Free equations of motion for all D = 6 supermultiplets

Nuclear Physics B298 (1988) 217-240 North-Holland, Amsterdam

FREE EQUATIONS OF MOTION FOR ALL D = 6 SUPERMULTIPLETS Aleksandar Depurtmetlt

R. MIKOVIC

of Ph.vsits ud

and Anton E.M. VAN DE VEN*

Astronomy,

Uniuersity

College Park. MD 20742,

Received

24 August

of Mu~~lund

at College Park,

D’SA

1987

We derive the complete set of free equations of motion for all D = 6 massless supermultiplcts. They are obtained by truncating the constraints for superconformal invariance and adding some new equations quadratic in superderivatives. We demonstrate the validity of the equations in the cases of (2.0). (4.0) and (2,2) supersymmetry.

1. Introduction Recently, a general formalism has been proposed for deriving the gauge-invariant free field theory for an arbitrary Poincare representation, including strings [l]. Starting from the conformal group in D dimensions one writes the set of equations of motion for its representations by applying conformal boosts to the equation P' = 0 [2]. The equations of motion for massless representations of the Poincare subgroup are then obtained by truncation of the conformal set. These equations apply to on-shell on-shell equations Then

by adding

field strengths. To get the off-shell formulation, one uses the to construct the PoincarC generators in the light-cone gauge. two commuting

and two anti-commuting

coordinates

one obtains

the expressions for the generators of OSp( D,2 12) which contains SO( D - 1,l) X OSp(1, 112) as a subgroup. The generators of OSp(1, 112) include the usual BRST (and anti-BRST) charge, and are used for constructing the gauge-invariant and gauge-fixed

actions

[l].

It is of particular interest to extend this formalism to include supersymmetric (string) field theories. This has recently been discussed by Siegel [3,4]. The first steps are: start from the superconformal group in D dimensions and write the superconformal set of equations of motion. Then, a truncation gives the set of equations of motion for all massless super-Poincare representations. This procedure has been shown to work for N = 1 supersymmetry in both D = 3 and D = 4 [5]. A *Work

supported

by PHY-85-17475

and PHY-86-19077.

0550-3213/88/$03.500Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V

218

natural

A. R. MikoviC, A. E. M. van de Ven /

next step would be to examine

extended

it is easier to study simple and extended spinors opposite

chirality

are related

In this paper,

supersymmetry

while the D = 4 spinors

chiralities,

by complex

in D = 4. However,

in D = 6. This is because

supersymmetry

in D = 6 come in two independent

just the one-chirality

D = 6 supermultiplets

conjugation

so it is impossible

of

to study

case.

we derive

the set of equations

of motion

for all Poincare

super

multiplets in D = 6. Following the procedure outlined in refs. [3,5], we obtain, in sect. 2, a set of equations of motion which is too strong, i.e. it overconstrains some known supermultiplets. We obtain a corrected set of equations of motion by trial and error. After explaining the use of the light-cone gauge and the choice of on-shell superfield strengths in sect. 3, we prove our equations of motion to be correct by examining in detail all supermultiplets with (2,0), (4,O) and (2,2) supersymmetry in sects. 4, 5 and 6. We present our conclusions in sect. 7 and include four appendices on our notation, covariant D-algebra, light-cone new equations quadratic in superderivatives.

D-algebra

and an analysis

of the

2. D = 6 equations of motion The superconformal group in D = 6 dimensions is OSp(6,2 1N,) X OSp(6,2 IN_), where N, and N_ are the numbers of left-handed and right-handed supersymmetric charges, respectively [4,8]. For the sake of simplicity we will consider first the case of only one chirality. The case of both chiralities will be treated in sect. 6. The generators of OSp(6,21N) are linear momentum Pap, supersymmetry charges Q$ angular momentum J!, dilatation A, second supersymmetry charges S”* (of opposite handedness of Qz), conformal boosts KaP and USp(N) generators U,,. Here SU*(4) (= SO(5,l)) indices are denoted by Greek letters (a = 1,. . . ,4), while USp( N) indices are denoted by lower case Latin letters (a = 1,. . . , N). One way to find the equations of motion is to (anti)commute the defining equation P* = 0 repeatedly with Sf [3,5], keeping only equations of dimension > 1. This procedure gives the following equations of motion for an arbitrary massless D = 6 superPoincarC

representation P2=0, paPDu = P

Pta’%,IP’+

(2.1)

(2.2)

0 ’

(&2)P”fl=O, P’“lYM IP) =

(2.3) 0)

(2.4)

Y

,;jab Here M,fl, d and u,~ are the spin (matrix)

+ uahPap = 0. parts of angular

(2.5) momentum,

dilatation

219

A. R. Mikovit, A. E. M. van de Ven / D = 6 supermultiplets

and internal generators respectively tors and superspace representations appendix vanish

and D$ab = iDi”Dp”). The necessary commutafor the relevant generators can be found in

A. At each step in the derivation upon

using

previous

equations.

PaPQ; = 0 by the substitution

(2.1)-(2.5)

are understood

of (2.1)-(2.5) For

instance

noncovariant

eq. (2.2) is obtained

Qz = i(D,” - 28”@Pap) and using

to apply to on shell superfield strengths.

