Supergravity Solution for Three-String Junction in M-Theory

arXiv:hep-th/9906247v3 6 Jun 2000

hep-th - 9906247

Supergravity Solution for Three-String Junction in M-theory P. Ramadevi

∗

Physics Department, Indian Institute of Technology Bombay, Mumbai - 400 076, INDIA

Abstract Three-String junctions are allowed configurations in II B string theory which preserve one-fourth supersymmetry. We obtain the 11-dimensional supergravity solution for curved membranes corresponding to these three-string junctions.

In the last few years, there has been lot of interest on three-string junctions in Type II B string theory [1]- [7]. It was orginally postulated by Schwarz that string theory can also admit multi-junction/multi-pronged strings as fundamental objects [1]. In Ref. [2], it has been shown that the gauge field configurations in F -theory on K3 for exceptional groups can be accounted only if multi-pronged strings besides the usual strings (two-pronged) are included as solutions in string theory. The one-fourth BPS nature of three-string junction was conjectured in [1] and proven in [3, 4]. These three-pronged strings/threestring junction connecting three D3-branes gives understanding of 14 -th BPS states of the ∗

E-mail: [email protected], [email protected]

1

SU(3) super yang-mills theory on the coincident three- D3-branes[5, 6] and their SU(N) generalisations presented in Ref. [7]. With enough evidence for such a junction configuration to be present in string theory, it is very essential that these junctions emerge as exact solution of supergravity field equations. Till date, attempts in 10-dimensions have failed due to the singular nature at the junction. The hope is that smoothening the junction would help in finding the solution. Hence, we look at curved membranes in M-theory corresponding to three-string junctions in II B theory. The supergravity solution for planar membranes corresponding to fundamental strings [8] gives us some idea in solving the curved case. The solution for such curved membranes representing three-string junction in IIB theory, after the forthcoming (tedious) calculations, are given in eqns.(61, 62)1 . It is important to stress that the mathematical limitations (analytic solution for a non-linear differential equation) enables us to obtain only an implicit supergravity solution for three-string junction. The two dimensional membrane corresponding to three-string junction is given by holomorphic curve [9]: f (u, v) =

q

(u − λ1 )(v − λ2 ) −

q

λ1 λ2 = 0 ,

(1)

where u and v are functions of complex coordinates U and V in R2 × T 2 . We begin by making an ansatz for the 11-dimensional metric gM N respecting SO(6) symmetry and gauge fields AM N P ( M, N, P = 0, 1, . . . 10) ds2 = −a0 (u, u ¯, v, v¯, r)dt2 + 2gµ¯ν (u, u ¯, v, v¯, r)dxµ dxν¯ +a1 (u, u ¯, v, v¯, r)

10 X

(dxm )2 ,

(2)

i=5

A = A0µ¯ν (u, u¯, v, v¯, r)dx0 ∧ dxµ ∧ ∧dxν¯ , where r 2 = 1

P10

m 2 m=5 (x )

(3)

and x0 = t, xµ = u, v; xν¯ = u¯, v¯.

We came across a recent paper [12] where a similar form appeared for localised orthogonal intersecting

membranes.

2

ˆ The vielbeins eaM and inverse vielbeins EaˆM for the metric ansatz in the upper triangular

form: ˆ

ˆ em n

evvˆ

√

a0 = (Eˆ00 )−1 √ = δmn a1 = (Enˆm )−1 , √ ˆ = evv¯¯ = gv¯v = (Evˆv )−1 = (Evˆ¯v¯)−1 ,

e00 =

euuˆ = = evuˆ = Euˆv = Euˆ¯v¯ =

s

gu¯u gv¯v − gu¯v gvu¯ gv¯v u −1 u ¯ −1 (Euˆ ) = (Euˆ¯ ) , gu¯v gv¯u ˆ ; evu¯¯ = √ , √ gv¯v gv¯v gu¯v −q , gv¯v (gu¯u gv¯v − gu¯v gvu¯ ) gvu¯ , −q gv¯v (gu¯u gv¯v − gu¯v gvu¯ ) ˆ euu¯¯

