Supersymmetric D0-branes in curved backgrounds

16 September 1999

Physics Letters B 462 Ž1999. 265–270

Supersymmetric D0-branes in curved backgrounds Ali H. Chamseddine

1

Center for AdÕanced Mathematical Sciences and Physics Department, American UniÕersity of Beirut, Beirut, Lebanon Received 12 July 1999; accepted 31 July 1999 Editor: R. Gatto

Abstract An action for supersymmetric D0-branes in curved backgrounds is obtained by dimensional reduction of N s 1 ten-dimensional supergravity coupled to super Yang-Mills system to 0 q 1 dimensions. The resultant action exhibits the SO Ž9 ,9q n . coset-space symmetry = UŽ1. where n s N 2 y 1 is the dimension of the SUŽ N . gauge group. q 1999 SO Ž9 .= SO Ž9q n .

Published by Elsevier Science B.V. All rights reserved.

1. Introduction Very little is known about D-branes w1–3x, in curved backgrounds w4x and what restrictions, if any, should be imposed on such backgrounds. Supersymmetry imposes constraints on the background metric. In the case of the superstring with world sheet supersymmetry, there is a direct relation between the number of supersymmetries and the background metric w5x. In flat background geometry the D0-brane action has 16 space-time supersymmetries w6x, and it is natural to ask for the type of curved backgrounds compatible with this symmetry. The action with 8 space-time supersymmetries Ž N s 2 in 4 dimensions. was shown to correspond to Kahler backgrounds w7x. ¨ In general it is difficult to construct such actions without having determined the underlying symmetry. There are few possible routes to handle this problem, the most obvious one is to quantize the D-brane

1

E-mail: [email protected]

action in the presence of a general superspace metric, thus keeping all supersymmetries, and to determine what are the required constraints on such a metric w8,9x. This is expected to be extremely complicated and only recently some work has been done in this direction w10x. The other possibility is to determine the necessary fields to make a supersymmetric multiplet and to find the corresponding invariant action under such transformations. We shall follow a simpler approach, which is straightforward but which the drawback that the underlying symmetry is not manifest. The idea is based on the observation that the supersymmetric D0-brane action was obtained by dimensionally reducing the super Yang-Mills action from ten to 0 q 1 dimensions w9x. The most general supersymmetric interaction in ten dimensions with N s 1 supersymmetry is that of supergravity coupled to super Yang-Mills with an arbitrary gauge group w11x. Dimensional reduction keeps maximal supersymmetry and rearranges the scalar fields to have the action of a non-linear sigma model on a coset space. Reducing to 0 q 1 dimensions have the peculiarity that there is no gravitational part and the only fields

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 9 3 8 - 7

A.H. Chamseddiner Physics Letters B 462 (1999) 265–270

266

1

coming from the gravity sector in ten-dimensions yields scalar fields. Similarly the vector fields coming from the Yang-Mills part give scalar fields taking values in the adjoint representation of the gauge group. As with compactification of supergravity theory it is expected that the coset space to be of the form w12x

q

2k 2

i 1 q xG M DM x q '2 E M fcM x 2 q ey2 f FMX N P FQX R S g M Q g N R g P S ik

SO Ž 9,9 q n . SO Ž 9 . = SO Ž 9 q n .

q

= U Ž 1.

24

where n s N 2 y 1 is the dimension of SUŽ N . gauge group. The absence of the gravitational sector in 0 q 1 dimensions also enables us to identify the gravitational coupling in higher dimensions with the string tension to insure that all rescaled scalar fields have the same dimensions and could serve as coordinates of the D-brane. The aim of this letter is to derive the D0-brane action with the most general curved background compatible with maximal space-time supersymmetry. This is done by compactifying the ten-dimensional theory and grouping all the resultant fields. The result obtained does not have manifest symmetry. Nonetheless, this suggests that a direct derivation in terms of supermultiplets, where some auxiliary fields as well as constrained variables are used, might drastically simplify the answer. This is the situation encountered in the derivation of the four-dimensional N s 4 supersymmetric action where superconformal methods were used to simplify the analysis w13x. This more systematic approach will be left for the future and our study here will be limited to the action obtained by dimensional reduction. The plan of this paper is as follows. In Section 2 we derive the bosonic dimensionally reduced action and in Section 3 we give the fermionic part. Section four includes comments on the results.

