The Minimal Unitary Representation of E8(8)∗

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by CERN Document Server

AEI-2001-108 hep-th/0109005

M. G¨unaydin ‡ Penn State University, Physics Department, University Park, PA 16802 E-mail: [email protected]

K. Koepsell, H. Nicolai Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut, M¨uhlenberg 1, D-14476 Golm, Germany E-mail: [email protected], [email protected]

Abstract We give a new construction of the minimal unitary representation of the exceptional group E8(8) on a Hilbert space of complex functions in 29 variables. Due to their manifest covariance with respect to the E7(7) subgroup of E8(8) our formulas are simpler than previous realizations, and thus well suited for applications in supergravity and superstring theory. One possible application arises in the context of the conformal quantum mechanics description of extremal black holes in maximally extended supergravity.

August 2001

∗ This work was supported in part by the NATO collaborative research grant CRG. 960188. ‡Work supported in part by the National Science Foundation under grant number PHY-0099548. 1 Introduction

In this paper, we present a new construction of the minimal unitary repre- 1 sentation of E8(8) on an Hilbert space of complex functions in 29 variables . The minimal realizations of classical Lie algebras and of G2 were given by Joseph a long time ago [20, 21]. These minimal realizations are related to the minimal orbits of the corresponding groups. The existence of the minimal unitary representation of E8(8) was first proved by Vogan [33] who located it within the framework of Langland’s classification. Later, the minimal unitary representations of all simply laced groups, including E8(8), E7(7) and E6(6) were constructed by Kazhdan and Savin [23], and Brylinski and Kostant [4, 5, 6, 7] by rather different methods. Gross and Wallach gave yet another construction of the minimal representation of E8(8) as well as for all exceptional groups of real rank four [15].

While formulas for G2 similar to the ones derived here for E8(8) can be found in [5], however, an explicit realization of the simple root (Chevalley) generators in terms of pseudo-differential operators for the simply laced exceptional groups was given only very recently [24], together with the spherical vectors necessary for the construction of modular forms. We present here an alternative – though presumably equivalent – realization of E8(8),which has the advantage of yielding very compact formulas, in contradistinction to the rather complicated expressions obtained by multiple commutation of the simple root generators [25]. The main reason for the relative simplicity of our final expressions (6) and (11) is their manifest E7(7) covariance, in spite of the fact that E7(7) is not realized linearly (it is realized linearly on the classical phase space, however). The compactness of our formulas makes them especially suitable for applications in string and M theory which have primarily motivated the present work. On the purely mathematical side, we calculate explicitly the quadratic Casimir operator of E8(8) and show that it reduces to a c-number for the minimal representation. This is consistent with the irreducibility of the minimal representation. Likewise the truncation of the minimal representation to subgroups of E8(8) that we study in this paper have, to our knowledge, not appeared in the literature so far. Our derivation is based on our previous work [16] where minimal nonlinear representations of the exceptional groups were constructed, which were conformal or quasiconformal in the sense that there exist generalized “light- cones” which are left invariant by the non-linear action. This construction relied essentially on the connection with Jordan algebras and Freudenthal triple systems. In particular, E8(8) can be realized in this manner as a group of non-linear transformations acting on R57, which is analogous to the non-linear action of the Möbius group on the real line.

1A preliminary account of some of the results presented in section 2 has already appeared in [27].

2 To find the minimal unitary representation of E8(8) over an Hilbert space of complex functions in 29 variables we first identify a phase space realization of the E7(7) subalgebra in terms of 28 coordinates and 28 momenta, such that all E7(7) variations can be realized via the canonical action δ X Q, X (1) Q ≡{ }P.B. where Q is the appropriate E7(7) charge. For the canonical realization of the full E8(8) we have to add the momentum conjugate to the 57th coordinate, and replace the Möbius action by the symplectic realization acting on a two dimensional phase space. The minimal unitary representation of E8(8) is then obtained by quantization, i.e. replacement of the classical momenta by differential operators. This requires in addition a prescription how to deal with the ordering ambiguities arising in the non-linear expressions for the R57 Lie algebra generators. The non-linear realization of E8(8) on is thereby converted to a unitary action on the infinite dimensional Hilbert space of square integrable complex functions depending on 29 real variables (in the coordinate representation). The elements of the E8(8) Lie algebra are then realized as self-adjoint operators on some dense subspace of L2(R29). A possible physical application of our results arises in the context of the (super)conformal quantum mechanics description of quantum black holes [10, 8, 28]. In that work, the theory of unitary representations of SL(2, R)played a crucial role in the classification of physical states. It is therefore an obvious idea to extend these concepts to maximal supergravity. Indeed, the candi- date Hamiltonian which we obtain as one of the E8(8) Lie algebra generators reads

1 2 2 2 L0 = 4 p + y +4y− I4(X, P) (2) which is precisely of the form studied by the authors of [10, 8, 28], with the only difference that they have a (coupling) constant instead of the differential operator I4(X, P). Here I4 is the quartic invariant of E7(7) when viewed as ij a function of the 28+28 variables X and Pij, and at the same time the quadratic E7(7) Casimir invariant when acting on one of the irreducible subrepresentations of E7(7) occurring in the decomposition of the minimal R representation of E8(8) under its subgroup E7(7) SL(2, ). As shown for cl × instance in [16], the classical invariant I4 can assume both positive and cl 1 1 negative values, with the special value I4 = 0 corresponding to 2 or 4 BPS black hole solutions of N = 8 supergravity [22, 13, 12]. It is therefore tempting to interpret (2) as the effective Hamiltonian de- scribing (in some approximation) N = 8 quantum black holes, such that every subrepresentation with a fixed eigenvalue of the operator I4 is iden- tified with the space of physical states associated with the corresponding black hole solution of N = 8 supergravity. While the manifest E7(7) invariance of N = 8 supergravity ensures that E7(7) transformations will not affect

