Physics Letters B 675 (2009) 387–392 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Classical and quantal ternary algebras ∗ Thomas Curtright a, , David Fairlie b, Xiang Jin a, Luca Mezincescu a, Cosmas Zachos c a Department of Physics, University of Miami, Coral Gables, FL 33124-8046, USA b Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK c High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439-4815, USA article info abstract Article history: We consider several ternary algebras relevant to physics. We compare and contrast the quantal versions Received 2 April 2009 of the algebras, as realized through associative products of operators, with their classical counterparts, Accepted 8 April 2009 as realized through classical Nambu brackets. In some cases involving infinite algebras, we show the Available online 11 April 2009 classical limit may be obtained by a contraction of the quantal algebra, and then explicitly realized Editor: L. Alvarez-Gaumé through classical brackets. We illustrate this classical-contraction method by the Virasoro–Witt example. © 2009 Elsevier B.V. All rights reserved. 1. Introduction pecially the Virasoro–Witt algebra [9], the classical limit may be obtained initially by a contraction of the quantal algebra, and then Ternary algebras can be realized in two ways: They may be explicitly realized through classical 3-brackets. built from antisymmetrized products of three linear operators – so-called quantal brackets – or they may be realized through a 2. Classical and quantal brackets generalization of Poisson brackets in terms of multivariable Jaco- bians – so-called classical brackets. Both constructions were intro- In this section, we define and compare the properties of var- duced in the physics literature by Nambu [1] after about twenty ious realizations of 3-brackets. These are just trilinear operations years of gestation. Lately, these algebras seem to be gaining in use- performed either on three operators or on three functions defined fulness and importance to physics. For example, there has been onamanifold. some progress in constructing a (2 + 1)-dimensional local quan- tumfieldtheorywithSO(8) superconformal symmetry [2–6] as 2.1. Three specific realizations of 3-brackets a stepping-stone to obtain a world-volume Lagrangian description for coincident M2-branes. The use of ternary algebras is crucial to The classical Nambu bracket [1] based on three variables (say the construction. x, y, z) is the simplest to compute, in most situations. It involves More generally, N-algebras involve antisymmetric operations on the Jacobian-like determinant of partial derivatives of three func- N entities at a time. The catchy name was introduced by Filip- tions A, B, C. pov in a paper that appeared just over a decade after Nambu’s [7], although Filippov does not seem to have been aware of Nambu’s ∂(A, B, C) abc {A, B, C}= = ε ∂a A∂b B∂c C, (1) earlier work.1 In any case, Nambu’s work motivated and inspired ∂(x, y, z) a lengthly mathematical development of these ideas by Takhtajan where a, b, c are implicitly summed over 1, 2, 3. The classical [10] (also see [11,12]) along with many other physics studies (see bracket is totally antisymmetric in the three argument functions. [13] and references therein, especially [14]), as well as much more (The { , } notation is here used to distinguish this classical case recent work [15–22]. from the quantal bracket, and should not be confused with anti- Here we compare classical and quantal realizations of ternary commutation.) algebras for several interesting cases relevant to physics. The quan- The operator or “quantal” 3-bracket was originally defined [1] tal and classical algebras are usually not the same, but for one very to be a totally antisymmetrized sum of trinomials, special case they are: Nambu’s su(2). For some infinite cases, es- [A, B, C]=ABC − BAC + CAB− ACB + BCA − CBA (2) * Corresponding author. regardless of the number of underlying independent variables. This E-mail address: [email protected] (T. Curtright). can be equivalently expressed as a sum of single operators multi- 1 Rather, Filippov was following up on earlier studies that had appeared in the mathematics literature, primarily by Kurosh [8]. This corrects an erroneous remark plying commutators of the remaining two, or as anticommutators on the history of Filippov’s work, as made in [9]. acting on the commutators, 0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.04.019 388 T. Curtright et al. / Physics Letters B 675 (2009) 387–392 [A, B, C]=A[B, C]+B[C, A]+C[A, B] where again, i1,...,i6 