The authors calculate the spectrum of the S^5 compactification of the chiral N=2, D=10 supergravity theory. The modes on S^5 fall into unitary irreducible representations of the D=5, N=8 anti-de Sitter supergroup U(2,2/4). These unitary supermultiplets involve field of spin <or=2 with quantised 'mass' eigenvalues. The massless multiplet contains fifteen vector fields, six self-dual and six anti-self-dual anti-symmetric tensor fields. The fields of the massless multiplet are expected to be those of a gauged N=8 theory in D=5 with a local gauge group SU(4)

Günaydin, M.

Marcus, N.

English

Caltech Authors - Main

Class. Quantum Grav. 2 (1985) Lll-L17. Printed in Great Britain 
LETTER TO THE EDITOR 
The spectrum of the S5 compactification of the chiral N = 2, 
D = 10 supergravity and the unitary supermultiplets of 
U(2,2/4)Jf 
M GunaydinS and N Marcus§ 
$ California Institute of Technology, Pasadena, CA 91 125, USA 
5 University of California, Berkeley, CA 94720, USA 
Received 23 October 1984 
Abstract. We calculate the spectrum of the S5 compactification of the chiral N = 2, D = 10 
supergravity theory. The modes on S5 fall into unitary irreducible representations of the 
D = 5, N = 8 anti-de Sitter supergroup U(2,2/4). These unitary supermultiplets involve 
fields of spin s 2  with quantised ‘mass’ eigenvalues. The massless multiplet contains fifteen 
vector fields, six self-dual and six anti-self-dual anti-symmetric tensor fields. The fields of 
the massless multiplet are expected to be those of a gauged N = 8 theory in D = 5 with a 
local gauge group SU(4). 
Recently there has been a surge of interest in Kaluza-Klein type theories in the context 
of supergravity. The main object of study so far has been the eleven-dimensional 
simple supergravity of Cremmer er a1 [ 11 and various compactifications of this theory 
down to four dimensions have been studied. A compactification is called spontaneous 
if, for example, the eleven-dimensional theory admits a vacuum solution leading to a 
four-dimensional spacetime M4 and a seven-dimensional compact internal manifold 
M,  parametrised by the extra seven coordinates. Then from a four-dimensional point 
of view the spectrum of the theory consists of massless modes and an infinite tower 
of massive modes at the Planck scale corresponding to the Fourier expansion on M,. 
In this letter we shall study the spectrum of the chiral N = 2, D = 10 supergravity 
theory [2] compactified on S5 down to five spacetime dimensions. This theory cannot 
be obtained from the eleven-dimensional supergravity by dimensional reduction in 
contrast to the non-chiral N = 2, D = 10 theory which can be. The non-chiral and 
chiral N = 2 supergravity theories can be regarded as the massless sector of type IIa 
and type IIb superstring theories in D = 1011. Both ofthese theories when dimensionally 
reduced to D = 4 give the N = 8 theory. 
The fields of the chiral N = 2, D =  10 supergravity are: a complex scalar cp, a 
complex Weyl spinor A, a complex anti-symmetric second rank tensor A,,, a complex 
Weyl gravitino (cIw, a real graviton (‘zehnbein’) e ;  and a real fourth-rank anti-symmetric 
tensor APupA. The field strength FpvpAm of the field A,,,, is self-dual in the free theory. 
The complex scalar field cp parametrises the non-compact coset space SU( 1, l ) /U( 1 )  
[4]. The covariant field equations of chiral N = 2, D = 10 supergravity were obtained 
t Work supported in part by the US Department of Energy under Contract No DE-AC-03-81ER40050 and 
by the Fleischmann Foundation. 
1 1  For a complete list of references on ten-dimensional string theories see the review article [3]. 
0264-9381/85/020011+07$02.25 @ 1985 The Institute of Physics L11 
L12 Letter to the Editor 
in reference [4]. So far no compactifying solution of this theory to four dimensions 
has been found that admits a non-Abelian isometry group G for the six-dimensional 
internal manifold. However, it is known that this theory can be compactified on S5 
leading to the five-dimensional anti-de Sitter space Ad& [5]. The isometry group of 
this vacuum solution is S0(4 ,2)  x SO(6) = SU(2,2) x SU(4) where S0(4 ,2)  is the 
five-dimensional anti-de Sitter group. This solution has maximal number of supersym- 
metries in Ads, and its full super-invariance group is U(2,2/4) whose even subgroup 
is SU(2,2) x U(4). Note that the extra U( l ) y  comes from the ten-dimensional theory 
itself. 