3 this eigenvalue, the remaining E8(8) transformations would relate extremal black holes to non-extremal ones and vice versa. As is evident from (2), the quadratic Casimir of the subgroup SL(2, R) is a function of I4 only. There- fore, depending on the eigenvalue of the I4(X, P), one obtains discrete series or continuous series unitary representations of SL(2, R). Interestingly, for the vanishing eigenvalues of I4 (and thus extremal quantum black holes), the Hamiltonian simpliﬁes drastically, and the state space reduces to the well known singleton representation of SL(2, R). Another (and possibly related) physical application of minimal representations has been outlined in [24]. That work evolved from an earlier attempt to determine the R4 corrections directly from the supermembrane and to understand them in terms of so-called “theta-correspondences” [29]; see also [32] for a related attempt to come to grips with the non-linearities of the supermembrane.

2 Coordinate Space Representation

We ﬁrst recall some basic features of the non-linear realization of E8(8) on 57 ij R coordinatized by 57 real variables X ,Xij ,y (i, j =1,...,8), see [16] for our notations and conventions, and{ further details.} A key ingredient in that construction was the 5-graded decomposition of E8(8) w.r.t. its subgroup E SL(2, R). Denoting its Lie algebra by e we have 7(7) × 8 2 1 0 +1 +2 e = g− g− g g g . (3) 8 ⊕ ⊕ ⊕ ⊕ An important property of this decomposition is the fact that the subspaces of grade 1and 2 together form a maximal Heisenberg subalgebra. The − − ij 1 2 corresponding generators E ,Eij g− and E g− obey the commutation relations ∈ ∈

[Eij ,E ]= 2i δij E. (4) kl − kl Obviously, this algebra can be realized as a classical Poisson algebra on a ij phase space with 28 coordinates X and 28 momenta Pij Xij,andone extra real coordinate y to represent the central term. We have≡

ij ij 1 2 E := yX ,Eij := yPij ,E:= 2 y . (5)

It is then straightforward to determine the generators of the e7 subalgebra in terms of these phase space variables (for instance by requiring that they reproduce the E7(7) variations via (1)). They are realized by the following ij 63+70 bilinear expressions in X and Pij

i ik 1 kl i G j := 2 X Pkj + 4 X Pkl δj ,

Gijkl := 1 X[ijXkl] + 1 ijklmnpqP P . (6) − 2 48 mn pq 4 To extend this canonical realization to the full E8(8) Lie algebra, we need one more variable, the momentum p conjugate to y. Combining the symplectic realization of SL(2, R) with the non-linear variations (23) and (24) of [16] we can then deduce the classical (phase space) analogs of the E8(8) generators. Owing to the non-linearity of the realization, the resulting expressions are not only quadratic, but go up to fourth order in the phase space variables ij X and Pij, and moreover contain inverse powers of y. Thus all generators of E8(8) can be realized on a 58-dimensional phase space coordinatized by ij X ,Pij ; y,p . The minimal unitary 29 dimensional representation of E8(8) is{ then obtained} by quantization, i.e. by introducing the usual momentum operators obeying the canonical commutation relations

ij ij [X ,Pkl]=iδkl , [y,p]=i.

In elevating the E8(8) generators to quantum operators the only problem vis-à-vis the classical phase space description is the non-commutativity of the coordinate and momentum operators, which requires some ordering prescription (we note, that the E7(7) generators (6) are insensitive to re-ordering the coordinates and momenta). Since we are interested in finding a unitary representation we insist that all operators are hermitean w.r.t. to the standard scalar product on L2(R29). We thus arrive at a unitary representation of the E8(8) Lie algebra in terms of self-adjoint operators acting on (some dense subspace of) a Hilbert space of complex functions in 29 real variables Xij,y . We emphasize that this realization requires complex functions, for the{ same} reason that the Schrödinger representation of the one-dimensional point particle requires complex wave functions 2. Before writing out the E8(8) generators, let us list the E7(7) commutation relations [Gi ,Gk ]=iδk Gi i δi Gk , j l j l − l j [Gi ,Gklmn]= 4i δ[k Glmn]i i δi Gklmn , j − j − 2 j ijkl mnpq i ijkls[mnp q] [G ,G ]=36 G s . (7) Observe that only the SL(8, R) subgroup acts by linear transformations; its maximal compact subgroup SO(8) is generated by Gij−,where

1 i j G± := G G . (8) ij 2 j ± i For later convenience, we also deﬁne

1 mnpq Gijkl := 24 ijklmnpqG . (9) 2 As we show in the next section the generators of E8(8) can be rewritten in terms of bosonic annihilation and creation operators. In the corresponding coherent state basis the Hilbert space will involve holomorphic functions in 29 variables

5 and the selfdual and anti-selfdual combinations

1 ijkl Gijkl± := G Gijkl (10) 2 ± + where the Gijkl− are compact and the Gijkl non-compact. The following generators extend this representation to the full E8(8) Lie algebra (as in [16] the E8(8) generators are designated in accordance with the 5-grading of E8(8)):

1 2 E := 2 y ,

Eij := yXij ,

Eij := yPij ,

1 H := 2 (yp+ py) ,

ij ij 1 ij F := pX +2iy− [X ,I (X, P)] − 4 1 ik lj 1 1 ij kl kl ij = 4 y− X P X y− (X P X + X P X ) − kl − 2 kl kl 1 1 ijklmnpq ij + y− P P P pX , 12 kl mn pq − 1 Fij := pPij +2iy− [Pij ,I4(X, P)] − 1 kl 1 1 kl kl =4y− P ikX Plj + 2 y− (PijX Pkl + PklX Pij) 1 1 kl mn pq y− X X X pP , − 12 ijklmnpq − ij 1 2 2 F := 2 p +2y− I4(X, P) . (11)

