N=1, D=10 tensionless superbranes1

12 November 1998

Physics Letters B 440 Ž1998. 35–42

N s 1, D s 10 tensionless superbranes P. Bozhilov

1

2, 3

BogoliuboÕ Laboratory of Theoretical Physics, JINR, Dubna, Russia Received 7 July 1998 Editor: P.V. Landshoff

Abstract We consider a model for tensionless Žnull. super p-branes in the Hamiltonian approach and in the framework of a harmonic superspace. The obtained algebra of Lorentz-covariant, irreducible, first class constraints is such that the BRST charge corresponds to a first rank system. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction The tensionless Žnull. p-branes correspond to usual p-branes with their tension y Ž pq1 .r2

T s Ž 2pa X . taken to be zero. This relationship between null p-branes and the tensionful ones may be regarded as a generalization of the massless-massive particles correspondence. On the other hand, the limit T ™ 0 Žbecause of 2 Ž a X .y1 A M Planck . corresponds to the energetic scale E 4 M Planck . In other words, the null p-brane is the high energy limit of the tensionful one. There exist also an interpretation of the null and free p-branes as theories, corresponding to different vacuum states of a p-brane, interacting with a scalar field background w1x. So, one can consider the possibility of tension generation for null p-branes Žsee w2x and references therein.. Another viewpoint on the connection between null and tensionful p-branes is that the null one may be interpreted as a ‘‘free’’ theory opposed to the tensionful ‘‘interacting’’ theory w3x. All this explains the interest in considering null p-branes and their supersymmetric extensions. Models for tensionless p-branes with manifest supersymmetry are proposed in w4x. In w1x a twistor-like action is suggested, for null super-p-branes with N-extended global supersymmetry in four dimensional space-time. Then, in the framework of the Hamiltonian formalism, the initial algebra of first and second class constraints is converted into an algebra of first class effective constraints only. The obtained BRST charge corresponds to second rank theory. It is proven that there are no quantum anomalies when the so called ‘‘generalized’’ qp ˆˆ operator ordering is applied. In the recent work w5x, among other problems, the quantum constraint algebras of the usual and conformal tensionless spinning p-branes are considered. 1

Work supported in part by the National Science Foundation of Bulgaria under contract f-620r1996. E-mail: [email protected] 3 Permanent address: Dept. of Theoretical Physics, ‘‘Konstantin Preslavski’’ Univ. of Shoumen, 9700 Shoumen, Bulgaria. E-mail: [email protected] 2

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 1 0 7 5 - 2

P. BozhiloÕr Physics Letters B 440 (1998) 35–42

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In a previous paper w6x, we announced for a null super p-brane model which possesses the following classical constraint algebra

 T0 Žs1 . ,T0 Žs2 . 4 s 0 ,  T0 Žs1 . ,TaA Žs2 . 4 s 0 ,

 T0 Žs1 . ,Tj A Žs2 . 4 s T0 Žs1 . q T0 Žs2 . E j d p Žs1 ys2 . ,  Tj A Žs1 . ,TkB Žs2 . 4 s d A B d jl TkB Žs1 . q d kl Tj A Žs2 . E l d p Žs1 ys2 . ,  Tj A Žs1 . ,TaB Žs2 . 4 s d A B TaA Žs1 . q TaA Žs2 . E j d p Žs1 ys2 . ,  TaA Žs1 . ,TbB Žs2 . 4 s y2 i d A BPˆa b T0 Žs1 . d p Žs1 ys2 . , Pˆab s Pm samb . Ž 1. Here s s Ž s 1 , . . . , sp ., Ž j,k s 1, . . . , p ., Ž A, B s 1, . . . , N ., where N is the number of super-symmetries, a , b are spinor indices and Pm is a Lorentz vector Ž m s 0,1, . . . , D y 1.. In Ž1. and below we do not write explicitly the dependence of the quantities on the time parameter t . In this letter, we consider the particular case of N s 1, D s 10 tensionless superbrane. We work in the Hamiltonian approach and in the framework of a harmonic superspace. Passing to a system of Lorentz-covariant, irredusible first class constraints, we obtain a BRST charge as for a first rank theory. 2. Hamiltonian formulation Let us begin with writing the initial Hamiltonian of the dynamical system under consideration H0 s d ps m0 T0 q u j Tj q m a Da