=

(4)

where the indices with hat refers to tangent space index distinguishing them from world volume indices. The arbitrary functions ai ’s and three-form components must be reduced to fewer number of unknowns by requiring that the field configuration (2, 3) preserve one-fourth supersymmetry. In other words, there must exist Killing spinors satisfying DM ǫ = 0

(5)

where DM is the supercovariant derivative appearing in the gravitino supersymmetry transformation, δψM = DM ǫ,

(6)

1 AˆBˆ DM = ∂M + ωM ΓAˆBˆ 4 1 S (ΓP QRS + 8ΓP QR δM )FP QRS − 288 M

(7)

where FM N P Q = 4∂[M AN P Q] . Here ΓAˆ are the D = 11 Dirac matrices obeying {ΓAˆ , ΓBˆ } = 2ηAˆBˆ , 3

(8)

where ηAˆBˆ = diag(−, +, + . . . +) and (9)

ΓAˆB... ˆ ˆ . . . ΓC] ˆ C ˆ = Γ[A ˆ ΓB

and Γ’s with world volume index can be converted to tangent space index using vielbeins. We will split the 11-dimensional Γ matrix respecting S0(6) symmetry in the following way: ΓAˆ = (iγαˆ ⊗ Γ7 , 1 ⊗ Σaˆ )

(10)

where γαˆ and Σaˆ are the D = 5 and D = 6 Dirac matrices and Γ7 = Σˆ5 Σˆ6 . . . Σ10 ˆ .

(11)

γˆ0 γuˆuˆ¯ γvˆvˆ¯ = i ; Γ27 = −1

(12)

satisfying the following properties:

To simplify the computations, we shall first impose the following constraints on ǫ γu ǫ = 0 ; γv ǫ = 0 ; Γ7 ǫ = −iǫ ,

(13)

which in the 11-dimensions gives one-fourth BPS nature of the curved membranes corresponding to three-string junction-viz., Γ1 Γ2 Γ5 . . . Γ10 ǫ = ǫ ,

(14)

Γ3 Γ4 Γ5 . . . Γ10 ǫ = ǫ .

(15)

With the above constraints, the spin connection on the spinor field can be simplified to ˆˆ

ˆˆ

(ωM )AˆBˆ ΓAB ǫ = EANˆ EBRˆ ∂R gN M ΓAB ǫ

(16)

In our background (2, 3), we shall now examine eqn.(5). After incorporating one-fourth BPS condition (13), we get 1 1 D0 ǫ = ∂0 ǫ + [ γ0 γ u (∂u ln a0 ) + γ0 γ v (∂v ln a0 ) 4 4 4

1 i q −1 − γ0 Σm (∂m ln a0 ) − a γ0 Σm H 4 6 0 i q −1 i q −1 a0 γ0 Iγ u + a0 γ0 Jγ v ]ǫ = 0 + 6 6 1 n n Dm ǫ = ∂m ǫ + [ (Σm Σ − Σ Σm )(∂n ln a1 ) 8 1 1 + Σm γ u (∂u ln a1 ) + Σm γ u (∂u ln a1 ) 4 4 i q −1 i q −1 − a0 (Σm Σn − Σn Σm )H − a0 Σm γ u I 24 12 i q −1 i q −1 v a Σm γ J + a H]ǫ = 0 , − 12 0 6 0 1 Du ǫ = ∂u ǫ + [ (g u¯u ∂u gu¯u + g v¯v ∂v gv¯u 4 q i a−1 0 +g u¯v ∂v gu¯u + g u¯v ∂u gv¯u ) + I]ǫ = 0 , 12 1 Dv ǫ = ∂v ǫ + [ (g v¯v ∂v gv¯v + g u¯v ∂u gv¯v 4 q i a−1 0 u ¯u vu ¯ +g ∂u gvu¯ + g ∂v gu¯v ) + J]ǫ = 0 , 12 1 Du¯ ǫ = ∂u¯ ǫ − [ (g u¯u ∂u¯ gu¯u + g u¯v ∂v¯ gu¯u + g v¯v ∂v¯gvu¯ 4 1 1 +g vu¯ ∂u¯ gvu¯ ) − Σm γ v ∂m gvu¯ − Σm γ u ∂m gu¯u 4 4 q i 1 uv + γ (∂v gu¯u − ∂u gvu¯ ) − Σm γ v a−1 (g v¯u P 4 12 0 i q −1 v¯v a (g P − g vu¯ Q) − g u¯u Q) + Σm γ u 12 0 i q −1 a (−Σm γ u ∂m A0u¯u + Σm γ v ∂m Ao¯uv + 6 0 i q −1 a K]ǫ = 0 , +γ uv {∂u A0¯uv + ∂v A0u¯u }) + 4 0 1 Dv¯ǫ = ∂v¯ǫ − [ (g v¯v ∂v¯ gv¯v + g u¯v ∂v¯ gu¯v + g u¯u ∂u¯ gu¯v 4 1 1 +g vu¯ ∂u¯ gv¯v ) − Σm γ v ∂m gv¯v − Σm γ u ∂m gu¯v 4 4 q 1 uv i − γ (∂u gv¯v − ∂v gu¯v ) − Σm γ u a−1 (g u¯v R 4 12 0 i q −1 u¯u a (g R − g u¯v S) − g v¯v S) + Σm γ v 12 0 i q −1 m u a (Σ γ ∂m A0¯v u − Σm γ v ∂m A0v¯v + 6 0 5

