Mass spectrum of D=11 supergravity on AdS2×S2×T7

Nuclear Physics B 563 Ž1999. 125–149 www.elsevier.nlrlocaternpe

Mass spectrum of D s 11 supergravity on AdS 2 = S 2 = T 7 Julian Lee, Sangmin Lee School of Physics, Korea Institute for AdÕanced Study, Seoul, 130-012, South Korea Received 1 July 1999; received in revised form 31 August 1999; accepted 15 September 1999

Abstract We compute the Kaluza–Klein mass spectrum of the D s 11 supergravity compactified on AdS 2 = S 2 = T 7 and arrange it into representations of the SUŽ1,1 <2. superconformal algebra. This geometry arises in M theory as the near horizon limit of a D s 4 extremal black hole constructed by wrapping four groups of M-branes along the T 7. Via AdSrCFT correspondence, our result gives a prediction for the spectrum of the chiral primary operators in the dual conformal quantum mechanics yet to be formulated. q 1999 Elsevier Science B.V. All rights reserved. PACS: 11.25.-w; 04.65.qe; 11.15.-q; 11.25.Hf Keywords: AdS; Supergravity; Spectrum; Compactification

1. Introduction Among all known examples of the AdSrCFT correspondence w1–4x, the least understood is the AdS 2rCFT1 case. The D s 1 conformal field theory ŽCFT., or conformal quantum mechanics ŽCQM., has not been formulated and therefore no quantitative comparison between the two sides of the duality has been made. See Refs. w5,6x for proposals on the CQM and Refs. w7–10x for progress made in the bulk theory. One of the most elementary check of the correspondence is to compare the spectrum of the two theories. In particular, the Kaluza–Klein ŽKK. mass spectrum of the supergravity ŽSUGRA. on AdS is identified with the spectrum of chiral primary operators in the dual CFT. One may hope that the KK spectrum of a SUGRA on AdS 2 may give a clue to formulate the dual CQM. The goal of this paper is to compute the KK spectrum in the cases where the AdS 2 is part of a stringrM theory vacuum. We specialize in the example of D s 11 SUGRA compactified on AdS 2 = S 2 = T 7.1 We consider only the zero-modes in T 7. From the string theory point of view, this theory is a valid approximation when R )) r,r, ˜ where 1

We thank Seungjoon Hyun for bringing our attention to this example.

0550-3213r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 5 9 8 - 2

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r,r˜ are the radius and the dual radius of T 7 respectively, and R is the radius of the sphere. In what follows, we will put R to 1 for simplicity. To obtain this geometry from M theory, one begin with compactifying M theory on T 7 with the following brane configuration w11x.2 Brane M2 M2 M5 M5

0 x x x x

1

2

3

4 x

5 x

6

7

x

x x

x x

x

8

9

10

x x

x x

x x

With suitable choice of the orientation of the branes, this configuration breaks N s 8 supersymmetry ŽSUSY. of the D s 4 theory to N s 1. When the number of branes in each group is all equal, the background metric describes a direct product of an extremal D s 4 Reissner–Nordstrom ¨ black hole and a T 7. See Section 3 of Ref. w11x for more 2 details. The AdS 2 = S space-time arising as the near horizon geometry of this black hole is known as the Bertotti–Robinson metric w12,13x. Note that the brane configuration at hand approaches the Bertotti-Robinson metric in the near horizon limit even when the four charges are not equal. The number of SUSY is doubled in the near-horizon limit as usual, so we have D s 4, N s 2 SUSY. The super-isometry group of the theory is SUŽ1,1 <2.. The KK spectrum form representations of the SUŽ1,1 <2. superalgebra. The methods of the computation used in this paper are well known from higher dimensional examples. There are two approaches to the problem; one is direct SUGRA calculation w14–18x, and the other uses representation theory of superconformal algebra together with duality symmetry of SUGRA w19–24x. We will adopt the first approach and explicitly calculate the spectrum, starting from the D s 11 SUGRA Lagrangian. Although we will be mainly interested in the modes which have bulk degrees of freedom. However, as was noted in Ref. w29x, we cannot ignore the boundary modes completely because one of them forms a multiplet with bulk modes. We will make further comments on this point later. This paper is organized as follows. In Section 2, we review the SUŽ1,1 <2. superalgebra and its representation theory following Refs. w19,20x. In Section 3, as a warm-up exercise we compute the spectrum of a toy model, namely the minimal D s 4, N s 2 SUGRA. This model illustrates many important aspects of the compactification on AdS 2 = S 2 in a simple setting. In Section 4, we present a summary of our main result. In Section 5 and 6, we sketch the computation of bosonic and fermionic mass spectrum of the ‘‘realistic’’ model obtained from the D s 11 supergravity. We focus on the reduction from D s 11 to D s 4 and how the N s 8 supermultiplet break into N s 2 multiplets. As this work was being completed, we received w29x which has overlap with Section 3 of this paper. While this paper was being submitted to the hep-th e-print archive, we received Ref. w30x which considered the same model in a manifestly U-duality covariant way. 2

There are many other brane configurations that are related to this one by U-duality. Three M5 branes intersecting over a line with momentum flowing along the line is one such example.

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2. Review of the SU(1,1 <2) superconformal algebra The SUŽ1,1 <2. superconformal algebra is defined by the following commutation relations:

w J a , J b x s i e a b c J c , L m , J a s 0,

w L m , L n x s Ž m y n . L mqn ,

a

aa L m ,Graa s Ž 12 m y r . Gmqr ,

J a ,Gra a s y 12 Ž s a . b Grba ,

 Gra a ,Gsbb 4 s e a b  e a b Lrqs y Ž r y s . Ž s ae . ab J a 4 .

Ž 2.1a . Ž 2.1b . Ž 2.1c .

and the hermiticity conditions L†m s Lym ,

† Ž J a. sJ a,

Ž Gra a .

†

bb s ea b ea b Gyr .

Ž 2.2 .

The bosonic generators Lq1 ,0,y 1 and J 0,1,2 generate the SLŽ2,R. conformal group and the SUŽ2. R-symmetry group, respectively. We have eight supercharges all together; 1 aa Ž1 1. Ž . Ž . G" 1r2 carry L 0 charge . 2 and transform in 2 , 2 representation of SU 2 R = SU 2 Aut , where the second SUŽ2. is a global automorphism. One explicit way to find the representations of a superalgebra is the oscillator construction w25–27x. The oscillator representation of the generators of SUŽ1,1 <2. is given by Ly1 s a†1 P a†2 ,

L0 s 12 Ž a1 P a†1 q a†2 P a 2 . ,

Jqs c 1† P c 2† ,

J 0 s 21 Ž c 1† P c 1 y c 2 P c 2† . ,

Gyy y1 r2 Gy1r2 s

Gq1 r2 s

Gyq y1r2 s

Gqy y1 r2

Gqq y1r2

a1 P c 1

ya 2 P c 2

a1 P c 2†

a 2 P c 1†

Lq1 s a 2 P a1 ,

a†2 P c 1

ya†2 P c 2

a†2 P c 2†

a†1 P c 1†

Jys c 2 P c 1

Ž 2.3a . Ž 2.3b .