E M fE N f g M N

eyfc M Ž G

M NP Q R

y6 g M PG Q g R N . c N FPX Q R y 14 eyf Tr Ž GM N GP Q . g M P g N Q i q Tr Ž lG M DM l . 2 ik y

2'2

Tr Ž lG MG

NP

c M GN P .

where FMX N P is the field strength of the antisymmetric tensor BM N modified by the gauge Chern-Simons three form, FMX N P s FM N P q v MŽCS. NP and FM N P s k3 E w M BN P x , while 2 v MŽCS. N P s 6 k Tr A w M E N A P x q 3 Aw M A N A

ž

Px

/

We can reduce this action from 10 to d dimensions with the following distribution of fields. The metric g M N gives a metric gmn , m vectors and 12 m Ž m q 1 . scalars in d dimensions, where m s 10 y d. The antisymmetric tensor BM N gives Bmn , m vectors and 1 i i 2 m Ž m y 1 . scalars. The gauge fields A M give Am , i and nm scalars A m , where n is the dimension of the gauge group. All in all we will have m Ž n q m . scalars which will span the coset space w12x SO Ž m,n q m .

2. Bosonic action of curved D0-branes Our starting point is the N s 1 supergravity Lagrangian in ten dimensions coupled to super YangMills system. This is given by Žup to quartic fermionic terms. w11x det Ž e MA .

y1

Lsy

1 4k

2

RŽ v . y

i 2

c M G M N P DN c P

SO Ž m . = SO Ž n q m . The case when d s 4 Ži.e. m s 6. is well established w11x. The case we are interested in have m s 9. To reduce this action to 0 q 1 dimensions we decompose: e MA s

ž

e 00˙ 0

B0˙a e ma

/

A.H. Chamseddiner Physics Letters B 462 (1999) 265–270

The inverse metric is ˙

e 00 0

ž

e AM s

e 0m e am

Grouping all terms together we obtain the bosonic part of the D0-brane action:

/

ey1 L b s y1

˙

˙

where e 00 s Ž e 00˙ . , e am e mb s d ab , e 0m s ye00 B0˙a e am . To evaluate y 14 det Ž e MA . RŽ v . we have w14x 1 4

det Ž e MA . R Ž v . s 161 e Ž V A B C V A B C y 2 V A B C V C A B y4V C A AV C BB .

C. where V A B C s ye AM e BN Ž E M e NC y E N e M . The only non-vanishing V A B C is

V 0 b c s ye00˙ e bn E 0˙ e n c

1 2 y E 0˙ g m n E 0˙ g m n y 2 Ž E 0˙ ln e . 8 2

ž

/

where e s detŽ e ma .. The antisymmetric tensor piece gives e 12

e 00˙ ey2 f Ž 3FaX b 0 FaX b 0 y FaX b c FaX b c .

where we are raising and lowering tangent space indices with the euclidean metric d a b , and FaX b c s 4k e am e bn e cp Tr Ž A w m A n A X

Fa b 0 s

1

k

˙ e 0 y 161 E 0˙ g m n E 0˙ g m n y 41 Ž E 0˙ ln e . k2 0 1 q E 0˙ fE 0˙ f q 14 ey2 f g m p g n q D 0˙ Bm n D 0˙ Bp q 2 k 2 yf m n q e g Tr Ž D 0˙ A m D 0˙ A n . 2 2

žž

/

ž

qe00˙ y

4k 2 3

ey2 f g m q g n r g p s

=Tr Ž A w m A n A p x . Tr Ž A w q A r A s x .

=Tr Ž w A m , A n x . Tr Ž A q , A r

y 14 det Ž e MA . R Ž v . e

1

y 14 eyf g m q g n r

Substituting this gives

s

267

px

.

.

//

where D 0˙ Bm n s E 0˙ Bm n y k 2 TrŽ A m D 0˙ A n y A n = D 0˙ A m .. To get the correct dimensions we identify the gravitational coupling k with the string tension a X and redefine the gauge fields Aim s a1 X mi , thus identifying them with the D0-brane coordinates. Multiplying the Lagrangian with an overall factor of Ž a X . 2 gives the bosonic part of the D0-brane action. The 82 fields g m n , Bm n and f are needed, beside the coordinates X mi , to provide coordinates for a D0brane action with a curved background. The rescaled bosonic Lagrangian becomes X

˙

žž

ey1 L b s e 00 y 161 E 0˙ g m n E 0˙ g m n y 14 Ž E 0˙ ln e .

e am e bn

= Ž E 0˙ Bm n y k 2 Tr Ž A m D 0˙ A n y A n D 0˙ A m . .

q 12 E 0˙ fE 0˙ f q 14 ey2 f g m p g n q D 0˙ Bm n D 0˙ Bp q

k2

ey f g m n Tr Ž D 0˙ X m D 0˙ X n .

