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Spectrum generating algebras in Kaluza-Klein theories
Volume 146B, number 6
PHYSICS LETTERS
25 October 1984
SPECTRUM GENERATING ALGEBRAS IN K A L U Z A - K L E I N THEORIES ¢~
M. GONAYDIN, L.J. ROMANS and N.P. WARNER a
California Institute of Technology, Pasadena, CA 91125, USA Received 25 June 1984
We describe the action of the euclidean conformal group on spheres. Using the example of S 7 compactification of the eleven-dimensional supergravity we show how the full spectrum provides unitary representations of SO(8,1). Our methods can be applied to compactification on spheres, products of spheres or any other manifold on which there is an action of a conformal group. We also make some conjectures concerning the relationship between the conformal group and supersymmetry.
One may define an action o f the euclidean conformal group SO(n + 1, 1) on the sphere, S n. While the conformal generators do not correspond to isometries and therefore do not commute with the Laplace and Dirac operators, their action simply induces a rescaling of the metric and this may be used to generate the spectrum of the Laplace and Dirac operators. We denote the cartesian coordinates in R n+l by x i and take S n to be the unit sphere z n + l x l x i = 1. Let ~7= diag(1,1 ..... 1 , - 1 ) a n d g be an element of SO(n + 1,1) such that grogT = 7. Define submatrices o f g according to
then the action o f g on S n is defined by
(gx) i = (Ail'x ] + Bi)(Ck x k + D) - 1 .
(2)
From this one can define the action o f g - 1 on functions f: S n --> C by (Vw(g - 1 ) f ) ( x i) = (Ck x k + D)W f((gx)i) ,
(3)
where co is an arbitrary complex number which we refer to as the weight. Consider the Hilbert space 1_2(Sn) of square inteWork supported in part by the US Department of Energy under Contract No. DEAC-03-81-ER40050 and by the Fleischmann Foundation. 1 Weingart Fellow. 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
grable (complex valued) functions over S n. The inner product in I_2(Sn) is simply
(I1, I2) =- f fl(X) I2(x)
dnx ,
(4)
Sn
where v g d nx is the SO(n + 1) invariant measure on S n. By a simple application of the change of variable formula for integration, one may show that this inner product is preserved under the action (3) of SO(n+ 1,1) provided one takes co = - n / 2 + ip where p is any real number. Thus the space L2(S n) provides a unitary representation o f S O ( n + 1,1) [1]. The unitary representations of a group are labeled by the eigenvalues of the Casimir operators. If one realizes a unitary representation of a group G in L2(M) where M is a rank one symmetric space, e.g., S n, then all but the quadratic Casimir operator vanish [2]. Such representations are referred to as " m o s t degenerate" [3]. Since the action o f S O ( n + 1,1) defines a diffeomorphism on S n, one may generalize its action to tensor fields in the obvious way [4]. However, for our purposes it is simplest to first define the infinitesimal action of this group. On the sphere, S n, there are (n + 1) scalar fields ~i which are the eigenmodes of the L a p l a c e - B e l t r a m i operator [] with eigenvalues - n . A useful basis for these functions is to take ~i = x i, for which one may derive the following identities: 401
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V ~ V ~ o i : - g a ~ i, ~oi~oi= 1, ~0iva~pi=0,
PHYSICS LETTERS (5,6,7)
(V,~0i)(vt~ oi) =gat3, ( V ~ 0 i ) ( v , ~ 0/) + ~oi~o / : 60.. (8,9) The vector fields K / = Vc~0 i are sometimes referred to as conformal Killing vectors since eq. (5) is equivalent to
LKigo<~ : -- 2~igo~ ,
(10)
where LKi is the Lie derivative. To define the action of SO(n + 1,1) on tensor fields of weight co we introduce the operator D / -= LKi co9 l. If one observes that the SO(n + 1) Killing vectors on Sn may be written as -
K~
-
= ~oivad
- - ~O/VoAO i ,
(11)
it is trivial to show that the operators L i/-~ LKi] and D / generate the Lie algebra o f S O ( n + 1,1). Thus one can also represent SO(n + 1,1) on the tensor fields on S n. The inner product is simply
(x,
r3=f vYdnx X m' ""mkYm 1 ...mk,
(12)
and the representation is unitary if and only if co = - n / 2 + io. Moreover, by combining the observation that we may t a k e ~oi = x i with the expressions for tensor harmonics given in ref. [5] one may use the identities eqs. ( 5 ) - ( 9 ) to explicitly calculate the action of D / on these modes. The only complication is that the action of D / does not preserve transverse gauge conditions. We may similarly define our action of SO(n + 1,1) on spinors X, by
LiJX = KiJavo~X + a ~ vijr,eCL,,
(13)
O i X = (VC~s0i)(V~X) + co~0in,
(14)
