The Worldvolume Action of Kink Solitons in AdS Spacetime
footnotetext: jkhoury@sas.upenn.edu,   ovrut@elcapitan.hep.upenn.edu,
   stokesj@sas.upenn.edu

A formalism is presented for computing the higher-order corrections to the worldvolume action of co-dimension one solitons. By modifying its potential, an explicit “kink” solution of a real scalar field in AdS spacetime is found. The formalism is then applied to explicitly compute the kink worldvolume action to quadratic order in two expansion parameters–associated with the hypersurface fluctuation length and the radius of AdS spacetime respectively. Two alternative methods are given for doing this. The results are expressed in terms of the trace of the extrinsic curvature and the intrinsic scalar curvature. In addition to conformal Galileon interactions, we find a non-Galileon term which is never sub-dominant. This method can be extended to any conformally flat bulk spacetime.

1 Introduction

There is a long literature on computing soliton solutions, of varying co-dimension, in both non-supersymmetric [1, 2] and supersymmetric/superstring [3, 4, 5, 6] theories of physical interest. This was followed, in each of these contexts, by computations of the lowest order Dirac-Born-Infeld (DBI) actions on the worldvolumes of these solitons [7, 8, 9, 10, 11, 12]. More recently, there has been interest in extending these calculations to include higher-dimensional operators, involving both extrinsic and intrinsic curvature, in these effective actions. This has been carried out with differing techniques, using both probe and back-reacted geometries, for bosonic [13, 14, 15, 16, 17, 18, 19] and supersymmetric branes [20, 21, 22, 23, 24, 25, 26, 27, 28].

Apart from the inherent interest in computing these higher-order corrections to the effective actions, the advent of Galileon theories has led to renewed interest in higher-derivative terms. First discovered in the context of the decoupling limit of the Dvali-Gabadadze-Porrati (DGP) brane-world scenario [29], where the Galileon scalar is related to the brane bending mode, Galileon theories have since been generalized [30]. Galileons have two remarkable properties: first, despite the fact that their interactions are higher-derivative, the corresponding equations of motion are nevertheless second order, and second, these interactions possess extended non-linearly realized symmetries. The original Galileons of [30] possess another important property that justifies treating them separately from the other possible non-Galileon higher-derivative terms which possess the same symmetries. There can be non-linear solutions and regions of momentum and field space for which Galileon terms are important relative to the kinetic terms, and yet the non-Galileon terms are unimportant, allowing us to work with only the finitely many non-linear Galileon terms rather than the whole effective field theory expansion [31, 32]. Furthermore, Galileons can lead to stable (that is, ghost-free) violations of the Null Energy Condition [33, 34], akin to the ghost condensate [35]. In fact, supersymmetric condensates naturally give rise to super-Galileons [36, 37]. This violation allows for non-singular bounces in the early universe [33, 38, 39, 40, 41, 42, 43] and cosmologies that expand from an asymptotically-flat past [44]. These interesting solutions generally require the Galileon-like terms to be important relative to the kinetic terms, so from an effective field theory point of view, if other unknown higher-order terms are to be neglected, it is crucial that the Galileons have the property described in the previous paragraph.

There are extensions of the original Galileons, the DBI Galileon, that arise naturally in describing the brane-bending mode of co-dimension one and higher brane worldvolumes [45, 46, 47]. They arise from Lovelock terms and their boundary terms in the worldvolume actions. The original Galileons are obtained after a certain small field limit. Non-Lovelock terms on the worldvolume lead to non-Galileon terms. Since the DBI Galileons arise from the point of view of a brane probing a relativistic spacetime, it is natural to ask whether there can be regions of momentum and/or field space in which the DBI Galileons are important relative to the DBI kinetic term, yet still dominant over the non-Galileon terms, as is the case for the original Galileons. If this is the case, then in these regimes, only a finite number of higher-derivative terms in the worldvolume action would have to be computed to completely determine the interesting non-linear dynamics. The DBI Galileons admit superluminal propagation around non-trivial solutions [48] which, along with other arguments, suggests that any UV completion, though not necessarily inconsistent, will not be in the form of a local Lorentz invariant quantum field theory or string theory [49]. Thus, if a low-energy worldvolume theory on a brane could be derived in which there were a sharp limit where only the Galileons are important, it would mean that either the theory describing the brane does not have a UV description as a local Lorentz invariant quantum field theory/string theory, or that the arguments concerning the connection between low-energy superluminality and UV physics are somehow evaded.