/

where D 0˙ A m s E 0˙ A m q AX0˙ , A m , and AX0˙ s A 0˙ y B0˙mA m . The redefinition of A 0˙ will insure that the field B0˙m will not appear in the action and therefore is irrelevant. The Yang-Mills part is

q

y 14 det Ž e MA . ey f Tr Ž GM N GP Q g M N g P Q .

=Tr Ž X w m X n X p x . Tr Ž X w q X r X s x .

e s y e00˙ ey f Tr Ž Ga b Ga b y 2Ga0 Ga0 . 4

2

ž

qe00˙ y

4 3Ž a X .

1 y 4Ž a X .

2

2

ey2 f g m q g n r g p s

eyf g m q g n r

where Ga b s e am e bn w A m , A n x ,

˙

Ga0 s ye am e 00 D 0˙ A m

2

=Tr Ž w X m , X n x . Tr Ž X q , X r

.

/

A.H. Chamseddiner Physics Letters B 462 (1999) 265–270

268

The scalar fields can be regrouped into a set X mR where R s 1, PPP ,9 q n plus an additional scalar field as a combination of the fields g m n , Bm n , f and X mi . Can one take the limit where g m n s dm n , Bm n s 0 and f s 0 ? This gives the flat background D0brane action plus the order 6 terms in X mi .This is usually incompatible with supersymmetry as we shall show later. The proper limit to flat backgrounds can be obtained by keeping the couplings k and a X independent, then taking the limit k ™ 0. The transformation law for e 00˙ with respect to time transformation is given by

d e 00˙ s E0˙ Ž j 0˙ e 00˙ . which would allow us to set e 00˙ s 1. In this action the coset space symmetry is not manifest. The coset space metric is a non-polynomial function of g m n , Bm n , X mi and f . To obtain manifest symmetry, one method would be to start with the symmetry SO Ž9,9 q n. using supersymmetric multiplets, and then gauge the SO Ž9. = SO Ž9 q n. subgroup. This will be the topic of a forthcoming project where a systematic analysis of all possible background symmetries would be carried out.

does not appear in the action. The nonvanishing components of the spin-connection are

v 0 b c s 12 e 00˙ Ž e bn E 0˙ e n c y e cnE 0˙ e n b . v a b 0 s y 12 e 00˙ Ž e bn E 0˙ e n a q e an E 0˙ e n b . The term 2i det Ž e MA . xG A DA x reduces to i y e Ž e 00˙ xG 0x Ž e an E 0˙ e n a . 4 y2 xG

i y eTr Ž e 00˙ lG 0l Ž e an E 0˙ e n a . 4 y2 lG

ik 2

y eŽ 8

Ž

y2 c 0 G

c

cc e dn E 0˙ e n d

Ž

ž

0bcde

cb

ž

DB cC

ca G b cd e bn E 0˙ e n d q e dn E 0˙ e n b

ey f ee00˙ 4 Ž ca G a b c d ec b q 6 c 0 G

q 3ca G a b c d 0c b y 2 c 0 Gc cd q 2 cc G 0 cd ˙

y2 cc Gd c 0 e cm e dn e 00

/

where cA s e AMc M , and DM c N s Ž E M q 14 v M A BGA B . =c N , gives upon compactification i

Ž D 0˙ q 14 e bn E 0˙ e n c G b c . l .

y6 cc Gd ce . Tr A w c A d A ex

The Rarita-Schwinger term A BC

0

Next, the fermi-bose interaction cc F X gives

3. The fermionic action

y det Ž e MA . cA G 2

Ž E 0˙ q 14 e bn E 0˙ e n c G b c . x .

Similarly the gaugino kinetic term gives upon reduction

24

i

0

.

ž

1

k

E 0˙ Bm n y k Tr Ž A m D 0˙ A n y A n D 0˙ A m .

//

Finally the cxEf coupling gives

..

1 2

i q e Ž ca G acG 0 E0˙ cc q 14 ca G acGd e G 0 cc Ž e dn E 0˙ e n e . . 2 i ˙ q ee00 ca G ac 0 Ž e bn E 0˙ e n b . 4

ž

1 q ca G b c 0 Ž e bn E 0˙ e n a q e an E 0˙ e n b . 2

=

/

where we have used c 0 s e00˙ Ž c0˙ y B0˙a e amcm . and ca s e amcm . Again, the definition of c 0 insures that B0˙a

eE0˙ fc 0x .