,

,

Ž 2.4 .

where a†i , a i are n-component vectors of bosonic creation and annihilation operators, and c i†, c i are the fermionic counterparts. It is straightforward to see that they satisfy all the commutation relations and hermiticity conditions Ž2.1a. – Ž2.2. except Ž2.1c., which is modified as

 Gra a ,Gsbb 4 s e a b  e a b Lrqs y Ž r y s . Ž s ae . ab J a q e a b I 4 ,

Ž 2.5a .

I ' 12 Ž a†1 P a y a†2 P a 2 . y 12 Ž c 1† P c 1 y c 2† P c 2 . .

Ž 2.5b .

The extra UŽ1. generator I must be added in order for the algebra to be closed. However, since I commutes with all the other generators, we may work in the restricted Fock space on which I s 0, where the algebra precisely reduces to that of SUŽ1,1 <2..3 3

We thank Jan de Boer for a correspondence on this point.

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Table 1 The short multiplets of SUŽ1,1 <2. superconformal algebra labeled by an integer n Lowest weight states

j

L0

Degeneracy

<0: < : qy < : Gqq y1 r2 0 ,Gy1r2 0 qq qy < : Gy1 G r2 y1r2 0

n r2 Ž ny1.r2 Ž ny2.r2

n r2 Ž nq1.r2 Ž nq2.r2

nq1 2= n ny1

For a given integer n, the oscillator vacuum is identified with the lowest J 0-weight q" state of a chiral primary operator. We act Gy1 r2 on the vacuum to obtain the lowest weight states of other primary operators in the supermultiplet. Higher weight states of a given operator are obtained by acting Jq on the lowest weight state. The quantum numbers of each state is easily computed using the explicit oscillator representation of the generators Ž2.3a., Ž2.3b.. The quantum numbers of the lowest weight state of each primary operator are summarized in Table 1. The total angular momentum j is defined by J 2 s jŽ j q 1.. The number of states for each primary operator, 2 j q 1, is also included in the table.

3. Toy model As a warm-up exercise, we compute the mass spectrum of the minimal D s 4, N s 2 SUGRA. It is the simplest SUGRA that admits the AdS 2 = S 2 solution with the SUŽ1,1 <2. superalgebra. The theory contains a single N s 2 gravity multiplet whose component fields are a graviton, a massless vector and a complex gravitino. 3.1. Result The computations in the following subsections show that the KK spectrum of the toy model contains the short multiplets in Table 1 for all even n. We have two copies of each multiplet for n 0 4 and one copy for n s 2. The result is depicted in Fig. 1. From the point of view of the SUGRA computation, each physical degree of freedom of the fields in D s 4 give a KK tower. That explains why we have four bosonic and

Fig. 1. The complete KK spectrum of the toy model. Each circle in the figure represents a state which has a definite value of h and j. The crossed circles correspond to the boundary states. The degeneracy Ž2 jq1. of each state is included in the circle. The states belonging to the same SUŽ1,1 <2. multiplet is connected by a dotted line. The two KK towers on the top row satisfy hs j and correspond to chiral primary states.

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four fermionic series of states in the spectrum. Depending on the spin and the polarization of a given field, the low lying modes Ž j s 0, 12 ,1. modes may be absent. Some of other low lying modes become massless and can be gauged away from the bulk spectrum. The absence of such modes is necessary in order for the KK spectrum to arrange itself into representations of SUŽ1,1 <2.. In addition to the bulk degrees of freedom, there may be modes that are pure gauge in the bulk but can live on the boundary. The authors of w29x showed that the boundary modes indeed exist and form one n s 2 and one n s 1 representations of SUŽ1,1 <2. algebra. We included these boundary modes in Fig. 1 for completeness. In particular, we cannot ignore them since one of them forms n s 2 multiplet with bulk modes, as can be seen from the figure. 3.2. Bosonic mass spectrum 3.2.1. Setup We normalize the fields such that the action reads

½

2 k 2 S s d 4 x'y G R y

H

1 4

F 2 y cm G

mn p

i =n cp y cm Ž F m n q 12 Fr s G 2

r sm n

. cn

5

,

Ž 3.1 . up to terms quartic in fermions that are irrelevant to our computation. The bosonic equations of motion consist of the Einstein and Maxwell equations in vacuum, R m n s 12 Fm l Fnl y

1 8

Gm n F 2 ,

= m Fm n s 0.

Ž 3.2 .

These equations admit dyonic Reissner–Nordstrom ¨ black-hole solutions. The near horizon geometry of an extremal black hole gives the AdS 2 = S 2 solution. The radius of the S 2 is equal to the Schwarzschild radius of the black hole. For simplicity, we consider an extremal electric black hole with unit radius only. Then the AdS 2 = S 2 solution reads Rmnls s y Ž gm l gns y gms gnl . , R abgd s g ag g bd y g a d g bg ,

Fmn s 2 emn , Fa b s 0.

Ž 3.3 .

where the Greek letters a , b . . . and mn . . . label two dimensional indices for AdS 2 and S 2 respectively. The fermions are set to zero. We are interested in the mass spectrum of the fluctuations of the fields around this background. We use the following parametrizations of the fluctuations: Gab s g a b q hŽ a b . q 12 h 2 g a b ,

Gmn s gmn q hŽ mn . q 12 h1 gmn ,

Gm a s hm a ,

Ž 3.4a . Fab s =a ab y =b aa ,

Fmn s 2 emn q =m an y =n am ,

where the parenthesis denotes the traceless part of a symmetric tensor.

Ž 3.4b .

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3.2.2. Spherical harmonics decomposition and gauge choice Each field can be decomposed into spherical harmonics. Unlike higher dimensional spheres, S 2 does not have genuine vector or tensor spherical harmonics. Vector and tensor fields are spanned by derivatives of the scalar spherical harmonics. hŽ a b . s f 1I=Ž a=b . Y I q f 2I eg Ž a=b .= g Y I hŽ mn . s hŽI mn . Y I ,

Ž j 0 2 . , h 2 s h 2I Y I ,

h1 s h1I Y I ,

hma s wmI=a Y I q ÕmI ea b= b Y I aa s q I=a Y I q b Iea b= b Y I

Ž j 0 1. , Ž j 0 1 . , am s amI Y I .

Ž 3.5 .

The composite index I specifies both the total angular momentum j and the J 0 eigenvalue m. The restrictions on the value of j for some fields are due to the fact that

=a Y Ž js0. s =Ž a=b . Y Ž js1. s eg Ž a=b .= g Y Ž js1. s 0.

Ž 3.6 .

Not all the modes in the above expansion are physical since the Ž D s 4. graviton and gauge fields are subject to the gauge transformations,

d h m n s =m L n q =n L m ,

Ž 3.7a .

d a m s yFm n L n q =m Ž S q L nA n . .

