The Self-Dual String Soliton ABSTRACT

Transcription

1 SLAC PUB KCL-TH hep-th/ August 1997 The Self-Dual String Soliton P.S. Howe N.D. Lambert and P.C. West Department of Mathematics King s College, London England WC2R 2LS ABSTRACT We obtain a BPS soliton of the M theory fivebrane s equations of motion representing a supersymmetric self-dual string. The resulting solution is then dimensionally reduced and used to obtain 0-brane and (p 2)-brane solitons on D-p-branes. phowe, lambert, pwest@mth.kcl.ac.uk

2 1. Introduction In recent years it has become clear that the still rather mysterious M theory governs many aspects of the lower dimensional string theories. What little is known of M theory is powerful enough to lead us to new phenomenon in string theory and indeed new string theories. In particular the M theory fivebrane is strongly believed hold a new kind of self-dual string theory on its worldvolume [1,2]. This new and somewhat elusive theory has also appeared in other contexts such as the compactification of type IIB string theory on K3 [3], M(atrix) theory on T 5 [4] and the S-duality of N =4,D= 4 super-yang-mills [3]. Thus one may hope that a greater understanding of this self-dual string may lead directly to a greater understanding of duality, string theory and M theory. In this paper we shall seek a supersymmetric string soliton solution to the M theory fivebrane s equations of motion [5,6,7]. In [6] a smooth string soliton solution of the field equations was found. However the solution obtained there takes all the scalar fields to be constant and so cannot preserve any supersymmetries, since there is nothing to cancel off the tensor field force (this will also be apparent from the supersymmetry transformation given below). Instead it represents a non-singular field configuration around a string, analogous to the Born-Infeld expression for the electric field of a charged point particle [6]. In the next section we will solve the field equations and Bogomoln yi conditions, following this in section three we will dimensionally reduce the string soliton to obtain solitons of the type II D-p-brane worldvolume theories. In the final section we will conclude with some comments on our solutions. 2

3 2. The Self-Dual String as a Soliton We use the conventions of [7] with the fivebrane embedded in flat eleven dimensional Minkowski space. In particular we denote tangent frame indices by a, b,... =0,1,2,..., 5 and world indices by m, n,... =0,1,2,..., 5 which are related by the sechsbein Ea m. The bosonic worldvolume theory of the M theory fivebrane contains five scalars X a,(a =1,..., 5) and a self-dual tensor h abc. Underlined indices refer to eleven dimensional quantities and a prime indicates the directions transverse to the fivebrane. This theory has six dimensional (2, 0) supersymmetry and therefore it also possesses the chiral fermions Θα i,whereα, β,... =1,..., 4and i, j,... =1,..., 4areSpin(1, 5) and USp(4) indices respectively. Let us look for a string soliton whose worldsheet lies in the x 0,x 1 plane and denote the remaining four transverse coordinates as a,b,..., m,n... =2,3,4,5. We break the SO(5) symmetry of the scalars down to SO(4) and take all fields to be independent of x 0 and x 1. Consider the ansatz X 5 = φ, h 01a = v a, (2.1) h a b c = ɛ a b c d vd, with the other components of h abc vanishing and the scalars X 1,X 2,X 3,X 4 constant. From the above we may determine the matrix m b a δ a b 2h acdh bcd 1+4v 2 0 = 0 1+4v 2 (1 4v 2 )δa b +8v a vb. (2.2) Following [7] we may now calculate the tensor H mnp = Em a En b Emm c b dm c e h ade (which is not necessarily self-dual). The only non-vanishing components are H 01m = v m, H m n p = 1 (1 4v 2 ) ɛ m n p q vq. (2.3) 3

4 In (2.3) and below all world tensors (including ɛ m n p q ) are expressed in terms of the Green-Schwarz metric 1 0 g mn = 0 1 which has its own sechsbein ea m metric on the fivebrane, (2.4) δ m n + m φ n φ =(m 1 ) ae b m. Finally we construct the induced b G mn = η ab Ea m Eb n =(1 4v 2 ) 2 g mn +16v a v b ea m e n b. (2.5) We may now write the purely bosonic field equations as [7] G mn m n φ =0, G mn m H npq =0, (2.6) where is the Levi-Civita connection of the Green-Schwarz metric (2.4). In addition to (2.6) we must also require that the three form H mnp is closed. Before proceeding it is necessary to obtain the supersymmetry transformation for the fermions Θα i. To do this we will follow closely the conventions and formalism of [5,7]. In the superembedding approach the superfields naturally have the superdiffeomorphism invariance δz M = v M, (2.7) where v M is a supervector field and z M =(x m,θ α ) is the superspace coordinate. To obtain the supersymmetry transformation it is sufficient to set v m =0. We assume that the fivebrane is in static gauge where the x m are identified with the first six coordinates of flat eleven dimensional Minkowski space z m. The thirtytwo dimensional odd coordinates α can be broken into two sixteen component parts 4