Here I4(X, P) is the fourth order diﬀerential operator I (X, P):= 1 (Xij P XklP + P XjkP Xli) 4 − 2 jk li ij kl 1 ij kl ij kl + 8 (X PijX Pkl + PijX PklX ) 1 ijklmnpqP P P P − 96 ij kl mn pq 1 XijXklXmnXpq + 547 . (12) − 96 ijklmnpq 16 ij As a function of X and Pij this operator represents the quartic invariant of E7(7) because

i ijkl [G j ,I4(X, P)] = [G ,I4(X, P)] = 0 . (13) We emphasize the importance of the ordering adopted in (12) for the vanishing of (the second of) these commutators. Orderings that diﬀer from (12) ij by a term containing the Euler operator iX Pij break E7(7) invariance. On the other hand, (13) is insensitive to re-orderings which diﬀer from (12) only by a c-number, and in the absence of a preferred ordering there is thus no

6 absolute signiﬁcance to the additive constant appearing in the deﬁnition of I4. For instance, an admissible re-ordering is

1 Gi Gj GijklG = I (X, P) 323 . (14) − 6 j i − ijkl 4 − 16

Incidentally, the latter relation shows that, when acting on a given representation of E7(7), I4 is just the quadratic E7(7) Casimir invariant, up to an additive constant. For the derivation of (11), we note that, as already pointed out in [4], the crucial step is the determination of the operators E and F corresponding to ij the lowest and highest root of E8(8), respectively. The expressions for F ij and Fij then follow by commutation with E and Eij. While the derivation of [4] relied on a generalization of the so-called “Capelli-identity”, the form of our operators E and F follows directly from E7(7) invariance and a scaling argument, up to the additive constant in (12). The latter originates from ij the re-ordering required to bring the commutator of Fij and F into the “standard form” deﬁned by the r.h.s. of (12).

The hermiticity of all E8(8) generators is manifest. As anticipated, all generators transform covariantly under the full E7(7) group.

[Gi ,Ekl]=iδk Eil i δl Eik i δi Ekl , j j − j − 4 j [Gi ,E ]=iδi E i δi E + i δi E , j kl k lj − l kj 4 j kl

[Gi ,Fkl]=iδk F il i δl F ik i δi F kl , j j − j − 4 j [Gi ,F ]=iδi F i δi F + i δi F . (15) j kl k lj − l kj 4 j kl

The remaining part of E7(7) acts as

[Gijkl ,E ]= i δ[ij Ekl] , mn − mn [Gijkl ,Emn]= i ijklmnpq E , − 24 pq [Gijkl ,F ]= i δ[ij F kl] , mn − mn [Gijkl ,Fmn]= i ijklmnpq F . (16) − 24 pq

The grading of the generators is given by the dilatation generator H

[H,E]= 2iE, [H,F]=2iF, − [H,Eij]= i Eij , [H,Fij]=iF ij , (17) − [H,E ]= i E , [H,F ]=iF . ij − ij ij ij 7 The remaining non-vanishing commutation relations are

[Eij ,Fkl]=12iGijkl , [Eij ,F ]=4iδ[i Gj] i δij H, kl [k l] − kl [E ,F ]= 12 i G , [E ,Fkl]=4iδ[k Gl] +iδij H, ij kl − ijkl ij [i j] kl ij ij ij ij [E ,Ekl]=2iδ E, [F ,Fkl]=2iδ F, kl kl (18) [E,Fij]= i Eij , [F,Eij]=iF ij , − [E,F ]= i E , [F,E ]=iF , ij − ij ij ij [E,F]=iH.

The E8(8) commutation relations are the same as in [16], except that the structure constants carry an extra factor of i. As can be veriﬁed by computation of the Cartan Killing form from the structure constants that can be extracted from the above commutation relations, the maximal compact subgroup SO(16) is generated by the following linear combinations of E8(8) generators

ij ij G− ,G− ,E + F ,E F ,E+ F. (19) ij ijkl ij − ij

This conﬁrms that we are indeed dealing the split real form E8(8). The quadratic E8(8) Casimir operator (in a convenient normalization) is a sum of three terms

[E ]= [SL(2, R)] + [E ]+ 0 (20) C2 8(8) C2 C2 7(7) C2 with

[SL(2, R)] := 1 (EF + FE 1 H2) , C2 2 − 2 [E ]:= 1 Gi Gj 3 GijklG , C2 7(7) − 2 j i − ijkl 1 ij ij ij ij 0 = E F + F E E F F E . (21) C2 4 ij ij − ij − ij Here the terms in the ﬁrst and second line represent the Casimir operators R of the SL(2, )andE7(7) subalgebras, respectively. When substituting the explicit expressions in terms of coordinates and momenta for the generators, we obtain

[SL(2, R)] = I 3 , C2 4 − 16 [E ]=3I 969 , C2 7(7) 4 − 16 237 0 = 4 I . (22) C2 − 4 − 4 ij Hence all terms containing the operators X ,Pij ,y and p actually cancel, leaving us with a constant value for the minimal representation

[E ]= 120 (23) C2 8(8) − 8 which is the E8(8) analog of the result

[SL(2, R)] = 1 g 3 , (24) C2 4 − 16 familiar from conformal quantum mechanics – except that there is no coupling constant any more for E8(8) that we can tune! Unlike for the group SL(2, R), this result does not imply the irreducibility of the minimal repre- 3 sentation of E8(8) . To show that, we would have to compute in addition the eigenvalues of the higher order E8(8) Casimir invariants. We expect, however, that these, too, will collapse to c-numbers when the coordinates and momenta are substituted, for the simple reason that we cannot build E8(8) invariants from the coordinate and momentum operators alone. This is in stark contrast to the singleton representation of E7(7) for which the coordinates and momenta do form a non-trivial (linear) representation of E7(7), permitting the construction of non-vanishing higher order E7(7) invariants. This provides an independent argument that the minimal representation of E8(8) is indeed irreducible. We conclude this section by giving the Chevalley generators corresponding to the eight simple roots of E8(8), with the labeling indicated in the ﬁgure.