Ž 3.7b .

The functions L m and S are also expanded in spherical harmonics. We need to make a choice of gauge. First, consider the case j 0 2. We can gauge away hŽ a b . completely by a suitable choice of La . We then use Lm to eliminate the wmI terms. Lastly, we use S to eliminate the q I terms. With this choice of gauge, we note that

= a hma s 0,

= a aa s 0.

Ž 3.8 .

For j s 1, hŽ a b . modes are absent, so La can be used to reduce other degrees of freedom. We find it convenient to parametrize L m and S as

LmŽ1. s Ž Km q =m X . P Y ,

Ž 3.9a .

LaŽ1. s P P ea b= b Y y X P =a Y ,

Ž 3.9b .

S Ž1. s Q P Y ,

Ž 3.9c .

where the dot product means the sum over the three components of j s 1 spherical harmonics. We can use X, Km and Q to gauge away h1 , wm and q, respectively. Under the gauge transformation by P, Õm is shifted by =m P. This indicates that Õm is a massless gauge field in AdS 2 . Indeed, the mass term for Õm is absent as we will see below. Being a gauge field in D s 2, Õm has no propagating degree of freedom in the bulk. Also h 2 can be locally gauged away by residual gauge symmetry. For j s 0, hŽ mn . , h1 , h 2 and am are the only modes that remain. The gauge parameter La is absent. We can use Lm to gauge away hŽ mn . . The vector am becomes a gauge field in AdS 2 with S being the gauge transformation parameter, and again has no bulk degree of freedom.

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3.2.3. Linearized field equations The linearized Einstein and Maxwell equations read l l kl kl R Ž1. m n Ž h . s Fn Ž =m a l y =l a m . q Fm Ž =n a l y =l a n . y g m n F =k a l y Fm k Fn l h

q 12 g m n Fj k F jl h k l y

1 4

hm n F 2 ,

= m Ž =m a n y =n a m . y 12 Ž 2= m h m l y =l h . F l n y =m h l n F m l s 0,

Ž 3.10 . Ž 3.11 .

where the linearized Ricci tensor is defined by 2 k k k R Ž1. m n Ž h . ' y= h m n y =m=n h k q = =m h n k q = =n h m k .

Ž 3.12 .

Upon spherical harmonics decomposition, the ab component of the Einstein equation yields the following three equations. They are the coefficients of g ab Y I, =Ž a=b . Y I, and eg Ž a=b .= g Y I, respectively.

=x 2 h 2I y j Ž j q 1 . Ž h1I q h 2I . s 4e mn =m anI y 2 h 2I q 4 h1I ,

Ž 3.13a .

h1I s 0,

Ž 3.13b .

= m ÕmI s 0.

Ž 3.13c .

We separate the trace and traceless part of the mn component of the Einstein equation. We replace hmn by hŽ mn . in all the equations below using the constraint Ž3.13b..

=x 2 h 2I y 2= m =n hŽI mn . s y4e mn=m anI ,

Ž 3.14a .

=x 2 hŽI mn . y j Ž j q 1 . hŽI mn . q 2 hŽI mn . s =m= l hŽIln . q =n= l hŽIlm . y gmn = l = s hŽIls . y =Ž m=n . h 2I

Ž 3.14b .

The ma component of the Einstein equation splits into two pieces. They are the coefficients of =a Y I and ea b= b Y I, respectively.

=m h 2I y 2= n hŽInm . s y4em n anI ,

Ž 3.15a .

=x 2 ÕmI y Ž j 2 q j y 3 . ÕmI s 2 emn= n b I .

Ž 3.15b .

The a component of the Maxwell equation splits in the same way,

= mamI s 0,

Ž 3.16a .

=x 2 b I y j Ž j q 1 . b I s 2 e mn=m ÕnI .

Ž 3.16b .

The m component of the Maxwell equation yields a single equation,

=x 2 amI y Ž j 2 q j y 1 . amI s emn= n h 2I .

Ž 3.17 .

3.2.4. Computation of the mass spectrum: j 0 2 Altogether, we have ten equations of motion Ž3.13a. – Ž3.17.. We already used Ž3.13b. to eliminate h1I . We also note that Ž3.15a. implies Ž3.14a. for j 0 1. So, the number of independent equations is eight. We put off the discussion of Ž3.14b. and Ž3.15a. to the end of this subsection.

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Among the other six equations, Ž3.13c. and Ž3.16a. are constraints, and the other four are dynamical equations for each physical field. We first use the constraints to set on shell.4 Õm s 2 emn= n Õ,

am s emn= n a.

Ž 3.18 .

To simplify notations, we are suppressing the superscripts I in the equations from here to the end of this subsection. Inserting these in Ž3.13a. and Ž3.16b. immediately yields

=x 2 h 2 y Ž j 2 q j y 2 . h 2 y 4=x 2 a s 0,

Ž 3.19a .

=x 2 b y j Ž j q 1 . b y 4=x 2 Õ s 0.

Ž 3.19b .

After some manipulations, the other two equations Ž3.15b. and Ž3.17. give

=x 2 Õ y Ž j 2 q j y 2 . Õ y b s 0,

Ž 3.20a .

=x 2 a y j Ž j q 1 . a y h 2 s 0.

Ž 3.20b .

They are diagonalized by the following linear combinations of the fields: s1 s b y 2 Ž j q 2 . Õ,

s2 s h 2 y 2 Ž j q 1 . a,

Ž 3.21a .

t 1 s b q 2 Ž j y 1 . Õ,

t 2 s h 2 q 2 ja.

Ž 3.21b .

They satisfy

= 2 si y j Ž j y 1 . si s 0,

Ž 3.22a .

= 2 t i y Ž j q 1 . Ž j q 2 . t i s 0.

Ž 3.22b .

In AdS 2 , the scaling dimension of the operator corresponding to a scalar field is given by w2,3x h s 12 Ž 1 q '1 q 4 m2 . .

Ž 3.23 .

This implies that the fields s1,2 have h s j and are chiral primaries, while t 1,2 have h s j q 2. It remains to analyze Ž3.14b. and Ž3.15a.. Inserting Ž3.15a. into Ž3.14b. and using Ž3.18., we find

=x 2 hŽI mn . y j Ž j q 1 . hŽI mn . q 2 hŽI mn . s 4=Ž m=n . a.

Ž 3.24 .

It is also possible to show that in two dimensions, Ž3.15a. implies

Ž = 2 q 2 . hŽ mn . s =Ž m=n . Ž h 2 q 4 a . .

Ž 3.25 .

It can be derived most easily in a light-cone coordinate and a conformal gauge. Combining these two equations, we find that hŽ mn . is algebraically determined by h 2 and hence has no degree of freedom. This argument is valid for j s 1 also, but not for j s 0. 4

This type of transformations appear in other compactifications with electric background field strength. See, for example, Ref. w16x.