The supersymmetry transformations in ten dimensions are

d e MA s yi keG Ac M k df s y

'2

ex

d BM N s k e f Ž i e G w M c

1

Nxq 2e

GM N x .

A.H. Chamseddiner Physics Letters B 462 (1999) 265–270

dc M s dx s

1

k i

2k

d AM s

DM e q 481 eyf Ž G

NP Q N PQ M q 9d M G

X

. e FN P Q

dx s

'2

q

1

G c 0˙ d A 0˙ s

a

G c 0˙

G cm

d Xm s

ex dl s

ž

d Bm n s k e f i e G w m c nx q dc 0 s

1

k

1 2'2

e Gm n x

/

˙

ž

1

k

E 0˙ Bm n

k y

/

Tr Ž X m D 0˙ X n y X n D 0˙ X m . e

2

Ž aX.

k q X

12 Ž a .

3

eyfG

=Tr Ž Xw m X n X

dca s

1

px

k

k 12 Ž a X .

3

G 0 e e bm e cn e dp

eyf Ž Ga b c d y 4d abGc d . px

. ˙

q 161 eyf Ž G 0 c d a q 34 G 0 c d da . e e cm e dn e 00

k y

Ž aX.

3

1

i X

a '2 iaX

'2 1 X

/

f

e 2 e G 0˙ l 1

f

e 2 e Gm l 1

˙

ey 2 f G a0e e am e 00 D 0˙ X m

SO Ž9 .= SO Ž9q n .

.

=e e bm e cn e dp Tr Ž Xw m X n X

3

The D0-brane action with maximal N s 16 space-time supersymmetry, derived here have the coset symmetry SO Ž9 ,9q n . which, however, is not

bcd

e 00 Ž e an E 0˙ e n b q e bn E 0˙ e n a . G b 0 e

y

Ž aX.

4. Comments

˙

4k

G a b 0e e am e bn e 00˙

a '2 From these transformations it should be clear that the truncation g m n s dm n , Bm n s 0, f s 0 is not consistent with supersymmetry because the fields X mi , g m n , Bm n and f are now mixed to form 9Ž9 q n. q 1 coordinates for the D0-brane. A proper way of going to the flat background limit is to keep k and a X distinct, and then take the limit k ™ 0.

e 00 Ž E 0˙ q 14 e amE 0˙ e m b G a b . e

q 121 eyfG c de e cm e dn

.q ' 4 2

=Tr Ž X m D 0˙ X n y X n D 0˙ X m .

a

'2

k

i px

E 0˙ Bm n y

0

k df s y

1

ž

=

The compactified supersymmetry transformations become after rescaling

d e 0˙a s yi ke

eyfG a b ce e am e bn e cp

=Tr Ž Xw m X n X

dl s ey 2 f G M Ne GM N

d e ma s yi e

3 3'2 Ž a . X

e feGM l

d e 00˙ s yi ke

G 0e e 00˙ E 0˙ f

k'2

ik

G Me E M f i

i

269

Tr Ž X m D 0˙ X n y X n D 0˙ X m .

/

ž

1

k

E 0˙ Bm n

manifest. The 81 fields which are not related to the SUŽ N . gauge group are essential to provide curvature for the background. We can say that curved backgrounds are only possible once n gauge fields are embedded into the above coset structure. One way to improve on this solution is to start with the light-cone formulation of the supermembrane in arbitrary background and find under what conditions the quantized action simplifies in such a manner as not to involve a square root. This is a difficult problem and the recent work of weakly coupling D0-branes to curved backgrounds may help to clarify the situation w10x. The action constructed in w10x contains higher order time derivatives indicating that higher deriva-

270

A.H. Chamseddiner Physics Letters B 462 (1999) 265–270

tive terms in ten-dimensional supergravity should also be included. It would also be very interesting to find out how the 82 scalar fields arise in this formulation. Another possibility is to study supersymmetry representations in 0 q 1 dimensions and form multiplets in complete analogy with the one obtained by superconformal methods in four dimensions. This would have the advantage of getting the coset space symmetry in a manifestly invariant way. In addition this would allow to investigate the general problem of finding the relation between the required degree of space-time supersymmetry and the nature of the curved background.

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