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The mass formula for a fundamental string as a BPS solution of a D-brane's worldvolume

We propose a (generalized) ``mass formula'' for a fundamental string described as a BPS solution of a D-brane's worldvolume. The mass formula is obtained by using the Hamiltonian density on the worldvolume, based on transformation properties required for it. Its validity is confirmed by investigating the cases of point charge solutions of D-branes in a D-8-brane (i.e. curved) background, where the mass of each of the corresponding strings is proportional to the geodesic distance from the D-brane to the point parametrized by the (regularized) value of a transverse scalar field. It is also shown that the mass of the string agrees with the energy defined on the D-brane's worldvolume only in the flat background limit, but the agreement does not always hold when the background is curved.


Introduction
To gain better understanding of string theories and M-theory, intersecting branes have played an important role, and worldvolume analyses have been powerful approaches to investigate the intersecting branes [1] [2][3] [4] [5]. In ref. [1][2] [3] solutions of worldvolume field theories of branes (in flat backgrounds) with nontrivial worldvolume gauge fields were obtained. In the case of D-branes, it was shown that an appropriate excitation of one of the transverse scalar fields is needed in order to obtain a supersymmetric (i.e. BPS) point charge solution of the worldvolume gauge field [1]. Each of the solutions was interpreted as a fundamental string ending on the D-brane on the basis of the fact that the (regularized) energy of each solution defined on the worldvolume is proportional to the (regularized) value of the scalar field, which is considered to be the length of the string. † This interpretation is also consistent with the charge conservation suggested in ref. [6] (see also ref. [7]). These analyses are very important in that they made clear "how one of intersecting two branes is described" from the viewpoint of another brane's worldvolume.
In fact, however, the energy defined on the brane's worldvolume does not always agree with the target-space mass of the string, though the agreement of the two holds for the case of ref. [1]. We can easily understand this by considering the fact that the worldvolume energy depends on the definition of the time of the worldvolume, while the mass of the string should not depend on it. That is, the derivation of the mass of the string must be considered more carefully. The main purpose of this paper is to present a generalized "mass formula" of a fundamental string described as a BPS solution of a D-brane's worldvolume, which holds independent of definition of its worldvolume time.
Our idea to obtain the mass formula is as follows: the mass of the string does not always agree with the worldvolume energy, to be sure, but it is also true that the two are very close to each other, since the two give the same results at least in the cases discussed in ref. [1]. So, we construct the mass formula in a heuristic way, by using the Hamiltonian density defined on the worldvolume, based on the invariance of the mass under the coordinate transformations of the worldvolume. Moreover, we also construct explicitly point charge solutions (with appropriate excitations of single transverse scalar fields) of branes' worldvolumes in a curved background. ‡ There are two advantages to consider these solutions: First, in each of the cases, the time component of the induced worldvolume metricg 00 (as well as spatial ones) becomes nontrivial, So, the differences † Since the energies of the point charge solutions are infinite, (UV) regularization is needed to calculate them.
‡ Some worldvolume solutions of branes in a curved (brane) backgrounds are discussed for other purposes in ref. [8] [9]. originating from the contribution ofg 00 become apparant in discussing the mass formula.
Second, in each case of the solutions, the mass of the corresponding fundamental string should be proportional to the geodesic distance from the brane to the point parameterized by the (regularized) value of the scalar field, and that the proportional coefficient should be an appropriate tension of the string, as discussed in ref. [10]. This requirement is very tight. So, once we find out the quantity which gives the mass stated above, it is expected to be the correct mass formula, even if it is constructed by hand. We will construct it based on this idea. (The discussion on the worldvolume interpretation of the string mass will be given finally in section 3.) The worldvolume theories we discuss here are the two cases: those of a test D-4brane and a test D-8-brane both embedded parallel to (a subspace of) the worldvolume of the D-8-brane background [11][12] (i.e. a massive IIA background). First, we present the two reasons to choose this background: One is that this background has only one transverse coordinate, leading to the fact that the harmonic function depends linearly on the coordinate. Only in this case, we can obtain explicitly the exact solutions of the worldvolumes without any extra assumption (as we will see later). We note that obtaining the explicit form is is crucial not only in order to find out the mass formula by hand and but for other discussions. Another reason is that choosing this background, we can see the supersymmetry preserved in the solutions by using superalgebras in a massive IIA background via brane probes" in ref. [13]. § From the supersymmetry, we can confirm that the solutions are BPS states and that their target-space interpretation is consistent.
Next, we explain why we choose the two worldvolume theories embedded in those ways: This is because at least one overall transverse space is needed after embedding (test) D-branes' worldvolumes in order to obtain supersymmetric point charge solutions. In the case of a D-2-brane (a D-6-brane), only the intersection with the background D-8-brane on a string (a 5-brane) leads to the preservation of supersymmetry, ¶ but, there is no overall transverse space. A D-0-brane is not adequate to this worldvolume description since there is no world space. So, we consider the above two cases, which preserve at this moment 1/4 and 1/2 supersymmetry, respectively.
