BPS Limit of Multi- D- and DF-strings in Boundary String Field Theory

A BPS limit is systematically derived for straight multi- D- and DF-strings from the D3D3bar system in the context of boundary superstring field theory. The BPS limit is obtained in the limit of thin D(F)-strings, where the Bogomolnyi equation supports singular static multi-D(F)-string solutions. For the BPS multi-string configurations with arbitrary separations, BPS sum rule is fulfilled under a Gaussian type tachyon potential and reproduces exactly the descent relation. For the DF-strings ((p,q)-strings), the distribution of fundamental string charge density coincides with its energy density and the Hamiltonian density takes the BPS formula of square-root form.


Introduction 2 BPS Multi-D-and DF-strings
In BSFT for superstrings, off-shell BSFT action S is obtained through an identification with worldsheet partition function Z, S = Z [22]. For the system of DpDp in their coincidence limit, the BSFT action of the tachyon field T and its complex conjugateT , coupled to an Abelian gauge field A µ with F µν = ∂ µ A ν − ∂ ν A µ , is given by [8] S = −2T p d p+1 x V (T,T ) − det(η µν + F µν ) F (y + )F (y − ), (2.1) where T p is tension of the Dp-brane. The runaway tachyon potential is Gaussian type, and functional form of the derivative term is where the variables, are expressed in terms of open string metric G µν and noncommutativity parameter θ µν as θ µν = 1 η + F [µν] . (2.5) Let us consider static multi-D(F)-strings from D3D3 (p = 3), which are stretched parallel to z-direction. An appropriate ansatz for the D-string and fundamental string is (2.6) where all the other components of the field strength are assumed to be vanishing. Substitution of the electric field E z (2.6) into the Bianchi identity, ∂ µ F νρ + ∂ ν F ρµ + ∂ ρ F µν = 0, forces E z to be constant. The static tachyon field (2.6) with constant electric field E z leads to tachyon equation, where y ± in (2.4) reduce to These field configurations automatically satisfy the equation of the gauge field A µ , ∂ µ Π µν = 0, where Π µν ≡ ∂L/∂(∂ µ A ν ). Since the momentum densities, T 0i and T 0z , and some off-diagonal stress components, T iz = T zi , are vanishing under the ansatz (2.6), the conservation of energymomentum tensor becomes ∂ j T ji = 0, (2.9) and it is equivalent to the tachyon equation (2.7) for nontrivial configurations.
To investigate the BPS limit of the D(F)-strings, we examine the pressure components perpendicular to the D(F)-strings where y ij ± are defined by As a necessary condition, pressure difference is required to vanish; We read first-order Cauchy-Riemann equation as Bogomolnyi equation from vanishing pressure difference (2.13) (∂ x ± i∂ y )T = 0, (∂ x ln τ = ±∂ y χ and ∂ y ln τ = ∓∂ x χ), (2.15) where T = τ e iχ . 1 By using (2.15), we easily check that the remaining off-diagonal stress component becomes automatically zero; For the n straight strings (anti-strings) spread arbitrarily on the (x, y)-plane, the ansatz on the tachyon phase χ is Then the tachyon amplitude τ is obtained as an exact solution of the Bogomolnyi equation ( where θ qr is the angle between two vectors, (x − x q ) and (x − x r ). Substituting (2.18)-(2.20) into the pressure components (2.10)- . Therefore, the pressure components (2.10)-(2.11) vanish only in the limit of zero thickness of each vortex, τ BPS → ∞, due to the rapidly-decaying tachyon potential V (τ ) (2.2) except for the site of each vortex x = x p , i.e., lim This nonvanishing pressure at each D(F)-string location is different from the character of BPS vortices in Abelian gauge theories with Higgs mechanism where the pressure components vanish everywhere including vortex points [4,23]. The stress component T x y also vanishes for the BPS configuration as shown in (2.16)-(2.17), and then the conservation of energy-momentum tensor (2.9) reduces to ∂ x T x x = 0 and ∂ y T y y = 0. For the aforementioned pressure components of the BPS D(F)strings in the infinite τ BPS limit, the equations hold when the derivatives are considered as weak derivatives [24]. As τ BPS → ∞, the static singular solution (2.18)-(2.19) of BPS equation satisfies the conservation of energy-momentum tensor (2.9), which is equivalent to the tachyon equation (2.7) for nontrivial tachyon configurations. In the section 3, we also show that the tachyon equation (2.7) does not support regular static straight D(F)-string solution.