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3.2.5. j s 1 The computation for j s 1 differs from that for j 0 2 in two ways. First, h1 is removed by a gauge choice rather than the constraint Ž3.13b. which is absent because =Ž a=b . Y Ž js1. s 0. Second, Õm and h 2 has no bulk degree of freedom and can be eliminated. The equations can be diagonalized as before, and the three eigenstates are identified with the j s 1 points of the t 1 , t 2 and s2 series. The absence of the corresponding point on the s1 series is a consequence of the fact that Õm is a gauge field. Note also that s2 can be gauged away on shell by a residual gauge degree of freedom. By X in Ž3.9b. with = 2 X s 0, s2 is shifted to s2 q X. Therefore it is also a boundary degrees of freedom. 3.2.6. j s 0 We have the field equations for h1 , h 2 and am

=x 2 Ž h1 q h 2 . s y4e mn =m an y 2 h1 ,

Ž 3.26a .

=x 2 h 2 s 4e mn =m an y 2 h 2 q 4 h1 ,

Ž 3.26b .

= n Ž =n am y =m an . s emn= n Ž h 2 y h1 . .

Ž 3.26c .

Recall that we gauged away hŽ mn . . Its equation of motion then gives a ‘‘Gauss law’’ constraint,

=Ž m=n . h 2 s 0.

Ž 3.27 .

One can easily show that Žin light-cone coordinates, for example. the only normalizable solution to the constraint is h 2 s constant. It is consistent to set h 2 to zero. We can eliminate the gauge field am from the h1 equation and find that m2 s 2. This is identified with the j s 0 point of the t 2 series. This completes the derivation of the bosonic spectrum in Fig. 1. 3.3. Fermionic mass spectrum The linearized field equation for the fermion reads

G

mn p

=n cp s y

i 2

Ž F m n q 12 Fr s G m n r s . cn .

Ž 3.28 .

The linearized SUSY transformation law plays the role of a gauge symmetry, that is, the following variation leaves the field equation invariant:

dcm s =m e y iFn l

ž

1 4

G ldmn y

1 8

Gm

nl

/

e.

Ž 3.29 .

It is convenient to separate the ‘‘trace’’ and the ‘‘traceless’’ part of cm and ca .

cm s cŽ m . q 12 Gm l ,

ca s cŽ a . q 12 Ga h

Ž G m cŽ m . s G a cŽ a . s 0 . . Ž 3.30 .

We decompose the D s 4 gamma matrices in terms of the D s 2 gamma matrices as follows:

G m s g m m 1,

G a sgmt a

Ž g s g 0g 1 . .

Ž 3.31 .

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Now we can split Ž3.28. into four components a

ž =u q g=u / h q g=u l y 2= c ž =u q g=u / l q =u h y 2= c x

y

x

y

Ž a . s yi gl ,

Ž 3.32a .

Ž m . s yi gh ,

Ž 3.32b .

y

m

x

=Ž m .h q g=uy cŽ m . s yi gcŽ m . ,

Ž 3.32c .

=Ž a . l q =ux cŽ a . s qi gcŽ a . ,

Ž 3.32d .

where the first two equations are the traceless parts of the m and a components of Ž3.28., respectively. The other two are the trace parts. Here, =ux ,=uy are the two-dimensional dirac operators. The gauge transformation law also divides into four pieces:

dcŽ m . s =Ž m . e ,

dl s =ux e y i ge ,

Ž 3.33a .

dcŽ a . s =Ž a . e ,

dh s g Ž =uy e q i e . .

Ž 3.33b .

Consider the spherical harmonics decomposition, I I l s lq SqI q ly SyI ,

cŽ m . s cŽ Im .q SqI q cŽ Im .y SyI ,

h s hqI SqI q hyI SyI ,

cŽ a . s cqI =Ž a . SqI q cyI =Ž a . SyI ,

e s eqI SqI q eyI SyI .

Ž 3.34 .

See Appendix B for the definition and properties of spinor spherical harmonics. For j 0 3r2, it is clear that one can gauge away h completely. Then Ž3.32c. sets cŽ Im . s 0. In turn, we find in Ž3.32b. that l satisfies the eom for a free massless spinor in d s 4. Finally, Ž3.32a. determines cŽ a . algebraically in terms of l. As a consistency check, we substitute it into Ž3.32d. and find the same eom for l. Thus all that remains is to find the mass spectrum of l. After the spherical harmonics decomposition, the equation reduces to

=ulqq i Ž j q 12 . glqs 0,

=ulyy i Ž j q 12 . glys 0.

Ž 3.35 .

The mass eigenstates are j 1 s Ž1 q i g . E and j 2 s Ž1 y i g . F both of which have m s j q 12 .5 The computation is slightly different for j s 1r2. To begin with, we note the following property of the j s 1r2 spherical harmonics: i =a S "s " ta S "´ =uy S "s "i S " . 2

Ž 3.36 .

It has three consequences. First, modes for cŽ a . are absent. Second, Ž3.32d. is trivially satisfied. Finally, the gauge variation of hy vanishes for arbitrary ey. We choose to gauge away hq and cŽ m .y using eq and ey, respectively. With these in mind, we 5

We may choose Ž1y ig . E and Ž1q ig . F. The two choices are not independent, since one can multiply either of them by g to get the other.

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analyze the three field equations. From the coefficients of Sq in Ž3.32a. and Ž3.32c., we find that

lqs 0,

cŽ m .qs 0.

Ž 3.37 .

The coefficients of Sy of the same equations yield

Ž =ux y ig . hys 0,

=Ž m .hys 0.

Ž 3.38 .

These two equations together imply that hy has no propagating degree of freedom and can be set to zero consistently. Finally Ž3.32b. gives

Ž =ux y ig . lys 0, which we recognize as the j s 1r2 point of the j 2 series.

Fig. 2. The KK spectrum of the Ds11 SUGRA on AdS 2 = S 2 =T 7.

Ž 3.39 .

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The scaling dimension of the operator corresponding to a spinor field in AdS 2 is given by h s < m < q 12 , Ž 3.40 . which implies that the fields j 1,2 have h s j q 1. 4. Summary of the main result We now turn to the model which is the main interest of this paper, namely, the one obtained from the low energy M-theory. We first dimensionally reduce D s 11 SUGRA to obtain D s 4. The resulting N s 8 SUGRA contains 1 graviton, 8 real gravitini, 28 vectors, 56 real spinors and 70 scalars. Compactification on AdS= S 2 keeps N s 2 SUSY unbroken. In the N s 2 language, we have 1 gravity, 6 gravitino, 15 vector and 10 Žcomplex. hyper multiplets. Each multiplet has 4 bosonic and 4 fermionic real degrees of freedom. The 4 q 4 KK towers arrange themselves into representations of SUŽ1,1 q 2. superalgebra. Fig. 2 describes the KK spectrum of each multiplet. The gravity multiplet is identical to that of the toy model. The vector multiplet is similar to the gravity multiplet, but it has two copies of the n s 2 representation. Gravitino multiplet contains two copies of representations for all odd n except for n s 1. Hyper multiplet includes the n s 1 representation. The analysis for the boundary modes is more complicated since one has to keep track of modes which may be removed by fixing gauges. We will concentrate on obtaining bulk modes. As in the toy model, we included the boundary degrees of freedom for the gravity multiplet in the figure. Boundary degrees of freedom can arise in the gravitino multiplet as well, but are not determined by the computation here. In Sections 5 and 6 we explain the dimensional reduction of the field equations from D s 11 to D s 4, how different fields fall into N s 2 multiplets and how each multiplet produces the KK spectrum given in Fig. 2. 5. Bosonic mass spectrum 5.1. Setup We normalize the fields such that the action reads 1 1 2 k 2 S s d 11 x'y G R y F 2 y CI G I JK=JCK q AnFnF 2 P 4! 3! 1 q d 11 x'y G C Ž G I JK L M NCJ FK L M N q 12 G K L G M J FKI L M . CJ . 4 P 4! I Ž 5.1 .

