp 2 u a + u b u )] au b a 2 1/2 2 ln b 2 [ ) ] a 2n 1 n + u a )n + u b ) n u a )n u b ) 13) b where T p is the tension of the Dp-brane. Here we set α = 2. Since we consider the case where both ends of the open string are attached on the same D-brane, u a = u b = u. Choosing b = 1 for the sake of simplicity, we obtain Z B u, a) =c B Tp ) 2u u 2 ln a n + u) 2 n u) 2 a2n 4

6 Closed string channel - Cylinder In the closed string channel, the one-loop diagram is a cylinder. A closed string comes out from the D-brane and propagates onto the D-brane. The interaction between the D-brane background and the string is encoded in the initial and final states of the closed string, which are termed the boundary states. Construction of the boundary state begins with defining the boundary states { X } which form a basis for the closed string states ˆX X = X X, 15) ˆXσ) = ˆx { an +ã n n)e 2inσ +a n +ã n )e 2inσ} Xσ) = x n xn e 2inσ + x n e 2inσ) where σ [0,π]and[a n,a m]=[ã n, ã m]=δ nm. For a given boundary action S bndy the boundary state is constructed as B = T p D[x, x]e isbndy[x, x] x, x. 16) Choosing S bndy Eq.6) as the boundary action in order to describe the tachyon background we obtain the boundary state [27] as B B = T p where x is defined as 1+ u ) 1 n { dxe 1 4 xux exp a n ) } n u ã n x 17) n + u ˆx 0 x = x x, a n x =ã n x =0, 18) where n =1, 2, 3,... Note that the tachyon profile parameter u in the ref. [27] differs from that in the ref. [21] by 1/4π.) We can calculate the partition function by making use of Eq.2) Z B u, τ) = Tp 2 B B exp [ τl 0 + L 0 ) ] B B = Tp 2 1+ u ) 2 { ) } dxdx x e 1 4 x ux n u exp ã n a n n n + u 5

7 { e τˆp2 +N+Ñ) exp = Tp 2 2π u + u 2 τ/2 n u n + u 1+ u ) 2 1 n a n ) } ã n e 1 4 xux x 1 e 2nτ n u n+u ) 2 19) where N = na na n, Ñ = nã nã n. Comparing the partition function 19) in the closed string channel with that 14) in the open string channel and using the zeta function regularization { 1 d n + ɛ =exp )} ζs, ɛ) ɛ s = ɛγɛ), 20) ds s=0 2π we find that they coincide if we choose τ = ln a, c B =4π 3/2. 21) Thus, the open-closed string duality is shown to be valid off-shell. 3 Superstring Field Theory at One-loop In this section, we extend our previous discussion on the open-closed string duality to the superstring theory. The bulk action in the supersymmetric theory is given as the superstring world sheet action S bulk = 1 d 2 z X µ X + ψ ψ + ψ ψ), 22) 8π Σ where ψ and ψ are the holomophic and antiholomophic fermionic fields. The interaction between the tachyon background and the superstring is described by the following boundary action S bndy = 1 [ dσ T X)) 2 +ψ T) 1 ψ T)+ ψ T) 1 8π Σ σ ψ T) ]. 23) σ 6

8 In order to produce the same tachyon profile in the bosonic sector as in the bosonic string theory we choose T X) =ux. The total world sheet action for the superstring is Open string channel - Annulus S = S bulk + S bndy. 24) The bosonic part is exactly the same as in the bosonic string theory discussed in the last section. The superstring has two different fermionic sectors, depending on the boundary conditions for the fermion fields; R-R sector and NS-NS sector. In this paper we only consider the NS NS sector. The equations for the fermion Green functions, following from the superstring action Eq.24) are 1 iy a 1 G F z, w) = i zwδ 2) z w), G F z, w) = +i z wδ 2) z w) 25) σ 1+iy b 1 σ ) G F r=a = ) G F r=b = 1+iy a 1 1 iy b 1 σ ) GF σ r=a ) GF where Gz, w) and G z, w) are the holomophic and antiholomophic fermion Green function, and y u 2. Explicit expressions of the fermion Green functions can be found in [21]. Then a simple integration over y yields the partition function in the superstring theory Eq.1). Up to a constant c S the partition function is obtained as [21] 1 Z S y,a) = c S 2T p ) 2 1 n +2y) 2 n 2y) 2 a n 2y y 2 ln a [n + y) 2 n y) 2 a 2n ] 2. 27) 1 This expression differs by a factor 4 y from that of ref. [21] due to a different choice of regularization scheme. Similar to the disk case, as mentioned in [15], this difference does not affect any physical quantities. However, we may easily remove this factor by choosing an alternative regularization scheme as following: In ref. [21], the divergences of bosonic Green function and fermionic Green function cancels each other and leaves a constant 8 ln 2), which leads to the 4 y factor in the partition function. If we follow the regularization scheme in [1] r=b Xθ)Xθ) Σ = lim ɛ 0 [ Xθ)Xθ) ln1 e iɛ ) ln1 e iɛ ) ] for both bosonic and fermionic Green functions separately, we will obtain the expression 27). 7, 26)

