The spacetime superalgebras from M-branes in M-brane backgrounds

29 October 1998

Physics Letters B 439 Ž1998. 12–22

The spacetime superalgebras from M-branes in M-brane backgrounds Takeshi Sato

1

Institute for Cosmic Ray Research, UniÕersity of Tokyo, Tanashi, Tokyo 188-8502, Japan Received 7 May 1998 Editor: H. Georgi

Abstract We derive the spacetime superalgebras explicitly from ‘‘test’’ M-brane actions in M-brane backgrounds to the lowest order in u via canonical formalism, and discuss various BPS saturated configurations on the basis of their central charges which depend on the harmonic functions determined by the backgrounds. All the 1r4 supersymmetric intersections of two M-branes obtained previously are deduced from the requirement of the test branes to be so ‘‘gauge fixed’’ in the brane backgrounds as to preserve 1r4 supersymmetry. Furthermore, some of 1r2-supersymmetric bound states of two M-branes are deduced from the behavior of the harmonic functions in the limits of vanishing distances of the two branes. The possibilities of some triple intersections preserving 1r4 supersymmetry are also discussed. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction The M-theory is currently a most hopeful candidate for a unified theory of particle interactions w1,2x and is extensively studied from various points of view w3,4x. Among them, the analysis via superalgebra is one of the most powerful approaches to investigate its various properties w5,6x. Since there are, of course, two kinds of supersymmetries, two kinds of algebras have been discussed so far: the spacetime superalgebra and the worldvolume ones. The former is initially constructed as the most general modification of the standard D s 11 supersymmetry algebra w7,8x, and then deduced explicitly from the M-5-brane action in the flat background via Žanti-.canonical commutation relations of the worldvolume fields w9xŽsee also Ref. w10x.. It is called M-theory superalgebra, from which it is shown that only five basic 1r2 supersymmetric constituents of M-theory can be permitted. And the latters, defined on the flat Žp q 1.-dimensional worldvolumes of p-branes, are the maximal extensions of the Žp q 1.-dimensional supertranslation algebras, from which all the 1r4 supersymmetric M-brane intersections as worldvolume solitons can be deduced w11–13x. Both of these analyses are also applied to D-branes w12,14x, although there are some subtleties in the worldvolume cases. In this way these discussions so far have been based only on the flat spacetime.

1

E-mail: [email protected]

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 9 9 8 - 8

T. Sato r Physics Letters B 439 (1998) 12–22

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Now, we will discuss the extension, in particular, of the former to those from branes in nontriÕial backgrounds. 2 One of our most important motivation for it is to get the superalgebra of the 10-dimensional massive IIA theory w15–17x. It has not been obtained yet since this theory does not admit the flat background because of the existence of cosmological constant w18,19x. This superalgebra is significant not only to investigate the properties of the theory like the above cases, but also to understand the 11-dimensional origin of the massive IIA theory, which is not yet known satisfactorily because of the no-go theorem presented in Ref. w20x, although several trials were made w21–23x Žand recently w42x.. Therefore, the extension is urgently necessary. Let’s reconsider the computation in Ref. w9x. We might be able to interpret it as follows: the M-5-brane action is originally invariant under local super-transformation before taking its background to be any specific geometry. Suppose we float the M-5-brane as a ‘‘test’’ brane in the flat background. Then, the system and hence the action have the supertranslation symmetry in all the directions for the following two reasons: first, the background has the supertranslation symmetry and the test brane is assumed to be so light that it has no effect on the background. Second, the configuration of the test brane including its orientation has not yet been fixed at this time. Therefore, we can define the Noether supercharge, compute its anti-commutator as the superalgebra and discuss the supersymmetric configurations permitted in the flat background on the basis of its central charges, which was done in Ref. w9x. This interpretation will suggest us to think of the following extension: if we float a test M-brane in a certain non-triÕial background which has some portions of supersymmetry, then the system and the action should have the same supersymmetry 3 because of the same reasons stated above. Therefore, in the same way, we can define the corresponding Noether supercharge, compute its commutator as the superalgebra, and finally, we are able to deduce from it the supersymmetric configurations of the test brane (including intersections with the background) permitted in the non-triÕial background. The aim of this paper is to examine this idea explicitly in the cases of a test M-2-brane and a test M-5-brane in an M-2-brane and an M-5-brane background, i.e. four cases in all. 4 The concrete procedures are as follows: we will take the background to be a M-brane solution which actually consists of such a large number of coincident M-5-branes that our ‘‘test brane’’ approximation is justified, and substitute the solution for the test brane action as was done in Ref. w25x. Then, we check the invariance of the action under the unbroken supersymmetry transformation, derive the representation of the supercharge in terms of the worldvolume fields of the test brane and their conjugate momenta, compute its algebra and discuss various supersymmetric configurations permitted in the background on the basis of the algebra. We note that our computations are performed only up to the low orders in u which might contribute to the central charges at zeroth order in u because they suffice to discuss the supersymmetric configurations which we want to know. The most important fact throughout our computations is that we can reduce the superspace with the supercoordinates Ž x, u . to that with coordinates Ž x, uq. , where the sign q of uq implies that uq has a definite worldvolume chirality of the background. The reason is the following: since half of supersymmetry is already not the symmetry of the system owing to the existence of the background brane, the corresponding parameter uy must not be transformed. Thus, the conjugate momentum of uy does not appear in the supercharge Qq, which means that the terms including uy do not contribute to the central charges at zeroth order in u . Therefore, we set u ys 0 from the beginning.