(17)

(18)

(19)

(20)

(21)

q

i a−1 0 L]ǫ = 0 , −γ uv {∂u A0v¯v + ∂v A0¯vu }) + 4

(22)

where H = g u¯u ∂m Au¯u0 + g u¯v ∂m Au¯v0 + g vu¯ ∂m Avu¯0 +g v¯v ∂m Av¯v 0

(23)

I = g v¯v ∂u Av¯v 0 − g vu¯ ∂u Au¯v0 + g v¯v ∂v Av¯u0 −g u¯v ∂v Au¯u0

(24)

J = g u¯u ∂u Au¯v0 − g u¯v ∂u Av¯v 0 + g u¯u ∂v Au¯u0 −g v¯u ∂v Av¯u0

(25)

P = (gu¯u ∂m Av¯v0 − gu¯v ∂m Au¯v0 )

(26)

Q = (gu¯u ∂m Au¯v0 + gu¯v ∂m Au¯u0 )

(27)

K = (g v¯v ∂v¯A0¯uv − g v¯u ∂v¯ A0u¯u + g v¯v ∂u¯ Av¯v0 −g v¯u ∂u¯ Av¯u0 )

(28)

R = (gv¯v ∂m Au¯u0 − gv¯u ∂m Avu¯0 )

(29)

S = (gv¯v ∂m Av¯u0 + gv¯u ∂m Av¯v 0 )

(30)

L = (g u¯u ∂u¯ Au0¯v − g u¯v ∂u¯ A0v¯v + g u¯u ∂v¯ Au¯u0 −g u¯v ∂v¯Au¯v0 )

(31)

From eqn.(17), equating the respective Γ terms we get, ∂0 ǫ = 0 , 1 i γ0 γ u ( ∂u lna0 + a−1 I) = 0 , 4 6 0 i q −1 1 a J) = 0 , γ0 γ v ( ∂v lna0 + 4 6 0 i q −1 1 a H) = 0 . γ0 Σm (− ∂m lna0 − 4 6 0 q

(32) (33) (34) (35)

Similarly equating the respective Γ terms in (18) we get: ∂n ǫ = − 6

i q −1 a Hǫ , 6 0

(36)

i q −1 1 a I , ∂u lna1 = 4 12 0 1 i q −1 a J , ∂v lna1 = 4 12 0 i q −1 1 a H . ∂n lna1 = 8 24 0

(37) (38) (39)

Comparing the above equations with eqns. (33, 34, 35), we deduce 1 (∂n lna0 )ǫ , 4

∂n ǫ = q

∂m ln a−1 = ∂m lna1 0 q

= ∂µ lna1 , ∂µ ln a−1 0

(40) (41) (42)

suggesting a relation a1 =

q

a−1 0 .