Motivated by these reasons, and simply by the desire to have a consistent and general method for calculating low-energy worldvolume actions for solitons in more general spacetimes, we present here a calculation of the leading and subleading corrections to the DBI worldvolume action of a scalar kink in anti-deSitter (AdS) spacetime. This is the first in a series of papers applying this extended method to effective string solitons of physical interest, which we hope to apply to UV complete systems like the heterotic string. In a series of papers [13, 16, 18], Gregory and collaborators presented a particularly compelling approach to the problem of computing higher-order corrections to worldvolume actions, within the context of flat space co-dimension one “kink” solitons. This involves a consistent series expansion in a parameter , the ratio of the kink thickness to the typical worldvolume fluctuation length. Using this formalism, the explicit worldvolume action of a probe kink in a flat background bulk space was computed [16]. In this paper, we modify and extend this formalism, using it to compute to second order in (and a second parameter ) the explicit worldvolume action of a kink soliton in AdS. This is carried out in two different ways, first with respect to the original AdS metric and second using a rescaled flat metric. Both lead to mutually consistent expressions for the worldvolume action including higher-order extrinsic and intrinsic curvature terms. Although three terms are indeed conformal DBI Galileons, a fourth term involving the square of the extrinsic scalar curvature explicitly is not—nor are there any momenta for which this term is sub-dominant.

The paper is structured as follows. In Section 2, we briefly review the formalism presented in [16] for computing the worldvolume action of a kink soliton of a real scalar field in flat spacetime. As a prelude to the AdS calculation, and to set our notation, we carry out the computation to second order in the expansion parameter . In Section 3, anti-deSitter spacetime is introduced, and the potential energy of the real scalar field is modified so that its equation of motion admits a kink soliton of the same functional form as in flat space. We then generalize the formalism of [16] so as to allow a calculation of the effective action on this kink worldvolume. The radius of AdS space introduces, in addition to , a second expansion parameter . The extrinsic and intrinsic curvatures, the generalized solution to the scalar equation of motion and the kink soliton worldvolume action are explicitly computed to second order in both parameters. Working in AdS spacetime introduces a number of technical issues, such as the appropriate “cut-off” of certain integrals, which are treated in detail. The conformally flat metric of AdS space leads to a non-vanishing constant extrinsic scalar curvature that greatly complicates the above analysis. In the second part of Section 3, we explore, in detail, the implications of working in a rescaled flat metric with respect to which the lowest order extrinsic curvature vanishes. It is shown how this simplifies the computation of the worldvolume geometric quantities, while leaving the analysis of the solution of the scalar equation of motion and the kink worldvolume action essentially the same. All these quantities are explicitly calculated to second order in . Using the direct relationship between the two metrics, we then compare the results of both approaches and show that they are identical, as they must be.

In Section 4, we analyze the worldvolume action computed in the previous section. Going to a conventional gauge, each term in the action is expressed as an explicit function of a real scalar field –the brane-bending mode. The relationship of these results to conformal Galileons [30] is discussed. In addition to the , and Galileons (the final Galileon appears at one order higher than our calculation), we find that there is a non-Galileon term proportional to the square of the extrinsic curvature scalar. Importantly, it is shown that that there is no region of momentum space for which this non-Galileon term is sub-dominant. We conclude that, although important contributions, Galileons are not the only relevant interactions on a kink/brane worldvolume. Finally, in Appendix A we prove that up to order , and to all orders with no worldvolume gradient operators, the worldvolume action can always be re-expressed purely in terms of Galileons by a specific field redefinition. However, the “non-Galileon” physics does not disappear — it is now non-trivially encoded in this field transformation.

2 Scalar Kinks in d=5 Flat Spacetime

The most general action for a real scalar field with minimal kinetic term coupled to gravity in spacetime is given by

(1)

where indices , is the five-dimension metric with signature , is the dimension Newton’s constant, is a cosmological constant and is an arbitrary potential. The associated Einstein equation is

(2)

where

(3)

Assuming that neither the temporal/spatial gradient nor the potential of depend on the Planck constant, in the limit that the dynamics decouples from gravity. Equation (2) then becomes

(4)

and the dynamics of the field can be consistently discussed in this background spacetime — the so-called “probe” limit — using the Lagrangian

(5)

In [16], Gregory and Carter used this probe limit to compute the induced worldvolume Lagrangian of the domain wall associated with the “kink” solution of the equation of motion in flat spacetime. In this section, we briefly review their formalism. Begin by setting

(6)

in (4) and taking the background spacetime to be be flat, denoting the metric by . In Cartesian coordinates , the metric takes the diagonal form . Now specify that

(7)

where is a positive constant of dimension , and is a constant of dimension , both independent of the Planck mass. The associated field equation is

(8)

Denoting the the fifth coordinate , we seek a solution for independent of the remaining coordinates. The equation of motion (8) then reduces to

(9)

Demanding that be positive for positive values of , this has the well-known “kink” solution

(10)

of width

(11)

Since this solution is independent of the remaining coordinates, it describes a static domain wall located at .