2 3 4 5678

Figure 1: Numbering of simple roots of E8(8)

They are

4567 1238 e1 =12G ,f1 =12G , 1 2 e2 = G 2 ,f2 = G 1 , 2 3 e3 = G 3 ,f3 = G 2 , 3 4 e4 = G 4 ,f4 = G 3 , 4 5 (25) e5 = G 5 ,f5 = G 4 , 5 6 e6 = G 6 ,f6 = G 5 , 6 7 e7 = G 7 ,f7 = G 6 , 78 e8 = E ,f8 = F78 .

3Modulo the subtlety that the two singleton irreps of SL(2, R) have the same eigenvalue of the Casimir

9 The generators of the Cartan subalgebra are given by

h = G4 + G5 + G6 + G7 = G1 G2 G3 G8 , 1 4 5 6 7 − 1 − 2 − 3 − 8 1 2 h2 = G 1 G 2 , h = G2 − G3 , 3 2 − 3 3 4 h4 = G 3 G 4 , − (26) h = G4 G5 , 5 4 − 5 h = G5 G6 , 6 5 − 6 h = G6 G7 , 7 6 − 7 h = G7 + G8 1 H. 8 7 8 − 2 Comparison with the formulas given in the appendix of [24] shows that the basis of coordinate vs. momentum variables used there diﬀers from ours by the choice of polarization (or “Fourier transformation”). Accordingly, the linearly realized subgroup exposed there is also diﬀerent from ours.

3 Oscillator Representation

In the coordinate representation, the linearly realized SL(8, R) subalgebra of E7(7) plays a distinguished role. As is well known, however, there is another basis of E7(7), which is equally important in supergravity and superstring theory, where SL(8, R) is replaced by SU(8) [9]. In this section, we demonstrate that the full E8(8) Lie algebra can be rewritten in terms of this complex basis. The change of basis is equivalent to the replacement of coordinates and momenta by creation and annihilation operators, and thus to the replacement of the coordinate representation by a holomorphic (Bargmann-Fock) representation. This will allow us to establish the connection with the oscillator realization of E7(7) discovered already some time ago [17]. For this purpose, we introduce the creation and annihilation (or raising and lowering) operators

aAB := 1 Γij (Xij i P ) 4√2 AB − ij ij a := 1 Γ (Xij +iP ) aAB † (27) AB 4√2 AB ij ≡ The normalization has been chosen such that

CD CD [ aAB ,a ]=δAB (28) Substituting these operators into (6) and deﬁning

A 1 ij 1 ijkl A G := i G−Γ + G− Γ = G , B − 4 ij AB 8 ijkl AB B ABCD 1 + 1 + ij kl G := 48 i Gikδlj 16 Gijkl Γ[ABΓCD] , (29) − 10 we obtain the singleton representation of E7(7) in the SU(8) basis of [17] GA := 2aACa 1 δAaCDa B BC − 4 B CD GABCD := 1 a[ABaCD] 1 ABCDEF GH a a (30) 2 − 48 EF GH A where G B now generates the SU(8) subgroup of E7(7). In deriving this result, we made use of the formulas (see e.g. the appendices of [9, 11]) Γijkl = 1 ijklmnpq Γmnpq (31) AB − 24 AB and

Γij Γkl = 2 δk[iΓj]m Γml +Γ[ij Γkl] [AB CD] − 3 [AB CD] [AB CD] ik kj 1 ik kj Γ[ABΓCD] = 24 ABCDEF GHΓEF ΓGH Γ[ij Γkl] = 1 Γ[ij Γkl] [AB CD] − 24 ABCDEF GH EF GH 1 ijklmnpq mn pq =+24 Γ[ABΓCD] (32) We note the complex (anti)self-duality relation

G GABCD † = 1 GEFGH (33) ABCD ≡ − 24 ABCDEF GH and the commutation relations [GA ,GC ]=δC GA δA GC , B D B D − D B [GA ,GCDEF ]= 4 δ[C GDEF]A 1 δAGCDEF , B − B − 2 B ABCD EFGH 1 ABCDI[EFG H] [G ,G ]= 36 G I . (34)

In order to render the remaining E8(8) generators SU(8) covariant, we deﬁne EAB := 1 Γij (Eij i E )=yaAB , √ AB ij 4 2 − (35) E := 1 Γij (Eij +iE )=ya , AB 4√2 AB ij AB The computation is more tedious for the generators which are cubic and quartic in the oscillators, and most conveniently done by checking the E8(8) algebra again. We have 1 2 2 F = 2 p +2y− I4(a, a†) (36) with the SU(8) invariant expression for I4 in terms of oscillators

I4(a, a†) I4(X, P) ≡ 1 AB CD BC DA =+2 (a aBCa aDA + aABa aCDa ) 1 (aABa aCDa + a aABa aCD) − 8 AB CD AB CD 1 ABCDEF GH + 96 aABaCDaEF aGH 1 AB CD EF GH 547 + 96 ABCDEF GHa a a a + 16 . (37)