We will denote

expressions from

P2 = 0. The

strengths,

eqs.

to be an referred

them by W,‘::f

L1,,h, which

to in the sequel

as superfield

can be assumed At this point

without loss of generality to be an irreps of SU*(4) X USp(N). we notice immediately that the superconformal method fails, since

eq. (2.4) implies that superfield strengths cannot carry SU*(4) up-indices. For any superfield strength that carries such indices eq. (2.4) would imply that it is a constant. That such indices are needed is clear from the example of (2,0) superYang-Mills, whose superfield strength is WE (see sect. 3). We therefore discard eq. (2.4). We will refer to the remaining these with 0; we obtain secondary something

equations equations.

as primary equations. By multiplying It turns out that only eq. (2.5) gives

new. Using eq. (B.14), we obtain

where D$‘;hc

= iDA”D,fD;),

etc. The precise numerical

prefactors

in this expression

are irrelevant for our purposes. We used eq. (2.2) written as P,,,D,“, the last two terms. Taking the [c$y]-piece of (2.6) we get ,;);b,

=

since Df3~~y~ a 1 = 0 and PraaDt, = 0. By taking

This implies

for an arbitrary

superfield

strength

tensor.

we find (2.8)

W

k(Uh“)

a

k is some constant

(2.7)

= 0.

U((lhD~)W=

where

0

the (abc)-piece

u”~D;‘P~,,,

= 0, to simplify

a

However,

(2.9)

3

by redefining

the superfield

strength (2.10)

we can set k in (2.9) to zero. Such redefinitions symmetry of eqs. (2.1)-(2.5). Therefore we have u(ubDd

This is a dimension-

$ secondary

equation.

u

=

are allowed

0.

It can be thought

since they form a

(2.11) of as a requirement

to

220

make

A.R. Mikovit,

the primary

conditions

equation

A.E.M.

vm de Vetz /

D = 6 supermultiplet.~

(2.5) is consistent.

for eq. (2.11) by multiplying

We can also derive

integrability

it with Dt. If we use eqs. (B.3) and (2.5) we

get

[ U(uhU’.M + &qp]

pas =

0,

(2.13)

Using the commutation relations of the u’s (see (A.7)) one can prove that (2.13) is equivalent to (2.12) and they imply, with a similar redefinition of the superfield strength

as in (2.10) n(Uh&)

= 0.

(2.15)

This tells us that the superfield strength has to be totally antisymmetric in its internal indices. Alternatively, the above equations can be derived starting from the superconformal tensor operator quadratic in (positive dimension) generators (see ref. [4]). So far we have a single dimension-t equation, (2.11), which is empty whenever the superfield strength carries no internal indices. In such cases we need a further dimension- $ equation which will eliminate auxiliary component field strengths contained

in DW. More specifically

we need an equation W”...A PI-..Y

D

[a

that implies

=O

(2.16)

’

D,W;.;f=O. An obvious

candidate

which may produce

(cf. sect. 3)

(2.17)

these equations

is

M/Da” + kD,” = 0.

(2.18)

This produces the desired equations (2.16) and (2.17) if we choose the coefficient k to be i(3m + n) where m and n are the number of down- respectively up- SU *(4) indices of the superfield strength. But consistency conditions tell us that even (2.18) is too strong. Namely, pieces we get

by multiplying

(2.18) with D and separating

Mc,‘Pp,u + M$Dj;;

+ 2kDc2’ + = 0’

uabMcaYPpjv = 0,

into irreducible

(2.19)

(2.21)

A. R. MikoviC, A. E. M. van de Ven /

271

D = 6 supermultiplets

and M, ayPply + M,/D,j;; eM[aYPplU

= 0,

(2.22)

- 2kPap) = 0,

(2.23)

- 2kP,,

M,$@ However

eq. (2.23) implies

h= 0 .

for a USp( N) non-singlet - 2kP,,

MtJ?lY

superfield

(2.24) strength

= 0,

(2.25)

which is consistent with (2.3) only if k - (d - 2) - m - n and this is not the case. Also it turns out that (2.22) and (2.24) are too strong (i.e. they constrain some physical fields) and we therefore discard them. The remaining equations, eqs. (2.19) (2.20) and (2.21), are in fact the equations we were seeking for. Actually, as we will demonstrate in sect. 3, eq. (2.20) can be derived from eq. (2.19) and is not an independent new equation of motion. We now give a preliminary discussion of the content of these additional equations. It is easy to see that (2.21) implies that any superfield strength is of one of the following two types W~~,~,.,~,p~~ )

VI...6

and similarly

While

these

a...8

(2.26)

(2.29)

7

eq. (2.20) implies

equations

iq”“Wy4.:-s”= 0)

(2.30)

h$~gu”w,;:;:; = 0.

(2.31)

are quadratic

in D’s they nevertheless

imply

the desired

A.R. MikoulC, A. E.M. vun de Ven /

222

constraints equations superfield

(2.16)

and (2.17).