5 α and α. Half of the worldsheet odd coordinates Θ α are then identified with half of the spacetime spinors z α. Here we have introduced a two step notation for the spinors indices: in intermediate formulae α and α are used but in the final six dimensional expressions they are decomposed into the pairs α αi and α i α when appearing as superscripts and α αi and α α i when appear as subscripts [7]. It should be clear whether we mean α to be sixteen or four dimensional depending on the absence or presence of i, j,... indices respectively. In addition to the worldvolume superdiffeomorphism (2.7) there are also flat eleven dimensional supersymmetry variations δθ α = ɛ α to consider. Combining these two transformations we find δθ α = v β Eβ α + ɛα, δθ α = v β Eβ α + ɛ α, (2.8) where E β α and E β α are the embedding matrices [7] Eα β = uα β + hα γ u β γ, Eα β = uα β + hα γ u β γ. (2.9) In (2.9) the tensor h β α is related to the three-form h abc [7] h β α h j αiβ = 1 6 δ j i (γabc ) αβ h abc, (2.10) and (u α µ,u µ α ) forms an element of Spin(1, 10). Now, to ensure that static gauge is preserved, δθ α = 0 we must set v β = ɛ α (E 1 ) α β. We may also assume for our purposes that ɛ α = 0. We then find the supersymmetry transformation for the fermions and we are left to evaluate (E 1 ) β α and E α β. δθ α = ɛ α (E 1 ) β α E α β, (2.11) We are assuming here that E β α is invertable, which is true at the linearised level. 5

6 It is instructive to first consider the lowest order approximation where Eα β = δα β, u β α = δ β α and This leads to the linearised supersymmetry δ 0 Θ j β u β α u j αiβ = 1 2 (γa ) αβ (γ b ) j i ax b. (2.12) ( 1 = ɛαi 2 (γa ) αβ (γ b ) j i ax b 1 ) 6 (γabc ) αβ δ j i h abc. (2.13) We remind the reader here that the matrices γ a are not the six dimensional Γ- matrices [7] but rather their chiral components. However the γ a can be thought of the as eleven dimensional Γ matrices in the dimensions transverse to the fivebrane. If we substitute the ansatz (2.1) into δ 0 Θα j = 0 we find 0=ɛ αi ( 1 4 δ γ α δ i k a φ (γ 01 ) γ α (γ 5) k i v a ). (2.14) In (2.14) we have used the fact that the fermions ɛα i are chiral. Therefore if we take v a = ± 1 4 a φ, (2.15) the solution will be invariant under linearised supersymmetries which satisfy ɛ βj 0 = ±(γ01 ) β α (γ 5) j i ɛαi 0. (2.16) Note that in terms of the Green-Schwarz metric, (2.15) implies that v m = ± 1 4 e m a n Ea n φ = ± 1 4 f m φ, (2.17) where f = 2 1+det(e 1 ). (2.18) Finally it remains to show that the full supersymmetry transformation (2.11) vanishes if (2.15) and (2.16) are satisfied. To this end one finds after some manipula- 6

7 tions that E γ α (u 1 ) β γ = 1 2 det(e 1 )δa m mx b (γ a ) αβ (γ b ) j i (1 + det(e 1 ))h abc (γ abc ) αβ δ j i. One can now substitute (2.15) into (2.19) to show that (2.19) provided that ɛ β = E β α ɛ α 0 where ɛα 0 δθ α = ɛ α (E 1 ) β α E α β =0, (2.20) satisfies (2.16). Let us now return to solving the equations of motion (2.6). Clearly the equation of motion for H 01m now reduces to that of φ. This leaves us with the equation of motion for H m n p ( ) n ɛ f r φ n p q r Gm m 1 4v 2 =0. (2.21) Finally we must also require that H mnp is closed, which is the case provided that ( ) f m φ m 1 4v 2 =0. (2.22) Remarkably (2.21) and (2.22) reduce to the same equation, which is itself equivalent to the equation of motion of φ. We are now left with only the scalar equation G mn m n φ =(1 4v 2 ) 2 δ m n m n φ=0. (2.23) Therefore we find the solutions are given by harmonic functions on the flat transverse space R 4.ForNstrings located at x m i the solution (2.3) reduces to H 01m = ± 1 4 m φ, H m n p =±1 4 ε m n p q q φ, φ=φ 0 + N 1 i=0 2Q i x x i 2, (2.24) where ε is the flat volume form on the transverse space, repeated indices are 7