H

H0 is a linear combination of the constraints T0 Žs.,Tj Žs. and Da Žs.. The latter are given by the expressions: T0 s pm pn h mn , diag Ž hmn . s Ž y,q, . . . ,q . , Tj s pn E j x n q pua E j u a , E j s ErEs j , Da s yipua y Ž puu . a . Ž 2. Here Ž x n , u a . are the superspace coordinates, u a is a left Majorana–Weyl spinor Ž a s 1, . . . ,16., pn , pua are the corresponding conjugated momenta, pu s pm s m, where s m are the 10-dimensional s-matrices Žfor our conventions see Appendix A.. H0 generalizes on the one hand the bosonic null p-brane Hamiltonian H B s d ps Ž m0 p 2 q m j pn E j x n . ,

H

and on the other - the N s 1 Brink–Shwarz superparticle with Hamiltonian H BS s m0 p 2 q m a Da . The constraints Ž2. satisfy the following Žequal t . Poisson bracket algebra  T0 Žs1 . ,T0 Žs2 . 4 s 0 ,  T0 Žs1 . , Da Ž s2 . 4 s 0 ,

 T0 Žs1 . ,Tj Žs2 . 4 s T0 Žs1 . q T0 Žs2 . E j d p Žs1 y s2 . ,  Tj Žs1 . ,Tk Žs2 . 4 s d jl Tk Žs1 . q d kl Tj Žs2 . E l d p Žs1 y s2 . ,  Tj Žs1 . , Da Žs2 . 4 s Da Žs1 . E j d p Žs1 y s2 . ,  Da Žs1 . , Db Žs2 . 4 s 2 ipua b d p Žs1 y s2 . , From the condition that the constraints must be maintained in time, i.e. w7x  T0 , H0 4 f 0,  Tj , H0 4 f 0,  Da , H0 4 f 0 , it follows that in the Hamiltonian H0 one has to include the constraints Ta s pua b D b

Ž 3.

P. BozhiloÕr Physics Letters B 440 (1998) 35–42

37

instead of Da . This is because the Hamiltonian has to be first class quantity, but Da are a mixture of first and second class constraints. Ta has the following non-zero Poisson brackets

 Tj Žs1 . ,Ta Žs2 . 4 s

Ta Ž s1 . q Ta Ž s2 . E j d p Ž s1 y s2 . ,

 Ta Žs1 . ,Tb Žs2 . 4 s 2 ipua b T0 d p Žs1 y s2 . .

In this form, our constraints are first class Žtheir algebra coincides with the algebra Ž1. for N s 1 and Pm s ypm . and the Dirac consistency conditions Ž3. Žwith Da replaced by Ta . are satisfied identically. However, one now encounters a new problem. The constraints T0 ,Tj and Ta are not irreducible, i.e. they are functionally dependent: a

Ž pu T . y D a T0 s 0 . It is known, that in this case after BRST-BFV quantization an infinite number of ghosts for ghosts appear, if one wants to preserve the manifest Lorentz invariance. The way out consists in the introduction of auxiliary variables, so that the mixture of first and second class constraints D a can be appropriately covariantly decomposed into first class constraints and second class ones. To this end, here we will use the auxiliary harmonic variables introduced in w8x and w9x. These are uma and Õa", where superscripts a s 1, . . . ,8 and " transform under the ‘‘internal’’ groups SO Ž8. and SO Ž1,1. respectively. The just introduced variables are constrained by the following orthogonality conditions uma u bm s C a b ,

um"u a m s 0 ,

umquym s y1 ,

where um"s Õa"smab Õb" , C ab is the invariant metric tensor in the relevant representation space of SO Ž8. and Ž u ". 2 s 0 as a consequence of the Fierz identity for the 10-dimensional s-matrices. We note that una and Õa" do not depend on s. Now we have to ensure that our dynamical system does not depend on arbitrary rotations of the auxiliary variables Ž uma ,um" .. It can be done by introduction of first class constraints, which generate these transformations 1 y ab y I a b s y Ž una pubn y unb puan q 12 Õq s a b pq pÕ . , Õ q 2Õ s

s a b s uma unb s mn ,

a y ya Iyqs y 21 Ž Õaq pq Õ y Õa pÕ . ,

I " a s y Ž um" pua m q 12 Õ . s " s a pÕ. . , puan

s "s un" s n ,

s a s una s n .

pÕ" a

Ž 4. una

In the above equalities, and are the momenta canonically conjugated to The newly introduced constraints Ž4. obey the following Poisson bracket algebra

and

Õa".