2 The possibility of this extension has been already pointed out in the earlier paper w10x for a different purposeŽrelated to nontrivial topologies., although it is not shown explicitly there. 3 In the middle of completing this work, this idea is pointed out and proved generically by Ref. w24x in some other context Žabout the new actions presented in it.. Our work will be worth doing in order to examine this idea explicitly to low orders in u . 4 We apply the method to the case in 10D massive IIA background in the next paper w40x, which will be completed soon.

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The consequence is that all the 1r4-supersymmetric intersections of two M-branes obtained previously both in 11D supergravity w25,26x and via worldvolume superalgebras w27x are deduced from the requirement of the test branes to be so ‘‘gauge fixed’’ in the backgrounds as to preserve 1r4-supersymmetry. In addition, one outstanding characteristic of the results is the dependence of Žthe r.h.s. of. the superalgebras on the harmonic functions determined by the backgrounds. By using this property we derive the following two kinds of bound states composed of two M-branes preserving 1r2 supersymmetry: the first ones are deduced from the configurations of the test M-p-branes parallel to the background M-p-brane with the converse orientation. The second is from the configuration of the test M-2-branes parallel to a two-dimensional subspace of the worldvolume of the background M-5-brane. In both cases all the supersymmetry is broken because there are only attractive forces between the two branes. Then, the ‘‘absorption’’ limits, namely the limits of zero distances are expected to lead to the restorations of 1r2 spacetime supersymmetry because the potential energies are minimized in the limit. It is shown that these are really the cases, which correspond to M-p-branerM-p-brane bound states and a M-2-branerM-5-brane bound state w28,9x Žpreserving 1r2 supersymmetry., respectively. Another merit of the results is that the possibilities of some supersymmetric triple intersections can be discussed directly on the basis of the spacetime superalgebras, although not systematically. This paper is organized as follows: in Section 2 we discuss the superalgebras from the test M-2-brane in an M-2-brane and an M-5-brane background. In Section 3 we discuss the superalgebras from the test M-5-brane in an M-2-brane and an M-5-brane background. In Section 4 we give short summary and discussions. Before presenting our results we give some preliminaries. We use ‘‘mostly plus’’ metrics for both worldvolume and spacetime. And we use Majorana Ž32 = 32. representation for Gamma matrices Gmˆ which are all real and satisfy  Gmˆ , Gnˆ 4 s 2hmˆ nˆ . G 0ˆ is antisymmetric and others symmetric. Charge Conjugation is C s G 0ˆ . We use the symbol to denote the number 10 as used in Ref. w6x . We use capital latin letters Ž M, N,... for superspace indices, small latin letters Ž m,n,... for spacetime vectors and early small greek letters Ž a , b ,... for spinors. Furthermore, we use late greek letters Ž m , n ,... for spacetime vectors parallel to the background branes ˆ ˆ ˆ aˆ ... for and early latin letters Ž a,b,... for spacetime vectors transverse to them. We use hatted letters Ž M,m,a, all the inertial frame indices and finally middle latin letters Ž i, j,... for worldvolume vectors.

h

2. The superalgebras from the M-2-brane in M-brane backgrounds In this section we will deal with the M-2-brane in M-brane backgrounds. 2.1. The M-2-brane in an M-2-brane background At first we will begin with the case of the test M-2-brane floating in an M-2-brane background. The M-2-brane action in a D s 11 supergravity background is w29x SM 2 s yT d j 3 y det g i j q T d j 3

H (

H

1 3!