(43)

Clearly, we have not used the actual form of gµ¯ν and A0µ¯ν in deducing the relation between a0 and a1 . We will see that the equations obtained by comparing Γ terms in (21, 22) will help us to determine three-form components and gµ¯ν . The set of equations we get from equating the respective γ terms in (21) are 1 ∂u¯ ǫ − [ (g u¯u ∂u¯ gu¯u + g u¯v ∂v¯gu¯u + g v¯v ∂v¯ gvu¯ 4 i q −1 a K]ǫ = 0 , +g vu¯ ∂u¯ gvu¯ ) + 4 0 i q −1 v¯u 1 a (g P − g u¯u Q) Σm γ v {− ∂m gvu¯ − 4 12 0 i q −1 a ∂m A0¯uv } = 0 , + 6 0 i q −1 v¯v 1 a0 (g P − g u¯v Q) Σm γ u {− ∂m gu¯u + 4 12 i q −1 a ∂m A0u¯u } = 0 , − 6 0 i q −1 uv 1 a (∂u A0¯uv γ { (∂v gu¯u − ∂u gvu¯ ) + 4 6 0 +∂v A0u¯u )} = 0 . Similarly, we get the following set from eqn.(22): 1 ∂v¯ǫ − [ (g v¯v ∂v¯ gv¯v + g u¯v ∂v¯ gu¯v + g u¯u ∂u¯ gu¯v 4 7

(44)

(45)

(46)

(47)

+g vu¯ ∂u¯ gv¯v ) +

i q −1 a L]ǫ = 0 , 4 0

(48)

1 i q −1 u¯v a0 (g R − g v¯v S) Σm γ u {− ∂m gu¯v − 4 12 i q −1 a ∂m A0¯vu } = 0 , + 6 0 i q −1 u¯u 1 m v a0 (g R − g v¯u S) Σ γ {− ∂m gv¯v + 4 12 i q −1 a ∂m A0v¯v } = 0 , − 6 0 i q −1 1 uv a (∂u A0v¯v γ {− (∂u gv¯v − ∂v gu¯v ) − 4 6 0 +∂v A0¯vu )} = 0 .

(49)

(50)

(51)

There are at least three solutions solving eqns. (32- 51). 1) For a sub-class of membranes satisfying N = |∂u f |2 + |∂v f |2 = const: ds2 = H

−2 3



(r, |f |) −dt2 + 2|du|2 + 2|dv|2

10 X 1 2 2 − |df |2 + H 3 (r, |f |) |df |2 + dx2m N N i=5   −1 1 iH 3 (r, |f |){−dt ∧ ∗ df ∧ d¯f¯ } A= 2N −1 ǫ = ǫ0 H 6 (r, |f |)



!

(52) (53) (54)

where the Hodge star operation ∗ is done on the R2 × T 2 space. The three-form potential in component form for the above metric, in the convention ǫu¯uv¯v = +1, simplifies to: 1 −1 H (r, |f |)|∂v f |2 , N 1 = −i H −1 (r, |f |)∂u f ∂v¯f¯ , N 1 −1 H (r, |f |)|∂u f |2 , = N 1 = −i H −1 (r, |f |)∂v f ∂u¯ f¯ N

A[0u¯u] = i A[0u¯v] A[0v¯v] A[0vu¯]

(55)

This restricted class includes holomorphic curves f = pu + qv representing planar membranes corresponding to (p, q) strings in IIB theory which preserve half supersymmetry. The membrane solution for f = u and f = v agrees with the results in Ref.[8]. 8

2) Intersecting M2 ⊥ M2 branes at a point [10, 11]: 1

1

ds2 = H13 (r)H23 (r){−H1−1(r)H2−1 (r)dt2 +2H1−1(r)|du|2 + 2H2−1 (r)|dv|2 +

10 X i=5

(dxm )2 }

A = iH1−1 dt ∧ du ∧ d¯ u + iH2−1 dt ∧ dv ∧ d¯ v

(57)

−1

−1

(56)

(58)

ǫ = ǫ0 H1 6 H2 6

where ǫ0 is a constant and H1 (r), H2 (r) are harmonic functions dependent only on the transverse coordinates common to both the branes. These solutions are meaningful only if U, V are compact coordinates with the charges being smeared over the branes. (a) For a coordinate transformation u = x1 + ix4 ; v = x2 + ix3 , the usual KaluzaKlein reduction along x3 on the above metric and T-duality along x4 gives the following ten-dimensional string metric: −1

−1

ds210 = −H1 2 (r)H2−1 (r)dt2 + H1 2 (r)dx21 1 2

+H1 (r)H2−1(r)dx22

1 2

+ H1 (r)

10 X

(dxm )2 .