with the obvious generalizations to higher spins. The spinor harmonics on spheres may be obtained from simple linear combinations of products of tensor harmonics, gamma matrices and the Killing spinors [6]. In this way one sees that the spinorial modes also form representations o f S O ( n + 1,1), and Dido raises and lowers the eigenvalues. To illustrate these techniques we consider the compactification of the eleven-dimensional supergravity on S 7 [7]. In order to obtain the physical spectrum it 402
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was found necessary to fix most of the eleven-dimensional diffeomorphism and gauge invariances [6,8-10]. After the gauge fixing one is left with four-dimensional gauge and diffeomorphism invariances, and the internal local SO(8) symmetry. Such gauge choices are not compatible with the action of SO(8,1), but it is elementary to reinstate this conformal symmetry. Before describing how this may be achieved we should point out that the massive gravitons and gravitinos form representations of SO(8, 1) for any weight including the unitary choice co = - } + ip. This is because these fields appear as Fourier coefficients of scalar and spinor modes on S 7. There is no gauge fixing for these modes. For the remaining bosonic fields, the most convenient gauge choices are the seven-dimensional deDonder and Lorentz gauges of ref. [8]. As was observed in ref. [8], the gauge choices for the (eleven-dimensional) graviton are compatible with conformal invariance which in our conventions corresponds to taking co = 5. It is also straightforward to show that the Lorentz gauge choices for AMN P commute with D / provided one assigns the weights co = 5, 3, 1 forAc, uv, Ac~u and A~t~,r, respectively (/~, v are d = 4 space-time indices). Observe tllat in those cases when it is necessary to mix fields in order to form mass eigenstates these fields may be given the same weight. Thus the bosonic spectlum does form representations of SO(8,1) with the foregoing gauge choices. The point at which the conformal symmetry is broken in ref. [8] is when a singlet and octet of scalar modes which arise from the trace of the graviton are gauged away. By keeping these modes we can restore the SO(8,1) symmetry. The problem is that the foregoing choice of weights is not unitary with respect to the inner product (12). To put the fields of given spin into unitary representations of SO(8,1) one has two options: either remove all gauge fixing or find a gauge choice which commutes with the unitary action of SO(8,1), if such exists. Since no such gauge choice is presently known we shall adopt the former course. From refs. [8,9] it is easy to read off the massive states that are pure gauge. Some of these pure gauge modes by themselves form unitary representations of SO(8,1). It is therefore not necessary to include all the gauge modes in order to complete the physical spectrum into unitary representations of S0(8, 1). For example, in the notation of ref. [8] the exact part of the pure gauge term a~uvl, i.e., ~ [ u < 1 can be set equal
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to zero, since it cannot arise from an SO(8, 1) transformation of the physical spectrum. The fermion spectrum was determined in refs. [6,9] using the gauge choice I"mt~M = 0. This gauge is not preserved by the action of SO(8, 1) defined by eqs. (13), (14), for any weight, co. As for the bosons, we are not aware of an SO(8,1) invariant gauge choice and will therefore remove the gauge fixing. This introduces two new infinite massive towers of spin-half modes, which are Fourier coefficients of the sevendimensional Rarita-Schwinger modes [6] : Xa = (Vc~ -- ~m7F~) X, where X is a Dirac mode. It should be noted that if X = ~7+, a Killing spinor, then the mode Xa vanishes. This corresponds to the fact that the octet of massless gravitini contain no gauge modes. The pure gauge modes that have to be added in order to complete the physical modes into unitary representations of SO(8,1) are given in the last column of table 1. The SO(8) content of the spectrum of the S 7 compactification of the eleven-dimensional supergravity Table 1 In this table we have only included those gauge modes necessary to form unitary representations of SO(8,1). The index l takes the values 0,1,2,.... JP
SO (8) Dynkin labels physical modes
gauge modes
2+
(/,0,0,0)
3/2
(l,0,0,1) (l,0,1,0)
1+
(l,0,1,1)
(/, 1,0,0)
1-
(/,1,0,0) (t,l,0,0)
(l+ 1,0,0,0) (l+ 1,0,0,0)
1/2
(l+ 1,0,1,0) (t,0,0,1) (/, 1,1, O) q,l,0,1)
(l+ 1,0,0,1) (/,0,1,0)
0+
(1,0,0,0) (l + 2,0,0,0) q,2,o,o)
(0,0,0,0) (1,0,0,0) (l+ 2,0,0,0) (/+ 1,1,0,0)
O-
(l,0,2,0) (l,0,0,2)
(/,0,1,1)
