Boson-Fermion Confusion: the String Path to Supersymmetry

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

provided by CERN Document Server

Boson-Fermion Confusion: The String Path To Supersymmetry

a P. Ramond ∗ aInstitute for Fundamental Theory, Physics Department, University of Florida P.O. Box 118440, Gainesville, Fl 32605

Reminiscences on the String origins of Supersymmetry are followed by a discussion of the importance of con- fusing bosons with fermions in building superstring theories in 9+1 dimensions. In eleven dimensions, the kinship between bosons and fermions is more subtle, and may involve the exceptional group F4.

1. Introduction progenitors of modern string theories. The generalization of Dual Models to include half-odd in- Although the idea of a symmetry between teger spins [6,7] followed soon after in early 1971. bosons and fermions must be very old, after all When expressed in terms of world-sheet symme- both are present in Nature, I am only aware of tries, both R and NS formulations were shown [8] non-relativistic attempts in that direction in the to be examples of supersymmetry on the 1 + 1 1960’s. Stavraki [1] (1967) proposed a current world-sheet. This in turn led to supersymme- algebra with both commutators and anticommu- try in 3 + 1 dimensions, with explicit local in- tators. Myazawa [2] (1968), then at the Univer- teracting field-theories [9]. It is only later that sity of Chicago, put together the fermions in the space-time supersymmetry between the R and 56 with the bosons in the 35 of SU(6) into one NS sectors was realized in 9 + 1 dimensions by algebraic structure, which led him to invent the the GSO [10] projection. Born in the context superalgebra we now call SU(6/21)! of Dual Resonance Models, relativistic supersym- In the Soviet Union, a brilliant generalization metry has now been found to play a central role of the Poincaré group to include anticommuting in the formulation of quantum field theories, and charges is proposed in 1971 by Golfand and Likht- perhaps even of Nature itself. man [3], realizing relativistic supersymmetry in 3 + 1 dimensions for the first time. This is followed in 1972 by a non-linear realization of this 2. Reminiscences symmetry by Volkov and Akulov [4]. This conference affected me like a madeleine, String Theory stems from the Dual Resonance and although still quite young, I will take a few Model, formulated to satisfy Dolen-Horn-Schmid lines to offer some recollections of the epic pe- duality [5], according to which, in pion-nucleon riod when the building blocks of supersymme- scattering, the averaged s-channel fermionic res- try were being laid down. Soon after graduat- onances were related to the t-channel boson ex- ing from Syracuse University in spring 1969, my changes. As such it implied a relation between wife and I sailed to Europe to spend the sum- bosons and fermions, although early workers seem mer at the ICTP, where J. Nuyts and H. Sug- to have put aside spin as an inessential com- awara introduced me to the joys of working on plication. Amplitudes involving only bosons, the Veneziano model. By the time we sailed back known today as the Veneziano (open string) and to the United States to join the Fermi National Virasoro-Shapiro (closed string) models, are the Laboratory (then known as NAL) as one of its ∗Research supported in part by The United States De- first postdocs, I had been hopelessly seduced by partment of Energy grant DE-FG02-97ER41029. their elegance and promise of simplicity. So much 2 so that, in the middle of the Atlantic, I found my- is during that visit that I met Mike Green, but I self in the reading room of the liner France, with had to wait until the spring of 1971 to meet John a seminal paper by Fubini and Veneziano on the Schwarz, and even longer L. Brink, and C. Thorn, harmonic oscillator formulation. I then went to my lifelong dual friends! my cabin to fetch postcards. Upon my return I saw the Fubini-Veneziano preprint on the table 3. The First Fermion-Boson Confusion I had vacated. Doubly puzzled, since I had just left it in my cabin, and it was, unlike my copy, Fermions and bosons cannot easily be confused, heavily annotated, I decided to wait for its owner. as they differ both by their quantization and In walks a rather lanky individual to reclaim his space-transformation properties, but the latter posession; it was André Neveu, on his way to join can be very similar in some dimensions. An early Joël Scherk in Princeton! example, where assigning bosonic quantum num- In the Fall of 1969, at NAL, I started working bers to fermions leads to a relativistic theory, is with Lou Clavelli on the group theoretical struc- the introduction of fermions to Dual Models [6]. ture of the Veneziano amplitudes. There were no The N-point (bosonic) Veneziano amplitude is a senior theorists at NAL; we were on our own, but sum of terms of the form for Nambu, who was very supportive of our work, < 1 V (2)V (3) V (N 1) N > , (1) and even invited us to lunch at the Quadrangle | · · · − | Club! I also met in that period, many of the where early luminaries: M. Virasoro, then at Madison, ikj Q(θj ) V (j) : e · : , (2) Scherk and his first wife (whom he had met at the ∼ Club Méditerranée!), S. Fubini, B. Sakita, and G. is the vertex for the emission of the jth particle Veneziano. of momentum kj , which resembles a plane wave In the spring of 1970, the Director of NAL, Bob with a generalized coordinate Wilson decreed: “All theorists must go to As- ∞ 1 (n) inθ (n) inθ pen”. I had never heard of the place, did not want Qµ(θ) = xµ+θpµ+ a †e− +a e .