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University of Groningen Supermembranes and Eleven-Dimensional Supergravity Bergshoeff, E.; Sezgin, E.; Townsend, P.K

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University of Groningen Supermembranes and Eleven-Dimensional Supergravity Bergshoeff, E.; Sezgin, E.; Townsend, P.K University of Groningen Supermembranes and eleven-dimensional supergravity Bergshoeff, E.; Sezgin, E.; Townsend, P.K. Published in: Physics Letters B DOI: 10.1016/0370-2693(87)91272-X 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: 1987 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Bergshoeff, E., Sezgin, E., & Townsend, P. K. (1987). Supermembranes and eleven-dimensional supergravity. Physics Letters B, 189(1), 75-78. https://doi.org/10.1016/0370-2693(87)91272-X 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). 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: 27-09-2021 Volume 189, number 1,2 PHYSICS LETTERS B 30 April 1987 SUPERMEMBRANES AND ELEVEN-DIMENSIONAL SUPERGRAVITY E. BERGSHOEFF l, E. SEZGIN International Centre for Theoretical Physics, 1-34100 Trieste, Italy and P.K. TOWNSEND DAMPT, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Received 6 February 1987 We construct an action for a supermembrane propagating in d= 11 supergravitybackground. Using the constraints of d= 11 curved superspace, we show that the action is invariant under Siegel-typetransformations recentlygeneralized by Hughes, Li and Polchinski. The transformation parameter is a world-volume scalar and d= 11 spacetime spinor. We also discuss the general problem of the coupling of n-dimensional extended objects to d-dimensional supergravity. 1. Now that we have become accustomed to the existence of a closed superspace five-form, in anal- notion that strings should replace particles, it is nat- ogy with the Henneaux and Mezincescu [6] con- ural to investigate the properties of higher-dimen- struction for the Green-Schwarz superstring action sional extended objects, in particular of membranes [ 7 ], where a closed superspace three-form is required. since they are the simplest extended objects, and they The novel feature of the theory of Hughes et al. is might describe strings in an appropriate limit. that the parameter of the Siegel-type transformation In 1962 Dirac [ 1 ] put forward a theory of an [8] is a scalar rather than a vector on the world extended electron based on the idea of a relativistic volume. membrane. In 1976, Collins and Tucker [2] studied The generalization of the Hughes et al. model to n- the classical and quantum mechanics of free relati- extended objects propagating in flat d-dimensional vistic membranes. A year later a locally supersym- superspace is evident. All that is required is the exis- metric and reparametrization-invariant action for a tence of a closer super (n + 2)-form given by spinning membrane was constructed by Howe and H =EoEaEa"...Ea' (Y ..... a,, ) or,8 , (1) Tucker [3]. The action describes anti de Sitter supergravity coupled to a number of scalar multi- where (E% E a) are the basis one-forms in super- plets in three dimensions. It is the membrane analog space. This form is closed provided that the follow- of the Neveu-Schwarz-Ramond formulation of the ing/'-matrix identity holds: spinning string theory. (ya) (o~ (y ......... )~) =0. (2) More progress towards the construction of a mem- brane theory was made by Sugamoto [4] in 1983. The purpose of this note is to construct Hughes et More recently, Hughes, Li and Polchinski [ 5 ] have al. -type actions describing the propagation of an n- constructed a Green-Schwarz-type action for a three- extended object in d-dimensional curved superspace. extended object propagating in flat six-dimensional We give a general formula for the action and the spacetime. The consistency of the action requires the transformation rules, whose consistency requires, among other things (see below), the existence of a Supported in part by INFN, Sezione di Trieste, Trieste, Italy. closed (n + 2)-form in curved superspace. Thus we 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. 