are implicitly summed from 1 to 6. For the classical bracket in (1), both left- and right-hand sides of the BI ac- =[B, C]A +[C, A]B +[A, B]C. (3) tually vanish. For the ALMY bracket, the identity follows by direct In principle, as well as in practice, it is necessary to have some calculation. information about products, hence about anticommutators as well For quantal 3-brackets the BI is a consequence of associativity. as commutators, to perform the actual evaluation of this quantal If one posits an operator 3-bracket based on associative products, bracket. Relatedly, the trace of [A, B, C] is non-trivial, in general. and it does not satisfy this identity, then one has erred. Thus the There is yet another definition of an operator 3-bracket, intro- BI is a necessary condition to realize a ternary algebra in terms of duced by Awata, Li, Minic, and Yoneya [23], which is valid for operator brackets. operators of trace class. It consists of re-packaging the commu- Finally, we note that the antisymmetrization of an operator 3-bracket acting on a second 3-bracket does not close to give a tators of any Lie algebra to define ··· i1 i5 [[ ] third 3-bracket, but rather a 5-bracket: ε Bi1 , Bi2 , Bi3 , Bi4 , B ]∝[B B B B B ]. The operator 5-bracket is defined in A, B, C=[A, B] Tr(C) +[C, A] Tr(B) +[B, C] Tr(A). (4) i5 1, 2, 3, 4, 5 general [13] as the totally antisymmetric signed sum over all 5! This is again a totally antisymmetric trilinear combination, but distinct words ABCDE,etc. it is a singular construction for finite dimensional realizations in the sense that TrA, B, C=0. For reasons to be discussed, this 3. Examples of ternary algebras ALMY bracket has properties intermediate between the classical 3-bracket and the full quantal bracket. When the 3-brackets close to yield other functions or opera- tors of a designated set, one is considering a ternary algebra. In 2.2. Properties of the various brackets this event, one may define structure constants to write, say for the quantal bracket, As mentioned already, all three are trilinear and totally anti- l symmetric. On the other hand, only two out of three automatically [Ai, A j, Ak]= fijk Al. (9) satisfy the Filippov condition, the so-called “FI” [7] (also see [10, Similarly for the classical and ALMY brackets. For the latter two 24]). Namely, types of brackets, the FI always holds, and this implies a bilinear consistency condition on the respective structure constants. This is A, B, {C, D, E} = {A, B, C}, D, E + C, {A, B, D}, E similar to the consistency condition imposed on the structure con- + C, D, {A, B, E} , stants of a Lie algebra by the Jacobi identity. However, this bilinear condition does not necessarily apply to the quantal structure con- A, B, C, D, E = A, B, C, D, E + C, A, B, D, E stants [13]. + C, D, A, B, E . (5) More generally, the BI gives a trilinear consistency condition on the structure constants for all three types of brackets. This trilinear So, for the classical and ALMY brackets, the FI is indeed an identity, condition must always be satisfied by any ternary algebra. For the hence it is a necessary condition to realize a ternary algebra in classical and ALMY brackets, the BI structure constant condition is terms of either of these 3-brackets. But in general the FI does not not independent of the FI condition. But for a quantal 3-algebra l hold for associative operator products. the BI condition is the only constraint imposed on the fijk by as- sociativity. It would be interesting to classify all quantal 3-algebras = [ ] − [ ] − [ ] l 0 A, B, C, D, E A, B, C , D, E C, A, B, D , E by constructing all solutions of the BI condition on the fijk .We leave this for the well-motivated reader to pursue. Here, we just − C, D, [A, B, E] consider various basic examples of intrinsic interest to physics. ≡ fi(A, B; C, D, E). (6) 3.1. Nambu’s su(2) That is to say, the Filippov condition, fi(A, B; C, D, E) = 0, is not an operator identity. It holds only in special circumstances. In this First, consider Nambu’s application to su(2). sense the FI differs from the Jacobi identity for associative operator products, a 2-bracket-acting-on-2-bracket situation. Looking ahead, [Lx, L y, Lz]≡Lx[L y, Lz]+L y[Lz, Lx]+Lz[Lx, L y] we write the Jacobi identity in the somewhat unusual form = 2 + 2 + 2 i Lx L y Lz . (10) 2 ij [A B ] B = ij A [B B ] (7) ε , i , j ε , i, j , To close the algebra, it is necessary to include the su(2) Casimir. where A is fixed but all indexed entries are implicitly summed But, having done so, one may rescale by a fourth root of the over i, j = 1, 2, hence antisymmetrized.