(3) √n µ µ to go, but one did not resist Bob very long. That X1 summer, at the Aspen Center for Physics, the Here xµ and pµ are the usual position and mo- break from grungy calculations, combined with mentum, and there are an infinite number of rel- the mountain air and the wonderful music, made ativistic harmonic oscillators me realize an easy way to look at these mod- (n) (m) nm els. I decided to use the generalized position [ aµ , aρ † ] = δ gµρ . (4) and momentum operators, introduced by Fubini This leads to a generalized momentum and Veneziano, in devising equations of motion. Upon my return at NAL, I wrote a short pa- dQµ(θ) ∞ (n) inθ (n) inθ Pµ(θ) = = pµ i √n a †e− a e .(5) dθ − µ − µ per on the Dual Klein-Gordon equation, where X1 I spoke of the correspondance between Dual sys- Both operators reduce to their point limit in the tems and point particles (later to be known as the (zero slope) limit zero-slope limit). To my dismay, it was rejected by Physics Letters, and I withdrew the paper. Qµ(θ) xµ ; Pµ(θ) pµ . (6) → → Shaken but undaunted, I proceeded to generalize Now if we apply this correspondance to the Klein- the Dirac equation. There I found to my surprise Gordon operator, a new algebraic structure, the square root of the µ µ Virasoro algebra. That Fall, I went to the Insti- pµ p Φ = 0 Pµ(θ) P (θ) Φ = 0 , (7) → tute in Princeton where Nambu was spending a with sabbatical. Again he was very supportive of my ideas, and encouraged me further, but I got stuck µ ∞ inθ inθ Pµ(θ) P (θ) = L0 + Lne− +L ne ,(8) because I did not know Grassmann variables! It − X1 3 where the L’s satisfy the Virasoro algebra. In group is SO(8), spinors and vectors have the same particular, L0 is, up to an additive constant, the number of degrees of freedom. equation of motion for the bosons, and the Ln In relativistic theories, bosons and fermions p a(n) are akin to the decoupling operators of∼ usually transform differently under space rota- QED.· Hence the analogy is established: take the tions. For massless particles the relevant group generalized quantities, and replace the product of of rotations is the light-cone little group. In the averages by the average of the products: 1 + 1 dimensions, there is no such group, and µ µ bosons differ from fermions only by quantiza- < Pµ(θ) > < P (θ) > < Pµ(θ) P (θ) > ,(9) → tion, and one can build bosons out of fermions where < . . . > denotes the average, or integra- without group-theoretical obstructions; the same tion over θ. applies to 2 + 1 dimensions. In 3 + 1 dimen- The same procedure can be applied to the Dirac sions, the little group is non-trivial and fermions equation are distinguished by their helicities– integer for bosons, half-integer for fermions. In higher di- γ p Ψ = 0 , (10) · mensions, fermions (bosons) transform according by imagining that the gamma matrices are them- to the spinor (tensor) representations of the Non- selves averages of something Abelian little group. In 9 + 1 dimensions, the massless little group is SO(8) and bosons and γµ Γµ(θ) = γµ + , (11) fermions have the same number of degrees of free- → · · · dom. This fact lies at the heart of the superstring leading to a generalized Dirac equation constructions. In 10 + 1 dimensions and above, < Γ(θ) P (θ) > Ψ = 0 , (12) they become different again. However, in special · numbers of dimensions, a strange kinship between together with spinor and tensor representations of the appropri- ate rotation group appears. In eleven dimensions, Γµ(θ), Γρ(θ0) = 2gµρδ(θ θ0) . (13) { } − it leads to the supergravity theory, and, as we will this requires the introduction of anticommuting show, possibly more. harmonic oscillators which carry vector indices 4. A Second Fermion-Boson Confusion? ∞ Γ (θ) = γ + γ b(n) e inθ + b(n)einθ , (14) µ µ 5 µ † − µ In 10 + 1 dimensions, there is no apparent kin- X1 ship between fermions and bosons. Yet there ex- where γ5 anticommutes with the Dirac matrices. ists a supersymmetric theory in eleven dimen- Then sions, M-theory, with supergravity as its local limit. The degrees of freedom of supergravity are ∞ inθ inθ Γ(θ) P (θ) = F0 + Fne + F ne− ,(15) massless particles, belonging to representations of · − X1 SO(9), the light-cone little group: where the F ’s satisfy the superVirasoro algebra. Graviton as a symmetric second-rank tensor This chain of reasoning covers the genesis of the • h(ij), with Dynkin label [2000], Dirac equation and the appearance of commut- ing and anticommuting structures in a relativis- Third-rank antisymmetric tensor, A , • [ijk] tic framework. It is merely a generalization of the with Dynkin label [0010], algebra of Dirac’s operator Rarita-Schwinger spinor-vector, Ψαi, with γ p , γ p = 2 p p . (16) • label [1001]. { · · } · This procedure, assigning a vector index to a Their group-theoretical properties are summa- fermion, might appear foolish at first glance, ex- rized in the following table of Dynkin indices of cept in 9+1 dimensions where the light-cone little different orders [11] 4

