Dark Solitons, D-branes and Noncommutative Tachyon Field Theory
DDark Solitons, D -branes and NoncommutativeTachyon Field Theory Stefano Giaccari a, b
Jun Nian c, d a Department of Physics, Faculty of ScienceUniversity of ZagrebBijeniˇcka 32, HR-10000 Zagreb, Croatia b Department of Physics & Center for Field Theory and Particle PhysicsFudan University200433 Shanghai, China c Institut des Hautes ´Etudes ScientifiquesLe Bois-Marie, 35 route de Chartres91440 Bures-sur-Yvette, France d C.N. Yang Institute for Theoretical PhysicsStony Brook UniversityStony Brook, NY 11794-3840, U.S.A.
E-mail: [email protected] , [email protected] Abstract:
In this paper we discuss the boson/vortex duality by mapping the (3+1)DGross-Pitaevskii theory into an effective string theory in the presence of boundaries. Viathe effective string theory, we find the Seiberg-Witten map between the commutative andthe noncommutative tachyon field theories, and consequently identify their soliton solutionswith D -branes in the effective string theory. We perform various checks of the duality mapand the identification of soliton solutions. This new insight between the Gross-Pitaevskiitheory and the effective string theory explains the similarity of these two systems at quan-titative level. Keywords:
Gross-Pitaevskii theory, effective string theory, boson/vortex duality, darksoliton, D -brane, noncommutative tachyon field theory, noncommutative soliton a r X i v : . [ h e p - t h ] N ov ontents D -branes 13 D -brane Interaction 21 It has been known for a long time that some excitations in Bose-Einstein condensates(BEC), e.g. vortex lines, vortex rings and dark solitons, are very similar to some basicingredients of string theory, such as open strings, closed strings and D -branes.In Refs. [1, 2], the comparison between dark solitons and D -branes has been madequantitatively. The authors studied a two-component BEC model, and found the energyof this system is the same as the one in a four-dimensional N = 2 supersymmetric sigmamodel [3]. Using the BPS procedure, they found the soliton solution and the vortex solutionof the system. However, the soliton solution that they found in this configuration is notthe standard dark soliton solution known in the Bose-Einstein condensates, instead it isthe boundary between two components of the BEC.More recently, some numerical work in BEC [4] has demonstrated that the vortexlines can be attached directly to dark solitons, which mimics the configuration of openstrings attached to D -branes in string theory. Hence, it is very conceivable that thereshould be some explanations about this similarity from theoretical point of view. If thiscorrespondence can be put on solid ground, one may expect to simulate string theory in a– 1 –EC system, and at the same time bring in some new ideas to the study of Bose-Einsteincondensates.We would like to explore this relation between BEC and string theory from theoreticalperspective. Our starting point is the following. The Gross-Pitaevskii equation is known asan effective theory to describe Bose-Einstein condensates [5], and it has been shown [6–8]that using the so-called boson/vortex duality for a spacetime without boundary the Gross-Pitaevskii equation can be mapped into an effective theory, which is very similar to thestandard string theory in the large B -field limit. In this paper, we will demonstrate thatthis duality can also be generalized to a spacetime with boundary, where certain solitonsolutions play the role of boundaries. By analyzing the effective string theory obtainedfrom the duality, eventually we would like to identify dark soliton solutions to the Gross-Pitaevskii equation with D -branes in the effective string theory. Relativistic Commutative Tachyon Field Theory Noncommutative Tachyon Field TheoryEffective String TheoryGross-Pitaevskii Theory
Figure 1 . The relation between different theories
Fig. 1 illustrates the relation among the theories and their soliton solutions, whichwill be discussed in this paper. As we discussed before, the Gross-Pitaevskii theory canbe mapped into an effective string theory. It can also be viewed as the non-relativisticversion of a relativistic commutative scalar field theory. Sometimes for convenience, we cangauge the relativistic commutative scalar field theory to obtain the commutative AbelianHiggs model. Although different in kinetic terms, the Gross-Pitaevskii theory and thecommutative scalar field theory share the same time-independent dark soliton solution.To identify dark soliton solutions with D-branes in the effective string theory, it ismore convenient to first relate them to solitons in the noncommutative tachyon field theory,which has a natural relation with string theory. The approach of applying noncommutativegeometry to string theory was first introduced by Seiberg and Witten [9]. They havestudied the Yang-Mills theory from the open string sector in the large B -field limit, andhave found that it admits both commutative and noncommutative descriptions, which are– 2 –elated by a nonlinear map of the corresponding gauge fields called the Seiberg-Wittenmap. A similar relation should be obtained for the tachyon field. If we think of therelativistic commutative scalar field theory as the commutative description of the tachyonfield, there should be a noncommutative tachyon field theory obtained from the effectivestring theory. The commutative and the noncommutative tachyon fields can be related bya Seiberg-Witten map, and in this way the dark soliton solution to the commutative scalarfield theory is related to the noncommutative soliton.To find the Seiberg-Witten map for the tachyon field, one can first consider the com-mutative and the noncommutative Abelian Higgs models. The commutative Abelian Higgsmodel has been studied in the literature in great detail. It is known that this theory hassome topologically nontrivial solutions such as the Nielsen-Olesen vortex line [10], which be-comes the vortex line solution in the Gross-Pitaevskii theory when the gauge field is turnedoff, and the endpoints of the Nielson-Olsen string have to terminate at (anti-)monopoles[11]. On the other hand, the (2+1)-dimensional noncommutative Abelian Higgs model andits vortex solutions have been studied in Ref. [12]. It is straightforward to generalize theanalysis in Ref. [12] to our case and find the Seiberg-Witten map between the tachyons.We then turn off the gauge fields to obtain the Seiberg-Witten map for pure scalar fieldtheories, which are just the commutative and the noncommutative tachyon field theoriesthat we are looking for. It can be shown that the noncommutative scalar field theory ob-tained in this way coincides with the noncommutative tachyon field theory obtained fromthe effective string theory at the leading order. For this theory, Ref. [13] has studied itssoliton solutions, and then Ref. [14] has shown that these noncommutative solitons can beidentified with the D -branes in string theoy.Hence, after all these steps above the circle shown Fig. 1 is closed, and under certainapproximations we can identify dark solitons in the Gross-Pitaevskii theory with D -branesin the effective string theory as well as solitons in the noncommutative tachyon field theory.This paper is organized as follows. In Section 2, we review the commutative AbelianHiggs model, the Gross-Pitaevskii theory and some solutions with nontrivial topology. InSection 3, we map the Gross-Pitaevskii theory into an effective string theory in the presenceof boundaries. As discussed above, we would like to identify dark solitons, noncommutativesolitons and D -branes. To test this identification, we compare different descriptions andtheir solitons in Section 4. The noncommutative Abelian Higgs model is discussed in Sub-section 4.1. By turning off the gauge field, we obtain a noncommutative scalar field theory,which can be interpreted as a noncommutative tachyon field theory. In Subsection 4.2, weobtain the noncommutative tachyon field theory directly from the effective string theory.The tensions of solitons are compared with the ones of D -branes from different descriptionsin Subsection 4.1 and 4.2. Moreover, the comparison of the D -brane interaction with thenumerical results of the dark soliton interaction is made in Subsection 4.3. Finally, futuredirections are discussed in Section 5. Some details of the computation will be presentedin a few appendices. In Appendix A we list some solutions with nontrivial topology tothe Gross-Pitaevskii theory. The derivations of the effective string theory with boundariesfrom the Gross-Pitaevskii theory are presented in Appendix B. The tachyon potential inthe presence of the B -field will be discussed in Appendix C.– 3 – Commutative Field Theories
In this section, we would like to review some known results of commutative scalar field the-ories. We first discuss the commutative Abelian Higgs model and the Nielsen-Olsen vortexline solution in Subsection 2.1. By turning off the gauge field, we obtain the relativisticcommutative tachyon field theory, whose non-relativistic version is the Gross-Pitaevskiitheory. We will discuss this theory and some solutions with nontrivial topology in Subsec-tion 2.2.
The commutative Abelian Higgs model in (3+1)-dimensions is given by the Lagrangian: L = − F µν F µν − | ( ∂ µ + ieA µ ) φ | − λ ( | φ | − | φ | ) , (2.1)where φ is a complex scalar, and A µ is the gauge field. The equations of motion are( ∂ µ + ieA µ ) φ − λ ( | φ | − | φ | ) φ = 0 , (2.2) ∂ ν F µν = i e ( φ ∗ ∂ µ φ − φ∂ µ φ ∗ ) − e A µ φ ∗ φ . (2.3)This theory has nontrivial classical solutions, including string-like Nielsen-Olsen vortexlines. The Nielsen-Olsen vortex line solution (or Nielsen-Olsen string) was found first byNielsen and Olsen in Ref. [10]. Under the gauge A = 0, considering a string-like solution,i.e. preserving a cynlindrical symmetry, one can assume the axis along the z -direction andapply the ansatz A ( r ) = r × e z r | A ( r ) | , (2.4)and the flux carried by the Nielsen-Olsen vortex line isΦ( r ) = (cid:73) A µ ( x ) dx µ = 2 πr | A ( r ) | , (2.5)Since the covariant derivative on the scalar field vanishes outside the vortex line, the phaseof φ defined by φ = | φ | e iη satisfies dη + eA = 0 for r → ∞ . (2.6)Hence, the property that φ should be single-valued requiresΦ = lim r →∞ (cid:73) r = r A = − lim r →∞ e (cid:73) r = r dη = n πe , (2.7)i.e. the magnetic flux carried by the Nielsen-Olsen vortex line is quantized.By plugging the ansatz (2.4) into the Lagrangian (2.1) and perfomring the variationof the fields, one can obtain the classical equations of motion for the configurations withcynlindrical symmetry, which can be solved numerically. The solutions have the asymptoticbehavior: | φ | = | φ | = const , | A | = 1 er + ce K ( er | φ | ) , for r → ∞ ; (2.8)– 4 – φ | = 0 , for r → . (2.9)As discussed in Ref. [11], an infinitely long vortex line has infinite energy, which isunphysical. In order to have a finite length, a vortex line can terminate at a magnetic(anti-)monopole, which has the magnetic charge g = n π/e . Hence, the magnetic fluxcarried by the Nielsen-Olsen vortex line can be absorbed by the monopole anti-monopolepair. Moreover, one can demonstrate that in this case the potential between the monopoleanti-monopole pair is linear in the distance between them, i.e., the Nielsen-Olsen vortexline realizes the confinement in the Abelian Higgs model. Schematically, the configurationof a finite Nielsen-Olsen vortex line with a monopole anti-monopole pair at two endpointsis shown in Fig. 2. rz Figure 2 . The sketch of the Nielsen-Olsen vortex line ending on a monopole anti-monopole pair
It is also known that the Nielsen-Olsen vortex line can be effectively described by theNambu string action [10, 11], which also confirms our following discussions that the scalartheory can be mapped into an effective string theory. As we will also see later, the vortexline solution to the Gross-Pitaevskii equation can be viewed as the Nielsen-Olsen vortexline solution when the gauge field is turned off.
