Volume 202, number 3 PHYSICS LETTERS B 10 March 1988

T H E S P I N N I N G SUPERPARTICLE. II . A S P E C T R U M IN T E N D I M E N S I O N S

J. K O W A L S K I - G L I K M A N NIKHEF-H, P.O. Box 41882. 1009 DB Amsterdam, The Netherlands

Received 19 November 1987

We present the quantization of the spinning superparticle model in ten space-time dimensions. The resulting wave function represents two spin 3/2 multiplets for type I and spin 5/2 multiplets for type II theories.

In a recent paper [ 1 ] (hereaf ter called I) we have formula ted a new superpart ic le model called the spinning superparticle. It can be considered as a global space - t ime SUSY extension of the massless spinning particle, or the local world-l ine SUSY extension of the massless superpart icle. The quant iza t ion of the model per formed in I was restr icted to four space - t ime dimensions .

The spinning superpart ic le model can be also con- s idered as a d imens iona l reduct ion of a d = 2 doubly graded sigma model ([ 2,3 ]) with rigid superspace as a target. Therefore one expects the spect rum of our model in ten space - t ime d imens ions corresponds to the massless part of the spectrum of the spinning superstring.

It is possible to quant ize the spinning superpart ic le covar iant ly by in t roducing auxil iary variables as in refs. [4,5 ], however since the main goal of this paper is to identify a spect rum of the theory, in what fol- lows I will restrict mysel f to the much s impler light- cone gauge quant izat ion.

The lagrangian o f the system reads as follows:

L = ½ [(1 / V) ¢oVkt, - iAuA" - (2 i /V)AMo,

+ 2 6 F . 0(¢o" - i~uAV) ] . (1)

where "1

oY ~ = X ~ - i 6 F " O , (2)

~ The conventions are listed in the appendix.

0370-2693/88/$ 03.50 © Elsevier Science Publ ishers B.V. (Nor th -Hol l and Physics Publishing Div i s ion)

and 0 are M a jo ra na -We y l spinors ~2, V and ~, are the einbein and one-dimensional gravit ino, respectively.

The lagrangian (1) is invar iant under local one-di- mensional supersymmetry, Siegel symmetry and its bosonic counterpar t as well as global space - t ime su- persymmetry. The explicit form of t ransformat ion laws has been presented in I.

F rom the canonical momenta corresponding to ( 1 )

0L=0J(" = (1 /V)(o) , - i ~ ' A , + V ~ F , O ) = p , , (3)

OL/O0" = - iO~ p , F }~ =_ N ° , (4)

a L / 0 0 ~ = 0 = 7 rg , ( 5 )

A OL/O]l ~ = ½iA I, = n , , (6)

O L / O f ' = O = n v , (7)

OL/O~=O=n ~' , (8)

one finds a p r imary hami l ton ian

H e = ½ Vp 2 + i ~ ' p , A ~ - V p ~ F ~ O

+ G p + z U G + f n ~ + s h Y + a n ~ , (9)

where

G , =n,- -o + i O a p , F } , , ~ 0 , (10)

,2 Here I consider only type-I theory. The quantization oftype-II theories in which 0 and 0 are two M-W spinors can be per- formed along exactly the same lines, and the result will be dis- cussed later.

343

Volume 202, number 3 PHYSICS LETTERS B 10 March 1988

A ~u=ztu -- ½iAu ~ 0 , (11)

are together with ~zv~0, ~ ' ~ 0 and no~0 the pri- mary constraints of the model.

The set of secondary constraints consist of

p Z ~ 0 , (12)

p ~ , A U ~ O , (13)

p~,~F" .~O , (14)

and there are no tertiary constraints. The constraints (11 ) are easily recognized as stan-

dard fermionic second-class constraints, and one can use a Dirac bracket instead of a Poisson bracket and put ~ strongly equal to zero. As a result one obtains the Dirac bracket

{A,,, A~} =iqu . . (15)

The constraint ~r°~ 0 and p~,~Fu 0 ~ 0 form a mixed first- and second-class set. As has been shown in I, one can remove the whole 0 dependence from the theory by adding the gauge fixing condition

P~,=(- -Po,P9,Pi ) , i = 1, ..., 8 . (16)

In fact (16) and (14) are equivalent to the single set of constraints 0.~0 which together with zro~0 form a second-class set of constraints. After imple- menting the Dirac bracket procedure one can make 0 and zr ~ strongly equal to zero.

