University of Groningen New spacetime superalgebras and their Kač-Moody extension Bergshoeff, Eric; Sezgin, Ergin Published in: Physics Letters B DOI: 10.1016/0370-2693(89)90564-9 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1989 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Bergshoeff, E., & Sezgin, E. (1989). New spacetime superalgebras and their Kač-Moody extension. Physics Letters B, 232(1), 96-103. https://doi.org/10.1016/0370-2693(89)90564-9 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license. More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne- amendment. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 03-10-2021 Volume 232, number 1 PHYSICS LETTERS B 23 November 1989 NEW SPACETIME SUPERALGEBRAS AND THEIR KA(~-MOODY EXTENSION Eric BERGSHOEFF ".theory Division, C~RN, CH- 1211 Geneva 23. Switzerland and Ergin SEZG1N International Center Jor Theoretical Physics, 1-34100 Trieste, Italy Received 31 July 1989 We present new spacetime algebras whose existence is duc to special l'-matrix idcntities which are also necessary for the exis- tence of super p-branes. They contain a pth-rank antisymmetric tensor, and a (p- 1)-rank antisymmetric tensor-spinor generator. Furthermore the translations do not commute with the supercharge. In the case of supermembrancs, we find a super-KaY-Moody extension of the new spacetime algebra. We give a realization of the algebra in terms of operators in d= 11 superspace, and find a connection with the d= 11 supermembrane action which leads to an elegant supergeometric formulation. By double dimensional reduction, we obtain a Ka~.-Moody algebra for the Type IIA Grccn-Schwar-z superstring. We also discuss the generalization of these results, including the construction of generalized super-Virasoro algebras, for super p-branes. 1. Introduction to give rise to an (N=8 or 16) rigid world-sheet supersymmetry. The two formulations are essentially equivalent in It is well known that superstrings admit two differ- the light-cone gauge [6,7]. However, for many pur- ent kinds of formulations as far as their supersym- pose it is desirable to have a covariant quantization merry properties are concerned. One of them is the scheme. The covariant quantization of the NSR spin- Ncvcu-Schwar-z-Ramond (NSR) formulation [1] in ning string is well understood. In many modem treat- which the world-sheet supersymmetry is manifest but ments of the theory, an increasingly central role is spacetimc supersymmetry is not. In this formulation, played by the NSR super-Virasoro algebra. In the case spacctime supersymmetry arises as a consequence of of GS superstring, one of the important questions Gliozzi-Scherk-Olive (GSO) projections [2 ] in the which naturally arises is to find the analog of the NSR Hilbert space. The field equations of the world-sheel super-Virasoro algebra. graviton and gravitino are the constraints which obey Siegel has argucd that [8,9] a generalized super- the supcr-Virasoro algebra. Virasoro algebra for the GS string can be arrived at In the other formulation, due to Green and Schwarz in two steps: ( 1 ) Find a set of operators ("superco- (GS) [ 3 ], the spacetime supersymmetry is manifest variant derivatives") satisfying a generalized KaY- but the world-sheet supersymmctry is not. In this case, Moody algebra. (2) Find a suitable set of constraints the theory has intercsting local world-sheet symme- quadratic in these operators satisfying a generalized try, known as K-symmetry, first discovered by Siegel Virasoro algebra ("Sugawara construction"). Siegel [4,5 ] for the supcrparticle, and later generalized by [8,9] achieved this construction for superparticles Green and Schwarz [3 ] for superstrings. In a light- and superstrings. Two salient features of his results cone gauge, a combination of the residual x-symme- are: (a) All constraints arc first class. In particular, try and rigid spacetime supersymmetry fuse together among the constrains is the one corresponding to the 96 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland) Volume 232, number 1 PHYSICS LETTERS B 23 November 1989 ~-symmetry. (b) The supercovariant derivative cor- of Siegel [8 ] and Green [ 10] for the GS superstring, responding to supertranslations does not commute pertaining to the new spacetime algebra, its Ka6- with the bosonic translations. Moody extension and the generalized