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can be given in the Gel'fand-Zetlin basis [9] as well as the Dynkin basis [6,9]. In what follows we denote the former by (ml, m2, m3, m4)G_ Z and the latter by (nl, n2, n3, n4)D. The unitary irreducible representations (UIR) of SO(n + 1,1) have been studied in refs. [ 1 1 - 1 3 ] . One important property of the UIR's o f S O ( n + 1,1) is that irreducible representations of SO(n + 1) occur with multiplicity one or zero inside a given UIR. In the sequel we shall follow the notation of ref. [13] for the labeling of UIR's of SO(8,1). J = 2 + modes. These modes fall into the (I, 0, 0, 0)G- z = (l, 0, 0, 0)D, l ~> 0 representations of SO(8). This tower forms the UIR of SO(8, 1) which is denoted as D3(Z9,4 = - ~ ) . The number Z9, 4 is the same as the weight ~ defined above. j = 3 modes. These modes fall into two infinite towers of SO(8) representations (l+ "~, 1 ~-, 1 ~, 1 ~)G-Z 1, _r ~ ' + l 1 1 1~ =(/,0,1,0)D,/~>0" (/,0,0,1)D andt~ 5,7,~,--~)G-Z These two towers together form the UIR labeled DO 1 1. Z9,4 _ 7 + ip). ~],~,~, J = 1 + modes. The physical modes lie in the (l + 1, 1,1,0)G-Z = (l,0, 1, 1)D, l~> 0 representations of SO(8). They, together with the pure gauge modes in the (l + 1,1,0, 0)G-Z = (1, l, 0, 0)D representations, constitute the UIR D0(0,1,1 ; Z9, 4 = - z) of SO(8,1). J = 1 - modes. The physical modes fall into two identical towers consisting of the representations (l + 1,1,0,0)G_Z = (l, 1,0,0)D, l ~> 0. Each tower of the physical modes, together with the gauge modes (l+ 1,0,0,0)G-Z = (l+ 1,0,0,0)D, l~> 0 form the UIR DI(0,1 ;Z9,4 = -27-). J - - ½ modes. The physical modes fall into the following representations: (l+ ~,~-,~,--~)G-Z 3 11 h -- ~.t r, + 1,0, 1,0)D, 1'+1~." ~,$,~',$)G-Z1 1 1~ =(I,O,O, 1)D,(I+~,~,.~,_~)G_Z331 1~. 3311, _~'1 = (l, 1,1,0)D, (l+ ~,~,-~,~)G-Z - it, 1,0, 1)D, If> 0. With the exception of the lowest (l = 0) mode in the second tower these towers form the UIR D(~,7,~,Z9, 4 1 ts. _-27-+ ip) of SO(8,1). The mode ~1 1 i 1~ t~,~,~,7)G-Z together 3 1 1 1"~ with the pure gauge modes (l+ ~,~,~,~)G-Z -- - "tl + 1,0, 0,1)D and (l+ .t2 , ~1, 2.1. 1- Z ,--~)G -_ (l,0, 1,0)D, l~> 0, form the UIR D(~,~,~,Z9, 11 1l 4 = _ _ ~ + ip). J = 0 + modes. The physical modes fit into the SO(8) representations: (l, 0, 0, 0)G-Z = (l, 0, 0, 0)D, l ~> 0, (/, 0, 0, 0)G-Z = (/,0,0,0)D, l ~> 2, (l+ 2,2,0,0)G_ z = (/, 2, 0, 0)D, l ~> 0. The first tower of SO(8) representations above fall into the UIR D3(Z9,4 = 7 ) of S0(8,1). The second tower falls into an identical lAIR =
403
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of SO(8, 1) provided one adds the gauge modes (0,0, 0,0) and (1,0,0,0). The third tower together with the gauge modes (l + 2, 0, O, 0)G-Z = (l + 2, O, 0, 0)D and (l + 2, 1,0,0)G-Z = (l + 1,1,0,0)D fall into the UIR Dl(0,2; Z9,4 = --27). J = O- modes. The physical modes fall into the SO(8) representations (l + 1, 1, 1,--1)G-Z = (I,0,2,0)D and (l + 1,1,1,+I)G-Z = (l, 0, 0, 2)O, l ~> 0. These towers together with the gauge modes (l + 1, 1,1,0)G-Z = (l,0, 1,1)O form the UIR D°(1,1,1 ;Z9, 4 = _ z ) of SO(8, 1). In this letter we have considered the unitary action of the conformal group SO(8,1) with respect to the inner product defined by eq. (12). Here the physical and gauge modes had to be considered together (for J = 0,1, 1) in order to form the UIR's of SO(8,1). However, the physical modes of a given spin by themselves have the same SO(8) content as certain classes of UIR's of SO(8,1). This suggests that there may exist a different inner product and a modified SO(8,1) action such that the conformal group is realized unitarily on the physical sector alone. Interestingly enough, the only difficulty in fitting the physical modes into UIR's of SO(8, 1) arises from the s = ½ mode (0,0,0, 1)D. This mode can be completed to the finite dimensional 16 of SO(8, 1) by the addition of an s = ½ singleton representation [ 14,15 ]. The s = 0 partner of this singleton representation is just the unphysical octet discussed earlier. This difficulty could also be alleviated by choosing a four-dimensional space-time which breaks all supersymmetries. The s = -~ octet will then be eaten by the gravitini through the super Higgs mechanism. The results presented here can be applied to compactification on any manifold which admits a conformal group action such as spheres and products of two spheres '1. The simplest case of compactification on S 1 of the five-dimensional pure gravity was studied in ref. [16], in which the conformal action of SO(2, 1) on the spectrum was given. Contrary to an assertion therein one can assign unitary weights to all the modes. We have also studied the compactification on S 2 of the simple (gauged) supergravity in d = 5. In this case the conformal group is SO(3,1) whose UIR's have been extensively discussed [ 17]. Many of the ideas de,1 We thank J. Dorfmeister and G.J. Zuckerman for discussions on this point. 404