For instance,

of motion into the decomposition strength of type (2.27)

D = 6 supermultiplets

inserting

(2.29), (2.31)

and previous

of DtDi

(cf. eq. (B.3))

we find for a

D:(D&~:::t1=o.

(2.32)

Dr”nWg;::.; = c,

(2.33)

Therefore

where the constant c can be set to zero by a redefinition of the superfield strength. Starting from the superconformal method, we have arrived at an ansatz for the complete set of equations of motion, namely (2.1)-(2.3) the next section we will prove by going to the light-cone correct.

(2.5) and (2.19), (2.21). In that this set of equations is

3. Light-cone In order to see what the equations of motion derived in the previous section imply for the components of the superfield strength we choose a special reference frame, such that (i=2

P_=P,=O

p++o,

,...A

(3.1)

where P,=

/QP()iP,).

This reference frame is known as the light-cone symmetry has been broken down to the subgroup SO(4) x SO&l)

= su(2)

x su(2)

(3.2) frame,

and

x SO(l,l)

manifest

.

Lorentz-

(3.3)

The advantage of going to this frame is that the number of components of a spinor gets reduced by a factor of two. Therefore superfields have considerably less component fields in the light-cone, which simplifies the analysis a lot. Spinors of SU *(4) decompose with respect to the subgroup in eq. (3.3) as

where the subscripts

represent

4 -+ (2, a,2

+ W)-l/2,

(3.4)

a-+ (I,%,,

+ (2,1)-i,&

(3.5)

SO(1, 1) weights. A vector decomposes

6+(W),+

(1,1)++(1,1)-.

as (34

A.R. MikoviC, A. E.M. van de Ven / D = 6 supermultiplets

For the momentum

223

we have the correspondence (2&J-&

OJ),--p,,

(3.7)

so that pap +

cGp+

)

(3.8)

PC+3+ CL&

)

(3.9)

where g, p are the first SU(2) indices, while 4, fi are the second SU(2) indices. For is equivalent to a Dirac-type an irreducible superfield strength W[l_:,-$;, eq.(2.3) equation

Therefore

for appropriate

choice of d, so that

the only non-zero

components

are

wg+-m .

(3.12)

Since (3.12) is an irrep of SU(2) x SU(2) it is completely symmetric in both the g’s and fi ‘s. This explains why it is sufficient to use field strengths which are completely symmetric with respect to SU *(4)-indices. Any other irrep is equivalent to some completely symmetric SU *(4) irrep on the light-cone. From now on we will omit underlines for SU(2) indices, but keep the dots for second SU(2) indices. When SU *(4) indices are intended we will state so explicitly. By using (3.8) in eq. (2.2) we get (3.13)

D,‘W= 0, i.e. the superfield The remaining

strength equations

is independent

of 132 in the light-cone.

are D(2)oh+ &‘p+=

0,

(3.14)

Map P + + McayD& + mD$ = 0,

(3.15)

Yjj(2)ah + ,fi(2)uh

(3.18)

and their implications

M

(a

P)Y

afl

--0.

224

A.R. MikoviC,A. E. M. vc~nde Ven / D = 6 supermultiplets

Eq. (3.15) implies

(3.18) since (3.19)

where the first step follows from group theory (see appendix C for the definition of Dc4)) and in the second step we used that D(” - P+D(‘) as follows from D-multiplication

on (3.14) (cf. (5.25)). To prove

associate

a superfield

eqs. (3.14)

strength

(3.15) eliminate

unconstrained components irreducible supermultiplet.

these equations

to an arbitrary

all auxiliary are

that

supermultiplet,

component

the physical

field

really

work

we will

and then show that

field strengths.

The remaining

strengths

they

and

form

an

Irreducible supermultiplets can be obtained by multiplying the smallest supermultiplet with the irreps of the little group x internal group [9]. For example, in the (N, 0) case, the smallest multiplet has dimension 2N and contains the following SU(2) X SU(2) X USp(N) irreps

(3.20) All other supermultiplets can be obtained by multiplying (3.20) with (p, 4; r) with r an irrep of USp(N). In the case of (N,, N_), the smallest multiplet is the direct product of (N,, 0) and (0, N_) smallest multiplets. Components of the supermultiplet (p, q; 1) @ 2” will be of the form (m, n; k) where k G $N denotes the irrep antisymmetric on k indices. The superfield strength will carry SU(2) x SU(2) x USp(N) indices of the lowest dimensional component in the multiplet, which is the one with the smallest value for m + n. We associate with (m, n; k) an SU *(4) field strength F{a~;:;fj{,,, _ail. In the case of a (p, q; nonsinglet) @J2N supermultiplet, it is impossible to describe it with just one superfield strength,

since there will be more than one component with the smallest value of therefore interpret such multiplets as a (p, q; 1) @J2N multiplet with some external index. For example (1,2; 3) 8 22 = (2,2; 3) @ (1,2; 2) @ (1,2; 4) can be interpreted as an USp(2) (2,0) Yang-Mills multiplet, its component field strengths being { Xe, F,“fi} where A is an USp(2) adjoint index. Notice also that any multiplet with extended supersymmetry can be thought of as being composed of simple (i.e. (2,0) and (0,2)) multiplets, with appropriate assignments to extended USp(N) irreps. However, this is just a kinematical fact and does not inform us about the equations of motion for such multiplets with extended supersymmetry. Given a general superfield strength W, we obtain new superfields by projection m + n. We