Concrete procedures are in the following: In each of the two cases we construct explicitly an point charge solution with an appropriate excitation of the only overall transverse scalar field. The consistency of the interpretation of each solution as a fundamental string is confirmed in two ways: by discussing its behavior in the flat background limit and by § In the previous paper [9] we have confirmed that the supersymmetry preserved in the solution of brane's worldvolume can be derived via "superalgebras in brane backgrounds" (see also [14]). ¶ We do not discuss bound states like (6|D6,D8) since the D-8-brane solution is "singular" just on the D-8-brane hyper-surface. checking its preserved supersymmetry. Then, we propose the generalized mass formula, and examine whether it gives the geodesic distance multiplied by an appropriate tension of the string. Moreover, we also discuss the condition that the mass of the string agrees with the energy defined on the D-brane's worldvolume.
The organization of this paper is as follows: In section 2 we construct the point charge solutions and discuss their mass formula. In section 3 we give the conclusion and some discussions on the consistency and the interpretation of the mass formula The notations in this paper is as follows: We use "mostly plus" metrics for both spacetime and worldvolumes. We denote coordinates of each p-brane's worldvolume as ξ i , ξ j , .. (i,j =0,1,..,p), those of 10D spacetime as x m , x n , .. (m,n=0,.., 9), fermionic coordinates as θ α , θ β , .., and those of superspace as Z M . We use hatted letters (M ,m,α..) for all the local Lorentz frame indices and under-barred letters (m, i) for spatial indices (but not time one), respectively. We denote gamma matrices as Γm, which are all real and satisfy {Γm, Γn} = 2ηmn. Γ0 is antisymmetric and others symmetric. Charge Conjugation is C = Γ0.

Point charge solutions of 10D IIA D-branes' worldvolumes and their mass formula
In this section we construct the point charge solutions corresponding to fudamental strings and discuss their mass formula.
The D-p-brane action in a general 10D massive IIA background [15][12] takes the form [16][17] EN M is the supervielbein. φ is the dilaton field and F ij are the components of a modified worldvolume 2-form field strength of the worldvolume 1-form gauge field V whereB 2 is the worldvolume 2-form induced by the superspace NS-NS 2-form gauge potential B 2 .C is a formal sum of worldvolume r-formsC (r) (r=1, 3,5,7,9) induced by the superspace R-R r-form gauge potentials C (r) . m is a mass parameter which is the dual of the 10-form field strength F (10) of a R-R 9-form C (9) [18] [12]. T p is the "formal" (but not physical) tension of the D-p-brane which is given We take the background of the action (2.1) to be the D-8-brane solution given by [11][12] where x µ and x ν (µ, ν=0,.. T-duality from the other D-p-brane solutions [12]. H = H(y) is a harmonic function on y. In this paper we set which means that the D-8-brane lies at y = 0. We choose c 1 > 0 and m > 0 to avoid a singularity at y = 0 and to get a real dilaton. We note that the solution (2.4) becomes the flat spacetime metric in the massless limit The Killing spinor of (2.4) has the form ε = H ǫ/8 ε 0 where ε 0 has a definite chirality, i.e.
We first consider a point charge solution of a D-4-brane worldvolume parallel to the background D-8-brane. Since the solutions we construct here is a bosonic one, we set fermionic coordinates θ to be zero. Moreover, we consider the ansatz where r is defined as r ≡ Σ 4 i=1 (ξ i ) 2 . We note that the upper two columns of (2.7) mean that the D-4-brane is embedded in the 1234-hyper-plane. That is, from target-space point of view, we consider the following intersection of three branes: * We choose α ′ = 1 in this paper. where y 0 is a positive constant (and x 9 = y).