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scribed here take on a simpler form for this compactification [18]. The action of SO(8, 1) on spinor fields defined by eq. (14) is not necessarily the most general. One might be able to add to VaX a term proportional to FaX and still obtain a realization of SO(8, 1). Such an expression would naturally follow from supercovariance. With this modified action one might obtain a unitary representation of SO(8,1) on the purely physical modes. Perhaps the most interesting question is the relationship between supersymmetry and the conformal action. At the naive level when one reduces the elevendimensional theory down to four dimensions, keeping all the massive states, one obtains a theory with two infinite towers of supersymmetries corresponding to the Fourier modes of the eleven-dimensional supersymmetry generator. The usual octet of four-dimensional supersymmetry generators are identified with the Fourier coefficients of the lowest modes, 77/, of one of the towers. The lowest modes of the second tower are denoted as r//_. The spinors r/+ and r/_ satisfy the Killing spinor equation with opposite signs. For an arbitrary weight, co, the SO(8,1) mixes all the modes in these supersymmetry towers. The SO(8,1) generators together with these supersymmetry towers, generate an infinite dimensional superalgebra. However, for an appropriate choice of the weight (r/I, r/J_) transform as the sixteen-dimensional spinor representation of SO(8,1). This situation is very reminiscent o f N = 8 conformal supergravity with (r//, r/J_) corresponding to the (QI, S J) supersymmetry generators. The reason why one does not get fields of spin greater than two is that one of the supersymmetry generators changes the mass as well as the spin and the superalgebra is anti-de Sitter and not Poincar& It is intriguing to conjecture that there may be a consistent truncation of the S 7 compactification which contains the massless gauged N = 8 supergravity and a finite number of massive fields in such a way that the resulting theory has "N = 16" supersymmetry generated by the Fourier coefficients of r/+ and r/_. Even if there is no such truncation, one can study the interplay between SO(8,1) and the " N = 16" supersymmetry generated by (r/+, r/_). We observe that the adjoint representation of the exceptional group F4(_20 ) decomposes into the adjoint and the sixteendimensional representation of SO(8,1). It could be
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that the conformal group, together with r/+, r/_ provide a realization of F4(_20 ) on the spectrum. One should bear in mind that r/+ and r/_ are commuting spinors. Another possible application of the conformal action on the compactifying spaces is that they might be related to the extended supergravity theories with noncompact gauge groups. Such theories may arise as different truncations of the spectrum, including both massive and massless states. It is also possible that some of the gauge modes may become physical in such a truncation. We would like to thank Gregg J. Zuckerman for useful discussions on the unitary representations of SO(n, 1).
References [1] N.Y~ Vilenkin, Special functions and the theory of group representations (American Mathematical Society, Providence, 1968). [2] S. Helgason, Lie groups, differential geometry and symmetric spaces (Academic Press, New York, 1978). [31 C.P. Boyer, J. Math. Phys. 12 (1971) 1599. [4] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge U.P., London, 1973) sect. 2.3. [5] G.W. Gibbons and M.J. Perry, Nucl. Phys. B146 (1978) 90.
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[6] A. Casher, F. Englert, H. Nicolai and M. Rooman, The mass spectrum of supergravity on the round seven-sphere CERN preprint TH-3794 (1984). [7] P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980) 233; M.J. Duff, in: Supergravity 81, eds. S. Ferrara and J.G. Taylor (Cambridge U.P., London, 1982); M.J. Duff and C.N. Pope, in: Supersymmetry and supergravity 82, eds, S. Ferrara, J.G. Taylor and P. van Nieuwenhuizen (World Scientific, Singapore, 1983); B. Biran, F. Englert, B. de Wit and H. Nicolai, Phys. Lett. 124B (1983) 45. [8] L. Castellani, R. D'Auria, P. Fr6, K. Pilch and P. van Nieuwenhuizen, The bosonic mass formula for FreundRubin solutions of d = 11 supergravity on general coset manifolds, Stony Brook preprint ITP-SB-83-61. [9] E. Sezgin, Phys. Lett. 138B (1984) 57. [10] B. Biran, A. Casher, F. Englert, M. Rooman and P. Spindel, Phys. Lett. 134B (1984) 179. [11] T. Hirai, Proc. Jpn. Acad. 38 (1962) 258. [12] U. Ottoson, Commun- Math. Phys. 8 (1968) 228. [13] F. Schwarz, J. Math. Phy~ NY 12 (1971) 131. [14] M. Giinaydin, Oscillator-like unitary representations of non-compact groups and supergroups and extended supergravity theories, Ecole Normale Sup6rieure preprint LPTENS 83/5 (January 1983). [15] H. Nicolai and E. Sezgin, Phys. 143B (1984) 389. [16] A. Salam and J. Strathdee, Ann. Phys. 141 (1982) 316. [17] See A. Held, E.T. Newman and R. Posadas, J. Math. Phys, NY 11 (1970) 3145, and references therein. [18] M. Giinaydin, L.J. Romans and N.P. Warner, in preparation.