We would now like to generalize this to kink solutions that depend on the remaining coordinates as well as . This will be achieved as follows. Let specify the typical fluctuation length of the new solution along the remaining coordinates and define

(12)

We will seek solutions for which and, hence, can be obtained from (10) by a perturbation expansion. As discussed in [16], this is most easily carried out in Gaussian normal coordinates, defined as follows. Let be the new solution and denote the associated defect worldsheet by . Let be a unit geodesic normal vector field to , and generalize to be the proper length along the integral curves of . The remaining four worldsheet coordinates of will be denoted by , . Each constant surface then has a unit normal , an intrinsic metric and an extrinsic curvature defined by

(13)

respectively. These two quantities are not independent, satisfying the constraints

(14)
(15)

where is the Lie derivative along the vector field. With respect to Gaussian normal coordinates, the equation of motion (8) can be written as

(16)

where

(17)

Scaling to dimensionless variables by setting

(18)

equations (14), (15) and (16) become

(19)
(20)
(21)

where .111 For notational simplicity, we henceforth drop the prime on , the dimensionality of being clear from the context. These equations can now be solved by expanding each dimensionless quantity as a power series in . That is, let

(22)
(23)
(24)

where each coefficient is generically a function of the coordinates . Substituting these into (19), (20) and (21), one obtains equations for each coefficient function order by order in .

Order : At this order and, hence, is an unspecified function of independent of . It follows that equation (19) implies vanishes and (20) is trivially satisfied. That is,

(25)

Here, and henceforth, any quantity that depends only on the coordinates with be denoted with a hat. The equation of motion (21) becomes

(26)

This is simply (9) written in the rescaled variable and, hence, has the solution

(27)

Order : At this order, it follows from (20) and then (19) that

(28)

with unspecified. At order , (21) becomes

(29)

where is arbitrary. Subject to the boundary conditions that , and, hence,

(30)
(31)

there is a unique solution of (29) given by

(32)

with

(33)

We conclude that to order , and restoring the dimensionful parameters,

(34)

As discussed in detail in [16], is continuous, but not continuously differentiable, across the wall surface.

Order : To this order, one need only know the metric and extrinsic curvature. Solving (20) and then (19), we find that

(35)

Note that none of the purely dependent quantities — that is, none of the hatted functions — have been determined by the above procedure. This will remain true to any order in the -expansion. There is a fundamental reason for this; namely, prior to computing the worldvolume action of the domain wall, one must leave unrestrained any degrees of freedom intrinsic to the wall itself. These can only be determined by varying the worldvolume action to get the equations of motion of the wall location. It follows that and are off-shell and, hence, arbitrary functions of the intrinsic coordinates at this stage of the calculation.

The worldvolume effective action can now be calculated to any required accuracy in the expansion. It is given by

(36)

where

(37)

and is the original Lagrangian density given in (5), (7) evaluated for the solution of the equation of motion given to order in (34). Taylor expanding around and using (19), (20) to second order, one finds

(38)

with

(39)

Similarly, inserting solution in (34) into (5),(7), going to dimensionless variables and using the equations of motion (26) and (29), it follows that

(40)

where

(41)

Multiplying (38) and (40) then gives

(42)

in each of the separate domains and . Note that the vanishing of allows one to equate

(43)

to this order in the -expansion, which we do henceforth. The above stated boundary conditions imply that when integrated over the contribution of the total divergence term vanishes. Furthermore, in (39) is odd in and also does not contribute. Hence, inserting (42) into (37) using (39), and (11) one finds

(44)

where

(45)

and

(46)

Rewritten in dimensionful variables, truncating the expansion at order and using the Gauss-Codazzi relation

(47)

the worldvolume Lagrangian is given by

(48)

where

(49)

This is the result presented by Gregory and Carter [16], after correcting some errors in their manuscript.

3 Scalar Kinks in d=5 AdS Spacetime

In this section we will use, and extend, the Gregory/Carter formalism to compute the worldvolume Lagrangian of a kink domain wall in anti-deSitter spacetime. In this case, we choose

(50)

Equation (4) then admits an AdS background solution with metric

(51)

where . The associated curvature tensors are given by

(52)

In the “probe” limit, the dynamics of the field can be consistently discussed in this AdS background using the Lagrangian

(53)

In order for the equation of motion to admit a kink solution, we must modify potential (7) to

(54)

The associated field equation now becomes

(55)

We seek a solution for that depends on the fifth coordinate but is independent of the remaining coordinates. The equation of motion (55) then reduces to

(56)

Despite the fact that we are now in AdS spacetime, (56) continues to admit the kink solution

(57)

of width

(58)

Since this solution is independent of the remaining coordinates, it describes a static domain wall located at .