irrep [1001] [2000] [0010] Its solutions consist of all triples, including the su- I0 128 44 84 pergravity multiplet. It is convenient to rewrite I2 256 88 168 the gamma matrices in terms of eight Grassmann I4 640 232 408 variables, and express the solutions as chiral su- I6 1792 712 1080 perfields in these variables, I8 5248 2440 3000 Ψ = ψ + ψiθ + ψij θiθj + , (20) 0 1 · · · These indices, except for I8, match between the and the supergravity solution corresponds to all fermion and the two bosons. It turns out that constant ψ. Under SO(9) SO(7) SO(2), these there are infinitely many trios of representations split as ⊃ × of SO(9) with similar group-theoretic relations among them. The simplest example is given by 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 , (21) the triplet made of fields with index structure reproducing the supergravity multiplet. Other h(ijk)l + A(ij)(kl)m + Ψα(ij)k , (17) solutions involve fields of higher spin. If these and group-theoretical properties fields are to be incorporated in a relativistic the- irrep [2100] [0110] [1101] ory, we must overcome the problem of massless spins with spins higher than two, bringing in well I 910 1650 2560 0 documented difficulties, with coupling spin-one I2 3640 6600 10240 current [14] and energy-momentum tensor [15,16] I4 19864 34920 54784 to massless particles of spin greater than one. I6 130840 217320 348160 I8 977944 1498344 2466304 5. A Simpler Example

They describe higher spin massless fields, with no A similar but much simpler construction can apparent supersymmetry. This is only one exam- be achieved for the coset SU(3)/SU(2) U(1). × ple of this infinite set, which can be obtained from At the lowest level, it leads to a triplet of repre- a character formula[12], traced to the three equiv- sentations on which N = 2 supersymmetry can alent embeddings of SO(9) inside the exceptional be realized in 3 + 1 dimensions. group F4! Under the embedding F4 SO(9), the ⊃ 5.1. The N = 2 Hypermultiplet 52 parameters of F4 contain the 36 generators of SO(9) and 16 parameters which transform as the We first recall the well-known light-cone de- SO(9) spinor representation, and label the coset scription of the massless N = 2 hypermultiplet in 3+1 dimensions [17], which contains two Weyl F4/SO9. Kostant[13] introduces over that space sixteen (256 256) gamma matrices which gener- spinors and two complex scalar fields, on which ate the Clifford× algebra the N = 2 SuperPoincaré algebra is realized. In- troduce the light-cone Hamiltonian γa , γb = 2 δab , a, b = 1, 2 . . . , 16 . (18) { } pp Note that the “vector indices” of these matrices P − = + , (22) actually transform as the spinor of SO(9)! This 2p is possible because of the anomalous embedding where p = 1 (p1 + ip2) . The front-form su- SO(16) SO(9), where the 16 vector of SO(16) √2 ⊃ persymmetry generators satisfy the anticommu- is the 16 spinor of SO9). Another example of tation relations fermion-boson confusion. Let T a be the genera- m n mn + tors of F4 not contained in SO(9), and form the + , + = 2δ p , Kostant equation {Q Q } − m n mn pp 16 , = 2δ + , m, n = 1, 2 ,(23) a a {Q− Q−} − p γ T Ψ / Ψ = 0 . (19) m n mn ≡ K + , = 2pδ . X1 {Q Q−} − 5