11 Note that the normal-ordered version of I4 (with all the annihilators to the right) is not E7(7) invariant, because

i I (a, a†)=:I (a, a†): 49 (38) 4 4 − 4 N− with the number operator

:= aABa (39) N AB which does not commute with GABCD. The remaining generators are now straightforwardly deduced by com- AB muting F with E and EAB: F AB := 1 Γij (F ij i F ) i[EAB ,F] 4√2 AB − ij ≡ AB i 1 AB CD CD AB = pa + y− (a a a + a a a ) − 2 CD CD 1 AC DB i 1 ABCDEF GH +4iy− a a a y− a a a , CD − 12 CD EF GH F := 1 Γij (F ij +iF ) i[E ,F] AB 4√2 AB ij ≡ AB i 1 CD CD = pa y− (a a a + a a a ) − AB − 2 AB CD CD AB 1 CD i 1 CD EF GH 4iy− a a a + y− a a a .(40) − AC DB 12 ABCDEF GH These generators now transform covariantly under the SU(8) group

[GA ,E ]=δA E δA E + 1 δA E , B CD C DB − D CB 4 B CD

[GA ,F ]=δA F δA F + 1 δA F , (41) B CD C DB − D CB 4 B CD and the remaining part of E7(7) acts as

[GABCD ,E ]= δ[AB ECD] , EF − EF [GABCD ,F ]= δ[AB F CD] . (42) EF − EF Furthermore,

CD CD [EAB ,E ]=2δAB E, CD CD [FAB ,F ]=2δAB F, [EAB ,FCD]= 12 i GABCD , − [E ,FCD]= 2iδ[C GD] δCD H. (43) AB − [A B] − AB It remains to discuss the SL(2, R) subgroup, for which we likewise switch to a complex basis (the SU(1, 1) basis)

1 1 2 2 2 L0 := 2 (E + F )= 4 (p + y )+y− I4(X, P) , 1 1 2 2 L1 := 2 (E F +iH)= 4 (y +ip) y− I4(X, P) , 1 − 1 2 − 2 L 1 := (E F iH)= (y ip) y− I4(X, P) , (44) − 2 − − 4 − − 12 2 In the absence of the term containing y− , it would again be convenient to employ creation and annihilation operators b := 1 (y + ip)andb = √2 † 1 (y ip) to recover the well known singleton representation of SU(1, 1), √2 − but for non-vanishing value of the quartic E7(7) invariantitisnotpossible 1 2 to switch this term oﬀ. The presence of y− and y− makes it somewhat awkward to express all generators in terms of creation and annihilation operators, so one might prefer to keep the coordinate representation in this sector. The commutation relations are, however, not aﬀected by the choice of variables.

[L0 ,L 1]= L 1 , [L+1 ,L 1]=2L0 . (45) ± ∓ ± − This basis is no longer hermitean, but

L0† = L0 , (L 1)† = L 1 (46) ± ∓ Diagonalizing the new hermitean generator 2L0 instead of H we obtain an alternative 5-graded decomposition 2 1 0 +1 +2 e = k− k− k k k . (47) 8 ⊕ ⊕ ⊕ ⊕ such that 0 0 1 1 2 2 (k )† = k , (k± )† = k∓ , (k± )† = k∓ (48) We note that, strictly speaking, the elements of kn by themselves do not belong to the real Lie algebra e8 as defined in section 2, but to its complex- ification. It is only the hermitean linear combinations which do. The subspaces of grade 2 are one-dimensional with generators L 1,the ± ∓ grade 0 space is spanned by the E7(7) generators (6) together with L0 and AB AB the subspaces of grade 1 are generated by , AB and , AB defined by ± E E F F AB := 1 (EAB +iF AB) , E √2 := 1 (E +iF ) , EAB √2 AB AB AB := 1 (EAB i F AB) , F √2 − := 1 (E i F ) , (49) FAB √2 AB − AB respectively. In terms of these generators all structure constants are real: AB AB [L+1 , ]=2 , EAB F AB [L 1 , ]= 2 , − FCD − E [ AB , ]=2L 1 , E E − [ , CD]=2L , FAB F +1 [ AB , CD]= 12 GABCD , E F − [ , CD]= 2 δ[C GD] +2δCD L . (50) EAB F − [A B] AB 0

13 4 Decompositions and Truncations

4.1 E SL(2, R) decomposition of the minimal unitary 7(7) × representation of E8(8) A non-compact group G admits unitary representations of the lowest weight (or highest weight) type if and only if the quotient G/K of G with respect to its maximal compact subgroup K is an Hermitean symmetric space [18, 19]. From this theorem it follows that the simple non-compact groups that admit lowest (highest) weight unitary representations are SO(n, 2), SU(n, m), R SO∗(2n), Sp(2n, ), E6( 14),andE7( 24). The unitary lowest (highest) weight representations belong− to the holomorphic− (anti-holomorphic) discrete series and within these representations the spectrum of, at least, one generator is bounded from below (above). More generally, a non-compact group G admits representations belonging to the discrete series if it has the same rank as its maximal compact subgroup4. Thus the non-compact group E8(8) as well as E7(7) admit discrete series representations. However, they are not of the lowest or highest weight type. In this subsection we will analyze the decomposition of the minimal rep- R resentation of E8(8) with respect to its subgroup E7(7) SL(2, ), but using the complex basis of the last section. The group SU(1×, 1) admits holomorphic and anti-holomorphic unitary representations, and they exhaust the list of discrete series representations for SU(1, 1). (This is not true for higher rank non-compact groups admitting such representations.) As mentioned earlier, the realization of the SU(1, 1) subgroup within the minimal unitary representation of E8(8) is precisely of the form that arises in conformal quantum mechanics [10]. This is perhaps not surprising since we obtained our realization from the geometric action of E8(8) as a quasi-conformal group in 57 dimensions [16]. By comparison of the SU(1, 1) subgroup (44) with that of [10] it follows that the coupling constant g in conformal quantum mechanics is simply

g =4I4(X, P) (51) in our realization. The quadratic Casimir of SU(1, 1) is

2 1 3 2[SU(1, 1)] = L0 (L1L 1 + L 1L1)=I4(X, P) . (52) C − 2 − − − 16

Thus for a given eigenvalue of I4(X, P), we are led to a deﬁnite unitary realization of SU(1, 1). As we showed above, I4(X, P) is simply the quadratic Casimir operator of E7(7) (cf. (14)) that commutes with SU(1, 1), up to an additive normal ordering constant. Hence classifying all the possible eigenvalues of I4(X, P) within the minimal unitary realization of E8(8) is 4For an excellent introduction the general theory of unitary representations of non- compact groups see [26].