D’“)W 5

(n=l,...,

4N),

D(r) zz D”,,

(3.21)

A.R. MikoviC, A.E.M. van de Ven /

where obtained

DC”)

is a generic

from D,,,,,

notation

for the set of irreducible

. . . D, ,Ia,,, (see appendix

225

D = 6 supermullipleis

B). Evaluation

nth

level operators

at 6’= 0 (denoted

by I)

yields the component field strengths. Once we show for the levels which contain physical fields that all auxiliaries can be eliminated, we are done, since then all the fields at higher levels are auxiliary, D(‘)D(“)

due to the symbolic

identity

= D()l+l)

_

+ aD(fl-1,

(3.22)

4. (2,0) supermultiplets The fundamental

multiplet

for left-handed

simple supersymmetry

2*=(2,1;1)$(1,1;2),

in D = 6 is (4.1)

and an arbitrary multiplet is obtained by multiplying this one will an irrep of SU(2) X SU(2) X USp(2) = SU(2) X SU(2) X SU(2). As explained in the previous section we will take an USp(2) singlet so that we need to consider the following class of supermultiplets

(p,q;1)@22=( P,4;2)~(P-l,q;l)~(P+l,q;l).

(4.2)

We distinguish two cases: p = 1 or p a 2. For p = 1 we have the irreps (4.3) in the multiplet,

which we associate (l,q;2) (2,4’

with component

field strengths

- Ffl .-B~-l(x),

(4.4)

AI’I. -L(x). a

(4.5)

1) -

For q = 1 this is a matter multiplet { c#I~,X,}, for q = 2 a super-Yang-Mills multiplet {At, Ff } (notice that the Fermi/Bose character of the fields in (4.4), (4.5) alternates with q). The SU *(4) superfield

strength

is therefore (4.6)

Since the second SU(2)-indices are inert we omit them from now on and study the case q = 1, without loss of generality. At the zeroth level we find only the physical field

W”I =q5”-

(1,1;2).

(4.7)

226

A. R. Mikouit

A. E.M. uan de Ven /

D = 6 supermultiplets

At the first level we have D,“W” = CubA, -t &, so besides due

the physical

to eq. (3.16)

auxiliary,

(4.8)

field, there is a triplet auxiliary field. This auxiliary vanishes implies DJ”W ‘) = 0 . At the second level, all fields are

which

since (4.9) (4.10)

D$W’=O,

as follows from eqs. (3.14) and (3.17) respectively. Notice that in the case of (2,0) supersymmetry, ij (2) does not exist. For this reason and because the superfield strength does not carry any first SU(2) indices, the remaining eqs. (3.15), (3.18) are empty. In the case p 2 2 the lowest dimensional component of the supermultiplets in eq. (4.2) is ( p - 1,q; 1). We therefore associate them with the SU *(4) superfield strengths (4.11)

The set of all (2,0) SU*(4)

superfield

w w, W,P WaP ... ... ...

strengths

WI

can be displayed

as a tower

woa

wa Wd ...

WuaP W;h ...

. .

On the right-hand margin appear the p = 1 multiplets discussed in the previous paragraph. The first three are the familiar matter-, Yang-Mills and mattergravitino-multiplets. Also for p > 2, the second SU(2) indices are inactive so we ignore

them and study only the case q = 1. In that case the physical

(k ,... ,_Z-(~-l,1;1),h”,l...,~,-(~~1;2)1F,1...,p-(p+l,~;l)}.

components

are (4.12)

The prototype p = 2, q = 1 is the (2,0) tensor-multiplet with scalar superfield strength W. (Another familiar supermultiplet is that with p = 2, q = 3, namely (2,O) supergravity. The Weyl tensor appears at the second level as D$W y8 - 84). At the zeroth level (4.13)

A. R. Mikovif,

A. E. M. van de Ven /

221

D = 6 supermultiplets

and at the first level &?I,

The auxiliary field strength (3.19, which implies

X, present

for p a 3, vanishes

D;W:...y which is the light-cone D(2k9,f,’

a, . ..a.,_2

Eq. (4.16) follows

version

as a consequence

of eq.

(4.15)

= Q,

of eq. (2.17). At the second

level we find

z 0,

(4.16)

from eq. (3.14)

and eq. (3.15) implies

X a,. .ap 2-P+k while leaving supermultiplets

(4.14)

,... olPmz = C, ,... a,_2 + Ca(alx:z...+2,.

(4.18)

ya ,... aP 4 =o,

,__.ap_z)

F unconstrained. This completes with (2,O) supersymmetry.

the

proof

for

the

case

of all

5. (4,0) supermultiplets The fundamental

multiplet

of (4,0) supersymmetry

is given by

24=(3,1;1)~(2,1;4)$(1,1;5) and a general

supermultiplet

(5.1)

by

(P,4;1)~24=(p-2,q;1)~(P-1,q;4)~(P,q;1)~(p,q;5)

(5.2)

~(p+l,q;4)$(p+2,q;l).

