IASSNS-HEP-92/86 Dec. 1992 Generalized Conformal and Superconformal Group Actions 3 9 9 and Jordan Algebras 1 n a J Murat Gu¨naydin∗ 3 1 School of Natural Sciences Institute for Advanced Study 1 Princeton, NJ 08540 v 0 and 5 Penn State University† 0 1 Physics Department 0 University Park, PA 16802 3 9 Abstract / h t We study the “conformal groups” of Jordan algebras along the lines suggested by - p Kantor. They provide a natural generalization of the concept of conformal trans- e formations that leave 2-angles invariant to spaces where “p-angles” (p ≥ 2) can h : be defined. We give an oscillator realization of the generalized conformal groups v i of Jordan algebras and Jordan triple systems. A complete list of the general- X ized conformal algebras of simple Jordan algebras and hermitian Jordan triple r a systems is given. These results are then extended to Jordan superalgebras and super Jordan triple systems. By going to a coordinate representation of the (su- per)oscillators one then obtains the differential operators representing the action of thesegeneralized (super)conformalgroupsonthecorresponding(super)spaces. The superconformal algebras of the Jordan superalgebras in Kac’s classification is also presented. (∗) Work supported in part by the National Science Foundation Grant PHY-9108286. (†) Permanent Address. e-mail: [email protected] or [email protected] 1 Introduction Conformal symmetry plays an important role in the formulation and un- derstanding of many physical theories. For example, the massless gauge theories in four dimensions are invariant under the fifteen parameter confor- mal group SO(4,2). The known string theories are all invariant under the infinite conformal group in two dimensions which can be identified with the reparametrization invariance of the string world-sheet. The two dimensional physical systems are known to exhibit conformal symmetry at their critical points. Theconformalinvarianceisnormallydefinedastheinvarianceofaquadratic form or a metric up to an overall scale factor which is a function of the lo- cal coordinates. This implies , in particular, the local invariance of angles defined by the metric or the quadratic form. It would be of physical inter- est to know if there exist generalizations of conformal invariance to spaces that are naturally endowed with higher order forms. For example, the p- brane theories are naturally endowed with a volume form which is of order p. Such a generalization was suggested by Kantor in his study of the in- variance groups of “p-angles” that can be defined over spaces with a p-form (not to be confused with a differential p-form) [1]. He studied , in particu- lar, the invariance groups of the “p-angles” defined by Jordan algebras with a generic norm of degree p. We shall refer to these groups generically as generalized conformal groups. In this paper we shall give a simple oscillator realization of the generalized conformal algebras of Jordan algebras. In this realization the Jordan triple product plays a crucial role. The fact that the Jordan triple product rather than the binary Jordan product is essential in our formulation allows us to extend our results to Jordan triple systems. We give a complete list of simple Jordan algebras and hermitian Jordan triple systems and their generalized conformal algebras. We then extend our re- sults tostudy thesuperconformal algebrasofJordansuperalgebras andsuper Jordan triple systems. The list of simple Jordan superalgebras as classified by Kac [2] and their generalized conformal superalgebras are also given. By going to the coordinate representation of the oscillators one obtains a dif- ferential operator realization of the action of the generalized conformal and superconformal algebras on the corresponding spaces and superspaces. 1 2 Linear Fractional Groups of Jordan Alge- bras as Generalized Conformal Groups The conformal transformations T on a Riemannian manifold with the metric g are defined such that under their action the metric transforms as µν T : g −→ φg (2.1) µν µν where φ is a scalar function of the coordinates. On a d-dimensional Eu- clidean space (d > 2) with a non-degenerate positive definite quadratic form (x,x) the conformal transformations leave invariant the following cross-ratio associated with any set of four vectors x,y,z,w: (x−z,x−z) (y −w,y−w) (2.2) (x−w,x−w) (y −z,y −z) as well as the quantity (x,y)2 (2.3) (x,x)(y,y) which is the cosine square of the angle between the vectors x and y. Over a Euclidean space one may use (2.1), (2.2) or (2.3) interchangeably to define conformal transformations. Using the condition (2.2) or (2.3) allows for an interesting generalization of conformal transformations as was suggested by Kantor [1] , which we shall briefly review below. Kantor considers an n-dimensional vector space V endowed with a non- degenerate form of degree p N(x) ≡ N(x,x,..,x) (2.4) To every ordered set of four vectors x,y,z and w in V one associates a cross- ratio N(x−z)N(y −w) (2.5) N(y −z)N(x−w) and for each set of p straight lines (p-angle) with direction vectors x ,...,x 1 p one defines the quantity N(x ,..x )p 1 p (2.6) N(x )N(x )···N(x ) 1 2 p 2 which is called the measure of the p-angle. Let us denote the invariance groups of the cross-ratio (2.5) and the measure of the p-angle (2.6) G and G˜, respectively. It can be shown that if G˜ is finite dimensional then it is isomorphic to G [1]. Some of the most interesting realizations of the above generalization of conformal transformations are provided by Jordan algebras with a norm form. If J is a semi-simple Jordan algebra with a generic norm form as defined by Jacobson [3] then considering J as a vector space one can study its generalized conformal transformation groups using the definitions (2.5) or (2.6). The corresponding groups G and G˜ coincide if and only if J contains no one dimensional ideals in the complex case and no one or two dimensional ideals in the real case. The action of G on J can be written as a “linear fractional transformation” of J. The linear fractional transformation groups of Jordan algebras were studied by Koecher [4] and Kantor [1]. The linear fractional transformation groups of Jordan superalgebras were studied in [5, 6, 7, 8]. 3 Conformal Algebras of Jordan Algebras and Jordan Triple Systems The reduced structure group H of a Jordan algebra J is defined as the in- variance group of its norm form N(J). By adjoining to it the constant scale transformations one gets the full structure group of J. The Lie algebra g of the conformal group of J can be given a three-graded structure with respect to the Lie algebra g0 of its structure group g = g−1 ⊕g0 ⊕g+1 (3.1) where g0 = h⊕E (3.2) with h denoting the Lie algebra of H and E the generator of the constant scale transformations. The Tits-Kantor-Koecher (TKK)[9] construction of the Lie algebra g establishes a one-to-one mapping between the grade +1 subspace of g and the corresponding Jordan algebra J: U ∈ g+1 ⇐⇒ a ∈ J (3.3) a 3 Every such Lie algebra g admits a conjugation (involutive automorphism) † under which the elements of the grade +1 subspace get mapped into the elements of the grade −1 subspace. Ua = U† ∈ g−1 (3.4) a One then defines [U ,Ub] = Sb a a (3.5) [Sb,U ] = U a c (abc) where Sb ∈ g0 and (abc) is the Jordan triple product a (abc) = a·(b·c)+(a·b)·c−b·(a·c) (3.6) with · denoting the commutative Jordan product. Under conjugation † one finds (Sb)† = Sa a b (3.7) [Sb,Uc] = −U(bac) a The Jacobi identities in g are satisfied if and only if the triple product (abc) satisfies the identities (abc) = (cba) (3.8) (ab(cdx))−(cd(abx))−(a(dcb)x)+((cda)bx) = 0 These identities follow from the defining identities of a Jordan algebra: a·b = b·a (3.9) a·(b·a2) = (a·b)·a2 (3.10) The elements Sb of the structure algebra g0 of J satisfy : a [Sb,Sd] = Sd −S(bad) = S(dcb) −Sb (3.11) a c (abc) c a (cda) Denoting as JA and as Γ(d) the Jordan algebra of n×n Hermitian matrices n over the division algebra A, and the Jordan algebra of Diracgamma matrices in d Euclidean dimensions, respectively, one finds the following conformal groups (G) and reduced structure groups (H) of simple Jordan algebras: 4 J H G JR SL(n,R) Sp(2n,R) n JC SL(n,C) SU(n,n) n JH SU∗(2n) SO∗(4n) n JO E E 3 6(−26) 7(−25) Γ(d) SO(d,1) SO(d+1,2) The symbols R, C, H, O represent the four division algebras. We should alsonotethatbytakingdifferentrealformsoftheJordanalgebrasoneobtains different real forms of the conformal and reduced structure groups. In the TKK construction only the triple product (abc) enters and the identities (3.8) turn out to be the defining identities of a Jordan triple system (JTS). Therefore, the TKK construction extends trivially to Jordan triple systems. Of particular interest are the hermitian JTS’s for which the triple product (abc) is linear in the first and the last arguments and anti-linear in the second argument. There exist four infinite families of hermitian JTS’s and two exceptional ones [10]. They are: Type I generated by P ×Q complex matrices M (C) with the triple P,Q P,Q product (abc) = ab†c+cb†a (3.12) where † represents the usual hermitian conjugation. Type II generated by complex anti-symmetric N ×N matrices A (C) N N with the ternary product 3.12. Type