14 equivalent to giving the decomposition of the E8(8) representation with respect to E SU(1, 1)5. Unitarity requires the eigenvalues of the Casimir 7(7) × operators to be real. Therefore all the eigenvalues of I4 must be real. As we showed above, the realization of E7(7) within E8(8) coincides with the singletonic oscillator realization of E7(7) [17]. The oscillator realization of E7(7) leads to an infinitely reducible unitary representation [17]. Hence the minimal representation of E8(8) will be infinitely reducible with respect to E7(7) SU(1, 1). Denoting× the eigenvalues of the quadratic Casimir operator SU(1, 1) as [SU(1, 1)] = j(j 1) (53) C2 − we find that

1 1 j = I4 + (54) 2 ± q 16 Depending on the eigenvalue of I4 we will be led to one of the well-known series of representations of SU(1, 1) [1, 26] (a) Continuous principal series:

j = 1 i ρ, 0 <ρ< , 2 − ∞ j(j 1) = ( 1 + ρ2) < 1 (55) − − 4 − 4

with the eigenvalues ` of L0 unbounded from above and from below

` = ` ,` 1,` 2 ,... (` R) (56) 0 0 ± 0 ± 0 ∈ (b) Continuous supplementary series:

j 1 < 1 ` , (j, ` R) (57) | − 2 | 2 −| 0| 0 ∈

again with unbounded eigenvalues ` of L0 in both directions. (c) Holomorphic discrete series D+(j) (lowest weight irreps):

j>0 ,`= j, (j R) 0 ∈ ` = j, j +1,j+2, ... (58)

j>0 ,`= j, (j R) 0 − ∈ ` = j, j 1, j 2, ... (59) − − − − − 5Here we should note that in certain exceptional cases the eigenvalue of the quadratic Casimir operator may not uniquely label the unitary irreducible representation of SU(1, 1). In such cases one needs to use additional labels such as the eigenvalues of L0.

15 For vanishing quartic invariant I4 we ﬁnd 1 3 j = 4 or j = 4 (60) corresponding to the two singleton irreps of SU(1, 1). Note that the eigenvalues of the Casimir operator for the two singleton irreps coincide. They are distinguished by the value of j which is the eigenvalue of L0 on the corresponding lowest weight vector.

4.2 The SU(2, 1) truncation of E8(8)

To give the decomposition of the minimal representation of E8(8) w.r.t. its E SL(2,R) subgroup we need to determine all possible eigenvalues 7(7) × of I4(X, P) within our realization. Since this determination appears to be rather complicated, we shall study this question in a somewhat simpler setting by truncating E8(8) to a special subgroup. E8(8) has the subgroup E6(2) SU(2, 1), where E6(2) has the maximal compact subgroup SU(6) SU×(2). It is instructive to truncate our realization to the SU(2, 1) subgroup,× which is one of the minimal subgroups that admit a non-trivial 5-grading. This is achieved by introducing coordinate x and momentum px ij corresponding to the symplectic trace components of X and Pij, respectively:

x := √2 Ω Xij , − 4 ij √2 ij px := 4 Ω Pij , (61) and throwing away the symplectic traceless components in our realization. The matrix Ωij is the symplectic metric deﬁned by 0 11 Ω := − (62) ij 110

ij jk k and Ω is its inverse: ΩijΩ = δi . The generators x and px obey the canonical commutation relation

[x,px]=i. (63) The resulting expressions for the generators of SU(2, 1) are

1 2 E := 2 y ,

E := yx, ↑

E := ypx , ↓ 1 H := 2 (ypy + py y)

A := 3 (x2 + p2) − 4 x 16 1 1 3 F := xpy y− (xpxx + px) , ↑ − − 2 1 1 3 F := pxpy y− (pxxpx + x ) , ↓ − 2 1 2 1 2 2 2 2 1 2 F := p + y− (x + p ) y− , (64) 2 y 8 x − 8 The 5-grading in the above basis is with respect to H which is the non- compact dilatation generator. To understand the resulting representation of SU(2, 1) it is useful to go to the basis in which the Lie algebra of SU(2, 1) has a 3-grading w.r.t. the compact U(1) generator that commutes with the SU(2) subgroup.