8 summed over and φ 0 and Q i are constants. Note that the solution is smooth everywhere except at the centres of the strings. Furthermore the presence of (1 4v 2 ) 2 in (2.23) implies that the equations of motion are satisfied even at the poles of φ, so that no source terms are required and the solution is truly solitonic. Clearly the string soliton (2.24) is self-dual even though in general the tensor H mnp need not be. If we consider a single string then we find it has the same electric and magnetic charges 1 Q E = Vol(S1 3) H = Q 0, S 3 1 Q M = Vol(S1 3) S 3 H = Q 0, (2.25) where S 3 is the transverse sphere at infinity, S1 3 the unit sphere and is the flat six dimensional Hodge star. To calculate the mass per unit length of this string one could wrap it around a circle of unit circumference and reinterpret the string as a 0-brane in five dimensions. The mass per unit length is then identified with the mass of the 0-brane. As we will see below this definition leads to an infinite tension, reflecting the infinite length of the membrane. Let us now consider the zero modes of a single string soliton. Given that we preserve the asymptotic form of the solution, there are four bosonic zero modes x m 0 coming from the location of the centre of the string. It can be seen that there are no other bosonic zero modes since these would come from non-zero expressions for h 0m n and h 1m n and would then break more supersymmetries and ruin the block diagonal form of ma b which was crucial for solving the field equations. This can also be seen from the fact that the solution preserves eight supercharges and so there is no room for any additional bosonic zero modes in the two dimensional worldsheet supermultiplet. The fermionic zero modes are generated by the broken supersymmetries ɛ i α which satisfy ɛ βj = (γ 01 ) β α (γ 5) j i ɛαi. (2.26) 8

9 If we call spinors with γ 01 ɛ = ɛ (γ 01 ɛ = ɛ) left (right) moving then we clearly have four left and four right moving fermion zero modes on the string world sheet, correlated with the eigenvalue of γ 5. At first sight the above counting appears to miss four bosonic zero modes coming from the scalars X 1,X 2,X 3,X 4. However these scalars do not lead to additional zero modes since they are fixed by their asymptotic values. This is analogous to the BPS monopole solutions in N = 4, D = 4 super-yang-mills where there are six scalars with an SO(6) symmetry relating them. A given BPS monopole will choose a particular scalar and break this symmetry down to SO(5). The monopoles obtained using different scalars are to be viewed as distinct, forming an SO(6) multiplet of monopoles. Similarly, here we obtain a SO(5) multiplet of self-dual strings. Thus the string has a (4, 4) supermultiplet of zero modes in agreement with that found in [2]. However, contrary to our solution, the strings described there do not carry any charge with respect to the H field. Therefore the string soliton found here might represent a D-string in the six dimensional self-dual string theory. From the M theory point of view the strings obtained here should be interpreted as the ends of infinitely extended membranes. The scalar X 5 = φ then corresponds to the direction of the membrane transverse to the fivebrane. Clearly the SO(5) symmetry rotates the choice of this direction. (2.4)The Self-Dual String as a SolitonSolitons on D-branes(2.4)(2.4)(2.4) 3. Solitons on D-branes It was shown in [7] that M theory fivebrane s equations of motion can be double dimensionally reduced to the Dirac-Born-Infeld equations of the type IIA D-fourbrane. It follows then that the self-dual string soliton constructed here can also be reduced to a 0-brane or 1-brane solution on the D-fourbrane worldvolume. By T-duality it follows that all of the type II D-brane worldvolume theories admit 9

10 0-brane and (p 3)-brane solitons preserving half of the supersymmetry, which can be obtained from the D-fourbrane solutions. Here we shall content ourselves with the p>3 branes since the lower dimensional branes do not admit asymptotically free solutions. In what follows below the background type II supergravity fields are those of flat Minkowski space and we set all but one of the scalar fields on the D-p-brane to be constant. First let us wrap the string around the compact dimension x 1 to produce a 0-brane BPS soliton on the D-fourbrane worldvolume with the two-form fieldstrength F mn = H mn1 [7]. Note that because of the self-duality condition the other components H mnp do not appear in the D-fourbrane s effective action as an independent 3-form [7]. In this case we obtain F 0m = 1 4 m φ, φ=φ 0 + N 1 i=0 (3.1) 2Q i x x i 2. We could also dimensionally reduce the self-dual string soliton by compactifying along x 5. In this case we obtain a string soliton in the D-fourbrane worldvolume. Defining F mn = H mn5 we obtain F m n = 1 4 ε m n p p φ, φ=φ 0 + N 1 i=0 (3.2) 4Q i x x i. The generalisation to p>4 branes is straightforward. One has 0-branes with F 0m = 1 4 m φ, φ=φ N 1 (3.3) Q i p 2 x x i=0 i p 2, 10