There are two kinds of commutative scalar field theory, relativistic and non-relativistic.Both of them will appear in the rest of this paper. If some solutions are time-independent,they are solutions to both the relativistic and the non-relativistic commutative scalar fieldtheory.Let us first look at the relativistic one. If we turn off the gauge field, the commutativeAbelian Higgs model (2.1) becomes the corresponding relativistic commutative scalar fieldtheory: L c = − | ∂ µ φ | − λ ( | φ | − | φ | ) , (2.10)– 5 –here the subscript “c” stands for the commutative theory, in contrast to the noncommu-tative theory that we will discuss later.The commutative scalar theory (2.10) can be viewed as the relativistic version of theGross-Pitaevskii (GP) theory, which is given by the following non-relativistic Lagrangian: L GP = iφ † ∂ t φ − m ( ∇ φ † )( ∇ φ ) − g | φ | − ρ ) . (2.11)Varying it with respect to φ † , we obtain the Gross-Pitaevskii equation: i ∂ t φ + 12 m ∇ φ − g ( | φ | − ρ ) φ = 0 . (2.12)The Gross-Pitaevskii equation has various solutions with nontrivial topology [5]. Let usbriefly summarize them in the following, and more details of these solutions can be foundin Appendix A. • Dark soliton:For the repulsive interaction, i.e. g > x normal to the plane it has the following profile: φ ( x ) = (cid:126) √ n √ m tanh (cid:20) x √ ξ (cid:21) , (2.13)where n is the density at infinity, and ξ ≡ (cid:126) / √ mgn is a parameter called healinglength. At the position x = 0 where the dark soliton is localized, the density is zero.Consequently, in cold atom experiments that simulate the Gross-Pitaevskii equation,this kind of solution appears dark. That is why it is called “dark soliton”. • Grey soliton:The grey soliton solution is similar to the dark soliton solution, but is a time-dependent moving plane extended in two spatial directions. From the dark solitonsolution discussed above, one can easily perform a Galilean boost to obtain a movinggrey soliton solution given byΨ( x − vt ) = (cid:126) √ n √ m (cid:32) i vc + (cid:114) − v c tanh (cid:34) x − vt √ ξ (cid:114) − v c (cid:35)(cid:33) , (2.14)where c and v stand for the speed of sound and the speed of the grey soliton respec-tively. At the center x = 0 the density is non-zero, which makes it appear grey inreal experiments. That is where its name comes from. • Bright soliton: For the attractive interaction, i.e. g < φ ( x ) = φ (0) 1cosh( x/ √ ξ ) . (2.15)In contrast to the dark and the grey soliton, the bright soliton solution has themaximal density at the center, which appears bright in real experiments.– 6 – Vortex line: Unlike the various plane-like solutions discussed above, the vortex lineis a string-like solution, which in the cylindrical coordinates ( η, ϕ ) is given by φ = √ n f ( η ) e isϕ , (2.16)where n is the density, and f ( η ) satisfies the following equation:1 η ddη (cid:18) η dfdη (cid:19) + (cid:18) − s η (cid:19) f − f = 0 . (2.17)For given boundary conditions and a fixed value of s , the equation above can besolved numerically. • Vortex ring: The vortex line solution has two endpoints. These two end points canjoin together, and the original vortex line solution becomes a ring-like object, whichis called vortex ring. Besides different shapes, it turns out that the vortex ring cannotbe at rest in contrast to the vortex line solution. The collision of two vortex ringshas been studied in Ref. [8] using the boson/vortex duality.We would like to emphasize that, although the Gross-Pitaevskii equation (2.12) is anon-relativistic version of the field equation for the commutative scalar field theory (2.10),they share the same time-independent dark soliton solution (2.13) discussed above.According to the well-known Derrick’s theorem, stable topologically nontrivial non-singular field configurations for a purely scalar field theory with second derivatives canexist only for d <
2, where d is the spatial dimension of the configuration. Hence, thedark soliton solution is an unstable configuration. In practice, one can restrict the size oftransverse dimensions of the dark soliton to make it stable.In principle, the solutions with nontrivial topology discussed in this section can beclassified using the homotopy group, which is beyond the scope of this paper but has beenstudied in great detail in the literautre. As we mentioned before, it has been found recently in Ref. [4] that vortex lines can beattached to the dark soliton planes to form a relatively stable configuration. If we treatthe vortex line solution as an effective string, we should be able to describe this novelstable configuration using an effective string theory, where dark solitons play the role of D -branes. In this section, we briefly review the mapping of the Gross-Pitaevskii equation intoan effective string theory in the presence of dark solitons as boundaries. The identificationof dark solitons with D -branes will be justified in later sections.It has been shown in Refs. [6–8] that without boundaries the (3+1)D Gross-Pitaevskiitheory (2.11) can be mapped into an effective string theory, and similar analysis can begeneralized to other dimensions (see e.g. Ref. [15]). In the presence of boundaries, somesubtleties and new features emerge and have to be paid special attention to.In this paper we consider the following simplest configuration. Two parallel darksolitons are placed in the bulk of the (3+1)D spacetime, which are separated in the spatial– 7 – -direction with a distance L . The dark soliton plane can be viewed as a (2+1)D spacetimewithout boundary. In the space between two dark soliton planes there can be closed vortexlines and open vortex lines, and the endpoints of the open vortex lines have to be attachedto one of the dark soliton planes.Now let us discuss the duality map in the presence of dark solitons, which is alsodiscussed in Ref. [4] using an alternative approach. For the configuration with boundaries,we can repeat the first few steps for the configuration without boundaries discussed inRef. [8]. Let us recall the Gross-Pitaevskii Lagrangian (2.11): L GP = iφ † ∂ t φ − m ( ∇ φ † )( ∇ φ ) − g | φ | − ρ ) . With the parametrization: φ = √ ρ e iη , (3.1)the original Gross-Pitaevskii Lagrangian (2.11) becomes L = i ˙ ρ − ρ ˙ η − ρ m ( ∇ η ) − ( ∇ ρ ) mρ − g ρ − ρ ) . (3.2)We can drop the first term as a total derivative and separate the phase-dependent partfrom the phase-independent part: L ≡ − ρ ˙ η − ρ m ( ∇ η ) , (3.3) L ≡ − ( ∇ ρ ) mρ − g ρ − ρ ) . (3.4)In the presence of a boundary, the phase η consists of two parts, one part η ( t, x, y, z )defined on the (3+1)D spacetime and the other part η ( t, x, y ) defined only on the (2+1)Dboundary, which can be the dark soliton plane. Both of them may contain singularities.We can perform the duality map on the (3+1)D and the (2+1)D spacetime separately, i.e., S = (cid:90) d x (cid:2) L D + L D (cid:3) + (cid:96) (cid:90) d x (cid:2) L D + L D (cid:3) , (3.5)where (cid:96) is a constant length scale due to the dimensional reason, which will be discussedlater in this subsection.The separation of the (3+1)D part and the (2+1)D part of the action can also beobtained in the following way. Instead of Eq. (3.1) let us use another parametrization: φ = p e iη , (3.6)where p = √ ρ . For a dark soliton given by Eq. (2.13), the background part of the factor p only depends on z : p = √ n tanh (cid:18) z √ (cid:96) (cid:19) . (3.7)It is usually said that there is a π -jump in the phase when across a dark soliton plane. Thisis just due to the fact that p changes sign from one side of the soliton to the other side, and– 8 –ne can absorb the sign into the phase using − iπ ). Equivalently, we can keep thesign change of p and consider the phase without a π -jump. Using the new parametrization(3.6), we can rewrite the GP Lagrangian (2.11) as L = ip ˙ p − p ˙ η − p m ( ∇ η ) − ( ∇ p ) m − g p − p c ) , (3.8)which can also be obtained by replacing ρ with p in Eq. (3.2), and p c = √ n is a constant.Again, the first term is a total derivative that can be dropped. In Appendix B, we analyzeEq. (3.8) term by term under the assumption that p consists of both the background andthe fluctuations, i.e., p = p + (cid:101) p , (3.9)where we take p to be the profile given by Eq. (3.7), which depends only on z , whilewe assume that (cid:101) p does not depend on z . The physical reason is that the dark soliton isvery heavy, so that its longitudinal position is fixed, and there are no fluctuations in thelongitudinal direction. As shown in Appendix B, in the background of a dark soliton, theaction can be expressed as (cid:90) d x L = (cid:96) (cid:90) d x (cid:20) − (cid:101) ρ ˙ η − m ( (cid:101) ρ + C ) ( (cid:101) ∇ η ) − m (cid:101) ρ (cid:16) (cid:101) ∇ (cid:101) ρ (cid:17) − g (cid:101) ρ − (cid:101) ρ ) (cid:21) , (3.10)where C is a constant. This expression corresponds to the 3D part of the action, andjustifies the separation presented in Eq. (3.5).As shown in Eq. (3.5), in the presence of boundaries the full theory should includeboth the (3+1)D duality and the (2+1)D duality. In the bulk one can still perform the(3+1)D duality is exactly the same as the case without boundaries discussed in Ref. [8].Hence, we focus on the (2+1)D duality map in the following. Note