For the static configurations ofṪ =˙T = 0 with constant E z , the conjugate momenta of the tachyon field and its complex conjugate vanish, Π T ≡ ∂L/∂Ṫ = 0 and ΠT ≡ ∂L/∂Ṫ = 0, and the conjugate momentum of the gauge field Π z is The Hamiltonian density obtained by a Legendre transform leads to the BPS formula for DFstrings ((p, q)-strings) [25], where the limit of D-strings, H| Πz=0 = 2T 3 V F (y + )F (y − ), is trivially involved in the absence of fundamental string charge density Π z = 0. Plugging the conjugate momentum (2.21), the Hamiltonian density (2.22) coincides exactly with the energy density −T t t , and, due to the boost symmetry along the z-direction, the multi-D(F)-string configuration satisfies Noticing easily that the energy density is proportional to the electric flux density as we read for the DF-strings that the charge distribution of fundamental string part is exactly proportional to the energy density of D-string part, which is confined at each string site in (x, y)plane.
If we require a BPS sum rule to the energy per unit D(F)-string length for the BPS configuration with y + = y − , the descent relation of a D(F)-string, is correctly reproduced, and a constraint condition for a BPS sum rule is achieved for the tachyon potential, In summary, the energy-momentum tensor of n D(F)-strings is in the BPS limit, where I(x − x p ) has unity at x = x p and zero at x = x p . Note that the pressure components orthogonal to the string direction vanish in the limit of critical electric field, |E z | → 1. From now on, let us perform the integration (2.27) with the tachyon potential (2.2) and show that it reproduces the required value for saturating the BPS sum rule. First, we consider a single D(F)-string of n = 1 at an arbitrary position. In this case, y is independent of x, y = 4τ 2 BPS , so is F . Then, a rescalingx = τ BPS x with a translation in (2.27) provides a definite integral without explicit dependence of τ BPS ; (2.30) If we perform the Gaussian integral for arbitrary τ BPS and take the limit of infinite τ BPS by using the asymptotic form of F (y) 2 , F (y) 2 = πy + π/8 + O(y −1 ), in (2.30), then value of the integral is 4π 2 , which satisfies the descent relation. Second, we consider the superimposed D(F)-strings of arbitrary |n|. Now y of F (y) has x-dependence as y = 4n 2 τ 2 BPS (τ BPS |x|) 2n−2 , and then we use the same rescaling of x as (2.32) As τ BPS increases, the integrand with explicit τ BPS dependence becomes with keeping |x| finite. For infinite |x|, the integrand vanishes due to the exponential term. Since F (y) 2 is analytic for every non-negative y, the integrand is finite atx = 0, and the asymptotic form of F (y) 2 guarantees finiteness of the integral (2.32) for finite τ BPS , we can take infinite τ BPS limit to F (y) 2 /πy part in (2.32). Therefore, value of the integral (2.32) is 4π 2 n which fits (2.27). Third, we consider the case of n separated D(F)-strings where the distance between any pair of D(F)-strings is much larger than 1/τ BPS . When x = x s (s = 1, 2, . . . , n), it is obvious that y in (2.20) diverges in the τ BPS → ∞ limit for any tachyon field. When x = x s , the term with p = q = r in (2.20) survives and hence y → ∞ in this BPS limit. Thus we see that y always becomes infinite in the τ BPS → ∞ limit. Accordingly, F (y) 2 in the integral (2.27) diverges everywhere. Let us examine the tachyon potential part in (2.27). When x = x p (p = 1, 2, . . . , n), τ in (2.19) vanishes and then the tachyon potential has unity, V (τ = 0) = 1. When x = x s , it vanishes in the infinite τ BPS limit and the integrand in (2.27) also vanishes due to the exponential damping of the tachyon potential despite of the leading divergent term of F , F (y) → √ πy . Therefore, among n 2 -terms in y (2.20) specified by the (q, r)-indices, the n-terms with q = r contribute to the integral (2.27). In addition, functional shape of the integrand diverges at each string site but vanishes away from the location of each D(F)-string. In what follows, we will show that the contribution of each term at x = x p to the integration is exactly the same as that of delta function given in single D(F)-string (2.30) as far as the distance |x p − x q | for any p and q (p = q) is sufficiently larger than 1/τ BPS . Since only the neighborhoods of D(F)-string sites, x = x p , contribute to (2.27) in performing the (x, y)-integration and become sufficiently small for infinite τ BPS , only the leading terms of V and F 2 can contribute nonvanishing value to the integral (2.27). To be specific, we can replace the integrand V F 2 and then perform the integration as follows, which is exactly the value in (2.27). Fourth, we consider the case of arbitrary BPS configuration where n p D(F)-strings among the n D(F)-strings are superimposed at an x p with n = p n p . If we replace the integration (2.30) by (2.31)-(2.34), the integration reproduces the value in (2.27) by applying repeatedly the above third argument. In synthesis, the aforementioned four arguments lead to a conclusion that the Gaussian type tachyon potential (2.2) fulfills the integration (2.27) in the thin BPS limit.