We need to distinguish three cases, namely p = 1, p = 2 and p > 3. For p = 1 we have the following supermultiplets (1, q; 1) @ 24 = (3,q; with corresponding

superfield

strength J.$I$..-“+

As in the (2,0)

case, the second

1) @ (2, q; 4) @ (1, q; 5)) (antisymmetric

on ab)

- (1, q; 5).

SU(2) indices

(5.3)

are inactive,

(5.4) so we drop them and

228

A.R. Mikouid, A. E.M. uun de Ven /

study

the case q = 1. The physical {+

components

(l,I;5),+-

D = 6 supermuhplets

are then

(2,I;4),

Q--

(3,l;l)j.

(5.5)

This is the (4,0) tensor-multiplet. For q = 2 we have a matter-gravitino and for q = 3 the (4,0) supergravity multiplet. At the first level we find ,z@‘r

of motion

fildl(~~hwc)lel a Inserting

eq. (5.6) into eq. (5.7), the physical fildl(ux;.

At the second

+ x,, hc ,

= QJW~; + $h”h”,

where x is a 16 of USp(4). The equation

()lel

multiplet

(5.6)

(3.16) implies

= 0. field strength

= 0 +

x”,.

(5.7) drops out and we get

hc = 0.

(5.8)

level we have

fiG$khw~d = (QuQ”d D;;),ub

+ 2pkG”lh)

Fap + ~n[4cxfjlK

+ ya”p cd,

= Z”” aB ’

(5.9) (5.10)

where X is a 10 and Y a 14 of USp(4). The fields D (2)uhWrdare immediately seen to be auxiliaries from eq. (3.14). Inserting eqs. (5.9), (5.10) into eq. (3.17) the physical field strength drops out again and we get a relation among the auxiliaries X, Y and Z. Since they carry different irreps of USp(4) it follows that X, Y and 2 vanish. For p = 2 the supermultiplet contains (2, q; I) @I24 = (1, q; 4) @ (2,q; Again

1) @ (2, q; 5) @ (3, q; 4) @ (4, q; 1).

we will omit the second SU(2) indices

and set q = 1. The superfield

(5.11) strength

is then W”and the physical

fields in the multiplet

(1,1;4),

(5.12)

are

{~~-(I,I;4),X,-(2,I;I),~‘bf-(2,1;5),F,”p-(3,1;4),A,p,-(4,1;1)}. (5.13) At the first level we have D”Wh= a The auxiliary

X - 10 vanishes

Qn”hhu + A”h a + x, oh .

(5.14)

due to eq. (3.16) which implies D’“wf” a

= 0

(5.15)

A. R. MikoviC, A. E. M. van de Ven /

At the second

229

D = 6 supermultiplets

level we find (5.16) (5.17)

Substitution

of these expressions

into eq. (3.17) yields X uh.c _-0,

(5.18)

Y,“p= E$ , so the physical

D(3)0,

a

field F appears

bcwd

Di3’“wb

where

2,

Multiplying

=

z

in both (5.16) and (5.17). Finally,

+

(abckd

+

2Q4b&d)

+

,,d,bc

+

at the third level

[Lb,

cldu,

@ya + fz” + sY,ab,

6, f are 5’s, 2,

(5.21) (5.22)

$, f are lo’s and

g, [ are 14 and

35, respectively.

eq. (3.14) with D, we obtain @)W”)c

Taking

fpdec

(5.19)

+ fDfDfJb)c

+ 3p,D;“@”

+ &‘p,D;

=

0.

(5.23)

the trace we get Di3ju = P, (0," +

UUbDba) .

(5.24)

From this equation we see that all l’s are indeed auxiliaries. Inserting eq. (5.24) back into (5.23) we discover that also all t’s are auxiliaries. Finally, by multiplying eq. (3.17) by D and taking the (c$y)-piece

where we have used eqs. (C.6) and (C.8). physical field X drops out, while i and To complete the proof we consider contain the full range of components strengths are given by w~1.::.$;

we find

Inserting (5.20) into (5.25) we find that the jj are constrained to vanish. p > 3 in which case the supermultiplets as in eq. (5.2). The associated superfield

- ( p - 2, q; 1) .

(5.26)

230

A. R. MikoviC, A. E. M. van de Ven /

For q = 1 the physical i %

a,,

1

components

-(P-2,1;1),%..

D = 6 supermultiplets

are

,_,,

a,,_,+-1,1;4),F,

cxp

,-(p,l;l),

‘Ih Fa,. “,’ I -(P,I:5),~~,..,~-(~+I,1;4),F,~...,~+,-(p+2,1;1)).