(59)

m=4

This represents delocalised solution for orthogonal intersection of fundamental string along x2 and D string along x1 . (b) For another coordinate transformation u = (z1 τ2 − τ1 z2 ) , v = z2 , where z1 = x1 + ix4 , z2 = x2 + ix3 , we obtain the following metric after dimensional reduction along x3 and T -duality along x4 : ds210



−1

−1

= B(r) −H1 2 (r)H2−1 (r)dt2 + H1 2 (r) 1

(dx1 τ2 − dx2 τ1 )2 + H2−1 (r)H12 (r)dx22 1 2

+H1 (r)τ2−2 dx24

1 2

+ H1 (r)

10 X

(dxm )

m=5

where B(r) =

q

2

!

,

(60)

1 + H1−1 (r)H2(r)τ12 . This solution is the simplest planar network of

F-strings and D-strings directed along (0, 1) and (−τ2 , τ1 ) in the (x1 , x2 ) plane. 9

General planar network involving [p1 , q1 ], [p2 , q2 ] strings can be similarly obtained using the coordinate transformation : u = p1 (τ2 z1 − τ1 z2 ) − q1 z2 , v = q2 z2 − p2 (τ2 z1 − τ1 z2 ). The equivalence of delocalised orthogonal intersecting strings with general planar network of strings is expected because the delocalised solution in M-theory (58) has no information about the intersection point or the subspace containing the two M2-branes. Hence eqn. (58) also represent delocalised solutions for the membrane corresponding to general planar network of strings. In order to distinguish the intersecting membranes from curved membranes corresponding to three-string junction, we have to look for fully or partially localised solutions. 3) Arbitrary membranes including those corresponding to three-string junction −2 3

ds2 = −H

1 3

+H

dt2 + 2H

10 X

−2 3

Gµ¯ν (u, u ¯, v, v¯, r)dxµ dxν¯

(dxm )2

(61)

m=5

A = iH −1 Gµ¯ν dx0 ∧ dxµ ∧ dxν¯ −1 6

ǫ = ǫ0 H

(62) (63)

,

where Gµ¯ν is Kahler and the function H is H = Gu¯u Gv¯v − Gu¯v Gvu¯ .

(64)

The metric in terms of the Kahler potential K is Gµ¯ν = ∂µ ∂ν¯ K .

(65)

The embedding of the membrane (1) in the metric can be seen by performing the following holomorphic coordinate tranformation with unit Jacobian: (u, v) → (α, β) ,

(66)

u − λ1 . v − λ2 √ The membrane surface in the new coordinate is f = α − λ1 λ2 . This membrane is where α =

q

(u − λ1 )(v − λ2 )

;

β = αln

different from the intersecting membranes given by u = λ1 ; v = λ2 . We hope to see this difference from partially or fully localised supergravity solution. 10

We are now left with the task of determining the form of K from the equations of motion for three-form gauge field in the presence of the curved membrane as the source: √ ∂M ( −gF M U V W ) +

1  U V W M N OP QRST ǫ 1152 FM N OP FQRST ) = J U V W (x) 1 δSmembrane =√ −g δAU V W (x)

(67)

where the membrane action is Smembrane = −T

Z

√ d3 ξ{ −deth

1 − ǫabc ∂a X M ∂b X N ∂c X P AM N P } . 6

(68)

Here hab = ∂a X M ∂b X N GM N is the induced metric on the membrane. ¯ we obtain the three-form current for the holomorphic Choosing ξ0 = t, ξ1 = β, ξ2 = β, membrane f to be T ¯ J 0β β = √ δ 2 (f )δ 6 (xm ) . −g

(69)

∂µ ∂ν¯ (2H + δ mn ∂m ∂n K) = jµ¯ν

(70)

Substituting the 3-form potential and metric (61, 62) respecting one-fourth supersymmetry, we get

where ′ ′

jµ¯ν = ǫµµ′ ǫνν ′ J 0µ ν

q

−detg .