that all the fields withtilde ( (cid:101) ) are defined in (2+1)D, i.e., they depend only on ( t, x, y ) but are independent ofthe longitudinal coordinate z .The steps are similar to the (3+1)D case. One can introduce a 3-vector f a = ( ρ, f ˆ a )with a ∈ { t, x, y } and ˆ a ∈ { x, y } . Assuming that (cid:90) D f ˆ a exp (cid:18) i(cid:96) (cid:90) d x m (cid:101) ρ (cid:48) f ˆ a f ˆ a (cid:19) = 1 , (3.11)we can rewrite the first two terms in the action (3.10) without changing the path integralin the following way: − (cid:101) ρ ˙ η − (cid:101) ρ (cid:48) m ( (cid:101) ∇ η ) + m (cid:101) ρ (cid:48) (cid:18) f ˆ a − (cid:101) ρ (cid:48) m (cid:101) ∇ ˆ a η (cid:19) = − (cid:101) ρ ˙ η + m (cid:101) ρ (cid:48) f ˆ a f ˆ a − f ˆ a ∂ ˆ a η = m (cid:101) ρ (cid:48) f ˆ a f ˆ a − f a ∂ a η , (3.12)where (cid:101) ρ (cid:48) ≡ (cid:101) ρ + C .Now one can separate the (2+1)D phase η ( t, x, y ) into the smooth part and the singularpart: − f a ∂ a η = − f a ∂ a η smooth − f a ∂ a η singular . (3.13)– 9 –he smooth part η smooth does not feel the (2+1)D vortices from the endpoints of vortexlines, and it is well-defined on the whole dark soliton plane. Integrating it out, we obtain ∂ a f a = 0 , (3.14)which can be solved by f a = 12 (cid:15) abc F bc with F ab = 12 ( ∂ a A b − ∂ b A a ) . (3.15)Consequently, the contribution of η smooth to the first two terms in the action (3.10) isexp (cid:20) i (cid:90) d x L D (cid:21) = (cid:90) D f a exp (cid:40) i(cid:96) (cid:90) d x (cid:20) − f a ∂ a η + m (cid:101) ρ f ˆ a f ˆ a (cid:21) (cid:41) = (cid:90) D f a exp (cid:40) i(cid:96) (cid:90) d x (cid:20) − f a ∂ a η singular + m (cid:101) ρ F a (cid:21) (cid:41) = (cid:90) D f a exp (cid:40) i(cid:96) (cid:90) d x (cid:20) − f a ∂ a η singular + m (cid:101) ρ (cid:101) F a (cid:21) (cid:41) , (3.16)and the remaining terms in the action (3.10) are L D = − ( ∇ (cid:101) F ˆ a ˆ b ) m (cid:101) ρ − g (cid:101) F a ˆ b , (3.17)where ˆ a, ˆ b ∈ { x, y } , and (cid:101) F denotes the fluctuation of the field strength F . Together, theyform (cid:90) D (cid:101) ρ D η exp (cid:40) i(cid:96) (cid:90) d x (cid:20) − (cid:101) ρ ˙ η − (cid:101) ρ m ( ∇ η ) − ( ∇ (cid:101) ρ ) m (cid:101) ρ − g (cid:101) ρ − ρ ) (cid:21) (cid:41) = (cid:90) D A a exp (cid:40) i(cid:96) (cid:90) d x (cid:34) − f a ∂ a η singular − g η ab η cd (cid:101) F ac (cid:101) F bd − ( ∇ (cid:101) F ˆ a ˆ b ) m (cid:101) ρ (cid:35) (cid:41) , (3.18)where η ab = ( − (cid:101) ρg/ (2 m ) , ,
1) is an effective 3D metric.Now let us consider the term − f a ∂ a η singular . Since we have found the solution f a = (cid:15) abc ∂ b A c , it can be plugged into the singular term, then we obtain i(cid:96) (cid:90) d x [ − f a ∂ a η singular ]= − i(cid:96) (cid:90) d x (cid:15) abc ( ∂ b A c ) ∂ a η singular = − i(cid:96) (cid:90) d x (cid:15) cab A c ( ∂ a ∂ b η singular )= − πi(cid:96) (cid:90) d x A c j c , (3.19)where we have defined a vortex current: j c ≡ π (cid:15) cab ∂ a ∂ b η singular , (3.20)– 10 –hich satisfies (cid:90) d x j = 12 π (cid:90) d x (cid:15) ˆ a ˆ b ∂ ˆ a ∂ ˆ b η singular = 12 π (cid:90) d x ∇× ( ∇ η singular ) = 12 π (cid:73) d(cid:126)x ·∇ η singular = 1 . (3.21)Because the vortices on the dark soliton plane can also be viewed as the endpoints of thevortex lines in the (3+1)D spacetime, we may also use the following relation in (3+1)D: (cid:15) λσµν ∂ µ ∂ ν η = − π (cid:90) d σ (cid:15) αβ ∂ α X λ ∂ β X σ δ ( x µ − X µ ) (3.22)to obtain i(cid:96) (cid:90) d x [ − f a ∂ a η singular ]= − i(cid:96) (cid:90) d x (cid:15) zcab A c ( ∂ a ∂ b η singular )= 2 πi(cid:96) (cid:90) d x (cid:90) dτ dσ A a (cid:15) αβ ∂ α X z ∂ β X a L δ ( x − X )= 2 πi(cid:96) (cid:90) dτ A a ∂ τ X a , (3.23)where the 3D δ -function is related to the 4D δ -function in the following way: δ ( x − X ) = 1 L δ ( x − X ) . (3.24)In the last step we have used the fact that only ∂X z ∂σ = ∂z∂σ is nonvanishing, when the vortexline is perpendicular to the dark soliton, i.e. z (cid:107) σ , hence, (cid:90) dσ∂ σ X z = (cid:90) dσ ∂z∂σ = (cid:90) dz = L , (3.25)where L is the distance between two parallel dark soliton planes.There are still two issues that we have to address carefully. One is the dimensionality.In the following we list the mass dimensions of various fields and parameters:[ ρ ] = 3 , [ η ] = 0 , [ m ] = 1 , [ g ] = − , (3.26)[ H ] = 3 , [ B ] = 2 , [ (cid:101) F ] = 3 , [ A ] = 2 . (3.27)Conventionally, the gauge field A has mass dimension 1, which can be achieved by absorbingthe length scale (cid:96) into A , i.e., (cid:96)A a → A a , (3.28)to make it of dimension 1. Also, the 2-form gauge field B µν is conventionally dimensionless.To achieve it, we can separate a dimensionful constant from it, i.e., B µν → πα (cid:48) B µν , (3.29)where α (cid:48) = (cid:96) s with (cid:96) s denoting the effective string length scale.– 11 –he other issue is that we treat the (3+1)D duality and the (2+1)D duality separately.Although they seem to be independent of each other, the dualities are related through aboundary integral in (3+1)D. Let us recall for the (3+1)D case: L D ⊃ − f µ ∂ µ η smooth − f µ ∂ µ η vortex . (3.30)We can partially integrate the first term and then integrate out η smooth to obtain theequation ∂ µ f µ = 0. During the derivation we drop a boundary term − (cid:90) d x ∂ µ ( f µ η smooth ) , (3.31)which vanishes when the dark soliton is absent. In the presence of the dark soliton, thisboundary term becomes − (cid:90) d x ∂ µ ( f µ η smooth ) = − (cid:90) d x f µ η smooth = − (cid:90) d x (cid:15) zabc
12 ( ∂ a B bc ) η smooth = (cid:90) d x (cid:15) abc B bc ∂ a η smooth , (3.32)which should be combined with the term − f a ∂ a η smooth in the (2+1)D duality. Hence,precisely speaking, for the (2+1)D duality instead of Eq. (3.13) there should be L D ⊃ − (cid:18) f a − (cid:15) abc B bc (cid:19) ∂ a η smooth − f a ∂ a η singular . (3.33)Partially integrating the first term, instead of ∂ a f a = 0 we will obtain ∂ a (cid:18) f a − (cid:15) abc B bc (cid:19) = 0 . (3.34)The solution to this equation is f a − (cid:15) abc B bc = 12 (cid:15) abc F bc with F bc = 12 ( ∂ b A c − ∂ c A b ) . (3.35)Consequently, f a = 12 (cid:15) abc ( F bc + B bc ) . (3.36)Hence, in the (2+1)D duality that we discussed above F should be replaced by F + B . Afterintroducing some length scales to match the conventional dimensions, the combinationshould be F ab + 12 πα (cid:48) B ab . (3.37)Therefore, the final expression of the (2+1)D dual theory is (cid:90) D A a exp (cid:40) − πi (cid:90) d x A a j a + i(cid:96) (cid:90) d x (cid:34) − g (cid:101) F + 12 πα (cid:48) (cid:101) B ) − ( ∇ (cid:101) F ˆ a ˆ b + πα (cid:48) ∇ (cid:101) B ˆ a ˆ b ) m (cid:101) ρ (cid:35) (cid:41) = (cid:90) D A a exp (cid:40) πi (cid:90) dτ A a ∂ τ X a + i(cid:96) (cid:90) d x (cid:34) − g (cid:101) F + 12 πα (cid:48) (cid:101) B ) − ( ∇ (cid:101) F ˆ a ˆ b + πα (cid:48) ∇ (cid:101) B ˆ a ˆ b ) m (cid:101) ρ (cid:35) (cid:41) , (3.38)– 12 –here we can drop the last term in the IR regime, and( (cid:101) F + 12 πα (cid:48) (cid:101) B ) ≡ η ab η cd (cid:18) (cid:101) F ac + 12 πα (cid:48) (cid:101) B ac (cid:19) (cid:18) (cid:101) F bd + 12 πα (cid:48) (cid:101) B bd (cid:19) . (3.39)Here again (cid:101) F and (cid:101) B are fluatuations of F and B respectively. The full theory should bethe combination of the (2+1)D action above with the one from the (3+1)D duality, whichhas the following expression in the IR regime: Z = (cid:90) DB µν DA a exp (cid:34) iη (cid:88) i (cid:90) Σ i dσdτ (cid:15) αβ ∂ α X µ ∂ β X ν B µν − ig (cid:90) d x h + iη (cid:88) j (cid:90) ∂ Σ j dτ A a ∂ τ X a − ig (cid:96) (cid:90) d x ( (cid:101) F + (cid:101) B ) (cid:35) , (3.40)where η = 2 π (cid:126) , α, β ∈ { τ, σ } , and Σ i is the worldsheet spanned by the i -th vortex linewith boundaries ∂ Σ j . The summation over ∂ Σ j includes all the endpoints X a = X a ( τ ) ofvortex lines attached to dark solitons. From the 2-form field B µν , one can define a 3-formfield strength: H µνλ ≡ ∂ µ B νλ + ∂ ν B λµ + ∂ λ B µν = H µνλ + h µνλ , (3.41)where H µνλ is the background field with H = ρ , and the fluctuations are given by h = h µνλ h µνλ / η µν = diag {− c s , , , } and the speed ofsound c s = (cid:112) gρ /m . (cid:101) F and (cid:101) B are the fluctuations of F and B on the soliton planerespectively.The first line of Eq. (3.40) is just the effective string action without boundaries dis-cussed in Refs. [6–8], while the second line of Eq. (3.40) is the contribution from boundaries.We would like to emphasize that the effective action (3.40) is invariant under the followinggauge transformations (see e.g. Ref. [16]): B µν → B µν + ∂ µ Λ ν − ∂ ν Λ µ ,A a → A a − Λ a , (3.42)where Λ µ are transformation parameters. This kind of gauge symmetry is also called theΛ-symmetry, which has been studied in many works, e.g. Ref. [17]. D -branes In Section 2, we have discussed the relativistic version of the Gross-Pitaevskii theory. Thistheory has the same time-independent dark soliton solution as the non-relativistic one,which can be mapped into an effective string theory, as discussed in Section 3.Since we are interested in the time-independent solutions, the Lorentz-violating term inthe non-relativistic Gross-Pitaevskii theory will be irrelevant. Moreover, we