(5.27)

At the first level

and the situation is similar to that in eq. (4.14): X, present for p > 4, is set to zero by eqs. (3.15) (3.18). At the second level we can copy eqs. (4.16) (4.17) adding an equation for b,“‘W identical to eq. (4.17) except for the addition of ab superscripts on the fields. Eqs. (3.15) (3.18) imply D$Wp 1 a2 ..apmi-P+Wa,.

and

it follows

that

X-

P++, while

Y, X”” and

D-multiplication, eqs. (5.29), (5.30) imply auxiliary fields. At the third level we find D$$W... D(3b.hCW a

= Gpv... ,,, = zgp+

D(3)“W a

={”

Yoh vanish.

that

a...

+

(5.29)

.np_,y

In fact, by further

all SU(2)-traces

of D’“)W

are

+ ... 7

(5.31)

. . . )

(5.32)

. . .

(5.33)

>

where the ellipses represent fields involving SU(2)-traces of Dc3)W’s and are therefore auxiliaries. From (5.23) (5.24) we see that all fields in (5.32) (5.33) are auxiliaries. Finally, at the fourth level D$\,W

=FaPys..

+ ...

D$juhW

= x$

ij$khW,,,

= j+ + ... afi...

+ ... )

D(4)Uh,Cdf,j,T > D(4)9,+f

=

yu'h

D’4’W

zz

2

>

(5.34) (5.35) (5.36) (5.37)

’

(5.38)

’

(5.39)

A. R. MikoviC,

A. E. M. vun de Ven /

for auxiliary

fields involving

Again,

dots stand

plying

eqs. (5.24) and (5.23) with

(5.37) respectively supermultiplets

SU(2)-traces

of Do W's. By multi-

D we see that (5.35), (5.36)

are all auxiliaries.

231

D = 6 supermuliiplets

This completes

(5.38), (5.39) and

the proof

for the case of all

both

and

with (4,O) supersymmetry.

6. (2,2) Supermultiplets In

this

section

supercharges

are

we consider present.

the situation

Besides

1 ,f.., N,), we have in addition anticommute with each other

the

the right-handed

{D:, The additional

equations

when

left-handed

of motion

D!}

left-

superderivatives

ones 0;

right-handed 0,”

(a =

(a’ = 1,. . . , N_), which

=o.

(64

are Pa0 D,4 = 0,

(6 J)

and their implications

where

k’ = - i(m + 3n).

analogues

Do not

confuse

Dc21ap with

D$).

We also have

the

of (2.15) and (2.23), namely u(,,yu,,&)

=

0,

(6.7) (6.8)

which together

inform

us that superfield

W;i:._.$

W?.-::u”fl *9

strengths

come in four kinds

W,qL,;:.a”‘, m

WBI ‘. lL a,...a,, 3

(6.9)

with total symmetry on Lorentz indices and total anti-symmetry on internal indices of both kinds (k < $N+, 1~ $N_). Concerning the D-algebra it seems at first sight that new DC") ‘s appear such as in DUD! = SflD$)’ + DA;?“. a a

(6.10)

232

A. R. MikoviE, A. E. M. vm de Ven / D = 6 supermultiplets

However,

upon

following

0"'

only 0,” and 0,“; survive

going to the light-cone

and we have the

‘s

0;;) , The fundamental

D(2)&

D(hh

U’h’)

multiplet

(2,O) and (0,2) fundamental

D(%

(2)

D

G

D,“Db

au’

of (2,2) supersymmetry

a’ .

is just the direct product

multiplet

of

multiplets

24=(2,2;1,1)~(1,1;2,2)~(1,2;2,1)~(2,1;1,2) and an arbitrary

(6.11)

is obtained

(6.12)

from

(6.13)

~(p,q-1;2,l)~(p+l,q;1,2)~(p-l,q;1,2). We distinguish three cases: p = q = 1, (p > 2, q = 1) and p, q > 2. For p = q = 1 we have a superfield strength w;with physical

{G-

(6.14)

(1,1;2,2),

components

This is the “N = 1” Yang-Mills

where the auxiliary

supermultiplet.

x in the first equation

F,p-(2,2;1,1)}.

(2,1;1,2),

(1,1;2,2),X”,-(1,2;2,1),h~-

(6.15)

At the first level we have

vanishes

the second equation vanishes due to the light-cone second level we need only to consider

due to (3.16) and similarly version

D;D$ W,l: = CuhCu>,,F,” + ’ . . ,

x in

of eq. (6.5). At the

(6.18)

where the auxiliaries indicated by ellipses vanish due to (3.16) and the light-cone version of (6.5), exactly as at the first level. This follows immediately, since D’s and D”s anticommute. The analysis of the remaining cases is very similar and we have shown that our equations are valid for these supermultiplets as well.

A.R. MikoviC, A.E.M.

van de Ven /

233

D = 6 supermultiplets

7. Conclusions We have derived super-Poincare

the complete

multiplets.

set of equations

superconformal

equations

while successful

in D = 3,4, does not work straightforwardly

equation, added

eq. (2.4)

to arrive

of motion,

by keeping

had to be discarded,

at a satisfactory

for all D = 6 massless

of motion

We have found that the procedure

only dimension

while others,

set of constraints

of truncating

> 1 equations,

in D = 6. A particular

eqs. (2.19)-(2.21),

for on-shell

the set of

had to be

superfield

strengths.