We have calculated the spectrum of the S5 compactification using the fact that the 
modes for different spin fields must fall into unitary representations of U(2,2/4) and 
that the graviton modes fall into symmetric tensor representations of S 0 ( 6 ) ,  correspond- 
ing to the scalar modes on s’. The modes for other spin fields are determined by the 
fact that together with the graviton modes they must form unitary irreducible representa- 
tions (UIR)  of U(2,2/4).  Since the highest spin field in the massive supermultiplets is 
s = 2 they correspond to short supermultiplets (in the sense of [6]) with quantised mass 
eigenvalues. 
We have constructed the UIRS of U(2,2/4) using the oscillator methods developed 
in [7-8]?. This method has recently been applied to the calculation of the spectra of 
S4 and S7 compactifications of the eleven-dimensional supergravity [ 10- 1 11 and agrees 
with the calculations previously done using coset space techniques. The advantage of 
the oscillator method is that it yields automatically the irreducible unitary supermulti- 
plets with quantised mass eigenvalues into which the spectrum must fit. The basic 
idea of the oscillator method is to identify a compact sub-supergroup K of a non- 
compact supergroup G and construct the UIRS of G in terms of finite dimensional 
unitary representations of K .  For the non-compact supergroup U(2,2/4) the relevant 
sub-supergroup is U(2/2) xU(2/2)  X U (  1) with respect to which the Lie superalgebra 
U(2, 2/4) has a Jordan structure [8], i.e., 
L =  L-OLooL+ (1) 
where Lo represents the generators of U(2/2) x U(2/2) x U( 1) and L - 0  L+ represents 
the non-compact generators of U(2,2/4) such that 
[LO, L+] = L+, 
[L+, L-] = LO, 
[LO, L-] = L-, 
[ L+, L+] = 0 = [ L-, L-1. 
The Lie superalgebra U(2,2/4) is realised in terms of bilinears of boson and fermion 
annihilation and creation operators tA, tA = ([A)t and vM, v M  = ( q M ) +  transforming 
covariantly and contravariantly under the two U(2/2) subgroups of U(2,2/4), respec- 
tively: 
i , j =  1,2 
p, U =  1,2 
[ a ,  a’] = s i  { C Y p ,  C Y ” } =  6;: 
(3) 
t For a review of the oscillator method and a discussion of its applicability to the calculation of the spectra 
of supergravity theories see [9]. 
Letter to the Editor L13 
Then the generators of U(2,2/4) are given in terms of the oscillators as 
L- = SA * iM, L -5 . S B @ j i M . r j N ,  L + + .  i" (4) 0- +A 
where the arrow over and fj means that we are taking an arbitrary number p of 
oscillators and the dot represents the summation over the generation index running 
from 1 to p .  Note that the three grading, (4), (Jordan structure) is given by the number 
operator of all the oscillators that generate an Abelian U(l )  belonging to Lo. Then 
starting from a 'ground state' in) in the super Fock space that transforms irreducibly 
under K and is annihilated by all operators in L- one can generate a UIR of G by 
repeated application of the operators in L+: 
R ={la), L'lSZ), (L')*ISZ), . . .}. ( 5 )  
Note that the irreducibility of the unitary representation of U(2,2/4) follows from the 
irreducibility of ISZ) under U(2/2) xU(2 /2 )0U( l ) .  The anti-de Sitter group S0(4 ,2)  
has a Jordan structure with respect to its maximal compact subgroup SU(2),, x SU(2),, x 
U( l ) E .  Positive energy UIRS of S0(4 ,2)  can be obtained by the oscillator method using 
the bosonic oscillators as above. They are also uniquely determined by a lowest state 
annihilated by L,, and which transforms irreducibly under SU(2),, x SU(2),, x U( l ) E .  
The SU(2),, x SU(2),, -- SO(4) is the rotation group in D = 5 and U( l )E  denotes the 
Abelian group generated by the energy operator. Similarly the internal symmetry group 
SO(6) = SU(4) has a Jordan structure with respect to its su(2)kl x su(2)k2 x U,( 1 )  
subgroup. Its representations can all be constructed by the oscillator method using 
fermion bilinears as above. The UIRS of SU(4) are also uniquely determined by the 
transformation properties of their lowest states annihilated by Lpr 
The short unitary supermultiplets whose highest spin is two, can all be constructed 
by taking as the ground state ISZ) the Fock vacuum IO). Even though the vacuum 10) 
is a singlet under SU(2/2) X SU(2/2) subgroup, it has definite U( 1 ) E  XU( 1 ) 1  quantum 
numbers that depend on p .  Therefore by varying the generation index p we obtain 
inequivalent unitary supermultiplets of U(2,2/4). By acting on the vacuum 10) with 
supersymmetry generators L'" and L'"' one generates new lowest states of both S0(4 ,2)  
and SU(4), i.e., states that are annihilated by both L,, and Lpx. Acting on such lowest 
states with L" and Lpx one generates UIRS of S0(4,2)  with definite SU(4) transforma- 
tion properties. Such lowest states are labelled by their transformation properties 
under the even subgroup SU(2),, x su(2)k, xSU(2),, xSU(2),, x U( 1 ) E  X U (  l )yOU( l)l. 