´ i jk CiŽ3. jk

Ž 2.1 .

where g i j s Eimˆ Ejnˆhmˆ nˆ is the induced worldvolume metric and CiŽ3. jk is the worldvolume 3-form induced by the ˆ ˆ superspace 3-form gauge potential. EiA s E i Z M EMA where EMA is the supervielbein. Note that at this time the action is invariant under local super-transformation. Let’s fix the background to an M-2-brane solution given by w30x 2 ds11 s Hy2 r3hmn dx mdx n q H 1r3dy ady bd a b

Cm1m 2m 3 s

´m1m 2m 3 det gmn

Hy1 ,

Ž the others. s 0

Ž 2.2 .

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where hmn is the 3-dimensional Minkovski metric with coordinates x m and H is a harmonic function on the transverse 8-space with coordinates y a, that is, H s 1 q q2 where y s y a y bd a b and q2 is a constant.

´m1m 2m 3 s gm 1n 1 gm 2n 2 gm 3n 3 ´

n 1n 2 n 3

and ´

012

y6

(

s 1. This background admits a Killing spinor ´ which satisfies

dcm s Em q 14 v mrˆ sˆGrˆ sˆ q Tmn1 n 2 n 3 n 4 Fn1 n 2 n 3 n 4 ´ s 0

ž

Ž 2.3 .

/

1 Ž Gmn1 n 2 n 3 n 4 q 8 G w n1 n 2 n 3dmn 4 x .. Then the Killing spinor has the form ´ s Hy1r6´ 0 where where Tmn1 n 2 n 3 n 4 s y 288 ´ 0 has the positive worldvolume chirality, i.e. G´ 0 ' G 012 ˆˆ ˆ ´ 0 s q´ 0 . T Since G satisfies G T s G and G 2 s 1, both 1 "2 G and 1 "2G are projection operators. So, if we denote 1 "2 G z " as z for a spinor z , the background is invariant under the transformation generated by the supercharge Qq and so is the system because the negligibly light test brane is assumed not to affect the background geometry and its configuration is not fixed yet at this moment. On the other hand, the background and the brane action are not invariant under the transformation by Qy, which means that we should set the corresponding transformation parameter ´y to be zero. Then, the conjugate momentum Py of uy does not appear in the Noether charge Qq only whose algebra we are interested in. Therefore, the terms including uy neÕer contribute to the central charges at zeroth order in u . Thus, we can set from the beginning

uys 0.

Ž 2.4 .

From now on, we will use these freely in all the cases we treat in this paper. Related with this, we exhibit the properties of G : C x s  G , Gaˆ 4 s 0. G , Gmˆ s w G ,C

Ž 2.5 .

Now, we are prepared to get the explicit representations of the superfields and the super-coordinate transformation in terms of component fields to low orders in u . By substituting Ž2.2. to the usual expressions w31x and using Ž2.4. and Ž2.5. we see that only the Eaaˆ has the nontrivial contribution from the background. ˆ ˆ From the results the superspace 1-form on the inertial frame E A s dZ M EMA is given by 5 E mˆ s dx n Hy1r3dnmˆ y i uq G mˆ duqq O Ž u 4 . E aˆ s dy b H 1r6d baˆ q O Ž u 4 . E aˆ s du aˆ qq 16 Hy1dHu aˆ qq O Ž u 3 . . Aˆ

Ž 2.6 . Aˆ

Since the 1-form E has no superspace Žcurved. indices, E , and hence the Nambu-Goto action, are invariant under the super-coordinate transformation w31x d Z M s J M in this background given by

J m s i ´q G muqq O Ž u 3 . J as0qO Žu 3. J a s ´ aqq O Ž u 2 . .