shall considerthe large noncommutativity limit, in which the kinetic term can actually be neglectedwith respect to the potential. For these reasons the difference between the relativisticcommutative scalar field theory and the non-relativistic Gross-Pitaevskii theory does not– 13 –ffect our discussions, as long as one focuses on the time-indepedent solutions. This simpleobservation enables us to use the more convenient relativistic theory to discuss some crucialissues of solitons.In this section, we first discuss the noncommutative version of the Abelian Higgs modeland justify the identification of their soliton solutions with the D -branes of the effectivestring theory introduced before. Our identification turns out to be valid in some limits,which we highlight by reviewing some crucial aspects of the noncommutative tachyon fieldtheory appearing in the string theory literature [14, 18, 19]. The commutative Abelian Higgs model has been discussed in Section 2, and it is given bythe Lagrangian (2.1). We are interested in the limit where the U (1) gauge field is vanishing,and in this limit we obtain a scalar field theory given by Eq. (2.10). We consider severalkinds of soliton solutions to this scalar field theory, in particular dark solitons (2.13) andvortex lines (2.16). In order to interpret these solitons from string theory’s point of view,we will consider their counterparts in the noncommutative scalar field theory obtained fromthe noncommutative Abelian Higgs model in the limit of vanishing gauge field. This is thesame strategy described in Ref. [12].As discussed in Ref. [9], the Seiberg-Witten map for the gauge field should preservethe gauge transformation relation, i.e.,ˆ A ( A ) + ˆ δ ˆ λ ˆ A ( A ) = ˆ A ( A + δ λ A ) , (4.1)where the hat (ˆ) denotes the noncommutative fields, and the gauge transformation forthe ordinary Yang-Mills theory is δ λ A i = ∂ i λ + i [ λ, A i ] , (4.2)while for the noncommutative Yang-Mills theory:ˆ δ ˆ λ ˆ A i = ∂ i ˆ λ + i ˆ λ (cid:63) ˆ A i − i ˆ A i (cid:63) ˆ λ , (4.3)with the star product (cid:63) between two noncommutative fields f and g defined by f ( x ) (cid:63) g ( x ) = e i θ µν ∂∂ξµ ∂∂ζν f ( x + ξ ) g ( x + ζ ) | ξ = ζ =0 , (4.4)and θ µν is given by [ x µ , x ν ] = iθ µν . (4.5)Solving the constraint (4.1), one obtainsˆ A i ( A ) = A i − θ kl { A k , ∂ l A i + F lk } + O ( θ ) , ˆ λ ( λ, A ) = λ + 14 θ kl { ∂ k λ, A l } + O ( θ ) . (4.6)– 14 –imilarly, for the scalar field we can also require that the gauge transformation relationshould be preserved under the Seiberg-Witten map for both the commutative and thenoncommutative fields, i.e.,ˆ φ ( φ ) + ˆ δ ˆ λ ˆ φ ( φ ) = ˆ φ ( φ + δ λ φ, A + δ λ A ) . (4.7)Solving this equation for the Abelian gauge field at the order θ , one obtains [12]:ˆ φ = φ − θ kl A k ∂ l φ + O ( θ ) . (4.8)Now, we can turn off the gauge field in the Abelian Higgs models to obtain the com-mutative and the noncommutative scalar field theory as follows: L c = −
12 ( ∂ µ φ )( ∂ µ φ ) − λ ( | φ | − v ) , (4.9) L nc = −
12 ( ∂ µ φ ) (cid:63) ( ∂ µ φ ) − λ ( φ (cid:63) φ − v ) , (4.10)where the subscript “c” and “nc” stand for the commutative and the noncommutativetheory respectively. The commutative theory is the same as the commutative scalar fieldtheory (2.10), which can be viewed as the relativistic version of the Gross-Pitaevskii theory(2.11), while the noncommutative one is just the noncommutative tachyon field theory thatwe are looking for. The Seiberg-Witten map for the scalar is trivial when turning off thegauge field: ˆ φ = φ . (4.11)However, the noncommutative scalar field still obeys the star product. The terms in (4.10)containing one single star product can be replaced by the corresponding ones with an ordi-nary product, because the difference is only given by total derivatives, which vanish afterthe spacetime volume integration. Therefore, star products in the quartic interaction termare the only difference between the ordinary action (4.9) and noncommutative field theoryaction (4.10). Our expectation is that the nonlocality introduced in this way may capturesome stringy behavior, in particular when static noncommutative solitons are considered.Noncommutative codimension-two soliton solutions have been found in the large non-commutativity limit in Ref. [13]. In the large noncommutativity limit, the kinetic term canbe neglected and the solitons need to satisfy dV (cid:63) dT = 0 , (4.12)where V (cid:63) ( T ) = λ ( T (cid:63) T + vT + vT ) is the potential expressed in terms of the shifited scalarfield T = φ − v . We restrict our discussions to real solutions. To find a such solution, oneneeds to first find a field φ satisfying φ (cid:63) φ = φ , (4.13)and then any function F with the form F ( x ) = (cid:80) ∞ n =1 a n x n has the property F ( λφ ) = F ( λ ) φ . (4.14)– 15 –ence, a potential of this form V ( T ) obeys dVdT (cid:12)(cid:12)(cid:12)(cid:12) T = λφ = (cid:18) dVdT (cid:12)(cid:12)(cid:12)(cid:12) T = λ (cid:19) φ . (4.15)The simplest function satisfying the condition (4.13) is given by [13]: φ ( r ) = 2 e − r /θ , r = x + x , (4.16)where θ ≡ θ in the large noncommutativity limit. The solution to equation (4.12) takesthe form T = − vφ ( r ) . (4.17)Note that this solution correctly interpolates between the value T = 0 where V ( T ) has alocal minimum and T = − v corresponding to an unstable local maximum. This construc-tion can be generalized to arbitrary even-codimension solitons by replacing r = x + x with r = x + x + · · · + x q . One can easily infer that a codimension-two soliton is justthe noncommutative counterpart of the modulus of the vortex line solution (2.16).Now we want to look into the possibility of identifying a such noncommutative solutionwith a D ( p − Dp -brane, and more generallywe would like to relate a codimension-2 q soliton with a D ( p − q )-brane. A crucial test isthe value of the soliton energy, which can be easily calculated, since in the large noncom-mutativity limit one can neglect the derivative term in the tachyon effective action (4.10),namely, S = − (cid:90) d p +1 x V (cid:63) ( T ) . (4.18)Inserting the soliton solution (4.17) into the equation above and integrating over x and x , we get S = − V ( − v ) (cid:90) d p − x (cid:90) d x φ ( r ) = − πθ V ( − v ) (cid:90) d p − x , (4.19)from which we can read off the tension T p − = 2 πθ V ( − v ) = 2 πθ λv . On the otherhand, the constant vacuum energy of the action (4.10) gives the value of the tension of theunstable Dp -brane T p = V ( − v ) = λv . (4.20)Therefore, we obtain the descent relation T p = (2 πθ ) T p − , (4.21)which is to be compared with the expected relation T p = (2 π ) α (cid:48) T p − , where α (cid:48) = − /m T = 1 /m H = 1 / (2 √ λv ).One can also construct codimension-one solitons. For a noncommutative field ˆ φ ( x )depending only on one coordinate x , such that ∂ t ˆ φ ( x ) = ∂ y a ˆ φ ( x ) = 0 for y a (cid:54) = x , one caneasily show that ˆ φ ( x ) (cid:63) ˆ φ ( x ) = ˆ φ . (4.22)– 16 –s this product is the only place where noncommutativity appears in the action (4.10),the same procedure that was used to get the commutative soliton (2.13) can be usedto prove that it is also a solution in the noncommutative case. However, interpreting acodimension-one soliton as a D ( p − Dp -brane, thefollowing descent relation for tensions has been derived in Refs. [18, 19]: T p = 8 √
23 1 m H T p − , (4.23)which differs from the expected one shown above. We will make more discussions on thispuzzle in the next subsection. In the previous subsection, we have seen a noncommutative scalar field theory obtainedby a generic noncommutative mapping of the relativistic Gross-Pitaevskii, whose solitonsolutions correspond to lower-dimensional branes in string theory. In order to set upthe correspondence at quantitative level, we need to investigate the relation between thenoncommutative scalar theory and the noncommutative tachyon theory, that is obtainedfrom string theory. The latter is known to have solitons reproducing the correct tensionsfor D -branes. We will see that the crucial ingredient to get them is to consider the specifickind of Seiberg-Witten map prescribed by string theory.First of all, we discuss the connection between the effective open string theory discussedin the previous section and the corresponding tachyon field theory. The effective open stringtheory (3.40) contains the standard bulk action of string theory in a background B -field,namely S = 14 πα (cid:48) (cid:90) Σ (cid:16) g µν ∂ a X µ ∂ a X ν − πiα (cid:48) B µν (cid:15) ab ∂ a X µ ∂ b X ν (cid:17) , (4.24)where µ, ν = 0 , , . . . , D , with D denoting the dimension of the space-filling D ( D − -brane,and B µν is an antisymmetric matrix of rank r = 4, so that we can assume B µν (cid:54) = 0 onlyfor µ, ν = 0 , . . . ,
3. On top of this we take B ν = 0 for any ν . We will also assume g µν = 0for µ = 0 , . . . , ν (cid:54) = 0 , . . . ,
3. Therefore, we have to take D ≥ D = 26, corresponding to critical strings. Furthermore, we assume constantbackground fields, i.e. g µν ≈ constant and B µν ≈ constant , such that H µνρ ≈
0. In theaction (4.24), we have included the standard kinetic term, which was not present in theaction (3.40), because it is suppressed in the large noncommutativity limit: α (cid:48) B µν → ∞ , g µν fixed , (4.25)with µ, ν = 0 , . . . ,