75 (North-Holland Physics Publishing Division) Volume 189, number 1,2 PHYSICS LETTERS B 30 April 1987 expect that the n-extended objects under considera- ventions as in ref. [11]. In (3) we have used the tion can consistently propagate only in d~< 11 super- notation gravities whose superspace formulation involves a Ei A = ( oiZM)EM A , (4) closed (n + 2)-form. We further expect that such forms exist in supergravity theories in which a closed where zM(~) are the superspace coordinates, and bosonic (n+2)-form occurs. As far as we know, the EMA(Z) is the supervielbein. following possibilites exist (we include the Yang- Note that the action has a cosmological constant Mills couplings whenever possible): with a fixed magnitude. This is so that the field equa- The dual formulation of d= 10, N= 1 supergravity tion of the metric go gives the embedding equation involves a closed seven-form. Its dimensional reduc- go =E,aEjbtlab = Tij . (5) tion on a (10-d)-dimensional torus leads to real closed (d- 3)-forms in d-dimensional supergravi- We require that the action S is invariant under a ties. (These are N=I supergravities in d=8, 9, 10; fermionic gauge transformation of the form [ 5 ] N=2 in d=7 and N=2 or4 in d=6) [9]. Apart from ~Ea=O, 6E'~=(l+F)"ax a , these, there is: (i) A real closed four-form in d= 11, N= 1 supergravity, (ii) a real closed three-form in non-chiral d= 10, N= 2 supergravity, (iii) a complex 6go=2[X o-gox~k/(n- 1 )1 closed three-form in chiral d= 10, N-- 2 supergravity. ( n = 2 for membrane ) , (6) Excluding Yang-Mills coupling, as is well known, closed super three-forms exist in d= 3, 4, 6, and 10. where the transformation parameter x~(~) is a 32 Considering the case of the membranes, from the component Majorana spinor, and a world-volume above list it follows that the candidate dimensions scalar, and are 7 and 11. Since the superspace formulation of ~E ~ = ~ ZMEM A, (7) d= 7, N= 2 supergravity is not known at present, we are led to consider the supermembrane propagating F'~a = (1/6~-g)ciJkEi'~EjbEkC(7,~bc)°'p . (8) in eleven-dimensional spacetime. Our main result is the construction of an action Here ~,a(a=0, 1, ..., 10) are the Dirac matrices in which describes a consistent coupling of d=ll eleven dimensions. Xij is a function of/?,. ~ which will supergravity to a supermembrane. In particular the be determined by the invariance of the action. The Kalb-Ramond-like third rank antisymmetic tensor choice of~gij is due to the fact that, given a variation field of d= 11 supergravity couples to the supermem- of the action of the form 5S= TijX °, and writing this bane via a Wess-Zumino term. variation as In the following we focus our attention on the ToX ~j =goX ° + (T~j-go)X ° , (9) description of the supermembane action in d= I 1. The extension to the case of n-extended objects is the second term on the right-hand side cancels given in the appendix. (5S/5g~j)Sg o. Thus we are left with the first term on the right-hand side, which equals the left-hand side 2. We propose the following action for a super- upon the use of (5). Effectively, this means that membrane coupled to d= 11 supergravity: whenever we encounter a variation of the form ToX ij, we can use eq. (5), provided that we add X~Jto fig o S= f d2 ~ ( ½x~--g g'JEiaEjbqab as in (6). The matrix F"p occurring in (8) satisfies the + CJkEiAEjBEk CBcBA -- ½~-~1. (3) property Here i--0, l, 2 labels the coordinates ~-- (z, (r, p) of F'~oF~,~=(Tit~TJjTkk])g°~,~=-F26~a . (10) the world volume with metric go and signature ( -, +, + ). The super three-form B is needed for the The normalization in (8) is chosen such that upon superspace description of d= I 1 supergravity [ l 0 ]. the use of the equation To=g o, the matrix F"a sat- For the Levi-Civita symbol ¢ok we use the same con- isfies the relation F% Fay = ~%. 76 Volume 189, number 1,2 PHYSICS LETTERS B 30 April 1987 Now using (6) the variation of the action (3) is an appropriate variation ofg o. [Using the argument (we consider a closed supermembrane and therefore below (8) once. ] In the remaining terms coming from discard the surface terms) (l-F2), we use the argument given below (8) repeatedly to compute further contributions to 8g o. ~iS= f d2~ [x/~ gO( _~E#E r T~ a)Ej~ Thus we find the result X,j= - ]eiktEkaEtb(~' ~b).~SEPES + ½x#En c~(yd),~pEndgitj( Tkk Tlq + t~k Tltl ) + ~ LikEi A Ej BE k C~E'~H,~csA +i,--,j. (19) _ ½,f-~ ~gO(T._ ½gJT- ½g")]. (I 1 ) In summary, the action (3) is invariant under (6) The torsion two-form T a and the four-form field provided that (13)-(17) hold, and X ij is given by strength H are defined by (our superspace conven- (19).
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