405
PHYSICS LETTERS
25 October 1984
SPECTRUM GENERATING ALGEBRAS IN K A L U Z A - K L E I N THEORIES ¢~
M. GONAYDIN, L.J. ROMANS and N.P. WARNER a
California Institute of Technology, Pasadena, CA 91125, USA Received 25 June 1984
We describe the action of the euclidean conformal group on spheres. Using the example of S 7 compactification of the eleven-dimensional supergravity we show how the full spectrum provides unitary representations of SO(8,1). Our methods can be applied to compactification on spheres, products of spheres or any other manifold on which there is an action of a conformal group. We also make some conjectures concerning the relationship between the conformal group and supersymmetry.
One may define an action o f the euclidean conformal group SO(n + 1, 1) on the sphere, S n. While the conformal generators do not correspond to isometries and therefore do not commute with the Laplace and Dirac operators, their action simply induces a rescaling of the metric and this may be used to generate the spectrum of the Laplace and Dirac operators. We denote the cartesian coordinates in R n+l by x i and take S n to be the unit sphere z n + l x l x i = 1. Let ~7= diag(1,1 ..... 1 , - 1 ) a n d g be an element of SO(n + 1,1) such that grogT = 7. Define submatrices o f g according to
then the action o f g on S n is defined by
(gx) i = (Ail'x ] + Bi)(Ck x k + D) - 1 .
(2)
From this one can define the action o f g - 1 on functions f: S n --> C by (Vw(g - 1 ) f ) ( x i) = (Ck x k + D)W f((gx)i) ,
(3)
where co is an arbitrary complex number which we refer to as the weight. Consider the Hilbert space 1_2(Sn) of square inteWork supported in part by the US Department of Energy under Contract No. DEAC-03-81-ER40050 and by the Fleischmann Foundation. 1 Weingart Fellow. 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
grable (complex valued) functions over S n. The inner product in I_2(Sn) is simply
(I1, I2) =- f fl(X) I2(x)
dnx ,
(4)
Sn
where v g d nx is the SO(n + 1) invariant measure on S n. By a simple application of the change of variable formula for integration, one may show that this inner product is preserved under the action (3) of SO(n+ 1,1) provided one takes co = - n / 2 + ip where p is any real number. Thus the space L2(S n) provides a unitary representation o f S O ( n + 1,1) [1]. The unitary representations of a group are labeled by the eigenvalues of the Casimir operators. If one realizes a unitary representation of a group G in L2(M) where M is a rank one symmetric space, e.g., S n, then all but the quadratic Casimir operator vanish [2]. Such representations are referred to as " m o s t degenerate" [3]. Since the action o f S O ( n + 1,1) defines a diffeomorphism on S n, one may generalize its action to tensor fields in the obvious way [4]. However, for our purposes it is simplest to first define the infinitesimal action of this group. On the sphere, S n, there are (n + 1) scalar fields ~i which are the eigenmodes of the L a p l a c e - B e l t r a m i operator [] with eigenvalues - n . A useful basis for these functions is to take ~i = x i, for which one may derive the following identities: 401
Volume 146B, number 6
V ~ V ~ o i : - g a ~ i, ~oi~oi= 1, ~0iva~pi=0,
PHYSICS LETTERS (5,6,7)
(V,~0i)(vt~ oi) =gat3, ( V ~ 0 i ) ( v , ~ 0/) + ~oi~o / : 60.. (8,9) The vector fields K / = Vc~0 i are sometimes referred to as conformal Killing vectors since eq. (5) is equivalent to
LKigo<~ : -- 2~igo~ ,
(10)
where LKi is the Lie derivative. To define the action of SO(n + 1,1) on tensor fields of weight co we introduce the operator D / -= LKi co9 l. If one observes that the SO(n + 1) Killing vectors on Sn may be written as -
K~
-
= ~oivad
- - ~O/VoAO i ,
(11)
it is trivial to show that the operators L i/-~ LKi] and D / generate the Lie algebra o f S O ( n + 1,1). Thus one can also represent SO(n + 1,1) on the tensor fields on S n. The inner product is simply
(x,
r3=f vYdnx X m' ""mkYm 1 ...mk,
(12)
and the representation is unitary if and only if co = - n / 2 + io. Moreover, by combining the observation that we may t a k e ~oi = x i with the expressions for tensor harmonics given in ref. [5] one may use the identities eqs. ( 5 ) - ( 9 ) to explicitly calculate the action of D / on these modes. The only complication is that the action of D / does not preserve transverse gauge conditions. We may similarly define our action of SO(n + 1,1) on spinors X, by
LiJX = KiJavo~X + a ~ vijr,eCL,,
(13)
O i X = (VC~s0i)(V~X) + co~0in,
(14)
with the obvious generalizations to higher spins. The spinor harmonics on spheres may be obtained from simple linear combinations of products of tensor harmonics, gamma matrices and the Killing spinors [6]. In this way one sees that the spinorial modes also form representations o f S O ( n + 1,1), and Dido raises and lowers the eigenvalues. To illustrate these techniques we consider the compactification of the eleven-dimensional supergravity on S 7 [7]. In order to obtain the physical spectrum it 402
