See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/304156453 Presentation for Supervirasoro Algebras Chapter · January 2003 DOI: 10.1007/1-4020-4522-0_395 CITATIONS READS 0 13 1 author: Cosmas Zachos Argonne National Laboratory 147 PUBLICATIONS 4,350 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Finite differences View project Phase space Quantum Mechanics View project All content following this page was uploaded by Cosmas Zachos on 05 May 2018. The user has requested enhancement of the downloaded file.
V *fhe subrniyed manuscript has been authored bv a contractor oi the U. S. Government unrtor contract No. W-31-109-ENG38. Accordingly, the U. S. Government retains a nonexclusive, royally-free license to publish or renioduce the published form oi this contribution, or allow others to do so, lor ANL-HEP-CP-88-49 U. S. Government purposes. A PRESENTATION FOR VIRASORO ALGEBRAS C.K. Zachos ANL-HEP-CP— 88-49 High Energy Physics Division * DE89 003883 Argonne National Laboratory, Argonne, IL 60439, USA Abstract The entire Virasoro, Ramond and Neveu-Schwarz algebras can each be constructed from a finite number of well-chosen generators satisfying a small number of conditions. Our most economical sets consist of just two starting generators in all cases, subject to no more than six conditions for the Virasoro case, five conditions for the Ramond case, and nine conditions for the Neveu-Schwan case. Consequently, the Virasoro algebra simply amounts to 6 equations in two operator unknowns, and correspondingly 5 and 9 equations for the foregoing superalgebras. Diff(S 1 ) The Virasoro algebra of [L m ,L n ] = (m — n)L m+n + c {m — m)S m+n fl , (1) appears like an infinity of variables L n constrained by an infinity of equations (1). However, in collaboration with David Fairlie and Jean Nuyts 1 ), we found that, after some of these equations are used to define all but two of the L n , all of the remaining ones follow from a small subset of (1): the algebra is completely specified by just 6 equations in 2 operator unknowns (5 and 9 equations for its Ramond and Neveu-Schwarz supersymmetrizations, respectively), and is thus a very tight, finite structure. To consider a familiar analog of such a reduction, recall that compass gimbal mountings require only two joints: the structure of 517 (2) \s determined as a solution of two of its commutation conditions, Ti = [T^, [Tz.Ti]] and Ti = [Ti, [Ti,Tj]], regarded as equations for just two cornerstone generators T\ and 7*2, the third generator T% simply being definedhy the third commutation relation Tz = —i[Ti,Tj]. The solutions to these two equations yield all representations of SU(2). (This analog is somewhat limited, since no other commutation relation is a mere consequence of these two; the case of SU(Z) would have been preferable, but to my knowledge this problem has not been solved). It is in this sense that (1) amounts to just two operators constrained by 6 nonlinear commutation conditions. To show this, I first construct all L n 's out of merely two, L% and L-2 for specificity; and then I derive all commutation relations out of an economical set of 6 conditions. In our published paper, the conditions are 8, but J. Uretsky 2 ! has been able to derive Conds.5,7 of our paper from the other six conditions. I retain the original numbering of conditions for reference to our paper, while you are invited to further reduce the number of sufficient conditions or to consider the minimal presentation problem. Starting Sets. To construct all L n 'a out of merely 2, it is necessary and sufficient for the indices of these to be: i) One positive and one negative; ii) Both larger than 1 in absolute value; 'Talk at the XVIIth International Colloquium on Group Theoretical Methods in Phyfics, June 1988, Ste-Adele, Quebec. Supported by the U.S.Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38. MASTER