We would now like to generalize this to kink solutions that depend on the remaining coordinates as well as . Specifying the typical fluctuation length along the remaining coordinates as , and defining

(59)

this will again be achieved as a perturbative expansion around (57) in the small parameter . As in the flat spacetime case, this is most easily carried out in the Gaussian normal coordinates defined in Section 2. Each constant surface has a unit normal , with an intrinsic metric and extrinsic curvature defined in (13). These two quantities are not independent. As in flat spacetime, the metric and extrinsic curvature continue to be related as

(60)

where is the Lie derivative along the vector field. However, in a general curved five-dimensional spacetime we note from a Gauss-Codazzi relation that

(61)

Using the expression for in AdS spacetime given in (52) and the definition of in (13), we find

(62)

and, hence, that

(63)

Note that in the limit , this equation reverts to the flat spacetime expression given in (15). In Gaussian normal coordinates, the equation of motion (55) becomes

(64)

where and are the extrinsic scalar curvature and worldvolume covariant derivative defined in (17). Scaling to dimensionless variables by setting

(65)

equations (60),(63) and (64) become

(66)
(67)
(68)

where and we have defined

(69)

These equations can now be solved by expanding each dimensionless quantity as a power series in . That is, let

(70)
(71)
(72)

where each coefficient is generically a function of the coordinates . Substituting these into (66),(67) and (68), one obtains equations for each coefficient function order by order in .

The non-zero curvature in the AdS analysis will require us to carefully examine the expansion of the and equations. First consider the equation (66). Substituting (71) and (72) into (66), we find to order and that

(73)
(74)

respectively. Now examine the the equation (67). Note that this can be written as

(75)

where

(76)

Expanding

(77)

it follows from (71), (76) that

(78)

Substituting (71), (72) and (77) into the equation (75), we find to order and that

(79)
(80)

Before continuing to the equation of motion, let us solve (73),(74) and (79), (80).

Order : Since at this order , it follows from (51) that

(81)

with unspecified. Hence, the equation (73) gives

(82)

For notational consistency, we write this as

(83)

Using the first relation in (78), we find that the equation (79) is trivially satisfied.

Order : Substituting (82) into the order equation (80), and recognizing that the second expression in (78) implies

(84)

we find that

(85)

This is solved by

(86)

with unspecified . Putting this result into the order equation (74) gives

(87)

This can be integrated to

(88)

To summarize: we have found that

(89)
(90)

where none of the hatted” functions of are specified.

Now consider the equation of motion. To proceed, one must substitute (70), (71), (72), (77) into (68) noting that it is the trace of defined by

(91)

that enters this equation. Expanding

(92)

we find using (89), (90) and (78) that

(93)

where . Inserting this along with (70), (71), (72) into (68), we find to order and that

(94)
(95)

Order : Using the first expression in (93), it is straightforward to show that (94) has the same solution as in (57). That is,

(96)

Order : Using (93), equation (95) becomes

(97)

Note that as and, hence, , this reverts to the flat space equation (29). Let us solve (97) using the ansatz

(98)

Inserting this into (97), the factor cancels and one is left with an equation for given by

(99)

Using (96), this becomes

(100)

This equation can be solved independently in each of the separate domains and . Corresponding to (30),(31), one must also impose the boundary conditions

(101)
(102)

Subject to these conditions, there is a unique solution of (100) which one can solve for numerically. For example, the solution with is presented in Figure 1.

Figure 1: Numerical solution for with .

We conclude that to this order in , and restoring the dimensionful parameters,

(103)

As in the flat spacetime case, is continuous, but not continuously differentiable, across the wall surface.

The worldvolume effective action can now be calculated to any required accuracy in the expansion using (36), where

(104)

is the AdS metric (51) and is the original Lagrangian density (53), (54) evaluated for the solution of the equation of motion given to order in (103). Taylor expanding around using (66),(67) and the fact that

(105)

one finds that

(106)

with

(107)

Similarly, inserting solution in (103) into (53),(54) and using the equations of motion (94),(97), it follows that

(108)

where

(109)

the are of the same functional form as the flat spacetime results in (41) and

(110)

Note that in (109) arises from the interaction term in the Lagrangian and not from a power series expansion in . Hence, , in both and contain explicit dependence on through, for example, the function . Multiplying (106) and (108) gives

(111)

where we have dropped three terms that are odd in that will vanish when integrated over the transverse coordinate. Note that the term proportional to is identical to the term in the flat spacetime result (42), and that the terms–which contain in –are subleading. To avoid having to calculate , we will, henceforth, work only to lower order. Inserting the solution (98) for and using the relation

(112)

we find that the remaining terms are

(113)

plus a total derivative term. We have also “integrated by parts” in anticipation of integrating over . Putting everything together, (111) becomes