25 October 1984
was found necessary to fix most of the eleven-dimensional diffeomorphism and gauge invariances [6,8-10]. After the gauge fixing one is left with four-dimensional gauge and diffeomorphism invariances, and the internal local SO(8) symmetry. Such gauge choices are not compatible with the action of SO(8,1), but it is elementary to reinstate this conformal symmetry. Before describing how this may be achieved we should point out that the massive gravitons and gravitinos form representations of SO(8, 1) for any weight including the unitary choice co = - } + ip. This is because these fields appear as Fourier coefficients of scalar and spinor modes on S 7. There is no gauge fixing for these modes. For the remaining bosonic fields, the most convenient gauge choices are the seven-dimensional deDonder and Lorentz gauges of ref. [8]. As was observed in ref. [8], the gauge choices for the (eleven-dimensional) graviton are compatible with conformal invariance which in our conventions corresponds to taking co = 5. It is also straightforward to show that the Lorentz gauge choices for AMN P commute with D / provided one assigns the weights co = 5, 3, 1 forAc, uv, Ac~u and A~t~,r, respectively (/~, v are d = 4 space-time indices). Observe tllat in those cases when it is necessary to mix fields in order to form mass eigenstates these fields may be given the same weight. Thus the bosonic spectlum does form representations of SO(8,1) with the foregoing gauge choices. The point at which the conformal symmetry is broken in ref. [8] is when a singlet and octet of scalar modes which arise from the trace of the graviton are gauged away. By keeping these modes we can restore the SO(8,1) symmetry. The problem is that the foregoing choice of weights is not unitary with respect to the inner product (12). To put the fields of given spin into unitary representations of SO(8,1) one has two options: either remove all gauge fixing or find a gauge choice which commutes with the unitary action of SO(8,1), if such exists. Since no such gauge choice is presently known we shall adopt the former course. From refs. [8,9] it is easy to read off the massive states that are pure gauge. Some of these pure gauge modes by themselves form unitary representations of SO(8,1). It is therefore not necessary to include all the gauge modes in order to complete the physical spectrum into unitary representations of S0(8, 1). For example, in the notation of ref. [8] the exact part of the pure gauge term a~uvl, i.e., ~ [ u < 1 can be set equal
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to zero, since it cannot arise from an SO(8, 1) transformation of the physical spectrum. The fermion spectrum was determined in refs. [6,9] using the gauge choice I"mt~M = 0. This gauge is not preserved by the action of SO(8, 1) defined by eqs. (13), (14), for any weight, co. As for the bosons, we are not aware of an SO(8,1) invariant gauge choice and will therefore remove the gauge fixing. This introduces two new infinite massive towers of spin-half modes, which are Fourier coefficients of the sevendimensional Rarita-Schwinger modes [6] : Xa = (Vc~ -- ~m7F~) X, where X is a Dirac mode. It should be noted that if X = ~7+, a Killing spinor, then the mode Xa vanishes. This corresponds to the fact that the octet of massless gravitini contain no gauge modes. The pure gauge modes that have to be added in order to complete the physical modes into unitary representations of SO(8,1) are given in the last column of table 1. The SO(8) content of the spectrum of the S 7 compactification of the eleven-dimensional supergravity Table 1 In this table we have only included those gauge modes necessary to form unitary representations of SO(8,1). The index l takes the values 0,1,2,.... JP
SO (8) Dynkin labels physical modes
gauge modes
2+
(/,0,0,0)
3/2
(l,0,0,1) (l,0,1,0)
1+
(l,0,1,1)
(/, 1,0,0)
1-
(/,1,0,0) (t,l,0,0)
(l+ 1,0,0,0) (l+ 1,0,0,0)
1/2
(l+ 1,0,1,0) (t,0,0,1) (/, 1,1, O) q,l,0,1)
(l+ 1,0,0,1) (/,0,1,0)
0+
(1,0,0,0) (l + 2,0,0,0) q,2,o,o)
(0,0,0,0) (1,0,0,0) (l+ 2,0,0,0) (/+ 1,1,0,0)
O-
(l,0,2,0) (l,0,0,2)
(/,0,1,1)