 I a b , I c d 4 s C b c I a d y C ac I b d q C a d I b c y C b d I ac ,  Iyq, I " a 4 s "I " a ,  I a b , I " c 4 s C b c I " a y C ac I " b ,  Iqa , Iyb 4 s C ab Iyqq I a b . This algebra is isomorphic to the SO Ž1,9. algebra: I a b generate SO Ž8. rotations, Iyq is the generator of the subgroup SO Ž1,1. and I " a generate the transformations from the coset SO Ž1,9.rSO Ž1,1. = SO Ž8.. Now we are ready to separate D a into first and second class constraints in a Lorentz covariant form. This separation is given by the equalities w10x: 1 a a D a s q Ž s a Õq . Da q Ž pu sq s a Õy . Ga , pqs p n unq , p D a s Ž Õq s a pu . b D b ,

G a s 12 Ž Õy s asq . b D b .

a

Ž 5.

a

Here D are first class constraints and G are second class ones:

 D a Žs1 . , D b Žs2 . 4 s y2 iC a b pqT 0 d p Žs1 y s2 . ,  G a Žs1 . ,G b Žs2 . 4 s iC a b pqd p Žs1 y s2 . . It is convenient to pass from second class constraints G a to first class constraints Gˆ a, without changing the actual degrees of freedom w10,11x: G a ™ Gˆ a s G a q Ž pq .

1r2

Ca

´

 Gˆ a Žs1 . ,Gˆ b Žs2 . 4 s 0 ,

P. BozhiloÕr Physics Letters B 440 (1998) 35–42

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where C a Žs. are fermionic ghosts which abelianize our second class constraints as a consequence of the Poisson bracket relation

 C a Žs1 . ,C b Žs2 . 4 s yiC a bd p Žs1 y s2 . . It turns out, that the constraint algebra is much more simple, if we work not with D a and Gˆ a but with Tˆ a given by Tˆ a s Ž pq .

y1r2

a a a 1r2 Ž s a Õq . Da q Ž pu sq s a Õy . Gˆa s Ž pq . D a q Ž pu sq s a Õy . Ca .

After the introduction of the auxiliary fermionic variables C a, we have to modify some of the constraints, to preserve their first class property. Namely Tj , I a b and Iya change as follows Tˆj s Tj q

i 2

C abCa E jC b ,

Iˆa b s I a b q J a b , Iˆya s Iya q Jya ,

j a b s 4i Ž Õy sc s a bsq sd Õy . C cC d ,

J a b s d ps j a b Ž s . ,

H

Jya s d ps jya Ž s . ,

jya s y Ž pq .

H

y1 ab

j pb .

As a consequence, we can write down the Hamiltonian for the considered model in the form: H s d ps l0 T0 Ž s . q l j Tˆj Ž s . q l a Tˆa Ž s . q l a b Iˆa b q lyq Iyqq lqa Iqa q lya Iˆya .

H

The constraints entering H are all first class, irreducible and Lorentz- covariant. Their algebra reads Žonly the non-zero Poisson brackets are written.:

½ T Žs . ,Tˆ Žs . 5 s Ž T Žs . q T Žs . . E d 0

1

j

2

0

1

0

2

j

p

Ž s1 y s2 . ,

½ Tˆ Žs . ,Tˆ Žs . 5 s ž d Tˆ Žs . q d Tˆ Žs . / E d j

1

k

l j k

2

l k j

1

½ Tˆ Žs . ,Tˆ Žs . 5 s Ž Tˆ Žs . q j

1

a

a

2

1

½ Tˆ Žs . ,Tˆ Žs . 5 s i s a

1

b

 Iyq,Tˆa 4 s

q p a b T0 d

2

Tˆ ,

1 2 a

½ Iˆ

ya

2

l

p

Ž s1 y s2 . ,

Tˆ Ž s2 . . E j d p Ž s1 y s2 . ,

1 2 a

Ž s1 y s2 . ,

,Tˆa s Ž 2 pq .

5

y1

p a Tˆa q Ž sq s a b Õy . aC b T0 ,

 Iˆa b , Iˆc d 4 s C b c Iˆa d y C ac Iˆb d q C a d Iˆb c y C b d Iˆac ,  Iyq, Iqa 4 s Iqa ,  Iyq, Iˆya 4 s yIˆya ,  Iˆa b , Iqc 4 s C b c Iqa y C ac Iqb ,  Iˆa b , Iˆyc 4 s C b c Iˆya y C ac Iˆyb ,  Iqa , Iˆyb 4 s C a b Iyqq Iˆa b ,

 Iˆya , Iˆyb 4 s yHd ps Ž pq . y2 j a b T0 .

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39

Having in mind the above algebra, one can construct the corresponding BRST charge V w12x

V s V min q p M P M ,  V , V 4 s 0 , V ) s V , where M s 0, j, a ,ab,yq,q a,y a. V min in Ž6. can be written as V min s V brane q V aux , V brane s d ps T0 h 0 q Tˆjh j q Tˆa h a q P0

H

½

Pa h jE jh a y 12 h aE jh j y qP

i 2

Ž 6.