Then, since V i = θ = 0, S W Z D4 does not contribute to the equations of motion, and the equations of motion to solve are given by So, let us examine more about L BI The induced worldvolume metricg ij is given bỹ arising in L BI is very complicated. However, setting the condition results in the simple form of the determinant: det(g ij +F ij ) = H 5ǫ/2 . † Then, the equations of motion (2.8) become the following two simple equations (2.12) ((2.11) arises from the 9th component of the former of (2.8), (2.12) from the time component of the latter of (2.8), and the others are solved.) So, requiring the two equations to be compatible, it needs to hold ǫ = −1, and the two are combined into one equation We note that the harmonic function gives nontrivial contribution to (2.13), which means that the equations of motion of the D-4-brane worldvolume are affected by the background D-8-brane. By using (2.5) and y > 0, the equation (2.13) is written as (2.14) We choose the boundary condition for r → ∞, which means that the D-4-brane lies at y = y 0 . Then, the solution is obtained, with the following unusual form, as where c 2 is a constant proportional to the electric charge of the gauge field. The electric charge Q 1 is defined as [3] where S n is the n-sphere, ⋆ is the worldvolume Hodge dual and D is the 2-form defined δF ij . Then, we have where Ω n is the volume of the unit n-sphere. We note that in this definition the string coupling g s = e φ is included in Q 1 since L BI D4 is proportional to the inverse of g s . Since the solution has been obtained explicitly, we next give some pieces of evidence that the solution corresponds to the fundamental string ending on the D-4-brane. First, we discuss the massless limit of the solution (2.16). We assume here that the charge Q 1 is independent of m. (The validity of the assumption is discussed later.) Then, in the massless limit (2.6), the solution (2.16) with (2.18) behaves as The right hand side of (2.19) is exactly the solution of a D-4-brane's worldvolume in the 10D flat spacetime, which corresponds to the fundamental string [1]. So, it is expected to correspond to a fundamental string. Next, we check the preserved supersymmetry of the solution by using "superalgebras in brane backgrounds via brane probes" [14][13] [9]. The superalgebra in a D-8 brane background via a D-4-brane probe is given in ref. [13] as where Q + α = 1+Γŷ 2 Q α is the supercharge preserved in (2.4) and M p is the worldspace of the D-p-brane. P (0)i is almost equivalent to the conjugate momentum of V i . (The contributions of the Chern-Simons term are subtracted.) Substituting the solution for the right hand side of (2.20), the superalgebra can be written as Since the three gamma matrix products Γŷ, CΓ1234Γ 11 and CΓŷΓ 11 (arising in (2.21)) commute with each other, all of them can be simultaneously diagonalized. Since the square of each matrix product is equal to the identity and each is traceless, both of the matrices For this purpose, we pass to the Hamiltonian formalism as done in ref. [1]. If we assume that V is purely electric and that only the scalar y is excited, S D4 reduces to (where H is the harmonic function). The canonical momenta of y and V i , are defined respectively as The HamiltonianH is constructed asH ≡ M 4 d 4 ξH where H is the Hamiltonian density given by We note that Π i is subject to the constraint ∂ i Π i = 0. Substituting the solution (2.16) forH, we can obtain the energy of the solution E defined on the worldvolume. We note that for a BPS solution like this case, it generically happens that the square root ofH becomes a perfect square and that the energy becomes a sum of the two parts: the part originating from the D-p-brane and that from the string. so, we denote the first part of H as H 1 and the second part as H 2 . Concretely, the energy in this case takes the form (2.25) The first term E 1 is the "energy" of the D-4-brane itself, and the second term E 2 is the energy of the excitation (i.e. the string), both defined on the worldvolume. Since we are interested in the second part, we compute only E 2 here. E 2 is infinite in this case, but if we regularize it by introducing a small parameter δ, we can get the energy for r ≥ δ as That is, the energy is (again) proportional to the difference of the coordinate. Thus, we conclude that the energy defined on the brane's worldvolume does not agree with the mass of the string in the case of D-branes in curved backgrounds. We note that this result is rather reasonable, in a sense, in the case ofg 00 = 1, because the energy has the same  [1] (and ref. [19]). Let us consider a triple junction of strings: a (0,1) string, a (n,0) string and a (n,1) string. If the string coupling g s is small, it holds ∆T where T (p,q) is the tension of a (p,q) string and ∆T is the additional tension ∆T ≡ T (n,1) − T (0,1) . On the other hand, the solution of a D-1-brane worldvolume corresponding to the above string junction is given in ref. [19]. In the flat background with the ansatz that x 0 = ξ 0 , x 1 = ξ 1 , x 9 = y(ξ 1 ), x m = constant for m=2,..,8, and θ = 0, the D-1-brane action is written as The solution of the D-1-brane's worldvolume as the triple string junction with an electric The energy of the solution can be computed by using the Hamiltonian, and the additional tension ∆T is also derived from this correctly. By taking into account the bending of the (n,1) string [19], it is obtained as ∆T /T (0,1) = (1/2)(F 01 ) 2 . Comparing this with (2.31), the "charge quantization condition" F 01 = q 1 = g s n is deduced for a point charge q 1 (for an integer n). By T-dualizing with respect to the directions of x m for m=2,..,8, the charge quantization condition for a electric point charge q 1 of the D-p-brane's worldvolume is shown to be (2.34) Next, we discuss the case which is related by T-duality to the case of (2.16). Let us suppose a D-1-brane parallel to a subspace of the worldvolume of a D-5-brane background, and that some number n of fundamental strings are absorbed in the D-1-brane. (This is also a BPS configuration since 1/8 spacetime supersymmetry is preserved [20].) If we consider a D-5-brane background solution, the string coupling g s becomes a local function on the transverse coordinates y a (g s = e φ = H −1/2 where H is the harmonic function on y a ), and so is the tention T (n,1) . If the "test" D-1-brane is put near the D-5-brane, the string coupling g s is considered to be sufficiently small around the D-1-brane. So, the equation (2.31) with g s = H −1/2 holds on the basis of the same discussion. On the other hand, suppose that a electric point charge q ′ 1 is added to the D-1-brane's worldvolume with an excitation of a scalar field y 9 . Then, the D-1-brane action is (2.35) If we assume the existence of the correponding point charge solution, the additional tension due to the field strength is derived in the same way, as So, comparing this with (2.31) we have the charge quantization condition H 1/2 F 01 = ng s for an integer n. By using the the "electric induction" D 01 = − 1 T 1 · δL D1 δF 01 (= HF 01 in this case) it can be rewritten in a more generic form as D 01 = n. (2.37) The higher dimensional D-brane cases are related to (2.37) by T-dualities. ‡ So, we have where r is defined as r ≡ 8 i=1 (ξ i ) 2 . Then, combined with θ = 0, only the term including C (9) in S W Z D8 does contribute to the equations of motion (C (9) 01..8 = H ǫ ). The expression of ‡ We note that the background fields should also be transformed by T-dualities simultaneously.