Nonexistence of Nonsingular D-and DF-string Solutions
In this section, we deal with the tachyon equation (2.7) and discuss nonexistence of the monotonicallyincreasing nonsingular D-vortex solution connecting the boundary conditions at the origin, τ (|x| = 0) = 0, and infinity, τ (|x| = ∞) = ∞. This perhaps supports uniqueness of the singular BPS multi-D(F)-string solutions obtained in the previous section. Suppose that we have n superimposed straight D(F)-strings stretched along the z-axis. Since the electric field E z is actually canceled in both sides of the tachyon equation (2.7), we have where y ± in (2.4) become The D(F)-string solutions of our interest are given by monotonically-increasing tachyon configurations connecting the boundary conditions, τ (r = 0) = 0 and τ (r = ∞) = ∞. Expansion of the tachyon amplitude τ near the origin is where τ 0 is an undetermined constant determined by the behavior at asymptotic region. Since the coefficient of subleading term τ 1 is always positive irrespective n, , (n = 1) 1 24 ln 2 64τ 2 0 8(ln 2) 2 − π 2 3 + 1 4 , (n = 2) 1 32(ln 2)(n + 1) , , (3.4) increasing tendency of the tachyon field τ (r) decreases as r increases. If we try expansion at asymptotic region by using a power law, τ ∼ τ ∞ r k , (k > 0), or a logarithmic increase, τ ∼ τ ∞ ln r, many possibilities are ruled out by the tachyon equation (3.1) and survived cases are where both τ ∞0 and τ ∞1 are not determined by the tachyon equation (3.1). The leading term is rapidly increasing since lim r→∞ τ ′ → ∞. Comparison of the power series solutions near the origin (3.3) and at the asymptotic region (3.5) suggests that smooth connection of both increasing tachyon profiles seems unlikely. Another possibility is the solution with maximum value, i.e., the tachyon amplitude increases near the origin, reaches a maximum value τ m at a finite coordinate r = r m , and then starts to decrease with d 2 τ /dr 2 | r=rm < 0. Expansion near r = r m gives where the coefficient τ m2 is In order to have the maximum τ m =τ (r m ), τ m and r m should satisfy the following inequality, Numerical works support that every regular solution with finite τ 0 has the maximum value τ m at a finite r m irrespective of n as shown in Fig. 1. Probably, there does not exist any static nonsin-

Conclusion
The system of D3D3 has been considered in the scheme of super-BSFT EFT (2.1) including a complex tachyon and U(1)×U(1) gauge fields. From the vanishing pressure difference, the firstorder Bogomolnyi equation (2.15) was derived and straight topological BPS multi-D(F)-string configurations were given as exact static solutions (2.18)-(2.19) which also satisfy the conservation of energy-momentum tensor (2.9). Since the forms of derived Bogomolnyi equation and singular BPS solutions coincide exactly with those in DBI type EFT, this BPS structure seems universal and is consistent with type II superstring theories. The expression of energy was rewritten by the BPS sum rule for the BPS multi-D(F)-string solutions (2.25), and reproduced the descent relation for codimension-two objects (2.26), which allowed to interpret the obtained vortex-strings as BPS D1-branes in IIB string theory. This results in a constraint condition for the BPS tachyon potential (2.27), and the Gaussian type potential of BSFT (2.2) fulfills the condition. Since it is nothing but making a sum of delta functions in the thin BPS limit (2.28), the uniqueness of BPS tachyon potential seems unlikely. When the z-component of constant electric field (2.6) is turned on, the conjugate momentum of the gauge field, the charge density of fundamental strings (2.21), is confined along the D-strings. In addition, the corresponding Hamiltonian density takes a BPS formula (2.22), p 2 + q 2 form for the D1-charge density q and the fundamental string charge density p, so that the configuration with constant electric field along the string direction is the DF-string (or (p, q)-string) from D3D3. Though we checked the conditions for BPS vortex configurations explicitly, the form of obtained BPS limit is different from the usual BPS bound for vortices, of which energy minimum is saturated only when the Bogomolnyi equations are satisfied. In this sense, the BPS bound for codimension-two branes from DD system needs further study. We also checked the possibility that the tachyon equation (3.1) could possess a nonsingular D(F)-string solutions and the analysis supported negative answer. Since we achieved a BPS limit of multi-vortex-strings, it may open systematic study of classical dynamics of BPS multi-D(F)-strings, particularly moduli space dynamics in the context of BSFT. Studies of the D(F)-strings in curved spacetime naturally have cosmological implication as candidates of cosmic superstrings.