We proved this modified set of equations to be correct by studying all supermultiplets with (2,0), (4,0) and (2,2) supersymmetry. The off-shell extension of D = 6 supermultiplets can now be derived along the lines of [4]. An important question is how to extend our work to the case D = 10. Notice that eq. (2.5) determines superfield strength. notation as

the superisospin, while (2.19) determines The eqs. (2.19) and (6.5) can be written -

the superspin of a together in SO(5,l)

-

M,a,Pc,+ M,adD%c,dD + k,Dyo,cD+ k2EahcdefBydefD=O,

(7.1)

(which is convenient for generalization to arbitrary D), where k, - (m - n) and k, - (m + n) and the first term is the usual Pauli-Lubanski tensor. We conjecture that in arbitrary D the complete set of equations of motion consists of p*=o,

(7.2)

$D=O, M,b&,+

(d-

(7.3)

+(D-2))P,=O,

(7.4)

plus two equations quadratic in superderivatives which determine superisospin and superspin. Study the situation in D > 6, presently in progress, will likely illuminate these issues. We would

like to thank W. Siegel and S.J. Gates for discussions

and remarks.

Appendix A NOTATION

AND

D = 6 SUPERCONFORMAL

D = 6 (N, 0) superspace

where

the Greek

indices

ALGEBRA

has the following

coordinates

are SU *(4) indices

and the Latin

indices

are USp( N)

234

A. R. MikoviC, A. E. M. vnn de Ven /

indices (N = 2,4,6,. . .). P airs of Lorentz of the SU *(4) antisymmetric tensor

while internal

indices

D = 6 supermultiplets

indices can be raised or lowered by means

are raised and lowered by means 8” =

of the symplectic

8, = 0”f2ub.

Pbll,,,

metric (‘4.3)

In the special case N = 2 we use the conventional notation C,, instead of 52,,. Symmetrization and anti-symmetrization on n indices are defined without factors of l/n !. Derivatives are defined by

The superconformal [Jf,J,6]

algebra

osp(6.2(N)

=i(@~--syBJ,F),

{CC,Q;} = 2QobPa,,

[ IJ”,

u’d]

is given by

= i,Qn(d(cUd)lb),

{

y”,

sb@} = - 2QahKa”,

(A.7)

A.R. MikoviC, A.E.M.

The superspace

representations

van de Ven /

D = 6 supermuhplets

of the relevant,

i.e. positive

dimension,

235

generators

are given by Pap = id,,

,

Qz = i( 8: - iB”pa,,), J,P= $i(x,,

ay~-~Pya,,)+i(8~a~-as,P~~a~)+iM,B,

A = - $i(x@

8,,+6~8~)

+id,

Uab = iea(aa,“) + iuab.

(A.@ Appendix B

COVARIANT

Covariant

DERIVATIVES

derivatives

0,” transform

as (4, N) of SU *(4) X USp(N)

and satisfy (B.1)

where

Ph = - Ghu is the symplectic

The product

metric,

satisfying

52 ah”= oh

-6’

of two D’s can be decomposed

as

D;D/

rr .

= flabPap + D;;“’ + b$“”

(B-2)

+ LnuhD;;j,

03.3)

where D;$kb = ID(uDph) >a 3

@4

iQph

@W

= +D[:D,:’ f -&v~D;;D~,~,

DA;) = - &

D;~D~,~.

(B.6)

So we have [D,“, Dj’] = 2,;juh Notice that antisymmetric

+ 2D$Jah + 2PhD3.

(B-7)

8)‘) is absent for N = 2. In general, to decompose a completely product of n D’s into irreps, it is sufficient to decompose it under

A. R. MikouiC, A. E. M. om de Ven /

236

SU *(4) [6]. The corresponding SU *(4) tableaus

USp(N)

and subtracting

D’s, we have the following

possible

D = 6 supermultiplets

expansion

is obtained

by transposing

traces. In the case of the product

the

of three

SU *(4) tableaus

(B.8) and the corresponding

USp( N) tableaus

are

We can subtract traces from the first and second tableau in (B.9), so that additional pieces, in the representation q, will appear. We therefore write 0,” [ Di, D;] + 2 perms. = D$,Ybc + (D$$’

two

+ cyclic perms.)

+ ( Da(f~y,s26’ + cyclic perms.)

+ D$yyahc + D$~i2bc], (B.lO)

where

Starting from (B.lO), one can derive the decomposition of DDc2) into rewriting the left-hand side of (B.lO) in terms of DDc2). It follows that DD”’ = Dc3’ + dD. Group

theory

tells D;D$;,

us what D (3) irreps may show up in (B.12) = D$;

B 0;;;

uj @ PacpD;, ,

D;D;;)fX = D~il)yUb~ @ Di’bu,‘“. c)$

Dc3)‘s, by

(B.12) so we can write (B.13)

Da(3$;Q+

(B.15)

A.R. Mikovif,

A.E.M.

van de Ven /

We do not need the precise numerical in (B.13)-(B.15)

although

231

D = 6 supermuliiplets

value of the coefficients

they may be easily calculated

multiplying

the terms

from (B.lO).