The kinematic supersymmetries are expressed as solved by the chiral superﬁeld

m ∂ + m ∂ + i m 12 + = m θmp , + = + θmp ,(24) Φ(y−, x , θm) = ψ0 + θmψ + θ1θ2ψ , (30) Q −∂θ − Q ∂θm while the kinematic Lorentz generators are given where the arguments of the ψ’s depend on by m y− = x− iθmθ , (31) 12 1 ∂ 1 m ∂ − M = i(xp xp) + θm θ m , − 2 ∂θm − 2 ∂θ and the transverse variables. Acting on this chiral + + i ∂ i m ∂ superfield, the constraint is equivalent to requir- M − = x−p θm θ m , ing that − − 2 ∂θm − 2 ∂θ + 1 +1 +2 + M (M + iM ) = xp , m + m ∂ ≡ √2 − + 2p θm , + , (32) ∂θ + Q ≈ − Q ≈ m M = xp+ , (25) − where the derivative is meant to act only on the naked θm’s, not on those hiding in y−. where the two complex Grassmann variables satisfy the anticommutation relations 5.2. Coset Construction The degrees of freedom of the N = 2 hy- ∂ m ∂ mn permultiplet in four dimensions appear as triv- θm, = θ , n = δ , { ∂θn } { ∂θ } ial solutions of the Kostant equation associated ∂ m ∂ with the coset SU(3)/SU(2) U(1). Let T A θm, n = θ , = 0 . × { ∂θ } { ∂θn } , A = 1, 2, . . . , 8, be the generators of SU(3). i 8 The (free) Hamiltonian-like supersymmetry gen- Among those, T , i = 1, 2, 3, and T generate its erators are simply SU(2) U(1) subalgebra. Introduce Dirac matrices over×the coset m p m m p m = + , = + , (26) a b ab Q− p+ Q Q− p+ Q γ , γ = 2δ , a, b = 4, 5, 6, 7 . (33) { } and the light-cone boosts are given by The Kostant equation over the coset SU(3)/SU(2) U(1) 1 p ∂ × M − = x−p x, P − + i + θm , (27) − 2{ } p ∂θm a / Ψ = γ TaΨ = 0 , (34) 1 p m ∂ K M − = x−p x, P − + i θ m , a=4X,5,6,7 − 2{ } p+ ∂θ has an infinite number of solutions which come in where 1 groups of three representations of SU(2) U(1), x = (x1 + ix2) . × √2 called Euler triplets. For each representation of SU(3), there is a unique Euler triplet, a , a : These generators represent the superPoincaré al- { 1 2} gebra on reducible superfields because the opera- 2a +a +3 a −a 2a +a +3 [a2] 1 2 [a1+a2+1] 1 2 [a1] 2 1 ,(35) tors − 6 ⊕ 6 ⊕ 6