Ž 2.7 . Aˆ

We can easily check the invariance of E explicitly up to second order in u . Note that the coordinates y a transverse to the background brane are not transformed Žat least up to the second order in u ., i.e. this is the supertranslation parallel to the background brane. ŽThus, we can define the corresponding Noether supercharge..

5

In fact we need to know the Žvanishing of the. contribution from Emnˆ at order u 2 . We can infer its vanishing in this specific simple background, but the expression of Emnˆ at order u 2 in general background was obtained recently w32x, by which our inference is confirmed.

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And it is also to be noted that G m s H 1r3G nˆdnˆm, i.e. the gamma matrices with the spacetime indices depend on the harmonic function. The remaining field is the super-3-form gauge potential. It is introduced by the gauge invariant 4-form field strength w31,33x i

R Ž4. ' dC Ž3. s

2

1

ˆ

E mˆ E nˆ E aˆ Ž Gnˆ mˆ . aˆ bˆ E b q

4!

$ $ $ $

Em 1 Em 2 Em 3 Em 4 F$$$$ m 4m 3m 2m1 ,

Ž 2.8 .

where F $ $ $ $ is the bosonic field strength which is in this case associated with the electric M-2-brane m 4 m 3 m 2 m1 background. From Ž2.8. we get 6 C Ž3. s

1

Hy1 Ž yemnr . dx m dx n dx r y

3!

i 2

q Hy2r3 dx rdrmˆ dx sdsnˆ uq Gmn ˆ ˆ du

i ˆ y H 1r3dx cdcaˆdx dd dbuq Gaˆ bˆ duqq O Ž u 4 . 2 and hence the supertransformation of C Ž3. :

Ž e 012 s y1. ,

Ž 2.9 .

7

i d C Ž3. s d y H 1r3dy cdcaˆdy dd dbˆ´q Gaˆ bˆ uqq O Ž u 3 . ' d Ž ´q D2 . . 2

ž

/

Ž 2.10 .

Thus, the M-2-brane action Ž2.1. is invariant under Ž2.7. up to total derivative, and we define the Noether supercharge Qaq in the Hamiltonian formulation as an integral over the test brane at fixed time M2 , given by w10x Qaq' QaqŽ0. y i

HM Ž CD . 2

2

as

2

q a y Pm

HM d j Ž i P

Ž CG muq . a . y 12 TH dy ady b Ž CGa b uq . a q O Ž u 3 . M2

2

Ž 2.11 . where Pm and Paq are the conjugate momenta of x m and uq, respectively, and QaqŽ0 . is the contribution from the Nambu-Goto action whose form is almost common to all the p-brane. In this way we get the superalgebra of Qaq :

 Qaq,Qbq 4 s 2H

M2

d 2jPm Ž CG m . ab q 2 P 12 T

a

HM dy dy

b

Ž CGab . ab q O Ž u 2 . .

Ž 2.12 .

2

Before discussing this result, we give the explicit expression of Pm :

Pm s

d L Ž0. d x˙

m

T q 2

Ž3. ´ 0 i jE i x n E j x r Cmnr q O Ž u 2 . ' PmŽ0. q PmW Z

Ž 2.13 .

where L Ž0. is the Nambu-Goto Lagrangian.

6 Although the aˆ of u aˆ q is the index of the inertial frame, u aˆ s u bdbaˆ q O Ž u 3 .. So, we need not distinguish the two indices in this paper. 7 Throughout this paper we make an assumption that all the fermionic Žbut not bosonic. cocycles in the superspaces are trivial. Then, the invariance of R Ž4. under the super-transformation means that d C Ž3. in any of the supersymmetric backgrounds under the assumption can be written as certain d-exact forms to full order in u .

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We now discuss the implications of this algebra. First, if the test brane is oriented parallel to the background brane, the term like a central charge arises from the PmW Z , although the original central charge vanishes. If we choose the static gauge E 0 x m s d 0m and E i x 0 s d i 0 , PmŽ0. and PmW Z are obtained respectively as 2

Ž0. < m sT

HM d jP 2

2

2

2<

Hy1dm0 q O Ž u 2 .

Ž 2.14 .

1

2

Hy1dm0 q O Ž u 2 . .

Ž 2.15 .