(71)

The above non-linear equation cannot be solved analytically. Perturbative approach over Minkowskian backgrounds gives integral representation for K which is dependent on jµ¯ν [12]. Since ju¯u = δ 2 (u)δ 6 (r), jv¯v = δ 2 (v)δ 6 (r) for the intersecting membranes is different from that of the curved membranes (69), the integrands of the integral representation for K are distinct. In this approach [12], we expand the Kahler potential K = metric Gm¯n =

P

l

(l)

P

n

K (n) and hence the

Gm¯n . Minkowskian background implies that we take the zeroth-order 11

(0)

K (0) = u¯ u + v¯ v so that Gm¯n = δm¯n . Further, comparing n-th order terms in eqn. (70) gives a set of differential equations. Using this set, we get a formal integral representation for K (n) involving the lower order metric components. However, the goal to obtain explicit closed form expression for localised or partially localised solution from the integral representation is still unsolved. It has been shown, for certain classes of orthogonal intersections and holomorphic curves, that the perturbation theory breaks down when there are more than three transverse dimensions [12]. This breakdown of the perturbation theory suggests that no such fully localized solutions exist with asymptotically flat boundary conditions. However, perturbation theory suggests that such solutions do exist when there are less than three transverse dimensions. So, for sufficiently smeared versions of the sources, one expects that the solutions could be obtained numerically even if an exact analytic form cannot be found. It is not clear at this stage whether perturbation theory over other backgrounds like planar membrane background would help in finding a closed form expression. We hope to pursue this isssue in future. The fully localised or partially localised supergravity solutions is also needed to understand the map of bulk parameters to boundary gauge theory parameters to prove AdSCFT correspondence. Such near-horizon or AdS metric for intersecting branes [13, 14] and intersecting M5-branes [15] corresponding to NS5-D4 branes in II A theory [16] has been obtained. The procedure elaborated for one-fourth BPS states can be generalised to construct supergravity solution for other non-planar networks [17, 18]. The challenging problem of finding closed form expression for localised/partially localised solutions still remains.

Acknowledgements: I would like to thank Alok Kumar for the fruitful discussions during the initial stages of this work. I am grateful to Ashoke Sen for his comments and

12

suggestions. I would like to thank Donald Marolf for explaining the difficulties in finding localised solutions. I would like to thank the organisers of the Black Hole Conference, Trieste where some of the issues on localised and delocalised solutions got clarified. I owe my thanks to Department of Theoretical Physics, TIFR for providing me local hospitality where part of this work was done. I would like to thank CSIR for the grant.

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[9] M. Krogh and S. Lee, String Network from M-theory, Nucl. Phys. B516 (1998) 241, hep-th/9712050. [10] A. A. Tseytlin, Harmonic Superposition of M-branes, Nucl. Phys. B475 (1996)149, hep-th/9604035; [11] J.P. Gauntlett, D. A. Kastor, J. Traschen, Overlapping Branes in M Theory, Nucl.Phys.B478:544-560,1996, hep-th/9604179. [12] A. Gomberoff, D. Kastor, D. orolf and J. Traschen, Fully Localized Brane Intersections- The Plot Thickens, hep-th/9905094. [13] D. Marolf, A. Peet, Brane Baldness vs. Superselection Sectors, hep-th/9903213. [14] D. Youm, Paritally Localized Intersecting BPS Branes, hep-th/9902208. [15] A. Fayyazuddin and D. J. Smith, Localized intersections of M5-branes and four dimensional superconformal field theories, hep-th/9902210. [16] E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl.Phys. B500 (1997) 3, hep-th/9703166. [17] O. Aharony, A. Hanany, Webs of (p,q) five-branes, five-dimensional theories and grid diagrams, JHEP 9801:002,1998, hep-th/9710116. [18] Sandip Bhattacharyya, Alok Kumar, Subir Mukhopadhyay, String Network and Uduality, Phys.Rev.Lett.81 (1998)754, hep-th/9801141; Alok Kumar, Subir Mukhopadhyay, Supersymmetry and U-brane Networks, hepth/9806126.

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