Appendix C LIGHT-CONE

D-ALGEBRA

Upon going to the light-cone and specializing following set of D’s up to the fourth level

to the case N = 4 we have the

D,"-•, D(%f'_ m 3 D(3h dY

D (4) uflyfi’

D(4)uh aP

_

The product

-

,-,

D(3)O. a

’

El

hC _

D(3kJ a

$, D(4)&,

m

of two

p$b _

-

Cd

(,

D(4bh

_

DC4'.

8,

D'sdecomposes

as

(C.2) which implies

To get the precise numerical generic identity

= DC3’ apy[o Qhc]

D,.A,,4,,

coefficients

+

{

G(j&l.

in the expansions

h<

+ 2 perms

+

of

DDc2"s, start from the

D$ :j,L? (+ {cu(P

2perms.)

/l

(C.4)

and substitute

D,,,D,,D,.,,

= 3D,,[ D,B> Dcy] + 3%,p,,D,p

- 3&,p,,D,.,

.

(C.5)

238

A. R. Mikooic!, A.

E.M. uun de Ven / D = 6 supermultiplets

Then we use (C.3) and after separating

the irreducible

pieces, we get

(C.8) Appendix D ANALYSIS

OF NEW D’” EQUATIONS

Equations (2.21) and (2.22) are non-empty only for superfield strengths carrying Lorentz-indices. In what follows we will omit internal indices since they are inactive. We can assume the superfield strength to be totally symmetric in both up- and down-indices, so it transforms as

WP, a ,..,

!A, _ n,,

&g

irrep of SU(4). It follows that m+l

(D.2) m+2 MC Y Dp’:$,‘& a a,...am

fin- [ ,. . . , , , ,

m + 1

m CD.31

where we omitted for simplicity the parts of the Young diagrams representing the up-indices. The terms indicated by “traces” in (D.2) vanish due to the equation of motion (2.3) (cf. (3.10), (3.11)). We choose the coefficient k in eq. (2.21) such that cancellation occurs between the “m + 2” diagrams in (D.3) and (D.4). Then the

A.R. MikoviC A.E.M. van de Ven / “m

+ 1”

and “m”

239

D = 6 supermultiplets

pieces of eq. (2.21) imply (D.5)

(D.6) respectively,

while the trace terms imply

(D-7) the “m + 1” and “m”

Similarly,

pieces of eq. (2.22) imply

(D-9) respectively,

while the trace terms imply @UhW,B:.~.~~ = 0.

Eqs. (D.5) and (D.6) together p and similarly

(D.10)

imply f,f,” ..x+D(2’WK...x=O

a[/1

y]___&

4P

(D.ll)

-/I...8

(D.8) and (D.9) imply

q~:““w;]-:; = 0.

(D.12)

Thus we have derived eqs. (2.28))(2.33). To show that the coefficients in eq. (2.19) are correct Applying

eq. (2.19) to an arbitrary

superfield

strength

takes some more work.

as in (D.l)

we get

k’

-

1 (n - l)!

D:~1Dp(f~W[2,:,,f;‘y

+ (2k - f(m

- H))

D$WPI...PJ~ n, a,, = 0, (D.13)

where we allowed for an undetermined coefficient k’. Cancellation symmetric pieces amongst the second and fourth term requires k=

:(3m+n).

of the totally

(D.14)

A. R. MikoviC, A. E. M. van de Ven /

240

Furthermore,

the decomposition

D = 6 supermultiplets

(D.4) can be written

as

(D.15)

+ traces. Rewriting

the first term in (D.13) as

p (N(a1 %*...a,),&=

[(

- : P(,,rpW,,],,,),,...,,~ + LY-

and collecting similar pieces in the second and fourth “m + 1” pieces in (D.2)-(D.4)) we find that -

P]+ 6%...%J1 (D.16) term of eq. (D.13) (i.e. the

k’ 2(m - I)! I q4rp

W alllQ%..%

+cxctp)

+(“l...a,,,)]

+ +m(m+l)(m-2)-2m+m(m+l) (m + 2)!

[(

x D&3% a will reproduce

1

]laz)a3.

a

m+a”P)+(
(D.17)

eq. (D.5) if we choose k’=

1.

(D.18)

References [l] W. Siegel and B. Zwiebach, Nucl. Phys. B282 (1987) 125 [2] A.J. Bracken and B. Jessup, J. Math. Phys. 23 (1982) 1925 [3] W. Siegel, Nucl. Phys. B263 (1986) 93 in Proc. of the Workshop for everything”, [4] W. Siegel, “Free field equations composite structures and cosmology, College Park MD (March 10-18, 1987) [5] A. MikoviC, Nucl. Phys. B298 (1988) 205 [6] J. Koller, Nucl. Phys. B222 (1983) 319 [7] P.S. Howe, G. Sierra and P.K. Townsend, Nucl. Phys. B22L (1983) 331 [S] A. Van Proeyen, K.U. Leuven preprint KUL-TF-86/9 (1986) [9] J. Strathdee, Int. J. Mod. Phys. A2 (1987) 273

on Superstrings,