2

WZ sT m

HM d jP

1

HM dx dx

HM dx dx 2

The symbol of the absolute value in P 0Ž0. is due to the Jacobian originated from the determinant in L Ž0.. Thus, we conclude that the parallel configuration with a certain orientation of the test brane has the 1r2 spacetime supersymmetry and the one with the other orientation breaks all the supersymmetry. Note that Ž2.14. and Ž2.15. are invariant under the 12-plane rotation and hence the discussion above holds, as it should be. Furthermore, even in the case that all the supersymmetry is broken, 1r2 supersymmetry is restored in the limit y ™ 0 Ži.e. H ™ `.. The reason is as follows: since P 0 A Hy1 and G m A H 1r3, the r.h.s. of the superalgebra Ž2.12. is proportional to Hy2 r3 , hence vanishes in the limit. In fact this restoration is reasonable because both of the forces via graviton and anti-symmetic tensor are attractive in this case and the potential energy is formally minimized in this ‘‘absorption’’ limit. This is the M-2-branerM-2-brane bound state preserving 1r2 supersymmetry. On the other hand, if the test brane is oriented orthogonally to the background brane, the central charge does have the nonzero value. In the static gauge with the test brane to be fixed, for example, to 34-plane, the algebra becomes

 Qaq,Qbq 4 s 2TH

M2

d 2j H 1r3 Ž 1 y G 34 ˆ ˆ . ab ,

Ž 2.16 .

which means that 1r4 spacetime supersymmetry is preserved in this configuration. We note that the r.h.s. of the superalgebra does not vanish completely in any limit if the test brane has at least one coordinate transverse to the background brane. This fact is common to all the other cases. The reason is the following: suppose that one of the worldvolume coordinates of the test brane j 1 equals to the transverse q coordinate yh. Then, H is expressed as H s 1 q where aX s 3,..,9. Thus, Hd 2j H K does not X

X

ŽŽ j 1 . 2 q Ž y a . 2 . 3

a .2

vanish for any constant K, even in the limit of Ž y ™ 0. In other words, the r.h.s. of the superalgebras do not vanish completely for the cases of the string intersection and the orthogonal orientation. Thus, we have derived the superalgebra from the M-2-brane action in an M-2-brane background. All the 1r4-supersymmetric intersections and a 1r2-supersymmetric bound state of two M-2-branes known before w27,34x have been deduced from the algebra. 2.2. The M-2-brane in an M-5-brane background Next, we will consider the M-2-brane in an M-5-brane background. The M-5-brane background solution is given by w35x 2 ds11 s Hy1 r3hmn dx mdx n q H 2r3dy ady bd a b

Fa b c d s y´ a b c d e Ee H

Ž 2.17 .

h

where m s 0,1,..,5 and a s 6,..,9, . The Killing spinor ´ has the form ´ s Hy1 r12´ 0 where ´ 0 has again the

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positive chirality of the worldvolume of the background: G X´ 0 ' G 012345 ˆˆ ˆ ˆ ˆ ˆ ´ 0 s q´ 0 . Since q

X 1" G ŽT .

are again

2

projection operators, only the supersymmetry corresponding to Q is the symmetry of the background. Thus, for the same reason stated in the case of the M-2-brane background, only Qq is the symmetry of the system and C 4 s  G X , Gmˆ 4 s w G X , Gaˆ x we set ´ys 0 and hence uys 0. We note that G X satisfies the Žanti-.commutators  G X ,C s 0. By using this relations and the formula presented in Ref. w31x the superspace 1-form on the inertial frame is given by E mˆ s dx n Hy1r6dnmˆ y i uq G mˆ duqq O Ž u 4 . E aˆ s dy b H 1r3d baˆ q O Ž u 4 . E aˆ s du aˆ qq 121 Hy1 dHu aˆ qq O Ž u 3 . .

Ž 2.18 .

The super-coordinate transformation is formally the same form as that in the M-2-brane background Ž2.7. except for the ranges of m and a. The superspace 3-form C Ž3. is introduced in the same way as Ž2.8.. Note that C Ž3. cannot be expressed globally in this case because it is associated with the magnetic M-5-brane solution. However, neither does it contribute to d L W Z nor the Pm up to first order in u , owing to the inertness of the transverse coordinates y a under the super-transformation. As a result,

d C Ž3. s d yiH 1r6 dx ndnmˆ dy bd baˆ´q Gmˆ aˆ uqq O Ž u 3 . ' d Ž ´q DX2 . .