9 The normalization of the one-loop partition function taken here is consistent with refs. [15, 33]. Closed string channel - Cylinder In the superstring theory the boundary state may be given as B = B B B F.Here B B, describing the interaction between the bosonic degrees of freedom of the superstring and the tachyon condensation background, is already given by Eq.17). The interaction between the fermion degrees of freedom and the tachyon condensation background is encoded in B F. Construction of the fermionic part of the boundary state is similar to that of the bosonic part. It begins with defining the following eigenstate in the NS-NS sector 1 ψ + ψ) θ, θ = θr e 2irσ + θ r e 2irσ) θ, θ. 28) 2 r=1/2 Here ψ = 2 br e 2irσ + b r e 2irσ), r=1/2 ψ = 2 r=1/2 br e 2irσ + b r e 2irσ), {b r,b s} = δ rs, { b r, b s} = δ rs. The fermionic part of the boundary state B F is written in terms of θ, θ as B F = D[θ, θ]e SF bndy [θ, θ] θ, θ. 29) For the background of the tachyon condensation we have Sbndy F = y dσ θ σ ) 1 θ 30) 4π θσ) = θr e 2irσ + θ r e 2irσ). r= 1 2 r=1/2 A simple algebra yields the explicit form of B F B F = 1+ y ) { r y exp b r r r + y 8 ) b r } 0 F 31)

10 where b r 0 F = b r 0 F =0. As in the bosonic sector the contribution of the fermionic sector to the partition function is written as the expectation value of the superstring propagator where Z F y,τ) = B F exp [ τl F 0 + L F 0 ) ] B F 32) = 1+ y ) { ) } 2 r y 0F exp br b r e τn F +ÑF ) r r + y r= 1 2 { ) } r y exp b r b r 0 F r + y = 1+ y ) ) 2 2 r y 1 e 2rτ r r + y r= 1 2 N F = r= 1 2 rb r b r, Ñ F = r= 1 2 Thus, the total partition function may be written as Z S y,τ) = 2 2 Z B y,τ)z F y, τ) = 2π 4Tp 2 y + y 2 τ/2 1+ 2y ) 2 1+ y n n r b r b r 33) ) 4 1 e 2nτ n 2y [ n+2y ) 2 ) ] 1 e 2nτ n y ) n+y whereweuseasimpleidentity fr) = fn/2) fn). 35) r= 1 2 At a glance, as in the bosonic case, we find that, the partition function in the closed string channel Eq.34) is the same as the partition function in the open string channel Eq.27) up to a constant. Comparison between two results yields τ = ln a, c S =4π 3/2. 36) 9

11 4 Discussion The open-closed string duality is one of the important features of the string theory. It is this duality that the boundary state formulation is entirely based on. It is also this duality that Polchinski [34] utilized to explore the fundamental properties of the D-branes. However, until recently use of the open-closed string duality has been limited to on-shell. Recently the boundary state formulation has been proven to be useful to discuss the tachyon condensation. So, it becomes important to understand the open-closed string duality off-shell. From the recent works we learn that the open-closed string duality holds off-shell at the tree level. In this paper we show that the duality is valid in the bosonic string theory at the one-loop level by comparing the one-loop partition function in the open string channel with that in the closed string channel. We also show that the openclosed string duality applies to the superstring off-shell. The one loop partition function of the superstring in the closed string channel is calculated explicitly by making use of the boundary state formulation. A direct comparison of it to the partition function obtained in the open string channel [21] confirms that the open-closed string duality is also effective for the superstring off-shell at the one loop level. We expect that the open-closed string duality holds at all higher loop levels of the superstring off-shell. So it helps us to study the tachyon condensation at higher loop levels. We conclude this paper with a remark on the infrared fixed point limit. In the infrared fixed limit, u and the unstable Dp-brane turns into a lower dimensional D-brane. It has shown at the tree level by explicit calculations of the partition function on a disk [15, 16, 27]. From the explicit calculation given in this paper we also observe this phenomenon at one loop level. It follows from Eq.19) that the partition function of the bosonic string theory reduces to lim Z Bu, τ) =T 2 4π π u p τ 1 1 e 2nτ 37) in the infrared fixed point limit. Since the propagator of the zero mode with initial and finalpointsfixedisgivenas x e τ ˆp2 x = 1 2 πτ, 38) 10