III generated by complex N×N symmetric matrices S (C) with N N the product 3.12. Type IV generated by Dirac gamma matrices Γ (C) in N dimensions N N with complex coefficients and the Jordan triple product ¯ ¯ ¯ (abc) = a·(b·c)+c·(b·a)−(a·c)·b (3.13) 5 where the bar − denotes complex conjugation. Type V generated by 1×2 complex octonionic matrices M1,2(OC) with the triple product (abc) = {(a¯b†)c+(¯ba†)c−¯b(a†c)}+{a ↔ c} (3.14) where † denotes octonion conjugation times transposition. Type VI generated by the exceptional Jordan algebra of 3×3 hermitian octonionic matrices JO(C) taken over the complex numbers with the triple 3 product 3.13. Below we tabulate the simple hermitian JTS’s and their conformal (G) and reduced structure groups (H): HJTS G H M (C) SU(P,Q) SL(P,R)×SL(Q,R) P,Q A (C) SO(2N)∗ SU∗(N) N S (C) Sp(2N,R) SL(N,R) N Γ (C) SO(N +1,2) SO(N,1) N M1,2(OC) E6(−14) SO(8,2) JO(C) E E 3 7(−25) 6(−26) 4 Oscillator Realization of the Generalized Conformal Groups In the TKK construction of the conformal algebras of Jordan algebras and Jordan triple systems the commutation relations are expressed in terms of the triple product (abc). Let us choose a basis e for the Jordan algebra a or the JTS and introduce the structure constants Σcd for the Jordan triple ab product (e e e ) = Σbde (4.1) a b c ac d 6 a,b,.. = 1,2,...D Using these structure constants one can give oscillator realizations of the generalized conformal algebras . Consider now a set of D bosonic oscillators A ,Ab(a,b,... = 1,2,...,D) that satisfy the canonical commutation relations: a [A ,Ab] = δb a a [A ,A ] = 0 (4.2) a b [Aa,Ab] = 0 The bilinears Sb = −ΣbdAcA (4.3) a ac d generate the structure algebra of the corresponding Jordan algebra or the Jordan triple system under commutation [Sb,Sd] = −ΣdeSb +ΣdbSe (4.4) a c ca e ce a If we further let U = −A (4.5) a a and define 1 Ua = ΣaeAcAdA (4.6) 2 cd e we find that they close into the generators of the structure algebra under commutation [U ,Ub] = Sb (4.7) a a Furthermore they satisfy [Sb,U ] = ΣbeU (4.8) a c ac e [Ua,Ub] = [U ,U ] = 0 (4.9) a b In proving some of these commutation relations we used the identity ΣdfΣbg −ΣbfΣdg = ΣdbΣfg −ΣdfΣbg (4.10) ce af ae cf cf ae ca fe 7 and the symmetry of the structure constants Σcd = Σcd , which follow from ab ba the defining identities (3.8) of JTS’s. Thus the operators U ,Ua and Sb a a generate the conformal algebra of the corresponding Jordan algebra or the JTS. To see how this realization is related to the action of the structure algebra on the JTS (or the Jordan algebra) expand the elements x ∈ J in the basis (e ): a x = xae x ∈ J (4.11) a Then the action of the conformal algebra on J is equivalent to the action of the differential operators on the “coordinates” xa obtained by realizing the operators A and Aa as a ∂ A = (4.12) a ∂xa Aa = xa (4.13) This leads to the differential operator realization ∂ U = − (4.14) a ∂xa 1 ∂ Ua = Σabxcxd (4.15) 2 cd ∂xb ∂ Sb = −Σbdxc (4.16) a ac ∂xd Thus we can interpret the action of the generalized conformal groups on Jor- dan algebras or JTS’s in the usual way [7, 8] i.e. the U and Ua are the a generators of translations and “special conformal transformations”, respec- tively. The Sb are the generators of “Lorentz transformations” and dilata- a tions [5, 7, 8]. 5 Oscillator Realization of the Conformal Su- peralgebras of Jordan Superalgebras Jordan superalgebras were defined and classified by Kac [2] using methods developed for Jordan algebras by Kantor [11]. An infinite family of Jordan 8 superalgebras was missed in the classification of Kac and was discovered by Kantor [12]. A Jordan superalgebra J is a Z graded algebra 2 J = J0 ⊕J1 (5.1) with a supercommutative product A·B = (−1)dAdBB ·A (5.2) d ,d ,... = 0,1 A B that satisfies the super Jordan identity: (−1)dAdC[L ,L }+(−1)dBdA[L ,L }+(−1)dCdB[L ,L } = 0 (5.3) A·B C B·C D C·A B where L denotes multiplication from the left by the element A. The mixed A bracket [,} represents an anticommutator for any two odd operators and a commutator otherwise. The super Jordan triple product is defined as [14, 6, 8]: (ABC) = A·(B ·C)−(−1)dAdBB ·(A·C)+(A·B)·C (5.4) The superconformal algebras g of Jordan superalgebras can be constructed in complete analogy with the TKK construction for Jordan algebras [14, 6, 7] . One defines a three-graded Lie superalgebra g = g−1 ⊕g0 ⊕g+1 (5.5) where the elements of the grade ±1 subspaces are labelled by the elements of the Jordan superalgebra J: U ∈ g+1 (5.6) A UA ∈ g−1 Their supercommutators give the generators SB belonging to the grade zero A subspace [U ,UB} = (−1)dAdBSB ∈ g0 (5.7) A A 9