In the next step we choose a noncovariant gauge to break the local supersymmetry associated with the first-class constraint (13):

A + ~ 0 . (17)

Using (15) we find that

{ p . A , A + } = - i x ~ p _ , (18)

and define a Dirac bracket

{A, B}* = (A, B}

+ ( l / i x / 2 p _ ) ( { A , p ~ , A ~ ' } { A + , B}

+{A, A + }{puA ~', B}) . (19)

After making a rescaling

A - - - - , . 4 - = A - + p ~ A q p _ , (20)

one finds that

{ A , , A j } * = i d , j . (21)

./i- has a vanishing Dirac bracket with all canoni- cal variables and therefore can be omitted in what follows; all other canonical brackets are not modi- fied. From (20) we see that At may be represented after quantization as D-- 8 ?)-matrices.

Having removed a gauge freedom associated with local supersymmetry and bosonic Siegel symmetry we turn to the constraints (10). It is well known that this set of constraints contains first- and second-class constraints, which can be separated only in a nonco- variant way. Using a fixed light-cone frame one can split a 16-component ten-dimensional Majorana -Weyl spinor into two eight-component eight-dimen- sional spinors:

0~--, (0 a, 0 ~) .

In terms of these spinors a first-class part of (10) is

i ob o pi y a/,Tr -- ~ p _ zr. ,.~ O , (22)

and the second-class one is

G,i _ ~o,~ _ ipi yiat, O b _ i x / ~ p _ O,i ,.~ O . ( 2 3 )

Since

{G a, G b } = 2,,/2ip_ , (24)

one defines a Dirac bracket

{A, B}** = {A, B}*

- ( 1 / 2 x / 2 i p _ ) { A , Ga}*{G a, B}*. (25)

In order to remove "off-symplectic" double-star brackets one makes a redefinition

n Oa ~'ffOa + iOapiyiaa ,

X i__, )~i + i0a faa 0, ,

7 ~ ° a - - . ~ ° a : G a ,

o ~ - - , ~ = ( , , /5p_ ) "2o ~ .

(26)

(27)

(28)

(29)

In terms of these variables one has canonical brackets

{)~', p/}**=d}, (30)

{z7 °", 0b}** = - d g , (31)

{ X - , p _ }**= 1. (32)

In the dotted spinors sector the only nonvanishing bracket reads

344

Volume 202, number 3 PHYSICS LETTERS B 10 March 1988

{tTa, fib}**= ½i~ab. (33)

ff~ has a vanishing bracket with all canonical vari- ables (which is obvious, since it is itself a second-class constraint) and therefore can be put strongly to zero.

The formula (33) can be understood in two equiv- alent ways: either (as it is usually done) one splits 0 ~ into two SU(4) spinors and interpret them as ca- nonical coordinates and momenta, or one realizes (33) after quantization as a Clifford algebra and represents

1( 0 ' 7 ) Oa--,(Oa)a~= ~ (7,,)~ . (34)

In our analysis of the spectrum we will follow this second possibility.

Now we remove the first-class constraint (22) by fixing the light-cone gauge: 0 ~ 0. After substituting definitions of tilde variables ( 2 6 ) - (29) into (22) we have

p2Oa-x f2p_ na ~O ,

0 ~ 0 ,

which is equivalent to

~r~.~0, (35)

0 ~ 0 , (36)

and ~" and 0 ~ are removed from the theory by using the Dirac bracket method (as it was in the case of the ¢~ variable).

To quantize the theory we replace Dirac brackets by i times (anti) commutators and find a represen- tation of canonical variables as operatos acting on the Hilbert space of wave functions. From the canonical (anti) commutators

[X", p~]_ = - i O ~ , (37)

[A,, AS] + = ~ o , (38)

[0 ~, 0hi+ =½oab , (39)

one finds

p~ = i0/0X ~' , (40)

1 ((yi)~b0 ( 7 ~ / ' ) A ' = --~ , (41)

and a representation of 0 a is given in (34).