the induced worldvolume metricg ij is the same as (2.9) except for the range of the indices (in this case i, j = 1, 2, .., 8). Setting the same condition as (2.10) makes the determinant of (g ij + F ij ) simple, such as det(g ij + F ij ) = H 9ǫ/2 . Then, we find the equations of motion where c ′ 2 is a constant proportional to the electric charge of the solution. By using the definition of the charge similar to (2.17), we have the electric charge Q ′ 1 : where Π (0) µ is the contribution of S BI D8 and the second term is that of S W Z D8 (where ǫ 01..8 = 1). Substituting the solution for the right hand side of (2.44), the superalgebra can be written as By the same discussion as done in the D-4-brane case, it is shown that 1/4 supersymmetry is preserved in this configuration, which is consistent with the spacetime interpretation.
So, we interpret the solution (2.42) as a fundamental string again.
Applying the mass formula (2.27) to this case, we obtain the result: Based on the same discussion as that done in the D-4-brane case, the unit electric charge Q ′ 1 for a (1,0) string ending on a D-8-brane can also be derived as Q ′ 1 = (2π) 7 . So, the tension of the string is reproduced correctly again, and (2.27) also gives the mass of the string for the solution (2.42) correctly. Therefore, we conclude that (2.42) is the correctly generalized mass formula, which holds when the background of the D-brane is curved. We note that the energy defined on the worldvolume again gives the difference of the coordinate y multiplied by T 8 Q ′ 1 . That is, this case is another example that the worldvolume energy does not agree with the mass of the string.

Summary and discussions
In summary, we have proposed (2.42) as the generalized mass formula which holds when the background of the D-brane is curved, and have shown explicitly using the two examples that the formula certainly gives the mass of a fundamental string described as a BPS solution of a D-brane's worldvolume. In addition, based on the obtained formula, we can see that the mass of the string agrees with the worldvolume energy only in the cases g 00 = −1 (whereg ij is the induced worldvolume metric). which include the case discussed in ref. [1].
Here, we discuss the consistency of the mass formula from another point of view, especially focusing on the factor √ −g 00 . Suppose we consider the D-4-brane embedded parallel to the D-8-brane background (2.4) with no excitation of worldvolume fields (i.e. x i = ξ i for i=0,..,4 and x 5 , .., x 8 , y : constants). Then, on the analogy of the mass formula of a (p,q) string given by Sen in ref. [10], the target-space mass m D4 of the D-4-brane should be proportional to its spatial volume element measured by the geodesic distances in the spacetime. So, it should be given by whereg ′ ij is the induced world-space metric of the D-4-brane. In fact, the mass m D4 obtained in this way can be shown to agree with the quantity M D4 defined as where H 1 is the first part (i.e. originating from the D-4-brane) of the Hamiltonian density of the solution (2.16) defined on the D-4-brane. This means that the information of the D-4-brane mass can also be extracted from the solution (2.16) by integrating the Hamiltonian density multiplied by the factor √ −g 00 with respect to world-space coordinates. Thus, the factor √ −g 00 arising in the formula is consistent in this sense, too.
Finally, let us discuss the worldvolume interpretation of the mass formula (2.27). If we define a worldvolume proper time τ as dτ ≡ √ −g 00 dξ 0 , the Hamiltonian density multiplied by √ −g 00 might be regarded as the energy density defined with respect to τ .
So, we might say that from worldvolume point of view, the mass of the string is interpreted as "the energy defined with respect to the worldvolume proper time".