L Ji L+ SU(2, 1) = − (65) K ⊕ J0 ⊕ K+ − The compact U(1) generator is

J = 1 (E + F 2 A)=L 1 A (66) 0 2 − 3 0 − 3 and the SU(2) generators are

1 J1 = (E + F ) , 2√2 ↑ ↓ 1 J2 = (E + F ) , 2√2 ↓ ↑ 1 J3 = 4 (E + F +2A) , (67) with the commutation relations

[Ji ,Jj]=iijkJk , (i, j, k =1, 2, 3)

[J0 ,Ji]=0. (68) The grade 1 generators are ± K =(E F ) i(E F ) , ± ↑ − ↓ ∓ ↓ − ↑ L = 1 (E F iH) , (69) ± 2 − ∓ satisfying

[J0 ,K ]= K , ± ± ± [J0 ,L ]= L . (70) ± ± ± More explicitly we have

1 1 2 2 K = (y ipy)+ y− [1 + (x + px)] (x ipx) , ± 2 { 1 ∓ 2 1 2 2 } 2 ∓2 L = (y ipy) y− [ 1+(x + px) ] (71) ± { 4 ∓ − 16 − }

The generator J0 that determines the 3-grading of SU(2, 1) w.r.t. maximal compact subgroup SU(2) U(1) is manifestly positive deﬁnite. In terms × 17 of the annihilation and creation operators ax,ax† the generator J0 takes the form:

1 2 2 2 J = y + p + y− N (N +1)+2N +1 (72) 0 4 { y x x x } where Nx = ax† ax is the number operator. This implies that the resulting unitary irreducible representations of SU(2, 1) must be of the lowest weight type (positive energy). Now the quadratic Casimir operator of SU(2, 1) is given by

[SU(2, 1)] = 1 (EF + FE) 1 H2 + 1 A2 C2 2 − 4 3 + 1 (E F + F E )+ 1 (E F + F E ) (73) 4 ↑ ↓ ↓ ↑ 4 ↓ ↑ ↑ ↓ Substituting the expressions for the generators we ﬁnd that the the quadratic Casimir of SU(2, 1) becomes simply a c-number, namely

[SU(2, 1)] = 3 (74) C2 − 16 which suggests that the representation may be irreducible. To have an irreducible representation the cubic Casimir must likewise reduce to a c- number. However, we can study the irreducibility of the representation without having to calculate the cubic Casimir. Since we know that our representation is of the lowest weight type we can use the fact that a unitary irrep of the lowest weight type is uniquely determined by a set of states Ω transforming irreducibly under the maximal compact subgroup SU(2) U|(1)i and that are annihilated by all the generators of grade 1 under× the 3- grading. Thus we need to ﬁnd all such states that are annihilated− by K and L : − − K Ω =2a + 1 y 1(N +1) a Ω =0 y √ − x x −| i 2 | i 1 2 1 2 L Ω = 2 ay 2 y− Nx(Nx +1) Ω = 0 (75) −| i − | i and that transform irreducibly under the maximal compact subgroup SU(2) U(1). The only normalizable state satisfying these conditions is the one par-× ticle excited state

a† 0 (76) y| i where the vacuum state 0 is annihilated by both annihilation operators | i a 0 =0 x| i a 0 = 0 (77) y| i This state is a singlet of SU(2) and has the U(1)charge1.Thisproves that the minimal realization of SU(2, 1) obtained by truncation of the minimal representation of E8(8) is also irreducible.

18 The subgroup E7(7) SU(1, 1) of E8(8) under truncation to SU(2, 1) reduces to U(1) SU(1, 1).× The generators of SU(1, 1) subgroup of SU(2, 1) are given by ×

1 L+ = 2 (E F iH) 1 − − 2 1 2 2 2 2 = (y ip ) y− [ 1+(x + p ) ] { 4 − y − 16 − x } L = 1 (E F + iH) − 2 1 − 2 1 2 2 2 2 = 4 (y + ipy) 16 y− [ 1+(x + px) ] {1 − − } L0 = 2 (E + F ) 1 2 2 1 2 2 2 2 = (y + p )+ y− [ 1+(x + p ) ] (78) { 4 y 16 − x } They are therefore the analogs of the generators(44). The generator of U(1) that commutes with SU(1, 1) is simply A. In this truncation the quartic E7(7) invariant reduces to

I = 1 [(x2 + p2 )2 1] (79) 4 16 x − which can be written in terms of the annihilation and creation operators as

1 I = (a† a )(a† a +1) (80) 4 4 { x x x x } √2 AB √2 AB with the (obvious) identiﬁcations a = Ω a and ax† = Ω a . x − 4 AB 4 AB Since the eigenvalues of the number operator a†a are non-negative integers n, 1 the eigenvalues of I4 are simply 4 n(n+1). In fact the realization of SU(1, 1) in this case leads to unitary lowest weight representations of the type studied in [10]. For n = 0 we get the singleton irreps of SU(1, 1) corresponding to 1 3 the values j = 4 and 4 . Thus the minimal unitary representation of SU(2, 1) decomposes into a discretely inﬁnite set of irreps of SU(1, 1) labelled by the eigenvalues of the U(1) generator which is the analog of the E7(7) subgroup of E8(8) for the SU(2, 1) truncation.

5 Outlook

The finite dimensional conformal group SU(1, 1) is well known to possess an infinite dimensional extension (the Witt-Virasoro group). One may therefore ask whether there exists a generalization of this fact to E8(8). In other words, does there exist an infinite dimensional Lie algebra (or Lie superalgebra) that contains the Witt-Virasoro algebra and E8(8) at the same time? While there appears to be no linear Lie algebra with this property (and no finite dimensional Lie superalgebra, either), an infinite dimensional non-linear algebra of -type does exist. It is a nonlinear quasi-superconformal algebra denoted W as QE8(8) [3]. The quasi-superconformal algebras in two dimensions were first introduced in[30] and further systematized in[2]. They were generalized in[31] where two infinite families of nonlinear quasi-superconformal algebras

19 were introduced. A classification of complex forms of quasi-superconformal algebras was given in [14]. In [3] a complete classification and a unified realization of the real forms of quasi-superconformal algebras were given. In the infinite central charge limit the exceptional quasi-superconformal algebra QE8(8) has the Lie algebra E8(8) as a maximal finite dimensional simple Lie subalgebra. The realization of QE8(8) given in[3] involves 56 dimension 1/2 bosons and a dilaton, which leads to a realization of E8(8) in the infinite central charge limit. It will be important to understand if the resulting realization can be related to the one given here. Furthermore one would like to know if one could use the methods of [3] to give a unified realization of the unitary representations of non-compact groups that act as quasi-conformal groups as formulated in [16], thus generalizing our minimal realization E8(8) to all such noncompact groups.