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iii) Relatively prime. The necessity of the last requirement is evident, while the sufficiency follows from reaching the raising and lowering operators L\, L-\ from L m , L- n for m,n > 1 relatively prime. If the last requirement fails, > 2 starting generators whose indices have no common divisor > 1 are needed to reach all L n 's; e.g. an irredundant starting set is L 2 , L~ 2 , £-1. To illustrate this construction, start with L3 and L- 2 and define: (Def.l) L 1 = $[L 3 ,L- 2 ), (Def.2) ZLi = ![£!, £_,!, (Def.3) L 2 = i[L 3 ,L_ 1 ], L Q =k\L 1 ,L- l ], i } (Def.4) (Def+) L n+1 = j^[Ln, L t ] for n > 2 , (Def-) £_„_! = ^[I-n.L-i] forn > 1 . Specification of the Algebra. To show that a small subset of commutation relations (1) guarantees the validity of all, commutator "unknowns" are correlated among themselves through the Jacobi identity J{m,n,p) : [[L m ,L n ],L p ] + p n , L p \,L m ] + [[L p ,L m \,L n ] = 0 . (3) At every step in the proof to follow, a given Jacobi identity can only be used if all three inner commutators, i.e. [L m , L n ] .. .etc., are already known by a preceding step in the recursion. In addition, a "parity" automorphism of the algebra helps shorten the proofs: L a -* - L- n , c~* -c . (4) All consequences of sets possessing this symmetry will also be automatically parity symmetric. An operator is said to be of level m if this is its eigenvalue under commutation with Lo (e.g. [L m ,Lo] = mL m ). From J(0,m,n), it follows that Level Lemma. Levels add under commutation, i.e. the commutator of two operators of level m and n is of level m + n. Thus, imposing (Cond.l) [L S ,L O ) ZL S (Cond.2) [Z,_2,J&O]= -2Z-2 specifies the level (commutation with Lo) of all L n conventionally. To completely determine (1), it suffices to impose 4 more conditions: (Cond.3) [L 2 ,Li) = L 3 (Cond.4) [L 2 , L- 2 \ = 4L 0 + 6c (Cond.6) [Ls,L 2 \ = 3L 7 ^' (Cond.8) [L_2,L_s]=3L_ 7 . c is defined as a mere number times the identity, so that it commutes with all the starting generators, and hence all operators. The relation (*): [Lz,L-i] = 3Li follows from .7(1,-2,2), Conds.3,4, the level lemma and the definitions. "Conds.5,7" of our original paper, namely [1/3,^2] = L5 and [Z/_2,i_ 3 ] = L_s, follow 2 ! from J(4,2,-1), 7(5,2,-1), ci) below, the above conditions, and the appropriate parity image statements under the automorphism (4).
A few known commutators \L m , £,„] are now tabulated with the entries indicating the source of specification; entries below the diagonal are omitted, since they are palpably identical to the ones above it: ||£-» Lo L 2 L 3 L-i Li 0 Def- Cond.2 Def.2 Cond.4 Def.l L-2 . 0 LL Def.4 * Def.3 L-i 0 LL LL Cond.l Lo . 0 Cond.3 Def+ Lx "Cond.5" . 0 L 2 0 L s - • Theorem. All commutators [L m , L n ] follow uniquely to comport with (1). Proof. Three cases are considered: a) m, n > 0; b) m, n < 0 will then hold automatically as the parity image of a); and c) mn < 0. a) Up to level k = m + n = 4, m, n > 0, all commutators are already known. Those for a general level k will be specified by induction on k. If all are known for all levels up to and including k - 1, the unknowns to solve for are X l k) = [Lt +l ,L k - t -i] , (7) t where t runs from t = 1 to t = r - 2, for k = 2r even, and to t = r - 1 for k = 2r + 1 odd. For even A : = 2r > 4, the r - 2 Jacobi identities J(l,s + l,k-s-2), s = 1,..., r-2, lead to a system of r — 2 equations in the r — 2 unknowns Xt , t = 1,..., r — 2: (8) where ( k-A 1 0 0 ... 0 0 ^ fc-5 0 2 0 ... 0 0 0 0 k-6 3 ... 0 0 0 0 Jfc-7 ... 0 0 0 (9) 0 0 0 0 ... r r - 3 0 0 0 ... 0 r - 1 j k . 0 and the vector m[ k ~ 2r) has components mi*~ 2r) = (2s + 3 - k){k - 2) . Now since detAf<* =2r ) = (k — 4)\/(r — 2)! is non-zero, (8) is invertible, and the X} ''s are uniquely determined in terms of L k as in (1). For odd k = 2r+l > 3, there are r - 1 unknowns, but the Jacobi identities J(l,s+1, k—s— 2), s = l , . . . , r - 2 , provide only r - 2 equations. The matrix in (9) now has to be supplemented by an extra row with the Jacobi identity J(2,3,k - 5), (and the last component of m( fc=2r+1 ) vanishes),
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