For example, for p = 1 we list below the possible lowest states, their SU(2),, x su(2)k, x 
sU(2),, X Su(2)k2 transformation properties (in Young tableau notation) and the SU(4) 
representations one obtains by the action of Lxp on them. 
The basis of a positive energy U I R  of S0(4,2)  can be taken to be the Fourier modes 
of a field in Ads5. The spacetime properties of this field is determined by the 
SU(2)j, x SU(2),,OU( l ) E  transformation properties of the lowest state of the UIR. The 
L14 Letter to the Editor 
anti-de Sitter ‘mass’ of a field of spin ( j l , j 2 )  is determined by the eigenvalue of the 
U( l ) E  generator on the lowest state. Therefore for p = 1 the unitary supermultiplet of 
U(2,2/4) consists of six scalar fields ( p u b  = -pbU (a,  b = 1,4), four complex left-handed 
spinors A: and four complex right-handed spinors A - ”  and an antisymmetric self-dual 
tensor field A,, and an anti-self-dual tensor field A@,. The number of bosonic and 
fermionic degrees of freedom do not match in this supermultiplet. We shall call this 
supermultiplet the doubleton of U(2,2/4) in analogy with the doubleton in D = 7 [ 101 
and singleton in D = 4  [ l l ,  12, 131. The doubleton representation does not have a 
PoincarC limit and it decouples from the spectrum of Ss compactification. This can 
best be understood by the fact that the six scalars of the doubleton multiplet are the 
conformal scalars (their gradients give conformal Killing vectors) and can be gauged 
away by appropriate conformal gauge choice. As in the case of D = 4 singleton [ 131 
and D = 7 doubleton [ 101 one may be able to construct a field theory of D = 5 doubleton 
in one lower spacetime dimension, i.e., in D = 4. The Ads group SU(2,2) will then 
be the conformal group in D = 4 and the resulting field theory will be conformally 
invariant. In fact, the unique candidate for this doubleton theory in D = 4 is the N = 4 
Yang-Mills theory which is known to be conformally invariant. The study of the 
doubleton field theory in D = 4 will be given elsewhere [ 141. 
For p = 2 one obtains the massless supermultiplet which contains the massless 
graviton. Forp 2 3 one gets massive supermultiplets of U(2,2/4).  We give the complete 
spectrum of the Ss compactification of chiral N = 2, D = 10 theory in table 1. Note 
that there are new massive towers of U(2,2/4) supermultiplets starting at p = 3 and 
p = 4 .  For p > 4  there are no new towers starting and one gets only the massive 
excitations of the towers starting at ~ ~ 4 1 .  In table 1 we also list the transformation 
properties of the lowest states under the rotation group SO(4) = SU(2),, XSU(2), and 
the corresponding fields. A, and A-  denote ‘left- and right-handed’ complex spinors. 
The chirality or handedness refers to the anti-de Sitter group SU(2,2) or to the rotation 
group SO(4). In five-dimensional Minkowski space one cannot define chiral spinors. 
Instead one can have symplectic Majorana spinors [ 161. The complex anti-symmetric 
tensors A,, and A,, are ‘self-dual’ and ‘anti-self-dual’ anti-symmetric tensor fields in 
the sense of [17]$. They satisfy the duality conditions 
A,, = (i/ 120 m ) &,,puhFPuh 
APY = (-i/ 
where FpuA(FpuA)  is the field strength of the field AuA(AuA) and m is the ‘mass’ of the 
respective field. 
The irreducible unitary supermultiplet for p = 2 is called the massless supermultiplet 
since it contains the massless graviton. In addition to_the singlet graviton it contains 
42 scalar fields, 48s = 1 fields, 15 vector fields, 6 A,,,, 6 A,, and 8 gravitini. If one were 
to count the degrees of freedom of this supermultiplet in the limit the anti-de Sitter 
group is contracted to the PoincarC group in D = 5 then one finds that they match 
precisely those of the ungauged N = 8 supergravity. The only subtlety in this limit 
involves the equivalence, via duality, of a vector field to an anti-symmetric tensor 
field. However, the fields that are dual to each other in Minkowski space are, in 
general, not equivalent to each other in the corresponding anti-de Sitter space. This 
t The spectrum of the bosonic sector of the Ss compactification has been independently obtained by Romans 
using differential geometrical techniques and agrees with our results [ 151. 