3, or equivalently g µν /α (cid:48) → B µν fixed. The directions µ = 4 , . . . ,
25 in the action (4.24) can be treated rather trivially, contributing just an overallfactor to the corresponding partition function. As we saw in the previous section, openstrings are characterized by a worldsheet with the topology of a disk, so we can in generalintroduce a term S (cid:48) = π (cid:90) dσ π V , (4.26)– 17 –here σ ∈ [0 , π ) is a parameter on the boundary ∂ Σ, and V is a general boundaryperturbation. In particular, in the previous section we considered V = − iA µ ( X ) ∂ σ X µ , (4.27)where A µ ( X , . . . , X ) is a U (1) gauge field. Using the Stokes theorem, for a slowly varying A µ or a constant F µν this boundary term can be written as S (cid:48) = − i π (cid:90) F µν X µ ∂ σ X ν . (4.28)Analogously, for a constant B -field we can rewrite − i (cid:90) Σ B µν (cid:15) ab ∂ a X µ ∂ b X ν = − i π (cid:90) B µν X µ ∂ σ X ν . (4.29)In this section, we are interested in the scalar effective action that can be obtained by con-sidering the slowly varying boundary deformation T ( X , X , X , X ) along the directions,in which B is non-null. In the infrared (IR) approximation under consideration, where allnon-convariant terms are neglected, this tachyon theory will be relativistic. In principle,it should include terms with derivatives of arbitrary order, which makes a direct study ofits solitons very daunting. On the other hand, in the large noncommutativity limit thetheory is dramatically simplified and its solitons can be related to the ones studied for theGross-Pitaevskii scalar theory in the previous subsection.The noncommutative tachyon field theory has its origin in string theory in the presenceof a constant B -field. In Ref. [9] it has been shown that, when the boundary perturbation(4.27) is considered with a slowly varying A µ , one can obtain an effective action of the formˆ S DBI = − G s (2 π ) p ( α (cid:48) ) p +12 (cid:90) d p +1 x (cid:113) − det( G µν + 2 πα (cid:48) ( ˆ F µν + Φ µν )) , (4.30)where the open string metric G µν , the open string coupling G s and the two-form Φ aredetermined by the formulae 1 G + 2 πα (cid:48) Φ = − θ πα (cid:48) + 1 g + 2 πα (cid:48) B , (4.31) G s = g s (cid:115) det( G + 2 πα (cid:48) Φ)det( g + 2 πα (cid:48) B ) . (4.32)In the first equation (4.31), G and Φ are determined in terms of the closed string metric g ,the closed string coupling g s , B and an arbitrary noncommutativity parameter θ , becausethey are symmetric and antisymmetric respectively. The gauge field ˆ A is related to thecommutative A by Eq. (4.6). The second equation (4.32) is motivated by demanding thatthe effective action (4.30) with ˆ F = 0 is the same as the usual commutative Dirac-Born-Infeld (DBI) action with F = 0. Actually, it has been shown that, using the transformationsof the fields ˆ A ( A ) given by Eq. (4.6), the two Lagrangians are related as L DBI = ˆ L DBI + (total derivative) + O ( ∂F ) . (4.33)– 18 –or θ = 0, Eq. (4.33) is obvious. For θ µν = − (2 πα (cid:48) ) (cid:18) g + 2 πα (cid:48) B B g − πα (cid:48) B (cid:19) µν . (4.34)the interpolating field Φ vanishes.In Ref. [18, 19] the same argument was used for the DBI-like effective action of thetachyon field for an unstable Dp -brane: S = − g s (2 π ) p ( α (cid:48) ) p +12 (cid:90) d p +1 xV ( T ) (cid:113) − det( g µν + 2 πα (cid:48) ( B µν + ∂ µ T ∂ ν T )) , (4.35)where V ( T ) measures the variable tension of the unstable D -brane, and is a runawaypotential, monotonically connecting its maximum coinciding with the tension of the Dp -brane and its minimum representing the vanishing unstable Dp -brane. In particular, onecan choose conventions in which V ( T = 0) = 1 and V ( T = ±∞ ) = 0. We can thereforeread off T p = 1 g s (2 π ) p ( α (cid:48) ) p +12 . (4.36)If we ignore terms with higher-order derivatives of the tachyon, the corresponding noncom-mutative tachyon action can be written asˆ S = − ˆ T p (cid:90) d p +1 x V (cid:63) ( ˆ T ) (cid:114) − det (cid:16) G µν + 2 πα (cid:48) ∂ µ ˆ T ∂ ν ˆ T (cid:17) , (4.37)where ˆ T is related to T in the trivial way (4.11), V (cid:63) ( T ) is obtatined from V ( T ) by replacingordinary products with (cid:63) -products, andˆ T p = 1 G s (2 π ) p ( α (cid:48) ) p +12 = T p (cid:112) det(1 + 2 πα (cid:48) g − B ) . (4.38)What is remarkable about the action (4.35), which is written in commutative formalism, isthat the Seiberg-Witten map replaces it with the completely equivalent noncommutativeexpression (4.37). This is different from the case discussed in Subsection 4.1, where thenoncommutative theory was different from its commutative counterpart. Nevertheless, itis not hard to see that, for slowly varying tachyons such that ∂ µ T is small, the actions(4.35) and (4.37) reduce to the more common tachyon actions of the form [20, 21]: S = Cg s (cid:90) d p +1 x L BI ( B ) (cid:20) f ( T ) g ij ∂ i T ∂ j T − V ( T ) + · · · (cid:21) , (4.39)and ˆ S = CG s (cid:90) d p +1 x √ G (cid:20) f (cid:63) ( T ) G ij ∂ i T ∂ j T − V (cid:63) ( T ) + · · · (cid:21) , (4.40)where C = g s T D p , and L BI ( B ) = (cid:112) det( g + 2 πα (cid:48) B ) is the usual DBI action for the vanish-ing gauge field. The factor f (cid:63) ( T ), like V (cid:63) ( T ), is simply obtatined from f ( T ) by replacingordinary products with (cid:63) -products. – 19 –everal computations of the tachyon potential are available in the literature using sometechniques from string field theory. To simplify the calculations, one can also consider thelarge noncommutativity limit. We will present some details in Appendix C, and the resultfor bosonic string theory is [21–26]: V ( T ) = ( T + 1) e − T . (4.41)It is explained in Ref. [24], that although suppressed the kinetic term in the action does nottake a canonical form in terms of the tachyon field T , and a change of variable exp( − T ) = φ will lead to the canonical kinetic term, and correspondingly the tachyon potential can beexpressed in the new variable as V ( φ ) = − φ log φ e (4.42)in Euclidean signature. One can expand it around the vacuum φ = 1. When only realfields are considered, the leading order expression of the potential in Minkowski signaturereproduces the shape of the potential in the noncommutative tachyon field theory (4.10)that has been discussed in Subsection 4.1.Now let us return to the effective string action (3.40) that is obtained from the non-relativistic Gross-Pitaevskii theory using the boson/vortex duality. Since there is a gaugesymmetry (3.42), we can choose a gauge A a = 0. Moreover, we focus on the region aroundthe dark solitons, where the B -field can be approximately viewed as constant, which impliesthat the field strength terms ∼ h and ∼ ( (cid:101) F + (cid:101) B ) in Eq. (3.40) can be neglected near darksoliton planes. Also, as discussed in Section 3, a term violating the Lorentz symmetry canbe dropped in the IR regime. Hence, under these approximations the effective action (3.40)is exactly the same as the standard string action in the large B -field limit, and the analysisabove holds in our case. It means that near dark soliton planes the effective string theory(3.40) can provide an effective noncommutative tachyon field theory of the form (4.40) withthe tachyon potential (4.41) or (4.42), which at the leading order coincides with the theory(4.10) obtained from the noncommutative Abelian Higgs model.We can consider the same kind of codimension-two solitons discussed in the previoussubsection, namely T ∗ φ ( r ), where T ∗ is the value at which V ( T ) has a local maximum.The tachyon potential V (cid:63) for such a solution has the value V ( T ∗ ) φ ( r ), where V ( T ∗ ) = 1in our conventions. Therefore, S = − C V ( T ∗ ) G s (cid:90) d p − x (cid:90) d x √ G φ ( r ) = − πθC V ( T ∗ ) G s (cid:90) d p − x √ G . (4.43)From Eq. (4.32) we obtain for the large B -field the relation between the open string coupling G s and the closed string coupling g s : G s = g s √ G πα (cid:48) B √ g , (4.44)where B = B . Taking into account θ = 1 /B , we obtain S = − (2 π ) α (cid:48) Cg s (cid:90) d p − x √ g . (4.45)– 20 –ence, the soliton tension is T sol = (2 π ) α (cid:48) Cg s = (2 π ) α (cid:48) T p = T p − , (4.46)where C = T p g s . This result is consistent with the one for the bosonic D ( p − q )-branes: T p − q = (2 π √ α (cid:48) ) q T p . (4.47)We can also consider codimension-one soliton solutions, whose energy is expressed asa one-dimensional integral in Eq. (A.10). As discussed in Ref. [4], to prevent the long-wavelength instabilities we have to restrict the sizes of the transverse dimensions to be ∼ π √ α (cid:48) , where (cid:96) s = √ α (cid:48) is the characteristic length of the effective strings, i.e. thelength of the vortex lines. Hence, we expect that the relation (4.47) also holds for q = 1,i.e. the codimension-one soliton solutions. On the other hand, it has been computed inRefs. [18, 19] using the tachyon field theory, that the relation for the tension of codimension-one soliton solutions coincides with the one for D ( p − T p − = (2 π √ α (cid:48) ) T p , which also supports the identification of dark solitons in Gross-Pitaevskii theory