The remaining constraint p2 ~, 0 is a first-class one and, on the quantum level correspond to the condi- tion on the wave function

[ ~ = 0 . (42)

Since the variable 0 a is represented as a matrix, a wave function is an element of the vector spaces on which this matrix acts. It follows from the represen- tation (34) that the wave function consists of two parts 7J, • - a vector in the 8s representation of SO(8) , and ~v _ an SO (8) spinor in representation 8c. This is exactly the spectrum of the N = 1 superparticle in ten dimensions. In our case, however, we should also take into account an action o f A i. Therefore the full wave function consists of four terms:

~Jai, ~-Idti, ~Jaa, ~Jha • (43)

The wave functions in (43) belong to reducible representations of SO(8) and we must decompose them into irreducible pieces. It can be done most eas- ily by using the rules of multiplication o f SO (8) rep- resentations (see refs. [6,7]). It is easy to see that the representation of (43) is nothing but the (8v+8c) × (8s+ 8c) representation (8c is a second inequiva- lent spinor representation of SO (8)), where the first multiplier corresponds to A ~ and the second to the 0 a action.

For the irrep decomposit ion one finds

(8v +8c) × (8s +8c)

= 1 + 2 8 + 3 5 c + 8 s +56~ +8c +56c +8~ + 5 6 v . (44)

The irreps (45) correspond to two chiral gravitino multiplets with opposite chiralities (see ref. [ 7 ]).

It is clear now how to extend the above analysis to type II theories. In those cases we have two 0 vari- ables which satisfy the following ant icommutat ion relations:

[0 'a, 0 '~] + =-~c~ ~/' , (45)

[02a, 02b] + =-½Oa~ , (46)

[0 ~, 02~'] + = 0 , (47)

for type IIA theory (0 variables have opposite chir- alities), or

[0,a, 0,b] + =-½~ab , (48)

345

Volume 202, number 3 PHYSICS LETTERS B 10 March 1988

[ 0 2 a 02/'~] + = l(~h[) , ( 4 9 )

[01~, 02~] + = 0 , ( 50 )

for type IIB theory (both 0 variables have the same chirality).

In type IIA theories the wave function is therefore in the representation ( 8v + 8c ) × ( 8s + 8c) × ( 8s + 8v ), where the first factor comes from the representation o fA ' and the second and third from the ones o f theta variables. Accordingly in type IIB theories the wave function belongs to the (8v+ 8c) X (8s+ 8c) × (8s+ 8c) representation. In both cases one finds fields o f spin higher then 2 (spin 5/2 multiplets) in the spectrum, which, however, may be not so disastrous in string theory as it is in the case o f point-like theories since string theories already contain massive higher spin excitations. More dangerous is probably a presence o f two gravitons in the spectrum, since there is no obvious way to understand which one corresponds to the gravitational force. It does not mean however that the spinning superparticle model is not interesting anymore - even if it is dubious whether it is realistic, it may provide a useful tool for deeper understanding of superstring theories [ 8 ].

I would like to thank Dr. J.W. van Holten for use- ful discussions and reading the manuscript. This work is part o f the research program of the stichting FOM.

Appendix

The F matrices satisfy

F ~ = ( 0

/-11 = ( 1;6

where

[ F u, F"] + =2~/l" ,

where q,J=Oo, i , j= 1 .... ,8, ~/+- = - 1. Our represen- tation of 3 2 × 32 F matrices is

a/~6 )

0 '

0 )

__116 '

~" ~,~ = (~, , . )~t

and

• iabyjl ,b + ~ jM,? ibb = 26o ~ab .

References

[ 1 ] J. Kowalski-Glikman, J.W. Van Holten, S. Aoyama and J. Lukierski, Phys. Lett. B 201 (1988) 487.

[2] R. Brooks, F. Muhammed and S.J. Gates Jr., Nucl. Phys. B 268 (1986) 599; Class. Quant. Gray. 3 (1968) 745.

[ 3 ] J. Kowalski-Glikman and J.W. van Holten, Nucl. Phys. B 283 (1987) 305; J. Kowalski-Glikman, Phys. Lett. B 180 (1986) 359.

[4] L. Brink, M. Henneaux and C. Teitelboim, Nucl. Phys. B 293 (1987) 505.

[5] E. Nissimov, S. Pachewa and S. Salomon, Nucl. Phys. B 296 (1988) 462.

[6] R. Slansky, Phys. Rep. 79 (1981) 1. [ 7 ] B. de Wit, in: Supersymmetry, supergravity and superstrings

'86 (World Scientific, Singapore, 1987). [8] J. Kowalski-Glikman, J.W. Van Holten, S. Aoyama and J.

Lukierski, in preparation.

346