ž

Ž 2.19 .

/

Thus, the action is invariant under the super-transformation up to total derivative. After the same procedures as in the M-2-brane background, we get the expression of the supercharge Qaq' QaqŽ0. y iHM 2Ž CDX2 .a . Then, the superalgebra is obtained as

 Qaq,Qbq 4 s 2H

M2

d 2jPm Ž CG m . ab q 2 P T

HM dx

m

dy a Ž CGm a . a b q O Ž u 2 . .

Ž 2.20 .

2

We conclude from the form of its central charge that the string intersection of the test brane with the background is the only 1r4-supersymmetric configuration permitted in this background. This is consistent with Ref. w27,34x, too. In this case the interpretation of the ‘‘boundary’’ of the test M-2-brane is as follows, as given in Ref. w36x: if we choose the gauge j 1 s x 1 and j 2 s yh, the intersection on the hypersurface y ' y a y bd a b s 0, Ži.e., j 2 s 0. does not correspond to any points of the M-2-brane because the proper distance on it is infinite. So, the ‘‘edge’’ of the M-2-brane is interpreted to disappear down the infinite M-5-brane throat. In other words the test M-brane has no boundary in this method. 8 On the other hand, if the test M-2-brane is parallel to any two-dimensional subspaces of the worldvolume of the background M-5-brane, all the supersymmetry is broken. However, 1r2 spacetime supersymmetry is restored in the limit y ™ 0 Ži.e. H ™ `. since P 0 CG 0 A Hy1r3. Thus, we can deduce the M-2-branerM-5-brane bound state with 1r2 spacetime supersymmetry given in Ref. w28,9x.

(

3. The superalgebras from the M-5-brane in M-brane backgrounds In this section we discuss the test M-5-brane in M-brane backgrounds. There are two new features which do not emerge in the previous cases of the test M-2-brane: one is the fact that the M-5-brane action contains worldvolume self-dual 2-form gauge potential A2 in addition to the usual scalar fields w37,38x. The super-trans-

8 If we set the worldvolume coordinate j 2 to take values in the open interval Ž0,`., it is interpreted as the M-2-brane ending on the M-5-brane w36x, on so large scales compared to that determined by M-5-brane tension that the background solution can be replaced with the M-brane source.

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formation of A2 is determined by the requirement of the invariance of the ‘‘modified’’ field strength w9x given by H s d A2 y C Ž3. .

Ž 3.1 .

The other is the introduction of the superspace 6-form field strength C Ž6. w39x whose field strength takes the form R Ž7. ' dC Ž6. y 12 C Ž3. R Ž4. s

i 5!

$

$

$ Em 1 ... Em 5 E aˆ Ž G$ m 5 .. m 1 .

ab ˆˆ

ˆ

Ebq

1 7!

$

$

Ž7 .

$ Em 1 ... Em 7 F$ m 7 .. m 1

Ž 3.2 .

where the 7-form F Ž7. is the Hodge dual of the bosonic 4-form field strength. 3.1. The M-5-brane in the M-2-brane background First, we will consider the test M-5-brane in the M-2-brane background Ž2.2.. This set-up might seem to be unreasonable because M-5-branes cannot have the boundary w36x. However, test branes have no boundary in this method as stated above in Ž2b. case. Thus, we can compute the superalgebra and discuss supersymmetric configurations in this case in the same way. The M-5-brane action is w37x SM 5 s yT d 6j

H

(y det ž g q H˜ / q 4'ŽyE ag. Ž E a. Ž H . ij

ij

2

) i jk

i

Hjk l Ž E l a . q

HŽ C

Ž6.

q 12 H C Ž3. .

Ž 3.3 .

X X X

j k 1 Ž H . ) i jkE k a and a is an auxiliary worldvolume scalar HiX jX k X , H˜ i j s ' 'yŽ E a.2 field. Since the superspace 1-form and the Žtransformation of . C Ž3. in the M-2-brane background are already given by Ž2.6. and Ž2.9., the transformation of A2 is

where Ž H . ) i jk s

´ i jk i

1

3! y g

i d A2 s y H 1r3dy cdcaˆdy dd dbˆ´q Gaˆ bˆ uq Ž ' ´q D2 . 2

Ž 3.4 .

and the transformation of C Ž6. is deduced in the same way as the 3-form C Ž3.. As a result, we get d L W Z ' dŽ ´q D . where 9

Da s y

i 4!