12 the infrared fixed point limit of Z B can be understood as lim u Z Bu, τ) =8π 2 T 2 p ) D e τl 0+ L 0 ) D 39) where D corresponds to the boundary state with the Dirichlet boundary condition. Thus, the one-loop partition function of the Dp-brane reduces to that of Dp-1)-brane. Then it implies that 8π 2 Tp 2 = T p 1 2, i.e., the descent relation between D-brane tensions T p 1 =2π α T p. 40) The infrared fixed point limit of the one-loop partition function of the superstring Eq.34) can be discussed in a similar way. Acknowledgements: This work is supported by a grant from Natural Sciences and Engineering Research Council of Canada. The work of TL was supported by KOSEF ) and was done during the PIMS-APCTP Summer Workshop 2001 held at Simon-Fraser University, Canada. References [1] E. Witten, Some Computations in Background Independent Off-shell String Theory, hep-th/ [2] S. Shatashvili, Comment on the background independent open sting field theory, hepth/ [3] J. A. Harvey, D. Kutasov and E. J. Martinec, On the relevance of tachyons, hepth/ [4] A. Gerasimov and S. Shatashvili, On exact tachyon potential in open string field theory, hep-th/ [5] D. Kutasov, M. Mariño and G. Moore, Some exact results on tachyon condensation in string field theory, hep-th/

13 [6] L. Cornalba, Tachyon Condensation in Large Magnetic Fields with Background Independent String Field Theory, hep-th/ [7] K. Okuyama, Noncommutative Tachyon from Background Independent Open String Field Theory, hep-th/ [8] O. Andreev, Some Computations of Partition Functions and Tachyon Potentials in Background Independent Off-Shell String Theory, hep-th/ [9] G. Arutyunov, S. Frolov, S. Theisen and A.A. Tseytlin, Tachyon condensation and universality of DBI action, hep-th/ [10] J. David, Tachyon condensation using the disc partition function, hep-th/ [11] T. Takayanagi, S. Terashima and T. Uesugi, Brane-Antibrane Action from Boundary String Field Theory, hep-th/ [12] G. Chalmers, Open string decoupling and tachyon condensation, hep-th/ [13] D. Kutasov, M. Mariño and G. Moore, Remarks on Tachyon Condensation in Superstring Field Theory, hep-th/ [14] D. Nemeschansky and V. Yasnov, Background Independent Open String Field Theory and Constant B-Field, hep-th/ [15] P. Kraus and F. Larsen, Boundary String Field Theory of the D D System, hepth/ [16] S.P. de Alwis, Boundary String Field Theory, the Boundary State Formalism and D-Brane Tension, hep-th/ [17] M. Mariño, On the BV formulation of boundary superstring field theory, hepth/ [18] V. Niarchos and N. Prezas, Boundary Superstring Field Theory, hep-th/

14 [19] R. Rashkov, K. S. Viswanathan and Y. Yang, Background Independent Open String Field Theory with Constant B Field On the Annulus, hep-th/ [20] T. Suyama, Tachyon Condensation and Spectrum of Strings on D-branes, hepth/ [21] K. S. Viswanathan and Y. Yang, Tachyon Condensation and Background Indepedent Superstring Field Theory, to apperar in Phys. Rev. D, hep-th/ [22] M. Alishahiha, One-loop Correction of the Tachyon Action in Boundary Superstring Field Theory, hep-th/ [23] S. Nakamura, Closed-string Tachyon Condensation and On-shell Effective Action of Open-string Tachyons, hep-th/ [24] K. Bardakci and A. Konechny, Tachyon condensation in boundary string field theory at one loop, hep-th/ [25] B. Craps, P. Kraus and F. Larsen, Loop Corrected Tachyon Condensation, hepth/ [26] G. Arutyunov, A. Pankiewicz and B. Stefański, jr., Boundary Superstring Field Theory Annulus Partition Function in the Presence of Tachyons, hep-th/ [27] Taejin Lee, Tachyon Condensation, Boundary State and Noncommutative Solitons, to appear in Phys. Rev. D, hep-th/ [28] Taejin Lee, Tachyon Condensation and Open String Field theory, hep-th/ [29] A. Fujii and H. Itoyama, Some computation on g Function and Disc Partition Function and Boundary String Field Theory, hep-th/ [30] E. T. Akhmedov, M. Laidlaw and G. W. Semenoff, On a Modification of the Boundary State Formalism in Off-shell String Theory, hep-th/

15 [31] C. G. Callan, C. Lovelace, C. R. Nappi and S. A. Yost, Nucl. Phys. B ) 525; Nucl. Phys. B ) 83; Nucl. Phys. B ) 221, Loop Sorrections to Superstring Equations of Motion. [32] W. Fischler and L. Susskind, Phys. Lett. B ) 262, Dilaton tadpoles, string condensates and scale invarinace II. [33] D. Ghoshal and A. Sen, JHEP ) 021, hep-th/ , Normalisation of the background independent open string field theory action. [34] J. Polchinski, Phys. Rev. Lett ) 4724, hep-th/