and D ( p − D -brane Interaction As another check of identifying dark solitons with D -branes, in this subsection we computethe interaction between two parallel D -branes in the effective string theory. A well-knowncomputation in the ordinary string theory has been done by Polchinski in Ref. [27] (seee.g. Ref. [28] for a summary).The calculation is essentially to evaluate the amplitude of exchanging a closed stringbetween two parallel D -branes (see Fig. 3), or equivalently to evaluate a 1-loop amplitudeof open strings. For the picture of closed strings in the NS-NS sector, only the gravitonand the dilaton were taken into account, because the antisymmetric B -field contributes athigher order. If one considers the type-II superstring, the contribution from the R-R sectorwill exactly cancel the one from the NS-NS sector, as discussed in Ref. [27]. However, fixingthe gauge A a = 0 and neglecting the fluctuations, the effective string theory (3.40) can bethought of as the large B -field limit of the ordinary string theory. Hence, in this case thecontributions from the graviton and the dilaton should be neglected, and only the B -fieldcontributes to the potential between two parallel D -branes.To compute the amplitude of exchanging the B -field between two parallel D -branes,we need the propagator of the B -field in the bulk and the coupling between the B -fieldand the D -brane. Following Ref. [28], we can read off the bulk propagator of the B -fieldfrom the effective action in the Einstein frame: S E = 12 κ (cid:90) d D x ( − (cid:101) G ) / (cid:34) (cid:101) R − D − ∇ µ (cid:101) Φ ∇ µ (cid:101) Φ − e − (cid:101) Φ / ( D − H µνλ H µνλ − D − α (cid:48) e (cid:101) Φ / ( D − + O ( α (cid:48) ) (cid:35) , (4.48)– 21 – igure 3 . The exchange of a closed string between two parallel D -branes where (cid:101) Φ = Φ − Φ , (cid:101) G µν = e − Φ) D − G µν , (4.49)and (cid:101) R is the corresponding Ricci scalar after the transformation. The terms relevant tothe bulk propagator of the B -field are S E ⊃ − κ (cid:90) d D x B µν (cid:3) B µν , (4.50)and the bulk propagator of the B -field in momentum space is [29]: (cid:104) B µν B ρσ (cid:105) = − iκ k ( δ µρ δ νσ − δ νρ δ µσ ) . (4.51)To obtain the coupling between the B -field and the D -brane, we expand the DBI-action inthe Einstein frame: S Ep = − τ p (cid:90) d p +1 ξ e − (cid:101) Φ (cid:115) det (cid:18) e (cid:101) Φ D − (cid:101) G ab + B ab + 2 φα (cid:48) F ab (cid:19) , (4.52)where the indices a , b run over the ( p + 1)-dimensions on the D -brane. The terms relevantto the coupling between the B -field and the D -brane are S Ep ⊃ − τ p (cid:90) d p +1 ξ B ab B ab . (4.53)From this coupling we see that the leading order contribution is already at 1-loop order,thus from field theory point of view we need to evaluate the following 1-loop graph:– 22 – kqk-q Figure 4 . The 1-loop Feynman diagram of the B -field coupled to D -branes The amplitude is i M ( k ) = 12 (cid:18) − iτ p (cid:19) (cid:90) d D − p − q (2 π ) D − p − (cid:18) − iκ q (cid:19) (cid:18) − iκ ( k − q ) (cid:19) · ∂X µ ∂ξ a ∂X ν ∂ξ b ∂X ρ ∂ξ c ∂X σ ∂ξ d ( δ µρ δ νσ − δ νρ δ µσ ) · ∂X ¯ µ ∂ξ a ∂X ¯ ν ∂ξ b ∂X ¯ ρ ∂ξ c ∂X ¯ σ ∂ξ d ( δ ¯ µ ¯ ρ δ ¯ ν ¯ σ − δ ¯ ν ¯ ρ δ ¯ µ ¯ σ )= 92 τ p κ (cid:2) ( p + 1) − p + 1) (cid:3) (cid:90) d D − p − q (2 π ) D − p − q ( k − q ) = 92 τ p κ ( p − i D − p − (4 π ) D − p − Γ (cid:16) − D + p (cid:17) Γ (cid:16) D − p − (cid:17) Γ( D − p −
3) 1 k − D + p . (4.54)To obtain the potential in the spacetime, we should apply the Born approximation andFourier transform the amplitude −M ( k ).In order to compare with the interactons between dark solitons, we consider the case D = 4, p = 0. As we discussed before, to make the dark soliton relatively stable andcompatible with the Derrick’s theorem, one has to restrict the size of transverse dimensionson the D -brane, i.e. to confine the system in a cylindrical geometry. Hence, in thiscomputation the D -branes in real BEC systems can be effectively thought of as D0-branes.After the Fourier transform of the amplitude, we obtain the potential between two parallel D -branes ( D = 4, p = 0): V ( x ) = i τ p κ (cid:90) d k (2 π ) e ik · x k = 9 τ p κ π δ ( x ) x . (4.55)We see that, strictly speaking the contribution of the B -field to the interaction betweentwo D0-branes is given by a Dirac δ -function, i.e. a contact interaction. However, in realitythe size of transverse dimensions on the D -brane is not zero, although negligible comparedto the distance between two D -branes. Hence, the Dirac δ -function in Eq. (4.55) can be– 23 –nderstood as lim (cid:96) → e − x /(cid:96) √ π(cid:96) , (4.56)where (cid:96) is proportional to the size of transverse dimensions on the D -brane. Therefore, weexpect the potential between two parallel dark solitons in BEC systems is a short-rangedrepulsion and exponentially decaying. As far as we know, there is no analytical expressionof the potential between two parallel dark solitons available in the literature, and thenumerical results [30] are consistent with our results from string theory computation atqualitative level. More interestingly, some recent studies [31] in optical systems confirmedexperimentally that, dark solitons can have attractions only when some nonlocal responseis turned on, which is also consistent with our expectation from string theory, i.e., in thepresence of a string tension term in the action, the exchange of the graviton and the dilatonwill induce an attrative interaction between two parallel D -branes. In this paper we have discussed the duality map between the Gross-Pitaevskii theory anda (3+1)D effective string theory. We generalize the previous works [6, 8] to the spacetimewith boundaries (see also Ref. [4]). As a consequence, we identify dark soliton solutionsin the Gross-Pitaevskii theory and D -branes in the effective string theory under certainapproximations, and various checks have been made to test this identification. With thisnew perspective, one has an opportunity to test many results and predictions of stringtheory in real experiments and on the other hand bring in new ideas to the study ofquantum fluids and cold atom systems.We would like to explore more aspects of this duality and its relation to a real cold atomsystem at quantitative level. For instance, Ref. [4] has started discussing the stability of theconfiguration of open vortex lines attached to the dark solitons, and we believe that a moredetailed analysis of this dual picture can help us study the time evolution of D -brane decay.More interestingly, by introducing some fermionic fields an emergent supersymmetry canbe realized in the cold atom systems. We hope that this can help stabilize dark solitons, inthe same way of stabilizing D -branes from the superstring theory, and eventually to helpsimulate superstring theory in real experiments.From more theoretical point of view, the boson/vortex duality discussed in this paperis also of great interest. As mentioned in Section 3, the duality can be generalized toother dimensions. Since the (1+1)D Gross-Pitaevskii equation, also called the nonlinearSchr¨odinger equation, is an integrable model, we expect the integrability should be main-tained in the dual theory [32]. Also, the (1+1)D nonlinear Schr¨odinger equation is dualto a 2D topological Yang-Mills-Higgs model at quantum level [33]. By constructing thegravity dual of the 2D topological Yang-Mills-Higgs model, we expect that the D -branesin the supergravity theory correspond to the soliton solutions of the (1+1)D nonlinearSchr¨odinger equation. These results will be presented elsewhere [34].More mathematically, the identification of solitons and D -branes discussed in thispaper can also be understood from the viewpoint of K-theory. As discussed in Ref. [28],– 24 –n the annihilation of a D p -brane and an anti- D p -brane, if the tachyon field is given by atopologically stable kink depending only on one of the dimensions inside the brane, then a D ( p − -brane will be left over after the annihilation. In our case, the (anti-) D p -brane canbe viewed as the space-filling (anti-) D -brane, while in the end we should see a D -braneleft, which can be identified as the dark soliton in the Gross-Pitaevskii theory. More detailsof this perspective and its applications to topological phases will be explored in the futurework.In stead of the boson/vortex duality, some recent works [35–37] have discussed a closelyrelated particle/vortex duality web, especially the dual of the fermionic field theory in(2+1)D. To apply these ideas to the (3+1)D Abelian Higgs model and understand thecorresponding web of dualities will help us understand the phase transition, the vacuumstructure and the renormalization group flow of the theory, which we would like to pursuesoon. Acknowledgements
We would like to thank Loriano Bonora, Ilmar Gahramanov, Steven Gubser, Song He,Antonio Mu˜noz Mateo, Nikita Nekrasov, Vasily Pestun, Vatche Sahakian, Ashoke Sen andXiaoquan Yu for many useful discussions. We also would like to thank M. M. Sheikh-Jabbari and J. Murugan for communications.
A Solutions to the Gross-Pitaevskii Equation
In this appendix we briefly summarize the solutions with nontrivial topology to the Gross-Pitaevskii theory (2.11).