$ For previous work on self-duality see [ 181. 
Letter to the Editor L15 
phenomenon has already been observed in D = 7 where anti-symmetric tensor fields 
of rank 2 and rank 3 are dual to each other in Minkowski space. However, a gauged 
N = 4 supergravity in D = 7 has been constructed involving the anti-symmetric tensor 
fields of rank 3 [ 191 whereas the construction using the fields of rank 2 has failed [20]. 
In fact, the gauged N = 4 theory in D = 7 involving tensors of rank 3 has no PoincarC 
limit [ 191. Thus, in analogy with D = 7, we expect that there be a gauged maximally 
extended simple supergravity in D = 5 whose fields are precisely those of the massless 
( p  = 2) supermultiplet of U(2,2/4).  The kinetic energy terms for the scalars of this 
theory will be a c+-model corresponding to the symmetric space E,,,,/Sp(8) as in the 
Table 1. We give the spectrum of the S5 compactification of the chiral N = 2,  D = 10 
supergravity. The states for a given p, together with their anti-de Sitter excitations, form 
an irreducible unitary supermultiplet of U ( 2 , 2 / 4 ) .  The p = 1 supermultiplet corresponds 
to the doubleton supermultiplet and decouples from the spectrum. The p = 2 supermultiplet 
is the ‘massless’ multiplet containing the singlet graviton. We also give the anti-de Sitter 
energies Eo of the lowest states of a given UIR of S O ( 4 , 2 ) .  The ‘mass’ of the corresponding 
field is determined by Eo and the SO(4)  quantum numbers of the lowest state. The last 
column gives the U (  l), quantum numbers of the fields. 
SU(4)  Trans- 
wa,, XSU(2) , ,  s o ( 4 )  formation U(1)Y 
x SU(2),, x SU(2)k2 Labels Anti-de Sitter Fields properties Quantum 
Young tableau of the energy Eo of the of the numbers 
of the lowest of the UIR of fields (Dynkin of the 
lowest states states lowest states U ( 2 , 2 / 4 )  labelling) fields 
2P 
2 p + l  
2p+  I 
2 p + 2  
2 p + 2  
2 p + 2  
2 p + 2  
2 p + 4  
2 p + 4  
2 p + 3  
2 p + 3  
2 p + 2  
2 p + 3  
2p+3  
2 p + 4  
0 
1 
I 
__ I 
1 
-1 
2 
1 
-1 
2 
-2 
3 
T 
-z 
0 
I 
-7 
3 
I 
1 
0 
L16 Letter to the Editor 
Table 1 (continued) 
~ ~~ 
SU(4) Trans- 
SU(2),, XSU(2),, ~ 4 )  formation U(1)Y 
x W 2 ) , ,  x SU(2)k, Labels Anti-de Sitter Fields properties Quantum 
Young tableau of the energy E,  of the of the numbers 
of the lowest of the U I R  of fields (Dynkin of the 
lowest states states lowest states U(2,2/4) labelling) fields 
2 p + 3  A(’) 
2 p + 3  A ‘3’ 
2 p + 5  A ‘4’ 
2 p + 4  Al” 
2 p + 4  ‘v 
2 p + 4  Al‘? 
2 p + 4  Al‘? 
2 p i 6  A;: 
2 p + 6  A;; 
2 p i 5  *YL 
2 p + 5  *?A 
2 p + 5  A ( 4 )  
1 
2 
1 
2 
3 
2 
3 
1 
-_ 
-- 
1 
-1 
0 
0 
- 1  
1 
1 
2 
1 
2 
-- 
0 
-1 
1 
0 
1 
1 
-5 
1 
1 
2 
1 
1 
_- 
n 
ungauged N = 8 theory [21,16]. However, both the E6(6) and Sp(8) symmetries will 
be broken by the other terms in the Lagrangian. It will have a local non-Abelian gauge 
group SU(4) whose gauge fields will be the 15 vector fields of the massless multiplet. 
We are presently working on the construction of this theory [22]. 
We would like to thank J Schwarz for useful discussions and the Aspen Center for 
Physics for providing the stimulating atmosphere in which this work was carried out. 
Letter to the Editor L17 
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The spectrum of the S^5 compactification of the chiral N=2, D=10 supergravity and the unitary supermultiplets of U(2,2/4)

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