$

$

a

$ q H 1r3dx ndnm dy b 1dba11 ... dy b 4 dba 44 Ž G$ a 4 .. a1 m u . y

i 4

$

$

a

q H 1r3dy b 1dba11 dy b 2 dba 22 d A2 Ž G$$ a 2 a1 u . .

Ž 3.5 .

The supercharge is given as before by Qaq' QaqŽ0. y iHM 5Ž CD .a where QaqŽ0 . takes the form w9x QaqŽ0 . s

5

HM d j 5

i Paqy Pm Ž CG muq . a q

i 2

Pi j Ž C Ž D2 . i j . a

Ž 3.6 .

where i is the space index of the test M-5-brane worldvolume and Pm , Paq and P i j are the conjugate momenta of x m, uq and A i j, respectively.

9

Because of the invariance of R Ž7. under the super-transformation, d C Ž6. y 12 C Ž3.d C Ž3. in the background can be written as a d-exact form dŽ ´qD5 .. Then, it is shown that it holds d L W Z s dŽ ´qD5 y 12 d A2 d A2 . to full order in u .

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Thus, the superalgebra is obtained as

 Qaq,Qbq 4 s 2H

d 5jPm Ž CG m . ab q 2

5

i

ij

HM d j 2 P H

M5

Eiy cdcaˆE jy dd dbˆ Ž CGaˆ bˆ . ab

1r3

5

1

y2P

4!

T

HM dx

m

dy a1 ... dy a 4 Ž CGm a1 .. a 4 . ab y 2 P 14 T

a

b

HM dy dy d A Ž CG

5

2

a b ab q O

.

Žu 2.

5

Ž 3.7 . The third term in Ž3.7. means the string intersection with the M-2-brane background leads to the preservation of 1r4 supersymmetry, which is again consistent with Refs. w27,34x and the result of Ž2b. case. And if we choose the temporal gauge aŽ j . s t, the second term including P i j turns to be the same as the last term as in Ref. w9x. Thus, these terms imply the possibility of the triple intersection with the configuration background M-2 test M-5 M-2

1 y y

2 2 y

y 3 3

y 4 4

y 5 y

y 6 y

y y y

y y y

y y y

y y y

where the third M-2 brane is within the M-5-brane Žto form the bound state. w9x. This configuration would preserve 1r4 spacetime supersymmetry since CG 23456 ˆ ˆ ˆ ˆ ˆ anticommutes with CG 34 ˆ ˆ . We note that the possibility of this triple intersection is discussed directly on the basis of the spacetime superalgebra, by making use of the fact that there exist an M-brane as a background from the beginning. Next, we will consider the M-2-branerM-5-brane bound state. Suppose j 1 s x 1, j 2 s x 2 Žthat is, a two-dimensional worldvolume subspace of the test M-5-brane is parallel to the background M-2-brane.. Then, if A2 s 0, all the terms except P 0 CG 0 ŽA Hy1 r6 . in the r.h.s. of Ž3.7. vanish in the static gauge. However, since the harmonic function depends on the ‘‘relatively transverse’’ coordinates j 3, j 4 and j 5, Hd 5j H K do not go to zero even in the limit of vanishing distance. This means that we cannot deduce the M-2-branerM-5-brane bound state from the algebra at least straightforward. But if we consider the background whose harmonic function does not depend on the ‘‘relatively transverse’’ coordinates as in Ref. w27x, the r.h.s. of the algebra vanishes in the limit, and we can obtain the 1r2-supersymmetric M-2-branerM-5-brane bound state. 3.2. The M-5 brane in the M-5-brane background Finally we will deal with the test M-5 brane in the M-5-brane backgroundŽ2.17.. Since almost all the preparations have been already given above and the procedures are similar to before, we present only the transformation of A2 , the result of the superalgebra as well as its interpretations in this case. The transformation of A2 in the M-5 brane background is

d A2 s yiH 1r6 dx ndnmˆ dy bd baˆ´q Gmˆ aˆ uqq O Ž u 3 . .