m ∂ + where a1, a2 are the Dynkin labels of the associ- + = m θmp , (28) D ∂θ − ated SU(3) representation. Here, [a] stands for the a = 2j representation of SU(2), and the sub- anticommute with the supersymmetry genera- script denotes the U(1) charge. Kostant’s operators. Irreducibility is achieved by acting on su- tor commutes with the SU(2) U(1) generated perfields for which by × m ∂ + + Φ = [ m θmp ]Φ = 0 , (29) D ∂θ − Li = Ti + Si , i = 1, 2, 3 ; L8 = T8 + S8 ,(36) 6 a sum of the SU(3) generators and the “spin” which change helicity by half a unit. Thus a nec- part, expressed in terms of the γ-matrices as essary condition for supersymmetry to be realized is to include all triplets, leading to an infinite- i ab i ab Sj = fjabγ , S8 = f8abγ , (37) component theory. −4 −4 We also note that there are states in the higher ab a b where γ = γ γ , a = b , and fjab , f8ab are Euler triplets with helicities larger than 2. If they structure functions of S6 U(3). The Euler triplet are to be interpreted as massive relativistic states, corresponding to a1 = a2 = 0, they must arrange themselves in SO(3) representations, which does not appear likely. Otherwise 0, 0 = [0] 1 [1]0 [0] 1 , (38) they must be interpreted as massless particles in { } − 2 ⊕ ⊕ 2 four dimensions, lealing with a theory of massless describes the degrees of freedom of the N = 2 states of spin higher than two. supersymmetric multiplet, when the U(1) is in- There are well-known difficulties with such the- terpreted as the helicity of the four-dimensional ories[15,16]. In particular, they do not have co- Poincaré algebra. variant energy momentum tensors, and it must be Is it possible to link this supersymmetric triplet that in the flat space limit they decouple from the to the others for which a = 0, while pre- 1,2 gravitational sector. Alternatively, the no-go the- serving relativistic invariance? 6 Not all triplets orems do not apply if there are an infinite num- can decribe relativistic particles, since their U(1) ber of such particles. The best argument against charges are in general fractional numbers, leading such theories is that no working example has yet to states that pick up strange phases after a space been produced, but we hope such a theory can rotation by 2π, while Fermi-Dirac statistics only be formulated with an infinite number of Euler allows states for which this phase is 1. Only multiplets. Euler triplets for which 5.3. Grassmann Numbers and Dirac Ma- a a = 3n , (39) 1 − 2 trices where n = 0, 1, 2, . . ., yield half-odd integer or In order to make contact with the supersymme- integer U(1) charges fit the bill. These Euler mul- try of the lowest Euler triplet, we represent [18] tiplets split into two groups, the self-conjugate, the γ-matrices in terms of Grassmann numbers and their derivatives as a, a : [a] a+1 [2a + 1]0 [a] a+1 , (40) 2 2 { } − ⊕ ⊕ 2 1 2 1 γ4+i5 = i + , γ4 i5 = i + which contain equal number of half-odd integer- rp+ Q − rp+ Q helicity fermions and integer-helicity bosons, and 2 2 2 2 γ6+i7 = i + , γ6 i7 = i + , naturally satisfy CPT. The other possible Euler rp+ Q − rp+ Q multiplets are of the form a, a + 3n with n = 1, 2, . . ., { } in terms of the kinematic N = 2 light-cone supersymmetry generators defined in the previous

a+2n+1 n a+n+1 section. It follows that the Kostant operator an- [a] [2a + 3n + 1] 2 [a + 3n] . − 2 ⊕ ⊕ 2 ticommutes with the constraint operators Since CPT requires states of opposite helicity, /, m = 0 , (41) these must be accompanied by their conjugates, { K D+ } a + 3n, a , with all helicities reversed. If both so that we can simplify its solutions to chiral su- { } n and a are even, each representation contains perﬁelds, on which these become (2a + 3n + 2) bosons and fermions, the fermions appearing in two diﬀerent SU(2) representations. + 2 ∂ γ4+i5 = 2i 2p θ1 , γ4 i5 = i − rp+ ∂θ The helicities within each triplet are separated − p 1 by more than half a unit, and they cannot be + 2 ∂ γ6+i7 = 2i 2p θ2 , γ6 i7 = i , related by operations, such as supersymmetry, − r + − p p ∂θ2 7