½

H

H

5

½

H

5

The terms quartic in CM are not relevant to this paper and have been omitted. Bosonic equations of motion consist of the Einstein and Maxwell equations in vacuum. 1 1 RMN s FM I JK FN I JK y G F2, Ž 5.2a . 2 P 3! 6 P 4! M N

J. Lee, S. Lee r Nuclear Physics B 563 (1999) 125–149

= M FM I JK s

1 2 P 4!P 4!

e I JK L 1 L 2 L 3 L 4 M 1 M 2 M 3 M 4 F L1 L 2 L 3 L 4 F M 1 M 2 M 3 M 4 .

137

Ž 5.2b .

The AdS 2 = S 2 = T 7 solution is given by 2 ds11 s gmn dx mdx n q g a b dx a dx b q d a b dz adz b q d st dw sdw t ,

Rmnls s y Ž gm l gns y gms gnl . , Fmn 45 s Fmn 67 s emn ,

R a bgd s g ag g bd y g a d g bg ,

Fa b 46 s Fa b 75 s ea b .

Ž 5.3 .

All other fields are set to zero. This is the near horizon geometry of the brane configuration shown in the introduction. The notational conventions for the indices are summarized in Appendix A. Note that the background fields are self-dual in the SO Ž4. for the four coordinates along which the branes lie. Equivalently, it transforms in the Ž0,1. representation of SO Ž4. , SUŽ2.q= SUŽ2.y. This follows from the requirement of partially unbroken supersymmetry. If any one of the relative sign for the gauge field is flipped, the supersymmetry is completely broken, even though it is still a solution to the equations of motion.6 This background breaks the SO Ž7. isometry of the internal T 7 down to SUŽ2.q= SUŽ2. 3 , where SUŽ2. 3 is the rotation of w 8,9,10 . These internal symmetries will play a crucial role in grouping the fields, as will be shown in the next section. 5.2. Linearized field equations and reduction to D s 4 We linearize the equations in D s 11 in fluctuations around the background, GM N s g M N q h M N ,

FI JK L s FI JK L q 4=w I a JK Lx ,

Ž 5.4 .

and then dimensionally reduce it to D s 4 by keeping only the zero-modes of the fluctuations in internal T 7 s T 4 = T 3. We then redefine some of the fluctuation fields, 1 Ž4. a s hŽ11. m n s h m n y 2 Ž Ba q Bs . g m n ,

aab c s C ab c ,

a m a b s A amb ,

h ab s B a b , a m n a s Dma n ,

h m a s Vma ,

Ž 5.5 .

The definitions of B,V,C, A, D remains valid when we replace the a,b indices by s,t indices. The shift in h m n is the linearized version of the Weyl rescaling which is necessary to absorb the volume factor of the internal dimensions and put the action into the standard Einstein–Hilbert form. Also, one can do Hodge dual transformation to reduce the indices of the tensor fields. The tensor field with three index, a m n l is the most trivial one, its dual field having rank y1 formally. This implies it has no dynamics. Indeed, one can show explicitly from its equations of motion that it has no degree of freedom and decouples from all the other 6

In fact, supersymmetry requires that the product of the signs of the gauge field be q1. We set all the signs to be q1 using coordinate redefinition and parity transformation.

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Table 2 Internal quantum numbers of the bosonic fields Field

SUŽ2.q

SUŽ2.y

SUŽ2. 3

Group

hm n Vma Vms BŽ a b. 3 Baa q2 Bss Ba s BŽ st . BSs A ambq A amby A ams A smt C abc C s a bq C s a by C ast C stu

0 1r2 0 1 0 1r2 0 0 0 1 1r2 0 1r2 0 1 1r2 0

0 1r2 0 1 0 1r2 0 0 1 0 1r2 0 1r2 1 0 1r2 0

0 0 1 0 0 1 2 0 0 0 1 1 0 1 1 1 0

F D B E A C A F F E C B D B A C F

Da Ds Fma nb

1r2 0 0

1r2 0 1

0 1 0

D A

fields. The next one we consider is the rank two tensor field Dma n whose linearized equation of motion is

=

m

½=

a ab b w m Di j x q Fw i j Vmx

5 s 0,

Ž 5.6 .

which turns into an identity if we introduce the dual scalar 3= w l D m nxa q 3Fwai bj Vmxb s e l m n k=k D a .

Ž 5.7 .

Then the Bianchi identity for the original D l m n a turns into the equation of motion for D a, 1 = 2 D a s e k l m n Fkalb Wmbn , Ž 5.8 . 4 where Wman is the field strength of Vman . The equation remains valid when a is replaced by s, except that in this case the right-hand side vanishes. The quantum numbers of the various fluctuation fields with respect to the internal symmetries are summarized in Table 2, along with that of the background gauge field Fmanb. Using this table, one can divide the fields into small groups, where the fields belonging to the same group can couple to each other. The fields within a group must have the same quantum numbers except the broken SUŽ2.y charge, which can be shifted by 1 by the background field. We label these groups by capital Roman letters. Note that we separated the self-dual and the anti-self-dual parts of the fields which are rank two tensors in SO Ž4. by A a b "' 12 Ž A a b " e a b c dA c d . .

Ž 5.9 .

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The linearized equations of motion in D s 4 are given by Group A

= 2 BŽ st . s = 2 C s a bys = 2 D s s = 2 Ž 3 Baa q 2 Bss . s 0.

Ž 5.10 .

Group B 1 = 2 C s a bqs 12 Fmanb Wms n y e k l m n Fkalb Fms n , 4 1 = n Fnsm s y e m n k l Fnakb =l C s a bq Ž Fms n ' 12 e st u Fmt un . , 4

Ž 5.11a . Ž 5.11b .

= n Wnsm s 12 Fmanb = n C s a bq.

Ž 5.11c .

Group C 1 = 2 C a s s e k l m n Fkalb Fmbns 4

Ž C a s ' 12 e st u C at u . ,

Ž 5.12a .

= 2 B a s s 12 Fmanb Fmbns ,

Ž 5.12b .

= n Fnams s yFmanb= n B b s y 12 e m

nkl

Fnakb =l C b s .