Dark Soliton
The classical soliton solutions can be found by solving the Gross-Pitaevskii equation di-rectly. A more field-theoretic way of finding the soliton solutions is to use the standardBPS approach, which we will briefly review now.Based on the famous Derrick’s theorem, a stable soliton solution for the pure scalartheory exists only for dimensions D ≤
2. Hence, we restrict our discussions to the (1+1)Dsolutions in the following, i.e., we assume that the soliton solutions are independent of twoother spatial dimensions in (3+1)D. The BPS procedure for the (1+1)D scalar field theorycan be summarized as follows.A general scalar field theory is given by L = −
12 ( ∂ x φ ) − V ( φ ) , (A.1)which leads to the field equation ∂ x φ − V (cid:48) ( φ ) = 0 . (A.2)If the potential V ( φ ) can be expressed as V = ( W (cid:48) ) , (A.3)– 25 –he energy of the system is given by E = (cid:90) ∞−∞ dx (cid:20)
12 ( ∂ x φ ) + ( W (cid:48) ) (cid:21) , (A.4)where W is a functional of the field φ , and W (cid:48) ≡ ∂W∂φ . (A.5)Consequently, E = (cid:90) ∞−∞ dx (cid:34)(cid:18) √ ∂ x φ − W (cid:48) (cid:19) + √ W (cid:48) ∂ x φ (cid:35) = (cid:90) ∞−∞ dx (cid:34)(cid:18) √ ∂ x φ − W (cid:48) (cid:19) + √ ∂W∂x (cid:35) = (cid:90) ∞−∞ dx (cid:34)(cid:18) √ ∂ x φ − W (cid:48) (cid:19) (cid:35) + √ W (+ ∞ ) − W ( −∞ )] . (A.6)If W (+ ∞ ) and W ( −∞ ) correspond to different vacua, the configuration provides a soli-ton with nontrivial topology, which is given by the solution of the first-order differentialequation ∂ x φ = √ W (cid:48) . (A.7)Eq. (A.7), which is also called the BPS equation, implies the field equation, since ∂ x φ = ∂ x ( √ W (cid:48) ) = √ W (cid:48)(cid:48) ∂φ∂x = √ W (cid:48)(cid:48) √ W (cid:48) = 2 W (cid:48) W (cid:48)(cid:48) , (A.8)which is exactly the field equation (A.2): ∂ x φ = V (cid:48) = 2 W (cid:48) W (cid:48)(cid:48) . (A.9)Now let us come back to the discussion of the soliton solutions to the Gross-Pitaevskiiequation. For the repulsive interaction, i.e. g >
0, the energy for the Gross-Pitaevskiiequation is E = (cid:90) ∞−∞ dx (cid:34) (cid:126) m (cid:12)(cid:12)(cid:12)(cid:12) d Ψ dx (cid:12)(cid:12)(cid:12)(cid:12) + g (cid:0) | Ψ | − n (cid:1) (cid:35) , (A.10)where Ψ = √ n f exp (cid:20) − iµt (cid:126) (cid:21) (A.11)with the chemical potential µ , and f is in general a complex function f = f + if . (A.12)We choose f = vc , and define φ ≡ (cid:126) √ m Ψ , (A.13)– 26 –hen the energy becomes E = (cid:90) ∞−∞ dx (cid:34) (cid:12)(cid:12)(cid:12)(cid:12) dφdx (cid:12)(cid:12)(cid:12)(cid:12) + g (cid:16) m (cid:126) | φ | − n (cid:17) (cid:35) . (A.14)Similar to what we discussed before, the BPS equation for the energy functional given byEq. (A.14) can be written as follows: dφdx = √ g (cid:16) n − m (cid:126) | φ | (cid:17) or dφdx = √ g (cid:16) m (cid:126) | φ | − n (cid:17) , (A.15)but the imaginary part of φ should be constant, in order that the energy functional hasthe expression of Eq. (A.6). The solutions to these two equations only differ by a minussign. Let us consider the first equation, which is equivalent to (cid:126) √ m dfdx = √ gn (1 − | f | ) ⇒ (cid:126) √ m df dx = √ gn (1 − v c − f ) , (cid:126) √ m i df dx = 0 . (A.16)For v = 0 the equations above simplify to √ ξ df dx = 1 − f , f = 0 , (A.17)where ξ ≡ (cid:126) / √ mgn is the healing length. The solution to these equations is the darksoliton: Ψ( x ) = √ n tanh (cid:20) x √ ξ (cid:21) . (A.18) Grey Soliton
If we perform a Galilean boost to the first one of Eqs. (A.16) using the method describedin Ref. [38], it becomes √ ξ df dx (cid:48) = 1 − v c − f , (A.19)where x (cid:48) ≡ x − vt . This new equation is exactly the same as Eq. (5.55) in Ref. [5] for anarbitrary constant v , and the solution to this equation isΨ( x − vt ) = √ n (cid:32) i vc + (cid:114) − v c tanh (cid:34) x − vt √ ξ (cid:114) − v c (cid:35)(cid:33) , (A.20)which includes both the dark soliton solution ( v = 0) and the grey soliton solution ( v (cid:54) = 0). Bright Soliton
When the interaction is attractive, i.e. g <
0, there is another kind of soliton solution tothe Gross-Pitaevskii equation, which is called the bright soliton and has the formΨ( x ) = Ψ(0) 1cosh( x/ √ ξ ) , (A.21)where n = | Ψ(0) | is the central density, and ξ ≡ (cid:126) / (cid:112) m | g | n .– 27 – ortex Line The Gross-Pitaevskii equation has another string-like solution called the vortex line. Itcan be viewed as the Nielsen-Olsen vortex line solution in the Abelian Higgs model in thelimit of vanishing gauge field. In this subsection, we follow Ref. [5] to review this kind ofsolution.To see the vortex line solution, we start with the Gross-Pitaevskii equation (2.12).Plugging the ansatz φ ( r , t ) = φ ( r ) exp (cid:18) − iµt (cid:126) (cid:19) (A.22)into Eq. (2.12), where µ is the chemical potential, we obtain (cid:18) − (cid:126) ∇ m − µ + g | φ ( r ) | (cid:19) φ ( r ) = 0 . (A.23)For a string-like solution, we can introduce the cylindrical coordinates ( r, ϕ, z ) and furtherparametrize φ as φ = √ n f ( η ) e isϕ , (A.24)where η = r/ξ with ξ ≡ (cid:126) / (cid:112) m | g | n , and s is an integer characterizing the angular mo-mentum carried by a vortex line. With this parametrization, one obtains the equation for f ( η ): 1 η ddη (cid:18) η dfdη (cid:19) + (cid:18) − s η (cid:19) f − f = 0 , (A.25)and the boundary conditions are f → , when η → ∞ ; f ∼ η | s | , when η → . (A.26)The equation above can be solved numerically for a given value of s . Once the solution f ( η ) is obtained, the energy of this configuration is E = Lπ (cid:126) nm (cid:90) R/ξ ηdη (cid:34)(cid:18) dfdη (cid:19) + s η f + 12 (cid:0) f − (cid:1) (cid:35) , (A.27)where L and R are the effective length of the vortex line and the radius of the systemrespectively. Vortex Ring
Similar to the vortex line solution discussed in the previous subsection, there is also thevortex ring solution, which does not have two endpoints, instead it is a closed string-like solution. In contrast to the vortex line solution, the vortex ring cannot be at rest.Moreover, as discussed in Ref. [5], the radius of the vortex ring can be either much largerthan the healing length ξ or comparable to the healing length ξ . Two parallel vortex ringswith opposite circulation can also form a vortex pair, which has been studied in Ref. [8]using the boson/vortex duality. – 28 – Some Details in the Boson/Vortex Duality Map
In this appendix, we present some details of the duality map in the presence of boundaries.Let us start with the Gross-Pitaevskii theory after the parametrization (3.1) given byEq. (3.8): L = ip ˙ p − p ˙ η − p m ( ∇ η ) − ( ∇ p ) m − g p − p ) , where the first term is a total derivative that can be dropped. In the following we analyzethe expression above term by term. • − p ˙ η :As we analyzed in Section 3, near the soliton plane p changes sign from one sideto the other side, while η does not have a π -jump, or in other words, we removethe π -jump of the phase and allow p to change sign, and now the phase η behavessmoothly. Hence, when we consider the limit that the healing length goes to zero,the smooth functions such as ˙ η and ( ∇ η ) will take their values at z = z , where z isthe longitudinal position of the dark soliton, i.e., the functions of the phase become z -independent in this limit. Therefore, − (cid:90) d x ( p + (cid:101) p ) ˙ η = − (cid:32)(cid:90) z + (cid:96)/ z − (cid:96)/ dz p (cid:33) (cid:18)(cid:90) d x ˙ η (cid:19) − (cid:32)(cid:90) z + (cid:96)/ z − (cid:96)/ dz (cid:33) (cid:18)(cid:90) d x (cid:101) p ˙ η (cid:19) − (cid:32)(cid:90) z + (cid:96)/ z − (cid:96)/ dz p (cid:33) (cid:18)(cid:90) d x (cid:101) p ˙ η (cid:19) , (B.1)where the last term vanishes due to p ( − x ) = − p ( x ), and for the first term (cid:90) z + (cid:96)/ z − (cid:96)/ dz p = n (cid:90) z + (cid:96)/ z − (cid:96)/ dz (cid:20) tanh (cid:18) z − z √ (cid:96) (cid:19)(cid:21) = n (cid:90) (cid:96)/ − (cid:96)/ dz (cid:20) tanh (cid:18) z √ (cid:96) (cid:19)(cid:21) = n(cid:96) (cid:90) / − / d (cid:101) z (cid:20) tanh (cid:18) (cid:101) z √ (cid:19)(cid:21) = n(cid:96) (cid:16) − √ / √ (cid:17) ≡ n (cid:101) (cid:96) . (B.2)We have defined (cid:101) z ≡ z/(cid:96) , and in the last step we have defined another length scaleof the order of the healing length. Therefore, − (cid:90) d x p ˙ η = − n (cid:101) (cid:96) (cid:90) d x ˙ η , (B.3)which is a total derivative, hence can also be dropped. What remains is − (cid:90) d x ( p + (cid:101) p ) ˙ η = − (cid:96) (cid:90) d x (cid:101) p ˙ η . (B.4)– 29 – − p m ( ∇ η ) :Like in the previous case, the smooth function ( ∇ η ) becomes z -independent in thesmall region around the dark soliton plane. Hence, − (cid:90) d x ( p + (cid:101) p ) m ( ∇ η ) = − m (cid:32)(cid:90) z + (cid:96)/ z − (cid:96)/ dz p (cid:33) (cid:18)(cid:90) d x ( (cid:101) ∇ η ) (cid:19) − m (cid:32)(cid:90) z + (cid:96)/ z − (cid:96)/ dz (cid:33) (cid:18)(cid:90) d x (cid:101) p ( (cid:101) ∇ η ) (cid:19) − m (cid:32)(cid:90) z + (cid:96)/ z − (cid:96)/ dz p (cid:33) (cid:18)(cid:90) d x (cid:101) p ( (cid:101) ∇ η ) (cid:19) , (B.5)where (cid:101) ∇ is the gradient operator on the coordinates ( x, y ). Similar to the previouscase, we