The “spin” parts of the SU(2) U(1) generators, and expressed in terms of Grassmann× variables, do not + 1 depend on p , T a1, a2 > = (a1 + 2a2) a1, a2 > . 8| 2√3 | 1 ∂ ∂ S = (θ + θ ) , 1 2 1 ∂θ 2 ∂θ The highest-weight states of each SU(3) represen- 2 1 tation are holomorphic polynomials of the form i ∂ ∂ S = (θ θ ) , 2 1 2 a1 a2 −2 ∂θ2 − ∂θ1 z1 z3 , 1 ∂ ∂ S3 = (θ1 θ2 ) , where a1, a2 are the Dynkin indices, since it is eas- 2 ∂θ1 − ∂θ2 ily seen to reproduce the above values for T3 and and T8. This describes all representations of SU(3) as √3 ∂ ∂ homogeneous holomorphic polynomials. Finally S8 = (θ1 + θ2 1) , 2 ∂θ1 ∂θ2 − we note that any function of the quadratic invari- identified with the helicity, up to a factor of √3. ant 5.4. Linear Realization of SU(3) Z2 z 2 + z 2 + z 2 , ≡ | 1| | 2| | 3| The SU(3) generators can be conveniently ex- can multiply these polynomials without affecting pressed on three complex variables and their con- their SU(3) transformation properties. jugates. Define for convenience the differential operators 5.5. Solutions of Kostant’s Equation Kostant’s equation ∂ ∂ ∂ , ∂ , etc. , 1 1 a ≡ ∂z1 ≡ ∂z1 / Ψ = γ T Ψ = 0 . K a a=4X,5,6,7 in terms of which the generators are given by now becomes two coupled systems of equations T +iT = z ∂ z ∂ , T iT = z ∂ z ∂ , 1 2 1 2− 2 1 1− 2 2 1− 1 2 (z1∂3 z3∂1)ψ1 + (z2∂3 z3∂2)ψ2 = 0 , T4+iT5 = z1∂3 z3∂1 , T4 iT5 = z3∂1 z1∂3 , − − − − − (z3∂1 z1∂3)ψ2 (z3∂2 z2∂3)ψ1 = 0 , T +iT = z ∂ z ∂ , T iT = z ∂ z ∂ , − − − 6 7 2 3− 3 2 6− 7 3 2− 2 3 and and (z ∂ z ∂ )ψ (z ∂ z ∂ )ψ = 0 , 1 3 1 − 1 3 0 − 2 3 − 3 2 12 T3 = (z1∂1 z2∂2 z1∂1 + z2∂2) , (z ∂ z ∂ )ψ + (z ∂ z ∂ )ψ = 0 . 2 − − 3 2 − 2 3 0 1 3 − 3 1 12 1 The homogeneity operators T = (z ∂ +z ∂ z ∂ z ∂ 2z ∂ +2z ∂ ) . 8 2√3 1 1 2 2− 1 1− 2 2− 3 3 3 3 D = z1∂1+z2∂2+z3∂3 , D = z1∂1+z2∂2+z3∂3 These act as hermitian operators on holomorphic functions of z and z , normalized with re- commute with /, allowing the solutions of the 1,2,3 1,2,3 K spect to the inner product Kostant equation to be arranged as irreps of the SU(2) U(1) generated by the operators × 2 3 3 zi L = T + S , i = 1, 2, 3 ; L = T + S . (f, g) d zd z e− i | | f ∗(z, z) g(z, z) . i i i 8 8 8 ≡ Z P The solutions for each triplet, conveniently writ- It is convenient to introduce the positive integer ten only for the highest weight states, are of the Dynkin labels a1 and a2, for which form

