Presentation for Supervirasoro Algebras Chapter January 2003 DOI: - - PDF document

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

13

1 author: Some of the authors of this publication are also working on these related projects: Finite differences View project Phase space Quantum Mechanics View project Cosmas Zachos Argonne National Laboratory

147 PUBLICATIONS 4,350 CITATIONS

SEE PROFILE

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

• r renioduce the published form oi this

contribution, or allow others to do so, lor

• U. S. Government purposes.

ANL-HEP-CP-88-49

A PRESENTATION FOR VIRASORO ALGEBRAS C.K. Zachos

High Energy Physics Division * Argonne National Laboratory, Argonne, IL 60439, USA

ANL-HEP-CP— 88-49 DE89 003883

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. The Virasoro algebra of Diff(S1) [Lm,Ln] = (m — n)Lm+n + c {m — m)Sm+nfl , (1) appears like an infinity of variables Ln constrained by an infinity of equations (1). However, in collaboration with David Fairlie and Jean Nuyts1), we found that, after some of these equations are used to define all but two of the Ln, 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 Ln'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

• ur published paper, the conditions are 8, but J. Uretsky2! 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

• ur 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 Ln'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

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac- turer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views am; opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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 Lm, 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 Ln's;

e.g. an irredundant starting set is L2, L~2, £-1.

To illustrate this construction, start with L3

and L-2 and define:

(Def.l)

L1 =

$[L3,L-2), (Def.2)

ZLi = ![£!, £_,!,

(Def.3)

L2 = i[L3,L_1],

(Def.4)

LQ=k\L1,L-l],

i }

(Def+)

Ln+1 = j^[Ln, Lt] 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) : [[Lm,Ln],Lp] + pn,Lp\,Lm] + [[Lp,Lm\,Ln] = 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. [Lm, Ln] .. .etc., are already known by a preceding step in the recursion. In addition, a "parity" automorphism of the algebra helps shorten the proofs:

La-*

• 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. [Lm,Lo] = mLm). 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) [LS,LO) ZLS

(Cond.2)

[Z,_2,J&O]= -2Z-2

specifies the level (commutation with Lo) of all Ln conventionally. To completely determine (1), it suffices to impose 4 more conditions: (Cond.3) [L2,Li) = L3 (Cond.4) [L2, L-2\ = 4L0

+ 6c

(Cond.6) [Ls,L2\ = 3L7

^'

(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, follow2! 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 \Lm, £,„] are now tabulated with the entries indicating the source

• f specification; entries below the diagonal are omitted, since they are palpably identical to the
• nes above it:

||£-»

L-2

L-i

Lo

Lx

L2 Ls

.

L-i

Def-

.

• Lo

Cond.2

LL

• Li

Def.2 Def.4

LL .

L2

Cond.4

* LL

Cond.3

L3

Def.l Def.3 Cond.l Def+ "Cond.5"

• Theorem. All commutators [Lm, Ln] 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 Xl

t k) = [Lt+l,Lk-t-i]

, (7) where t runs from t = 1 to t = r - 2, for k = 2r even, and to t = r - 1 for k = 2r + 1

• dd.

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

k .

1

fc-5

2 k-6 3

Jfc-7 ... 0 ... 0 ... 0 ... 0 ... r ... 0 0 ^

r - 3

r - 1 j

(9)

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

Lk 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),

so it becomes (r — 1) x (r - 1): ( k

M<k=2'+1> =

• 4

k

k

k

1 —

• 5

7 2

k-6

3

ifc-7

1

0 . . . ... 0 . . . 4 . . . ... 0 ... 0 . . . r - 3

r

^ r - 2 ) (10) For r = 4, the bottom row is (-1,2, -1) instead. Again,

• 7)(* -
• !

(ii) for r > 3. For r = 2,3, i.e. i f c = 5,7, the extra Jacobi identity does not exist to provide the additional row, and (Cond.6) has to be imposed: based on it, "Cond.5" is then derived through the mixed-index Jacobi identities listed before. c) To solve for [Lm, L_n], specify i) [Lm, L-i] for m > 2 inductively in m, through J[—l, l,m), with basis m = 2. ii) The parity image of the above, namely [L-m,L\], through the automorphism. iii) [Lm,L-m] and [Lm,Li-m] for m > 2, inductively in m, through sequential use of J(l,m,-m) and J(-l,m + 1,-m), with basis m = 2. iv) [Lm, L-n] Vn < m inductively in m, with basis m = 2: for a given m, assume all [Lm, £_n] are known Vn < m. Then J(l,m,-n) and iii) specify [Lm+i, L_rt] Vn < m + 1. v) Likewise for the parity image of iv), i.e. n > m. Thus the entire Virasoro algebra (1)

• holds. •

A corresponding strategy also applies to the NS/R superalgebras with the results cited. You mny wish to explore similar questions for N > 1 superalgebras, the SU{2) Kac-Mcody algebra, or even ordinary Lie algebras, and to determine a minimal set of commutation relations implying the

• rest. You may further attempt to to relax or modify some of the above conditions, in search of new

consistent algebras. I have not addressed the open problems of determining the complete necessity

• f condition sets (i.e. ensuring that each condition in them is independent), varying the starting

generators to possibly reduce the size of the above set, or systematically interchanging (Tietze) selected definitions with conditions, and monitoring variations in the size of the set. References

• 1. D. Fairlie, J. Nuyts, and C. Zachos, "A Presentation for the Virasoro and Super-Virasoro

Algebras", Comm. Math. Phys. 117, 595 (1988).

• 2. J. Uretsky, "Redundancy of Conditions for a Virasoro Algebra", ANL-HEP-PR-88-41, July

1988, unpublished.

View publication stats View publication stats