Acknowledgments: We are grateful to B. Pioline and A. Waldron for discussions, and for sending us an advance copy of ref. [24]. We would also like to thank R. Brylinski for bringing the references [33] and [15] to our attention and for helpful discussions.

References

[1] V. Bargmann. Irreducible unitary representations of the Lorentz group Ann. Math. 48, 568 (1947).

[2] M. Bershadsky. Conformal ﬁeld theories via Hamiltonian reduction Comm. Math. Phys. 139, 67 (1993).

[3] B. Bina and M. G¨unaydin. Real forms of non-linear superconformal and quasi-superconformal algebras and their uniﬁed realization. Nucl. Phys. B502, 713 (1997).

[4] R. Brylinski, B. Kostant. Minimal representations of e6, e7,ande8 and the generalized Capelli identity. Proc. Natl. Acad. Sci. 91, 2469 (1994)

[5] R. Brylinski, B. Kostant. Minimal representations, geometric quantization, and unitarity. Proc. Natl. Acad. Sci. 91, 6026 (1994)

[6] R. Brylinski, B. Kostant. Lagrangian models of minimal representations of e6, e7 and e8. Prog. Math. 131, 13 (1995) [7] R. Brylinski, B. Kostant. Geometric quantization and holomorphic half-form models of unitary minimal representations I Preprint (1996)

[8] P. Claus, M. Derix, R. Kallosh, J. Kumar, P. Townsend, A. Van Proeyen. Black holes and superconformal mechanics. Phys. Rev. Lett. 81, 4553 (1998)

20 [9] E. Cremmer, B. Julia. The SO(8) supergravity. Nucl. Phys. B159, 141 (1979)

[10] V. de Alfaro, S. Fubini, G. Furlan. Conformal invariance in quantum mechanics. Nuovo Cim. A34, 569 (1976)

[11] B. de Wit, H. Nicolai. d = 11 supergravity with local SU(8) invariance Nucl. Phys. B274, 363 (1986)

[12] S. Ferrara, M. G¨unaydin. Orbits of exceptional groups, duality and BPS states in string theory. Int.J.Mod.Phys., A13, 2075 (1998) [hep-th/9708025]

[13] S. Ferrara, J. Maldacena. Branes, central charges and U-duality invariant BPS conditions. Class. Quant. Grav., 15, 749 (1998) [hep- th/9706097]

[14] E.S. Fradkin and V. Yu. Linetsky. Phys. Lett. B275, 345 (1992); ibid. B282, 352 ( 1992).

[15] B. Gross and N. Wallach. A distinguished family of unitary representations for the exceptional groups of real rank =4, in Lie Theory and Geometry: in Honor of B. Kostant, J.-L. Brylinski, R. Brylinski, V. Guillemin, V. Kac, eds, Progress in Mathematics 123,Birkhauser, Boston (1994), 289-304.

[16] M. G¨unaydin, K. Koepsell, H. Nicolai. Conformal and quasiconformal realizations of exceptional Lie groups (2000). Commun. Math. Phys. 221, 57 (2001), hep-th/0008063 [17] M. G¨unaydin, C. Sa¸clio˘glu. Oscillator like unitary representations of noncompact groups with a Jordan structure and the noncompact groups of supergravity. Commun. Math. Phys. 87, 159 (1982)

[18] Harish-Chandra. Representations of semisimple Lie groups. V. Am. J. Math. 78, 1 (1956)

[19] Harish-Chandra. Representations of semisimple Lie groups VI. Inte- grable and square- integrable representations. Am.J.Math.78, 564 (1956)

[20] A. Joseph. Minimal realizations and spectrum generating algebras. Commun. Math. Phys. 36, 325 (1974)

[21] A. Joseph. The minimal orbit in a simple Lie algebra and its associated maximal ideal. Ann. Sci. Ec. Norm. Super., IV. Ser. 9, 1 (1976)

[22] R. Kallosh, B. Kol. E(7) symmetric area of black hole horizon Phys. Rev. D53, 5344 (1996)

21 [23] D. Kazhdan, G. Savin. The smallest representation of simply laced groups Israel Math. Conf. Proceedings, Piatetski-Shapiro Festschrift 2, 209 (1990)

[24] D. Kazhdan, B. Pioline, A. Waldron. Minimal representations, spherical vectors, and exceptional theta series I. hep-th/0107222

[25] See: http://www.lpthe.jussieu.fr/ pioline/minrep/E8.h. ∼ [26] A. Knapp. Representation theory of semisimple groups, an overview basedonexamples.Princeton Mathematical Series 36.

[27] K. Koepsell. PhD Thesis, Hamburg University (June 2001)

[28] J. Michelson, A. Strominger. Superconformal multi-black hole quantum mechanics. JHEP 09, 005 (1999)

[29] B. Pioline, H. Nicolai, J. Plefka, A. Waldron R4 couplings, the funda- mental membrane and theta correspondence JHEP 0103, 036 (2001)

[30] A. Polyakov. Int.J.Mod.Phys.A5, 833 (1990).

[31] L.J. Romans. Nucl. Phys. B357, 549 (1991).

[32] F. Sugino, P. Vanhove U-duality from matrix membrane partition function hep-th/0107145

[33] D. Vogan. Singular Unitary Representations in Non-commutative Harmonic Analysis and Lie Groups, J. Carmona and M. Vergne, eds. Springer Lecture Notes 880, Berlin, 1981, 506-535.