a1 a1 a2 2a +a +3 T3 a1, a2 >= a1, a2 > , Ψ = z3 z2 : [a2] 1 2 , | 2 | − 6 8

a1 a2 i a −a Ψ = θ1 z1 z2 : [a1 + a2 + 1] 1 2 , the form Ψ(y−, x , za, za, θi), which satisfy both 6 a1 a2 the chiral and Kostant constraints. Work is in 2a +a +3 Ψ = θ1θ2 z1 z3 : [a1] 2 1 , (42) 6 progress to see if one can introduce interactions where we have indicated their SU(2) Dynkin la- among these light-cone superfields . bels. All other solutions in the same Euler triplet can be obtained by repeated action of the lower- 6. Outlook ing operator This example only serves to introduce the ∂ method we intend to pursue. The interesting L1 iL2 = θ2 + (z2∂1 z1∂2) . case in eleven dimensions singles out the coset − ∂θ1 − F4/SO(9). A similar procedure will lead us to We now see how the triplets arise as polynomials consider light-cone superfields over 4 copies of 26 of the same degree. real variables[19], and eight Grassmann variables. The light-cone Lorentz group generators are now 5.6. The Poincaré Algebra built out of the SO(9) generators, of the form It is easy te represent the Poincaré algebra on the Euler triplets. Starting from the more general Lij = T ij + Sij , i, j = 1, 2 . . . , 9 , (45) representation in which T ij generate the SO(9) little group in 12 1 ∂ 1 m ∂ 12 the particular representation associated with each M = i(xp xp) + θm θ m + S , − 2 ∂θm − 2 ∂θ Euler triplet, and they act along the 256 256 unit ij × + + i ∂ i m ∂ matrix. In particular, T = 0 for the supergrav- M − = x−p θm θ m , − − 2 ∂θm − 2 ∂θ ity multiplet, and the “spin” part is + + + + M = xp , M = xp , 16 − − ij i ij ab ab 1 p ∂ S = f γ . (46) 12 − 4 M − = x−p x, P − + i + (θm + S ) , a,bX=1 − 2{ } p ∂θm 1 p m ∂ ij ab − 12 Here the f are the structure functions of the M = x−p x, P − + i + (θ m S ) , − 2{ } p ∂θ − exceptional group F4! In closing we note that and identify the rest of the helicity generator as the same coset plays a prominent role in the pro- jective geometry associated with the Exceptional 12 1 Jordan Algebra, leading one to hope in a new in- S = T8 . (43) √3 terpretation with SO(9) as a space group. Details These act on chiral superfields whose entries are will be presented elsewhere. polynomials in z and z. I wish to think Professors Shifman and Vain- There is no difficulty in writing a free La- shtein for their warm and wonderful hospitality, grangian for these solutions, but one could con- and the opportunity to meet some pioneers of su- template writing a free Lagrangian for all the so- persymmetry, and re-acquaint myself with many lutions at one fell swoop. For this, we need to old friends. consider a chiral superfield whose entries are ar- bitrary polynomials in the complex variables, but subject to the Kostant equation as a constraint. The action would be of the form

2 3 3 S = d xdx− d zd z , (44) Z Z L suggesting that the classical space includes the group manifold as well. The Lagrange density would then be built out of chiral superﬁelds of 9

REFERENCES 19. T. Fulton, J. Phys. A: Math. Gen. 18, 2863(1985) 1. G. L. Stavraki, International School of The- oretical Physics, Yalta (1966); Physics of High Energy and Theoretical Elementary Par- ticle Physics. Ukrain. Acad Sci./JINR, Kiev (1967) 2. H. Miyazawa, Phys. Rev. 170, 1958(1968) 3. Yu.A. Golfand and E.P. Likhtman,JETP Lett. 13, 323(1971; Pisma Zu. Eksp. Teor. Fiz. 13 452(1971) 4. D.V. Volkov and V. P. Akulov, JETP Lett. 16, 438(1972). 5. R. Dolen, D. Horn, C. Schmid, it Phys. Rev. 166, 1768(1968) 6. P. Ramond, Phys. Rev D3, 988(1971) 7. A. Neveu and J. H. Schwarz, Phys. Rev. D4, 1109(1971) 8. J. L. Gervais and B. Sakita, Phys. Rev. D4, 2291(1971) 9. J. Wess, B. Zumino, Phys. Lett. B49, 52(1974) 10. F. Gliozzi, J Scherk, and D. Olive Nucl.Phys. B122, 253(1977) 11. See, W. G. McKay, J. Patera, Tables of Di- mensions, Indices, and Branching Rules for Representations of Simple Lie Algebras, Mar- cel Dekker (New York and Basel), 1981. 12. B. Gross, B. Kostant, P. Ramond, and S. Sternberg, Proc. Nat’l Acad. Sci. USA 95, 8441(1998). 13. B. Kostant, Duke J. Math. 100, 447(1999) 14. Loyal Durand III, Phys. Rev. 128, 434(1962); K. Case and S. Gasiorowicz, Phys. Rev 125, 1055(1962) 15. E. Witten and S. Weinberg, Phys. Lett. B96, 59(1980) 16. C. Aragone and S. Deser, Nuovo Cim. 57B, 33(1980) 17. Anders K.H. Bengtsson, Ingemar Bengtsson, Lars Brink, Nucl. Phys. B227, 31(1983) 18. This work has been done in collaboration with L. Brink; see also Lars Brink, Pierre Ramond, Dirac Equations, Light Cone Supersymmetry, and Superconformal Algebras. In *Shifman, M.A. (ed.): The many faces of the super- world* 398-416. [HEP-TH 9908208], and references therein.