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can be given in the Gel'fand-Zetlin basis [9] as well as the Dynkin basis [6,9]. In what follows we denote the former by (ml, m2, m3, m4)G_ Z and the latter by (nl, n2, n3, n4)D. The unitary irreducible representations (UIR) of SO(n + 1,1) have been studied in refs. [ 1 1 - 1 3 ] . One important property of the UIR's o f S O ( n + 1,1) is that irreducible representations of SO(n + 1) occur with multiplicity one or zero inside a given UIR. In the sequel we shall follow the notation of ref. [13] for the labeling of UIR's of SO(8,1). J = 2 + modes. These modes fall into the (I, 0, 0, 0)G- z = (l, 0, 0, 0)D, l ~> 0 representations of SO(8). This tower forms the UIR of SO(8, 1) which is denoted as D3(Z9,4 = - ~ ) . The number Z9, 4 is the same as the weight ~ defined above. j = 3 modes. These modes fall into two infinite towers of SO(8) representations (l+ "~, 1 ~-, 1 ~, 1 ~)G-Z 1, _r ~ ' + l 1 1 1~ =(/,0,1,0)D,/~>0" (/,0,0,1)D andt~ 5,7,~,--~)G-Z These two towers together form the UIR labeled DO 1 1. Z9,4 _ 7 + ip). ~],~,~, J = 1 + modes. The physical modes lie in the (l + 1, 1,1,0)G-Z = (l,0, 1, 1)D, l~> 0 representations of SO(8). They, together with the pure gauge modes in the (l + 1,1,0, 0)G-Z = (1, l, 0, 0)D representations, constitute the UIR D0(0,1,1 ; Z9, 4 = - z) of SO(8,1). J = 1 - modes. The physical modes fall into two identical towers consisting of the representations (l + 1,1,0,0)G_Z = (l, 1,0,0)D, l ~> 0. Each tower of the physical modes, together with the gauge modes (l+ 1,0,0,0)G-Z = (l+ 1,0,0,0)D, l~> 0 form the UIR DI(0,1 ;Z9,4 = -27-). J - - ½ modes. The physical modes fall into the following representations: (l+ ~,~-,~,--~)G-Z 3 11 h -- ~.t r, + 1,0, 1,0)D, 1'+1~." ~,$,~',$)G-Z1 1 1~ =(I,O,O, 1)D,(I+~,~,.~,_~)G_Z331 1~. 3311, _~'1 = (l, 1,1,0)D, (l+ ~,~,-~,~)G-Z - it, 1,0, 1)D, If> 0. With the exception of the lowest (l = 0) mode in the second tower these towers form the UIR D(~,7,~,Z9, 4 1 ts. _-27-+ ip) of SO(8,1). The mode ~1 1 i 1~ t~,~,~,7)G-Z together 3 1 1 1"~ with the pure gauge modes (l+ ~,~,~,~)G-Z -- - "tl + 1,0, 0,1)D and (l+ .t2 , ~1, 2.1. 1- Z ,--~)G -_ (l,0, 1,0)D, l~> 0, form the UIR D(~,~,~,Z9, 11 1l 4 = _ _ ~ + ip). J = 0 + modes. The physical modes fit into the SO(8) representations: (l, 0, 0, 0)G-Z = (l, 0, 0, 0)D, l ~> 0, (/, 0, 0, 0)G-Z = (/,0,0,0)D, l ~> 2, (l+ 2,2,0,0)G_ z = (/, 2, 0, 0)D, l ~> 0. The first tower of SO(8) representations above fall into the UIR D3(Z9,4 = 7 ) of S0(8,1). The second tower falls into an identical lAIR =
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of SO(8, 1) provided one adds the gauge modes (0,0, 0,0) and (1,0,0,0). The third tower together with the gauge modes (l + 2, 0, O, 0)G-Z = (l + 2, O, 0, 0)D and (l + 2, 1,0,0)G-Z = (l + 1,1,0,0)D fall into the UIR Dl(0,2; Z9,4 = --27). J = O- modes. The physical modes fall into the SO(8) representations (l + 1, 1, 1,--1)G-Z = (I,0,2,0)D and (l + 1,1,1,+I)G-Z = (l, 0, 0, 2)O, l ~> 0. These towers together with the gauge modes (l + 1, 1,1,0)G-Z = (l,0, 1,1)O form the UIR D°(1,1,1 ;Z9, 4 = _ z ) of SO(8, 1). In this letter we have considered the unitary action of the conformal group SO(8,1) with respect to the inner product defined by eq. (12). Here the physical and gauge modes had to be considered together (for J = 0,1, 1) in order to form the UIR's of SO(8,1). However, the physical modes of a given spin by themselves have the same SO(8) content as certain classes of UIR's of SO(8,1). This suggests that there may exist a different inner product and a modified SO(8,1) action such that the conformal group is realized unitarily on the physical sector alone. Interestingly enough, the only difficulty in fitting the physical modes into UIR's of SO(8, 1) arises from the s = ½ mode (0,0,0, 1)D. This mode can be completed to the finite dimensional 16 of SO(8, 1) by the addition of an s = ½ singleton representation [ 14,15 ]. The s = 0 partner of this singleton representation is just the unphysical octet discussed earlier. This difficulty could also be alleviated by choosing a four-dimensional space-time which breaks all supersymmetries. The s = -~ octet will then be eaten by the gravitini through the super Higgs mechanism. The results presented here can be applied to compactification on any manifold which admits a conformal group action such as spheres and products of two spheres '1. The simplest case of compactification on S 1 of the five-dimensional pure gravity was studied in ref. [16], in which the conformal action of SO(2, 1) on the spectrum was given. Contrary to an assertion therein one can assign unitary weights to all the modes. We have also studied the compactification on S 2 of the simple (gauged) supergravity in d = 5. In this case the conformal group is SO(3,1) whose UIR's have been extensively discussed [ 17]. Many of the ideas de,1 We thank J. Dorfmeister and G.J. Zuckerman for discussions on this point. 404