obtain − (cid:90) d x ( p + (cid:101) p ) m ( ∇ η ) = − n (cid:101) (cid:96) m (cid:90) d x ( (cid:101) ∇ η ) − (cid:96) m (cid:90) d x (cid:101) p ( (cid:101) ∇ η ) = − (cid:96) m (cid:90) d x (cid:32)(cid:101) p + n (cid:101) (cid:96)(cid:96) (cid:33) ( (cid:101) ∇ η ) . (B.6) • − ( ∇ p ) m : − (cid:90) d x ( ∇ p ) m = − m (cid:90) z + (cid:96)/ z − (cid:96)/ dz (cid:90) d x (cid:34)(cid:18) ∂ (cid:101) p∂x (cid:19) + (cid:18) ∂ (cid:101) p∂y (cid:19) + (cid:18) ∂p ∂z (cid:19) (cid:35) = − m (cid:32)(cid:90) z + (cid:96)/ z − (cid:96)/ dz (cid:33) (cid:90) d x (cid:34)(cid:18) ∂ (cid:101) p∂x (cid:19) + (cid:18) ∂ (cid:101) p∂y (cid:19) (cid:35) − m (cid:34)(cid:90) z + (cid:96)/ z − (cid:96)/ dz (cid:18) ∂p ∂z (cid:19) (cid:35) (cid:18)(cid:90) d x (cid:19) , (B.7)where the second line in the expression above contributes a constant, which can bedropped from the action, and the first line gives − (cid:96) m (cid:90) d x (cid:16) (cid:101) ∇ (cid:101) p (cid:17) . (B.8) • − g ( p − p c ) :( p − p c ) = (cid:0) ( p + (cid:101) p ) − p c (cid:1) = p +4 p (cid:101) p +6 p (cid:101) p +4 p (cid:101) p + (cid:101) p − p p c − p (cid:101) pp c − (cid:101) p p c + p c . (B.9)After neglecting the terms that have odd powers in p , we obtain the relevant terms p + 6 p (cid:101) p + (cid:101) p − p p c − (cid:101) p p c + p c , (B.10)– 30 –here the terms independent of (cid:101) p contribute only constants after the integration overspacetime, which can be dropped from the action. The remaining terms are (cid:101) p − (cid:101) p ( p c − p ) . (B.11)Hence, − g (cid:90) d x ( p − p ) = − g (cid:32)(cid:90) z + (cid:96)/ z − (cid:96)/ dz (cid:33) (cid:18)(cid:90) d x (cid:101) p (cid:19) + g (cid:32)(cid:90) z + (cid:96)/ z − (cid:96)/ dz ( p c − p ) (cid:33) (cid:18)(cid:90) d x (cid:101) p (cid:19) = − g(cid:96) (cid:90) d x (cid:101) p + g ( p c (cid:96) − n (cid:101) (cid:96) ) (cid:90) d x (cid:101) p = − g(cid:96) (cid:90) d x (cid:32)(cid:101) p − n (cid:32) − (cid:101) (cid:96)(cid:96) (cid:33)(cid:33) + g(cid:96) (cid:90) d x (cid:32) n − n (cid:101) (cid:96)(cid:96) (cid:33) , (B.12)where we used p c = √ n , and the second term above is a constant, that can be droppedfrom the action. What remains after the integration is − g(cid:96) (cid:90) d x (cid:32)(cid:101) p − n (cid:32) − (cid:101) (cid:96)(cid:96) (cid:33)(cid:33) , (B.13)where (cid:101) (cid:96) ≡ (cid:96) (cid:0) − √ / √ (cid:1) , hence (1 − (cid:101) (cid:96)/(cid:96) ) is a positive constant. We candefine (cid:101) p c ≡ (cid:118)(cid:117)(cid:117)(cid:116) n (cid:32) − (cid:101) (cid:96)(cid:96) (cid:33) . (B.14)Combining all the terms together, we obtain the action around a soliton plane (cid:90) d x L = − (cid:96) (cid:90) d x (cid:101) p ˙ η − (cid:96) m (cid:90) d x (cid:32)(cid:101) p + n (cid:101) (cid:96)(cid:96) (cid:33) ( (cid:101) ∇ η ) − (cid:96) m (cid:90) d x (cid:16) (cid:101) ∇ (cid:101) p (cid:17) − g(cid:96) (cid:90) d x (cid:0)(cid:101) p − (cid:101) p c (cid:1) = (cid:96) (cid:90) d x (cid:20) − (cid:101) p ˙ η − m (cid:0)(cid:101) p + C (cid:1) ( (cid:101) ∇ η ) − m (cid:16) (cid:101) ∇ (cid:101) p (cid:17) − g (cid:0)(cid:101) p − (cid:101) p c (cid:1) (cid:21) = (cid:96) (cid:90) d x (cid:20) − (cid:101) ρ ˙ η − m ( (cid:101) ρ + C ) ( (cid:101) ∇ η ) − m (cid:101) ρ (cid:16) (cid:101) ∇ (cid:101) ρ (cid:17) − g (cid:101) ρ − (cid:101) ρ ) (cid:21) , (B.15)where C ≡ n (cid:101) (cid:96)/(cid:96) is a constant. In the last line we rewrite the theory in the variable (cid:101) ρ = (cid:112)(cid:101) p .This action is very similar to the 3D part in the action (3.5) by restricting the Lagrangian(2.11) on a 3D space. The only difference is an additional term − ( C/ m )( (cid:101) ∇ η ) , but itdoes not affect the (2+1)D duality. The reason is following. In the duality map, we willintroduce an auxiliary field f a = ( ρ, f ˆ a ) with ˆ a ∈ { x, y } , and for (cid:101) ρ (cid:48) ≡ (cid:101) ρ + C : − (cid:101) ρ ˙ η − (cid:101) ρ (cid:48) m ( (cid:101) ∇ η ) + m (cid:101) ρ (cid:48) (cid:18) f ˆ a − (cid:101) ρ (cid:48) m (cid:101) ∇ ˆ a η (cid:19) = − (cid:101) ρ ˙ η + m (cid:101) ρ (cid:48) f ˆ a f ˆ a − f ˆ a ∂ ˆ a η = m (cid:101) ρ (cid:48) f ˆ a f ˆ a − f a ∂ a η , (B.16)– 31 –here in the path integral (cid:90) D f ˆ a exp (cid:18) i(cid:96) (cid:90) d x m (cid:101) ρ (cid:48) f ˆ a f ˆ a (cid:19) = 1 . (B.17)Hence, (cid:101) ρ (cid:48) or consequently the constant C does not show up in the action after the dualitymap. C Tachyon Potential
In this appendix we discuss how to compute the tachyon potential V ( T ) appearing in theeffective action (4.40) from string field theory [21–26].Consider the string action defined on the unit disk Σ given by S = S + S (cid:48) , (C.1)where S is a bulk action and S (cid:48) is a boundary term. In particular, S = 14 πα (cid:48) (cid:90) Σ (cid:16) g µν ∂ a X µ ∂ a X ν − πiα (cid:48) B ij (cid:15) ab ∂ a X i ∂ b X j (cid:17) , (C.2)where µ, ν = 0 , , . . . , D and i, j = 0 , , . . . , p ≤ D . For the time being we can ignore thedirections µ = p + 1 , . . . , D . For a constant B -field, − i (cid:90) Σ B ij (cid:15) ab ∂ a X i ∂ b X j = − i (cid:90) ∂ Σ B ij X i ∂ t X j , (C.3)where ∂ t is a tangential derivative along the boundary ∂Σ . The boundary conditionsdetermined by the equations of motion are g ij ∂ n X j + 2 πiα (cid:48) B ij ∂ t X j (cid:12)(cid:12) ∂Σ = 0 , (C.4)where ∂ n is a normal derivative to ∂ Σ. For B = 0 these are Neumann boundary conditions,corresponding to open strings, and that is why we can refer to g ij as the closed string metric.When B has rank r = p and B → ∞ , or equivalently g ij →
0, along the spatial directionsof the brane, the boundary conditions become Dirichlet, i.e. ∂ t X j (cid:12)(cid:12) ∂Σ = 0. Therefore, thephysical picture is that for B = 0 the ends of the open string are free to move, and thePolyakov action describes the space-filling D p -brane.In the following discussions, it is more convenient to use the two open string parameters G − ≡ (cid:18) g + 2 πiα (cid:48) B (cid:19) S , Θ ≡ (cid:18) g + 2 πiα (cid:48) B (cid:19) A , which are symmetric and antisymmetric respectively. In Ref. [22, 23] it has been arguedthat the open strings are described by the boundary term S (cid:48) , which has the form S (cid:48) = π (cid:90) dσ π V , (C.5)– 32 –here σ is a parameter on the border ∂ Σ, and V is a general boundary perturbation thatcan be parametrized by couplings λ i : V = λ i V i . (C.6)Defining the ghost number-one operator O = c V , the spacetime string field theory action S is defined by ∂S∂λ i = 12 π (cid:90) dσ π π (cid:90) dσ (cid:48) π (cid:10) O i ( σ ) (cid:8) Q B , O (cid:0) σ (cid:48) (cid:1)(cid:9)(cid:11) λ , (C.7)where Q B is the BRST charge. For the tachyon field O = cT ( X ), one can find {Q B , cT ( X ) } = c∂ t c (1 − ∆ T ) T ( X ) , (C.8)where ∆ T = − α (cid:48) G ij ∂ ∂X i ∂X j . (C.9)The general form for the action satisfying this equation is [39, 40]: S = − β i ∂Z∂λ i + Z , (C.10)with Z the partition function and β i the beta function for the coupling λ i . In particular,for the following explicit form of the tachyon profile [21, 24]: T ( X ) = a + 12 α (cid:48) u ij X i X j , (C.11)one can rewrite the action as S ( a, u ) = (cid:20) tr (cid:0) G − u (cid:1) − a ∂∂a − tr (cid:18) u ∂∂u (cid:19) + 1 (cid:21) Z ( a, u ) , (C.12)where Z ( a, u ) = e − a + γtr ( G − u ) det ( Γ ( E + u ) Γ (1 + E − u )) , (C.13)with E ± = G − ± Θ .There are two equivalent descriptions of the action above. We can describe the D p -brane in a constant B -field background by treating the B -term as a boundary interactionterm. In this approach, the boundary conditions are Neumann and by direct computationone can find Z ( a, u ) = T D p (cid:90) d p +1 xe − T (cid:112) det ( g + 2 πα (cid:48) B ) (1 + . . . ) , (C.14)where the dots stand for the higher-order terms in u or the higher-derivative terms of T .One can therefore reconstruct the following shape for the action S = T D p (cid:90) d p +1 xe − T (cid:112) det ( g + 2 πα (cid:48) B ) (cid:0) T + α (cid:48) G ij ∂ i T ∂ j T + . . . (cid:1) . (C.15)This form, even if reproducing the standard gauge symmetries and showing the connectionto the well-known DBI action, is not particularly helpful in studying the soliton solutions.– 33 –lternatively, the action takes a much more convenient form in the large noncommutativitylimit studied by Seiberg and Witten [9], i.e. G − (cid:28) Θ , in which the partition functionbecomes lim G Θ →∞ Z ( a, u ) = e − a det (cid:16) π sin πΘu (cid:17) . (C.16)As shown in Refs. [21, 26], this can be written conveniently as (cid:90) d p x Pf (2 πθ ) exp (cid:63) ( − T ( x )) , (C.17)with θ ij = 2 πα (cid:48) Θ ij . It is clear that in this limit the kinetic term is suppressed, and theaction is dominated by the potential term: S = (cid:90) d p x Pf (2 πθ ) ( T ( x ) + 1) (cid:63) exp (cid:63) ( − T ( x )) . (C.18)From this expression, we can read off the well-known form of the tachyon potential [22–25]: V ( T ) = ( T + 1) e − T . (C.19) References [1] K. Kasamatsu, H. Takeuchi, M. Nitta, and M. Tsubota, “Analogues of D-branes inBose-Einstein condensates,”
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