Ž E jh j . h 0 q Ž E jh 0 . h j

q Pk Ž E j h k . h j

5

P0 h asaqb h b ,

b b V aux s Iˆa bha b q Iyq hyqq Iqahqa q Iˆyahya q Ž P ach .cb .y P b ch .ca.q 2 Pqahq q 2 Pyahy . ha b a q Ž Pqahqa y Pyahya . hyqq Ž Pyq hy q P a bhyb . hqa q 12 d ps Pa h ahyq

½

H

q Ž pq .

y1

p a Pa y Ž sq s a b Õy . aC b P0 h ahya y Ž pq . brane

y2 ab

5

j P0 hybhya .

aux

These expressions for V and V show that we have found a set of constraints which ensure the first rank property of the model. V min can be represented also in the form

V min s d ps

H

Ž T0 q 12 T0g h . h 0 q ž Tˆj q 12 Tj g h / h j q Ž Tˆa q 12 Tag h . h a

q Iˆa b q 12 Igahb ha b

ž

/

qa ˆya q 12 Iya q Ž Iyqq 12 Iyq q 12 Iqa g h . hyqq Ž I g h . hqa q I g h hya

ž

/

q d psE j Ž 12 Pk h kh j q 14 Pa h j . .

H

Here a superŽsub.script gh is used for the ghost part of the total gauge generators G tot s  V , P 4 s  V min , P 4 s G q G g h . We recall that the Poisson bracket algebras of G tot and G coincide for first rank systems. The manifest expressions for G g h are: T0g h s 2 P0 E jh j q Ž E j P . h j , Tj g h s 2 P0 E jh 0 q Ž E j P0 . h 0 q Pj E k h k q Pk E jh k q Ž E k Pj . h k q 32 Pa E jh a q 12 Ž E j Pa . h a , Tag h s y 32 Pa E jh j y Ž E j Pa . h j y i P0 saqb h b q 12 Pa hyqq Ž 2 pq .

y1

p a Pa y Ž sq s a b Õy . aC b P0 hya ,

b a Igahb s 2 Ž P achPbPc y P b chPaPc . q Ž Pqahq y Pqbhq . q Ž Pyahyb y Pybhya . , 1 qa ya p a Iyq g h s P hqa y P hya q 2 d s Pa h ,

H

qb aP qa yq a Iqa hy q P a bhyb , g h s 2 P hP b y P hyqq P yb aP ya yq a Iya hq q P a bhqb g h s 2 P hP b q P hyqy P

q d ps Ž 2 pq .

H

½

y1

p a Pa y Ž sq s a b Õy . aC b P0 h a y Ž pq . M

y2 a b

5

j P0 hyb . M.

Up to now, we introduced canonically conjugated ghosts Žh , PM ., ŽhM , P and momenta p M for the Lagrange multipliers l M in the Hamiltonian. They have Poisson brackets and Grassmann parity as follows Ž e M is the Grassmann parity of the corresponding constraint.:  h M , PN 4 s dNM , e Ž h M . s e Ž PM . s e M q 1 ,

 hM , P N 4 s y Ž y1. e e dMN , e Ž hM . s e Ž P M . s e M q 1 ,  l M ,p N 4 s dNM , e Ž l M . s e Ž p M . s e M . M

N

P. BozhiloÕr Physics Letters B 440 (1998) 35–42

40

The BRST-invariant Hamiltonian is Hx˜ s H min q  x˜ , V 4 s  x˜ , V 4 ,

Ž 7. min

because from Hcanonical s 0 it follows H s 0. In this formula x˜ stands for the gauge fixing fermion Ž x˜ ) s yx˜ .. We use the following representation for the latter.

x˜ s x min q hM Ž x M q 12 rŽ m.p M . ,

x min s l M PM ,

where rŽ M . are scalar parameters and we have separated the p M-dependence from x M. If we adopt that x M does not depend on the ghosts Žh M , PM . and ŽhM , P M ., the Hamiltonian Hx˜ from Ž7. takes the form Hx˜ s Hxmin q PM P M y p M Ž x M q 12 rŽ M .p M . q hM

 x M ,GN 4 h N q 12 Ž y1. e

N

PQ  x M ,UNQP 4 h Ph N ,

Ž 8. where Hxmin s  x min , V min 4 , and generally  x M ,UNQP 4 / 0 as far as the structure coefficients of the constraint algebra UNMP depend on the phase-space variables. One can use the representation Ž8. for Hx˜ to obtain the corresponding BRST invariant Lagrangian Lx˜ s L q LGH q LGF . Here LGH stands for the ghost part and LGF – for the gauge fixing part of the Lagrangian. If one does not intend to pass to the Lagrangian formalism, one may restrict oneself to the minimal sector Ž V min , x min , Hxmin .. In particular, this means that Lagrange multipliers are not considered as dynamical variables anymore. With this particular gauge choice, Hxmin is a linear combination of the total constraints min min Hxmin s H brane q Haux ab yq qa ya s d ps L0 T0tot Ž s . q L j Tjtot Ž s . q L a Tatot Ž s . q L a b Itot q Lyq Itot q Lqa Itot q Lya Itot ,