11 and (p 3) branes with F m n = 1 4 ε m n p p φ, φ=φ 0 + N 1 i=0 (3.4) 4Q i x x i. In the above solutions m,n,p... refer to the transverse space, which is p dimensional for (3.3) or three dimensional for (3.4). The scalar field only depends on the transverse coordinates and ε is the flat volume form on the three dimensional transverse space to the (p 3)-brane. These solutions preserve half of the worldvolume supersymmetry and in the case of a single soliton carry the electric charge ±Q 0 (for (3.3)) or magnetic charge ±Q 0 (for (3.4)) with respect to F mn. These solutions represent an infinitely long string or (p 2)-brane respectively, streching out from the D-p-brane along the direction specfied by the scalar φ. As with the self-dual string this transverse direction is acted on by the SO(9 p) symmetry which rotates the scalars. These solutions are infinitely massive, similar to a point-like charge in Maxwell s theory. However this is to be expected given the interpretion as the end of an infinitely long string or (p 2)-brane. One can also explicitly verify that these solitons preserve half the worldvolume supersymmetry at the linearised order by compactifying D = 10super-Maxwell theory to p+1 dimensions, viewed as the lowest order approximation to the D-brane effective action. However our construction from the self-dual string solution shows that (3.3) and (3.4) are solutions to the non-linear Dirac-Born-Infeld equations of motion preserving half of the supersymmetry to all orders. The D-threebrane is a special case and so we shall treat it separately here. By wrapping the self-dual string on a torus to four dimensions we obtain dyonic 0-brane solitons on the D-threebrane worldvolume. Now x 1 and x 5 are compact so that φ is depends only on x m = x 2,x 3,x 4. Let us consider new coordinates (u, v) 11

12 related to (x 1,x 5 )by ( ) ( )( ) u a b x 1 = v c d x 5, (3.5) with ad bc = 1. If we define the two-form field strength F mn = H mnu then we find the dyonic solutions F = af E + cf M, φ = φ 0 + N 1 i=0 4Q i x x i, (3.6) where F E and F M are the purely electric and purely magnetic field strengths with the components F0m E = 1 4 m φ, (3.7) Fm M n = 1 4 ε m n p p φ. The solution corresponding to a single dyon then has the electric and magnetic charges (Q E,Q M )=±Q 0 (a, c). (3.8) As is well known the SL(2, Z) S-duality symmetry of N =4,D= 4 theory acts as ( ) ( a b a A 1 c d c ) b A (3.9) d where A SL(2, Z) and is simply the modular group acting on the torus [8,9]. 12

13 (2.4)The Self-Dual String as a SolitonDiscussion(2.4)(2.4)(2.4) 4. Discussion In this paper we have obtained a self-dual string soliton as a BPS solution of the M theory fivebrane s equation of motion. Furthermore we showed that the full non-linear equations of motion and Bogomoln yi equation were satisfied to all orders. We also used the self-dual string soliton to obtain soliton states on type II D-p-branes for p>2. Finally we would like to close with some comments. It is interesting to note that at the linearised level, where we the two metrics G mn and g mn are flat and the tensor ma b is the identity, the field equations (2.6) become non-interacting. Thus even the theory of a free tensor multiplet contains a non-trivial BPS soliton in its spectrum. In fact this statement deserves some qualification since the theory isn t completely free at the linearised level because the fermions still interact with the bosons (although the fermions have been set to zero in the solution). However, this does help to clarify why the analysis of the M theory fivebrane s BPS states carried out in [2] was so successful, even though it assumed that one could treat the self-dual string theory as non-interacting. This unusual situation is what one might expect given the interpretation of the self-dual string as limit in which a type IIB D-threebrane wrapped around a 2-cycle of K3 becomes light and decouples from gravity [3]. If one considers the supergravity field equations for a generic p-brane there are equations for the field strength and for a scalar field, similar to (2.6) (with G mn = g mn now identified with the spacetime metric). However, in supergravity what makes these equations interacting and non-linear is the fact that all these fields couple to the spacetime metric, even though they don t couple directly to each other (at least to lowest order in the supergravity effective action). Thus in a limit where gravity is decoupling one may expect the lowest order bosonic field equations to become those of a free theory. The type IIB threebrane (and hence the related six dimensional self-dual string also) is actually unique in this respect because the dilaton is constant [10]. 13

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