Ž 5.12c .

Group D 3 = 2 C a b c s Wmw an Fmbncx , 2 1 = 2 D a s e k l m n Fkalb Wmbn , 4

= n Wnam s y 12 e m

nkl

Ž 5.13a . Ž 5.13b .

Fnakb =l D b q 12 Fmbnc= n C a b c .

Ž 5.13c .

Group E 1 1 cb b cy c a ac b d Ž c d . = 2 B Ž a b. s 12 Fmacy , n Fm n q 2 Fm n Fm n q 2 Fm n Fm n B

Ž 5.14a .

= n Fnamby s Fmbnc= n B Ž c a. y Fmacn= n B Ž cb. .

Ž 5.14b .

Group F 1 = 2 C s e k l m n Fkalb Fmabq n , 8 1 = 2 Bss s Fmanb Fmanbq y 12 Fmakb Fmalb h k l , 2

Ž 5.15a . Ž 5.15b .

= n Fnambq s Fkalb =k h l m y 12 Fmanb Ž 2= k h k n y =n h kk y =n Bss . q 12 e m

k ln

Fkalb =n C,

Ž 5.15c . 1 a b a bq R Ž1. q 12 Fnakb Fmakbq y m n Ž h . s 2 Fm k Fn k

1 6

g m n Fkalb Fkalbq y 12 Fmakb Fnal b h k l

1 1 1 q g m n Fkajb Fkalb h jl q Fmakb Fnakb Bss y = 2 Bss . 6 4 6

Ž 5.15d .

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5.3. Computation of mass spectrum in each multiplet We have separated fields which decouple from one another using their internal quantum numbers. We should now disentangle the field equations further and find out which field belong to which N s 2 multiplet. Obviously, the bosonic fields in the same multiplet satisfy the same field equations, and the same for the fermions. In this section, we jump to solve the equations of motion of fields in each multiplet, except for the gravity multiplet which has been analyzed in detail in Section 3. The reduction of the equations obtained in the previous subsection to the final form require somewhat lengthy algebra, and we put it off until the next subsection. The complication partly arises from the fact that we chose a specific D s 11 configuration from the beginning. Although the M theoretic origin of the geometry is manifest in this framework, the U-duality invariance of the D s 4 theory and its symmetry breaking pattern is obscured. A manifestly duality invariant approach sketched in w21x could simplify the process to a large extent. 5.3.1. Hyper multiplet Minimally coupled scalars in D s 4 belong to this multiplet. Clearly, the KK modes have m2 s jŽ j q 1.. It follows that h s j q 1. There is no gauge symmetry associated with the scalars. 5.3.2. Vector multiplet A vector multiplet contains a vector A m and two real scalars f 1 , f 2 . In the simplest case, f 1 couples to Aa only and f 2 to Am . The field equations for the first group are

Ž =x 2 q =y 2 y 2 . f 1 s

1 2

e a b Fa b ,

Ž 5.16a .

b

m

= Fm a s 4ea b= f 1 .

Ž 5.16b .

In the same gauge as in the toy model, Aa is expanded in the spherical harmonics as Aa s b Iea b= b Y I .

Ž 5.17 .

We then get the equations

ž

=x 2 y j Ž j q 1 . y 2

yj Ž j q 1 . 2

=x y j Ž j q 1 .

y4

f1 s 0. b

/ž /

Ž 5.18 .

along with the constraint

=m A m s 0.

Ž 5.19 .

For j 0 1, one finds that the mass eigenvalues are m2 s j Ž j y 1 . , Ž j q 1 . Ž j q 2 . ´ h s j, j q 2.

Ž 5.20 .

2

For j s 0, b is absent and f 1 has m s 2,h s 1. The field equations for f 2 and Am are

Ž =x 2 q =y 2 q 2 . f 2 s n

n

= Fn m s 4emn= f 2 .

1 2

e mn Fmn ,

Ž 5.21a . Ž 5.21b .

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As in the toy model, one can use the constraint = mAm s 0 to set Am s emn= n a. The equations then become,

ž

=x 2 y j Ž j q 1 . q 2

y=x 2

y4

=x 2 y j Ž j q 1 .

/ž

f2 s 0. a

/

Ž 5.22 .

For j 0 1, one finds the same mass eigenvalues as for f 1 and Aa . For j s 0, Am is a gauge field in D s 2 and can be eliminated, leaving f 2 with m2 s 2,h s 1. 5.3.3. GraÕitino multiplet Minimally coupled vectors in D s 4 belong to this multiplet. One obtain two D s 2 scalars with m2 s jŽ j q 1. for all j 0 1. For j s 0, the mode for Aa is absent and Am becomes a gauge field in D s 2, so there is no bulk degree of freedom. 5.3.4. GraÕity multiplet This multiplet was analyzed for the toy model case. Conformal weights of the bosonic states satisfy h s j, j q 2. 5.4. Grouping D s 4 fields into N s 2 multiplets Group A All the fields in this group are minimally coupled scalars in D s 4 and belong to the hypermultiplet. Group B It is convenient to dualize Fms n by defining F˜ms n ' y 12 e m n k l Fksl y Fmanb C s a bq.

Ž 5.23 .

Then the equation of motion of F s becomes the Bianchi identity for F˜ s, and the Bianchi identity for F s become

= n F˜nsm s Fmanb= n C s a bq.

Ž 5.24 .

Note that the right-hand side is the same as that of Ž5.11c.. In terms of F˜ s, the field equation for C becomes

= 2 C s a bqs 12 Fmanb F˜ms n q Wms n q Fmc nd C sc dq .

ž

/

Ž 5.25 .

Clearly, F˜ s y W s decouple from C and contribute to the 3r2 multiplet. Since C is coupled to F˜ s q W s by F a b , we find that C s47q decouples and contribute to the hypermultiplet. The remaining fields belong to the vector multiplet. In particular, C s45q couples to A˜ms q Vms and C s46q couples to A˜as q Vas Group C Writing down all the components of the field equations and collecting those which couple to one another, one finds twelve identical copies of the following set of coupled equations:

= 2 B s y= 2 C s 12 Ž ye mn Fmn q e a b Fa b . ,

Ž 5.26a .

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= n Fn m s emn= n Ž yB q C . ,

Ž 5.26b .

= n Fn a s ea b= b Ž B y C . .

Ž 5.26c .

As before, we set Aa s b Iea b= b Y I ,

Am s AmI Y I ,

Am s emn= n a.

Ž 5.27 .

It is then easy to show that Ž B q C . belong to the hypermultiplet, Ž a q b . to the gravitino multiplet, and Ž B y C . and Ž a y b . together to the vector multiplet. Group D If we define Cas

1 3!

e abcdC bcd ,

we find that the field equations have exactly the same structure as those in the previous group. Group E First, B45 q B67 , B46 y B75 and B44 y B55 y B66 q B77 decouple and contribute to the hyper multiplet. The other equations fall into six groups each of which contains one scalar and one vector. Each group gives the spectrum for half a vector multiplet. Explicitly, the six groups are

Ž B 44 q B 55 y B 66 y B 77 , Am45y . ,

Ž B 44 y B 55 q B 66 y B 77 , A46y ., a

Ž B 46 q B 75 , Am47y . ,

Ž B 45 y B 67 , A47y ., a

Ž B 47 y B 56 , Am46y . ,

Ž B 47 q B 56 , A45y .. a

Ž 5.28 .