Ž 3.8 .

The superalgebra is give by

 Qaq,Qbq 4 s 2H

M5

d 5jPm Ž CG m . ab q 22

5

ij

HM d j i P

H 1r6Eix ndnmˆE jy bd baˆ Ž CGmˆ aˆ . ab

5

y 122 T

HM dx

m1

... dx m 3 dy a1 dy a 2 Ž CGm1 .. m 3 a1 a 2 . ab y

5

y 22 T

a

HM dy dx 5

m

d A2 Ž CGm a . ab q O Ž u 2 . .

2 4!

T

HM dx

m

dy a1 ... dy a 4 Ž CGm a1 .. a 4 . a b

5

Ž 3.9 .

T. Sato r Physics Letters B 439 (1998) 12–22

21

In fact only the fourth term is difficult to derive straightforward in this case because the magnetic 3-form potential which cannot be globally expressed contributes to d L W Z. However, since they can be expressed locally in a certain gauge, we can confirm in a gauge that some of these terms do not vanish, although we do not exhibit the computation because is very primitive and awkward to present. Then, we reproduce the term on the ground that it should be gauge invariant and Lorentz ŽSOŽ1,5. = SOŽ5.. covariant. Now, we will interpret the result. If the test M-5-brane is oriented parallel to the background M-5-brane, perfectly the same circumstances occur as in the case of the M-2-brane parallel to the background M-2-brane. So, we present only the result, without repeating the explanations. The configuration of two parallel M-5-branes and an M-5-branerM-5-brane bound state, both of which preserve 1r2 spacetime supersymmetry, are deduced. If the test 5-brane intersects on 3-brane with the background M-5-brane, 1r4 spacetime supersymmetry is preserved because of the third term Žas in Ref. w27x.. The fourth term shows that the string intersection of the two M-5-brane leads to the preservation of 1r4 supersymmetry w28x. The last term, together with the second term with the temporal gauge condition aŽ j . s t, implies the possibility of the existence of the triple intersections preserving 1r4 supersymmetry with the following configurations: background M-5 test M-5 M-2

1 1 y

2 2 y

3 3 3

background M-5 test M-5 M-2

1 y y

2 y y

3 y y

4 y y

5 y y

y 6 6

y 7 y

y y y

y y y

y y y

5 5 5

y 6 6

y 7 y

y 8 y

y 9 y

y y y

and 4 y y

where each of the third M-2 branes is within the test M-5-brane.

4. Discussion In summary we have discussed the method of computing explicitly the spacetime superalgebras from the test M-brane actions in M-brane backgrounds to the lowest order in u . As the consequences we have deriÕed all the 1r4-supersymmetric intersections of the two M-branes known before from the central charges of the spacetime superalgebras, as the supersymmetric ‘‘gauge fixing’’ of the test brane permitted in the background. We have also deduced some of 1r2 supersymmetric bound states of two M-branes from examining the behavior of the harmonic functions in the limit of vanishing distances. In addition, the possibilities of some triple intersections preserving 1r4 supersymmetry have been discussed. In order to obtain the 1r2 supersymmetric bound state ŽM-2-brane within M-5-brane., we need to assume only in the case of the M-5-brane in the M-2-brane background that the hamonic function is independent of the ‘‘relatively transverse coordinates. Thus, when we deal with the system composed of a p-brane and a q-brane with the inequality q G p, it is more suitable for this method to discuss the system as the p-brane in the q-brane background. Except for this subtlety, the method discussed in this paper is confirmed to be consistent with those presented before, and hence reliable. Having confirmed its reliability, we will apply this method to the p-branes in 10-dimensional massiÕe IIA backgrounds w40x as stated in the introduction, in which case the background haÕe to be nontriÕial w18,19x. It will also be interesting to apply this to the cases of other backgrounds w41x, or use it to the new supersymmetic invariant p-brane action presented recently w24x.

T. Sato r Physics Letters B 439 (1998) 12–22

22

Acknowledgements I would like to thank Prof. J. Arafune for careful reading of the manuscript and useful comments. I am grateful to Akira Matsuda for many useful discussions and encouragement. I also would like to thank Taro Tani for stimulating discussions.

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x w34x w35x w36x w37x w38x w39x w40x w41x w42x

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