H

and we can treat here the Lagrange multipliers L 0 , . . . , Lya as constants. Of course, this does not fix the gauge completely. 3. Comments and conclusions The introduced harmonic variables uma , Õa" helped us to construct SO Ž1,1. = SO Ž8. covariant quantities Ž s a, pq, etc.. from SO Ž1,9.-covariant ones. We note here that this is an invertible operation. For any Lorentz vector Am we have An s yunqAyy unyAqq una A a , where A a s una An , A"s un"A n . For Lorentz spinors the reversibility is demonstrated in the equality Ž5. for example. To ensure that the harmonics and their conjugate momenta are pure gauge degrees of freedom, we have to consider as physical observables only such functions on the phase space which do not carry any SO Ž1,1. = SO Ž8. indices. More precisely, these functions are defined by the following expansion F Ž y,u,Õ ; p y , pu , pÕ . s Ý una11 . . . unakk puankqkql l . . . puankqkql l SO Ž8. singlet Õaq1 . . . Õaqm Õaymq 1 . . . Õaymq n b1 b r yb rq 1 b my nq r a 1 . . . a mq n , n 1 . . . n kq l = pq . . . pq pÕ . . . py Fb 1 . . . b my nq r Ž y, p y . , Õ Õ Õ

where Ž y, p y . are the non-harmonic phase space conjugated pairs. In this letter we begin the investigation of a p-brane model with N s 1 supersymmetry in 10-dimensional flat space-time. Starting with a Hamiltonian which is a linear combination of first and mixed Žfirst and second.

P. BozhiloÕr Physics Letters B 440 (1998) 35–42

41

class constraints, we succeed to obtain a new one, which is a linear combination of first class, irreducible and Lorentz-covariant constraints only. This is done with the help of the introduced auxiliary harmonic variables. Then we give manifest expressions for the classical BRST charge, the corresponding total constraints and BRST-invariant Hamiltonian. It turns out, that in the given formulation our model is a first rank dynamical system. The problems of Lagrangian formulation, finding solutions of the classical equations of motion and quantization will be considered in the second part of the paper.

Acknowledgements The author would like to thank A. Pashnev and B. Dimitrov for critical reading of the manuscript.

Appendix A We briefly describe here our 10-dimensional conventions. Dirac g- matrices obey

Gm Gn q Gn Gm s 2hmn and are taken in the representation

G ms

ž

˙

Ž s m . ba

0 b

Ž s˜ m . a˙

0

/

.

G 11 and charge conjugation matrix C10 are given by G 11 s G 0G 1 . . . G 9 s

ž

dab

0

0

yda˙b

˙

/

,

C10 s

ž

˙

C ab

0 ab

Ž yC . ˙

0

/

,

and the indices of right and left Majorana–Weyl fermions are raised as

c a s C a b˙cb˙ ,

f a˙ s Ž yc .

ab ˙

fb .

We use D s 10 s-matrices with undotted indices

Ž s m.

ab

b

s C a a˙ Ž s˜ m . a˙ ,

y1

˙

Ž s m . ab s Ž yC . bb˙ Ž s m . ba ,

and the notation

s m1 . . . m n ' s w m 1 . . . s m n x for their antisymmetrized products. From the corresponding properties of D s 10 g-matrices it follows: s

gb gb ab ba Ž s m . ag Ž s n . q Ž s n . ag Ž s m . s y2 dabh mn , Ž sm 1 . . . m 2 sq 1 . s Ž y1. Ž sm 1 . . . m 2 sq 1 . ,

n

s ms n 1 . . . n n s s mn 1 . . . n n q

Ý Ž y1. k h mn s n k

1

. . . n ky 1 n kq 1 . . . n n

ks1

The Fierz identity for the s-matrices reads:

Ž sm .

ab

ga

gd bg ad Ž s m . q Ž sm . Ž s m . q Ž sm . Ž s m . bd s 0 .

.

P. BozhiloÕr Physics Letters B 440 (1998) 35–42

42

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