Group F First, h 2 ' h a a , Bss, Am45q and A46q belong to the gravity multiplet. Second, hm a , a 46q Am and A45q ,C belong to the vector multiplet. Finally, Am47q and A47q decouple and a a contribute to the gravitino multiplet Žsee Table 3.. Table 3 Summary of the number of degrees of freedom a group of bosonic equations contribute to each of the four multiplets Group A B C D E F Sum

grav 0 0 0 0 0 4 4

3r2

vec

hyper

Total

0 4 12 6 0 2 24

0 8 24 12 12 4 60

18 4 12 3 3 0 40

18 16 48 21 15 10 128

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6. Fermionic mass spectrum

6.1. Linearized field equations and reduction to D s 4 The linearized field equation for the gravitino in D s 11 reads

G I JK=JCK s

1 4 P 4!

G I JK L M NCJ FK L M N q

1 8

G JKC L F I JK L .

Ž 6.1 .

Throughout this section we suppress the bar on the background field strength. The linearized local SUSY transformation law plays the role of gauge symmetry for fermions:

dCM s =M e q

1 12 P 4!

FI JK L Ž 8 d MI G

JK L

yG

I JK L M

.e.

Ž 6.2 .

In dimensional reduction to D s 4, it is convenient to define

l s G a Ca , x s G sCs ,

CŽ a. s Ca y CŽ s. s Cs y

1 3

1 4

Ga l ,

Ga x .

Ž 6.3 .

The following shift in the D s 4 spin 3r2 fields bring their kinetic term into the standard form,

CmŽ11. s CmŽ4. y 12 Gm Ž l q x . .

Ž 6.4 .

We then decompose the fermion into chiral and anti-chiral components with respect to SO Ž4. of T 4 ,

C "' 12 Ž 1 " G . C ,

Ž 6.5 .

where G ' 4!1 e a b c d G a b c d. As in the previous section, we can divide the field equations into a few groups using the internal symmetry Žsee Table 4.. After some gamma matrix

Table 4 Internal quantum numbers of the fermionic fields Field Cmq Cmy CŽqa. CŽya. Ž3 l q2 x .q Ž3 l q2 x .y CŽqs. CŽys. xq xy

SUŽ2.q 1r2 0 1 1r2 1r2 0 1r2 0 1r2 0

SUŽ2.y 0 1r2 1r2 1 0 1r2 0 1r2 0 1r2

SUŽ2. 3 1r2 1r2 1r2 3r2 1r2

Group I J H I G H G H I J

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algebra, one finds that the field equations and gauge transformation laws in D s 4 are given by Group G q

G n =nCŽqs.s G n =n Ž 3 l q 2 x . s 0,

Ž 6.6a .

q

dCŽqs.s d Ž 3 l q 2 x . s 0.

Ž 6.6b .

Group H

G n =nCŽqa.s

1

Fiaj dG i j CŽqd . ,

4

1

y

G n =n Ž 3 l q 2 x . s 1

G n =nCŽys.s y

16

16

Ž 6.7a .

Ficj dG

ij

y

G cd Ž 3l q 2 x . ,

Ficj dG i jG c dCŽys.,

Ž 6.7b . Ž 6.7c .

y

dCŽqa.s d Ž 3 l q 2 x . s dCŽys.s 0.

Ž 6.7d .

Group I 1 G n =nCŽya.s y Fiaj bG 8

G

mnk

b

 G i jxqq Ž G i jk y 2 G id jk . Cqk 4 ,

1 ab =nCq F ŽG k sy 8 ij

G n =n xqs dCŽya.s

1 8

1 4

mi j

y 2 d m iG j . G a CŽyb.,

Ž 6.8b .

Fiaj bG i jG a CŽyb.,

Ž 6.8c .

Fiaj bG bG i jeq,

dCmqs =m eq,

Ž 6.8a .

Ž 6.8d .

dxqs 0.

Ž 6.8e .

Group J

G n =n xys y G

mnk

=nCy ks

1 16

Ficj dG c d Ž G

i jk

y 2 G id jk . Cy k ,

1

Ž 6.9a . 1

Ž F a bG i jm n q 2 Fmanb . G a b Cnyq 32 Fiaj b Ž G m i j y 2 d m iG j . G a bxy, 16 i j Ž 6.9b .

dCmys =m eyy dxys y

1 16

1 32

Fiaj b Ž G

mi j

Fiaj bG a bG i jey.

y 2 d m iG j . G a bey,

Ž 6.9c . Ž 6.9d .

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6.2. Computation of the mass spectrum in each multiplet 6.2.1. Hyper multiplet Spinors with non-zero mass generated by the background gauge field belong to this multiplet. After diagonalizing the mass matrix, they satisfy the equation of motion of the form

G m =m c q i G 01c s 0.

Ž 6.10 .

1 2

One finds that < m < s j q " 1, which implies that h s j, j q 2. There is no gauge symmetry acting on the spinors in this multiplet. 6.2.2. Vector multiplet Minimally coupled massless spinors in D s 4 belong to this multiplet. One easily finds that h s < m < q 12 s j q 1 for all j 0 12 . There is no gauge symmetry acting on the spinors in this multiplet. 6.2.3. GraÕitino multiplet A gravitino multiplet contains a gravitino c and a spinor x . In the same notation as in the toy model, their coupled equations of motion break up as follows:

ž =u q g=u / x s yi g Ž h y l. , ž =u q g=u / h q g=u l y 2= c ž =u q g=u / l q =u h y 2= c

Ž a . s yi gx ,

Ž 6.11b .

Ž m. s i g x ,

Ž 6.11c .

y=Ž m .h q g=uy cŽ m . s 0,

Ž 6.11d .

y=Ž a . l q =ux cŽ a . s 0.

Ž 6.11e .

x

y

x

y

x

y

a

y

m

x

Ž 6.11a .

where we expressed the four-dimensional gamma matrices as tensor products of two dimensional ones as in the case of the toy model, and =ux ,=uy are the two-dimensional dirac operators. The gauge transformation laws are given by

dcŽ m . s =Ž m . e ,

dl s =ux e ,

dcŽ a . s =Ž a . e ,

dh s g=uy e ,

dx s yige .

Ž 6.12 .