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scribed here take on a simpler form for this compactification [18]. The action of SO(8, 1) on spinor fields defined by eq. (14) is not necessarily the most general. One might be able to add to VaX a term proportional to FaX and still obtain a realization of SO(8, 1). Such an expression would naturally follow from supercovariance. With this modified action one might obtain a unitary representation of SO(8,1) on the purely physical modes. Perhaps the most interesting question is the relationship between supersymmetry and the conformal action. At the naive level when one reduces the elevendimensional theory down to four dimensions, keeping all the massive states, one obtains a theory with two infinite towers of supersymmetries corresponding to the Fourier modes of the eleven-dimensional supersymmetry generator. The usual octet of four-dimensional supersymmetry generators are identified with the Fourier coefficients of the lowest modes, 77/, of one of the towers. The lowest modes of the second tower are denoted as r//_. The spinors r/+ and r/_ satisfy the Killing spinor equation with opposite signs. For an arbitrary weight, co, the SO(8,1) mixes all the modes in these supersymmetry towers. The SO(8,1) generators together with these supersymmetry towers, generate an infinite dimensional superalgebra. However, for an appropriate choice of the weight (r/I, r/J_) transform as the sixteen-dimensional spinor representation of SO(8,1). This situation is very reminiscent o f N = 8 conformal supergravity with (r//, r/J_) corresponding to the (QI, S J) supersymmetry generators. The reason why one does not get fields of spin greater than two is that one of the supersymmetry generators changes the mass as well as the spin and the superalgebra is anti-de Sitter and not Poincar& It is intriguing to conjecture that there may be a consistent truncation of the S 7 compactification which contains the massless gauged N = 8 supergravity and a finite number of massive fields in such a way that the resulting theory has "N = 16" supersymmetry generated by the Fourier coefficients of r/+ and r/_. Even if there is no such truncation, one can study the interplay between SO(8,1) and the " N = 16" supersymmetry generated by (r/+, r/_). We observe that the adjoint representation of the exceptional group F4(_20 ) decomposes into the adjoint and the sixteendimensional representation of SO(8,1). It could be
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that the conformal group, together with r/+, r/_ provide a realization of F4(_20 ) on the spectrum. One should bear in mind that r/+ and r/_ are commuting spinors. Another possible application of the conformal action on the compactifying spaces is that they might be related to the extended supergravity theories with noncompact gauge groups. Such theories may arise as different truncations of the spectrum, including both massive and massless states. It is also possible that some of the gauge modes may become physical in such a truncation. We would like to thank Gregg J. Zuckerman for useful discussions on the unitary representations of SO(n, 1).
References [1] N.Y~ Vilenkin, Special functions and the theory of group representations (American Mathematical Society, Providence, 1968). [2] S. Helgason, Lie groups, differential geometry and symmetric spaces (Academic Press, New York, 1978). [31 C.P. Boyer, J. Math. Phys. 12 (1971) 1599. [4] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge U.P., London, 1973) sect. 2.3. [5] G.W. Gibbons and M.J. Perry, Nucl. Phys. B146 (1978) 90.
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[6] A. Casher, F. Englert, H. Nicolai and M. Rooman, The mass spectrum of supergravity on the round seven-sphere CERN preprint TH-3794 (1984). [7] P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980) 233; M.J. Duff, in: Supergravity 81, eds. S. Ferrara and J.G. Taylor (Cambridge U.P., London, 1982); M.J. Duff and C.N. Pope, in: Supersymmetry and supergravity 82, eds, S. Ferrara, J.G. Taylor and P. van Nieuwenhuizen (World Scientific, Singapore, 1983); B. Biran, F. Englert, B. de Wit and H. Nicolai, Phys. Lett. 124B (1983) 45. [8] L. Castellani, R. D'Auria, P. Fr6, K. Pilch and P. van Nieuwenhuizen, The bosonic mass formula for FreundRubin solutions of d = 11 supergravity on general coset manifolds, Stony Brook preprint ITP-SB-83-61. [9] E. Sezgin, Phys. Lett. 138B (1984) 57. [10] B. Biran, A. Casher, F. Englert, M. Rooman and P. Spindel, Phys. Lett. 134B (1984) 179. [11] T. Hirai, Proc. Jpn. Acad. 38 (1962) 258. [12] U. Ottoson, Commun- Math. Phys. 8 (1968) 228. [13] F. Schwarz, J. Math. Phy~ NY 12 (1971) 131. [14] M. Giinaydin, Oscillator-like unitary representations of non-compact groups and supergroups and extended supergravity theories, Ecole Normale Sup6rieure preprint LPTENS 83/5 (January 1983). [15] H. Nicolai and E. Sezgin, Phys. 143B (1984) 389. [16] A. Salam and J. Strathdee, Ann. Phys. 141 (1982) 316. [17] See A. Held, E.T. Newman and R. Posadas, J. Math. Phys, NY 11 (1970) 3145, and references therein. [18] M. Giinaydin, L.J. Romans and N.P. Warner, in preparation.
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