super-Virasoro The feature (b) above is rather interesting: Re- algebra. After that we present our results for super- cently, Green [ 10 ] pointed out that the global limit membranes, and Type IIA superstrings. Generaliza- of the Siegel algebra is a new (differing from the tions to other super p-brancs arc also discussed. super-Poincar6) superalgcbra in which the bosonic translations do not commute with supersymmetry. This new supcralgcbra can bc naturally described by 2. The Green-Schwarz superstring an extended superspace which contains ncw fer- mionic coordinates. An attractive feature of this su- To describe the classical mechanics of the GS su- peralgcbra is that its associated super-Killing form is perstring, Siegel introduces the operators P~,, and D,, nondegenerate. This fact has been exploited by Green corresponding to bosonic and fermionic translations [ 10 ] to find a (2 + 1 )-dimensional Chern-Simons in d= 3, 4, 6 or 10 superspace respectively, and a new formulation of the GS superstring. operator D,, which obeys the following KaY-Moody In this notc, generalizing Green's result, we find superalgebra [ 8 ] new spacctime superalgebras in all dimensions where supcr p-branes exist [ 11-13 ], i.e. in addition to the {D~(a), Dp( a' ) } = F~,~/fl~,( a)5( a-a ' ) , superstring casc (p= 1; d=3, 4, 6, 10), we find that new spacctime superalgebras also exist for the super- [D,,(~),/'~(~' ) ] =F~#P(~)5(~- ~' ), mcmbrancs (p= 2; d=4, 5, 7, 11 ), and the other supcr {D~(a), .QtJ(a' )}=i5~ Old(a-~' ), p-brancs: (p=3; d=6, 8), (p=4; d=9) and (p=5; [Pu(o'), P~(o" ) ] =ir//,,0~ d(a-a' ) , (1) d= 10). In particular, we shall see that (a) the new superalgebras contain a pth-rank antisymmetric ten- with other commutators and anticommutators van- sor, and a (p- 1 )-rank antisymmetric tensor-spinor ishing, and 01-0/0e. generator, (b) the bosonic translations do not com- One can check that the Jacobi identities arc satis- mute with supersymmetry, and (c) the existence of fied by (1), provided that the /'-matrix identity the algebras rely on /'-matrix idcntities which are Fu(~aF~')a=O holds. This is possible only in d=3, 4, precisely the ones needed for the existence of the super 6 and 10. The algebra ( 1 ) can be realized in terms of p-branes. the following operators [ 8,9 ]: In this paper, we shall focus our attention on the most interesting case of the eleven-dimcnsional su- Pu(a) = 1-,6-X~ + o~x ~ -iOF~O,O, permembrane, and construct for that case a general- ) ized KaY-Moody superalgebra. We shall also find the realization of the algebra in terms of operators in D~(cl) = -~ ~- -- F~flO ~ 1 ~-X~ "]- 01 x/~ d= 1 1 superspace. Here, a surprising connection with the d= 1 1 supermembrane action emerges in that the same super three-form which plays a central role in -- ½iFU aO/~OF~Ol O, the construction of the ~c-invariant supermembrane [2~( a)=iO~ O~ , (2) action [ 12] also arises in the superspace realization of the KaY-Moody generators. In fact, this leads to a where OFUO - O"( F u ) ~0 a. The operators are invar- remarkably simple and elegant supergeometric for- iant under translations and supersymmetry which are mulation of the full KaY-Moody algebra. generated by [ 8,9 ] We shall also suggest a way to construct the con- straints (including those of reparametrizations and x-symmetry) which are quadratic in the KaY-Moody generators, but we shall not computc their full alge- bra. In the next section we describe briefly the results 97 Volume 232, number 1 PHYSICS LETTERS B 23 November 1989 f i 5 .~¢= ½t'2+~'~D,~, .~'~=F~*Spt'D*8, Pu = da 5~ X ' %<B=D~Dp, c4,=il-'<~I*sD<~OiDp . (6) [ ) The generators .¢/, ,~, '$ and ci, obey the sought gen- eralized Virasoro algebra [8]. In particular, the gen- erator .4 obeys the usual Virasoro algebra: +~lFa*801" u ,8--OK,,O~O ] . (3) [,~(a), .~(a' ) ] =iOi6(a-a')[,~'(a) +.~(a' ) ] . (7) These generators obey the usual super-Poincar6 al- gebra. However, Green [ 10] has recently observed The remaining part of the algebra is very' compli- that, one can write down a new spacetimc superalge- cated. It has been studied in detail also by Romans bra with the lbllowing nonvanishing commutators [ 14 ]. Schematically, it is given by and anticommutators: [.~,.~]~a",w, [d,~/,q~a'~, [.~,~]~6'¢, {Q,~, Q*8}=F'~/flS,, [Q,~, l'u] =I'.,wK*8, (4) {,~, .~} ~d' ~+d(P,~¢ + ~ +~,~) , where K,~ is a new fermionic generator.