One can always gauge away h. Then Ž6.11d. sets cŽ m . to zero. For j 0 3r2, Ž6.11b. determine cŽ a . algebraically. The only independent equations that remain are Ž6.11a. and Ž6.11c. with h and cŽ m . removed. The mass eigenvalues are the same as those of hyper multiplet: < m < s j q 12 " 1, h s j, j q 2. For j s 1r2, the modes for cŽ a . are absent and Ž6.11e. is trivially satisfied. Eq. Ž6.11b. gives an algebraic relation between l and x . So the number of degrees of freedom is reduced by half. One finds h s j q 2 for all modes. 6.2.4. GraÕity multiplet The equations satisfied by this multiplet was analyzed for the toy model case. One finds h s j q 1, with number of degrees reduced by half for j s 1r2.

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6.3. Grouping D s 4 fields into N s 2 multiplets Group G All the spinors in this group are massless and minimally coupled in D s 4 and belong to the vector multiplet. Group H We choose the following basis for SO Ž4. gamma matrices:

G 4s

ž

0 1

1 , 0

/

G 5,6,7 s i

ž

0

s 1,2,3

ys 1,2,3 , 0

/

Ž 6.13 .

where s i are the Pauli matrices. In this basis, an SO Ž4. spinor splits into two chiral spinors as

hs

hy . hq

ž /

Ž 6.14 .

Consider Ž3 l q 2 x .y first. To simplify the equations, we use the letter C to denote Ž3 l q 2 x .y in the equations to follow. In the basis we choose, the field equation reduces to

G n =nC s

i 2

Ž s 1 m G 01 q s 2 m G 23 . C .

Ž 6.15 .

We can further decompose the equation by setting

Cs

C1 . C2

ž /

Ž 6.16 .

After splitting each of C 1,2 into two pieces according to their chirality in the non-compact D s 4 space-time, and recombining those pieces which couple to each other, one finds that one linear combination belongs to the vector multiplet and the other one to the hyper multiplet. The spectrum is exactly the same for CŽqs. except that it has twice as many degrees of freedom as 3 l q 2 x . We find the same result even for CŽqa. again except for the degeneracy. In counting the degeneracy, one should remember the constraint G aCŽ a. s 0. In the basis we chose above, it reduces to

CŽ4. y i s 1CŽ5. y i s 2 CŽ6. y i s 3 CŽ7. s 0.

Ž 6.17 .

Group I Doing the same sort of recombination of spinors as above, one finds the following results. 1. A third of CŽya. decouple from all the other fields. They belong to the vector multiplet. 2. Another third of CŽya. couple to xq. They contribute to the hypermultiplet. 3. The last third of CŽya. couple to Cmq. They belong to the spin 3r2 multiplet.

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Table 5 Summary of the fermionic spectrum Group

grav

G H I J Sum

0 0 0 4 4

3r2

vec

hyper

Total

0 0 16 8 24

24 24 8 4 60

0 24 16 0 40

24 48 40 16 128

Group J 1. A half of xy decouple and belong to the vector multiplet. 2. A half of Cmy decouple and satisfy the same equation as the gravitino in the toy model. Therefore they belong to the gravity multiplet. 3. The other components of xy and Cmy couple to each other. They contribute to the spin 3r2 multiplet. Table 5 summarizes the result of this section.

Acknowledgements We are grateful to Seungjoon Hyun for many helpful discussions, Youngjai Kiem for carefully reading the manuscript and making useful comments, and Jan de Boer for a correspondence. S.L. would like to thank Jeremy Michelson for discussions at Cargese ` ’99 ASI.

Appendix A. Notations and conventions We consider D s 11 SUGRA on AdS 2 = S 2 = T 4 = T 3. Each manifold in the product is parametrized by x m Ž m s 0,1., y a Ž a s 2,3., z a Ž a s 4,5,6,7. and w s Ž s s 8,9,10., respectively. We use the indices Ž M, N, . . . . to label all eleven coordinates together and Ž m,n, . . . . to label the coordinates of AdS 2 = S 2 . The signature of the metric is Žyq . . . q.. The field strength of a p-form potential in any dimension is defined by FM 0 . . . M p s p=w M 0 A M 1 . . . M p x .

Ž A.1 .

Appendix B. Spherical harmonics The spherical harmonics form a basis for the fields living on a sphere. In this appendix we consider only the case of S 2 . We can construct them by considering the eigenstates of maximal commuting subalgebra of SUŽ2. group, which are the total angular momentum J 2 s jŽ j q 1., its z component Jz s m, the orbital angular momen-

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tum L2 s l Ž l q 1., and the spin S 2 s sŽ s q 1.. The case for the scalar is easiest since s s 0 and J 2 s L2 , which we identify with the Laplacian on the sphere, =y 2 , where y indicates two-dimensional coordinates parametrizing the two-sphere. This expression for L2 can be obtained by embedding S 2 into three-dimensional space, writing down L2 in terms of the Cartesian coordinates, which is quite well known, and reexpressing these in terms of polar coordinates. Therefore, by construction, we have

=y 2 Y Ž j, m. Ž y . s yj Ž j q 1 . Y Ž j, m. Ž y . ,

Ž B.1 .

where Y Ž j, m. Ž y . denotes the eigenstates with eigenvalues Ž j,m., which were defined above. Next consider the spinor spherical harmonics where s s 1r2. The easiest way to consider it is to embed the sphere in the three-dimensional space and use cartesian coordinates. We construct them by taking the tensor product of scalar spherical harmonics with 2-component spinor, and taking appropriate linear combinations. We then get the expression w28x

S ljsl " 1r2 , m s

1

'2 l q 1



( (

"

l"mq

l.mq

1 2

1 2

Yl my 1r2 Ž u , f .

Yl

my 1r2

Ž u ,f .

0

,

Ž B.2 .

where the labels indicates the eigenvalues as usual, and all the harmonics are normalized to unity. For given j, the only possible values of l are j " 12 , so the degeneracy is 2Ž2 j q 1.. However, it is convenient for our purpose to group the spherical harmonics of given j according to the eigenvalue of J˜' =uy rather than m,l, where =uy ' t a =a is the two-dimensional Dirac operator on the sphere with ta given by the usual Pauli matrices. One can show that J 2 s =uy 2 y 14 ,

Ž B.3 .

by comparing the two-dimensional operator with the expression in the embedding three-dimensional cartesian coordinates, so it is obvious that J˜ commutes with J 2 , and its eigenvalues are "Ž j q 12 .. However, it turns out that neither of L2 nor Jz commutes with J.˜ Therefore we have to find two other operators which commute with J 2 , J˜ in order to distinguish linearly independent spherical harmonics. We will not identify them since they are not needed for the present discussion. Note that the chirality operator t ' 12 eab t at b anticommutes with J.˜ Therefore given an eigenstate with J˜) 0, which we denote by Sqj , we have a counterpart Syj ' tSqj and vice versa.7 This immediately implies that there are same number of Sqj states and Syj states for given j. Since the degeneracy of total states is 2Ž2 j q 1., we have 2 j q 1 Sqj Žor Syj . states. We also state without proof that the lowest spinor spherical harmonics are killing spinors, satisfying the relation

ž 7

=a .

i 2

1r2 ta S " s 0.

/

Our notations closely mimic those in w17x, but the convention for the " sign is flipped.

Ž B.4 .

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149

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x

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