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I.C/82/9U

INTERNATIONAL CENTRE FOR THEORETICAL

MAXIMAL EXTENDED IK SEVEI DIMENSIONS

E. Sezgirt

and INTERNATIONAL ATOMIC Atdus Salam AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1982 Ml RAM ARE-TRIESTE

IC/82M I. Supergravity in higher dimensions are of interest because International Atomic Energy Agency they can lead to the construction of more complicated four-dimensional and theories, the discovery of their hidden hints for the auxiliary United Nations Educational Scientific and Cultural Organization 2) 3) fields and for the breaking of the . Examples studied INTERNATIONAL CENTRE FOB THEORETICAL PHYSICS so far are theories in eleven , ten and five dimensions. Viewed purely as Kaluza-Klein theories, supergravity theories in higher dimensions have the merit of possessing within them a unique embedding of fermions and scalars . They also offer the possibility of studying varieties of spontaneous compactifications ' (possibly hierarchical), thereby MAXIMAL EXTENDED SUPERGRAVITY THEORY IN SEVEN DIMENSIONS accommodating progressively larger and physically acceptable unifying groups.

In this note ve present construction of the maximal extended super- Lagrangian in seven dimensions . The content of the theory we work with is derived from the pioneering work of Cremraer and Julia ' who on E, Sezgin the basis of their construction of N = 1 supergravity in d = 11, have International Centre for Theoretical Physics, Trieste, Italy, given the possible content of such maximally extended supergravity theory in d = 7 and conjectured that the theory would possess USp(U) local and **) and SL(5,R) global symmetries . We construct a Lagrangian which actually exhibits these (off-shell) symmetries. We have discovered that the anti- symmetric tensor fields in the theory possess a generalized gauge invariance, consistent with . Of all lower than 11-dimensional theories, International Centre for Theoretical Physics, Trieste, Italy, and the one in d = 7 is unique because only in this theory the count (10) of Imperial College, London, England. vector fields matches the count needed to gauge the maximal compact subgroup, S0(5), of the global SL{5,R) of the Lagrangian. Such gauging might Jbe carried out in 7-dimensions (analogous ABSTRACT to the Hicolai-de,Wit construction which however exists in It-dimensions only). In this case the theory may exhibit an overall Yang-Mills S0(5) x S0(5) A maximal extended supergravity Lagrangian is constructed in seven symmetry in d = 7 , with the first S0(5) gauged by the ten vector fields dimensions, which exhibits USpCO •& S0(5) local and SL(5,R) global invariances. in the theory and the second S0(5) gauged by composites of scalar fields.

We find that the antisymmetric second rank tensor fields must possess a The theory contains five second-rank generalized gauge invariance in order that the theory is consistent and super- fields. We conjecture also that after a suitable compactification from symmetric. We conjecture that the theory might admit S0(5) * S0(5) Yang-Mills d = 7 to d = It these fields may provide the 15-fold of gauge fields needed for a symmetries in d = 7 and SO(M * S0(5) x S0(5) in d = h (with the last S0(5), gauging composite fields) after suitable compactification. *) "Maximal extended", in the sense that, the particle content is read off from the maximal N=l, d=ll theory when reduced to d=7• MIFAMARE - TRIESTE **) In this context, one may recall that non-trivial solutions of 11- July 1982 dimensional equations are known, which lead to a spontaneous compactification of d = 11, to anti-de Sitter d = k x S or anti-de Sitter d = 7 x S . Which one of these solutions is energetically the more favourable is not known at present. Even without this motivation, * To be submitted for publication. ve believe that the richness of the 7-dimensional supergravity structure warrants an independent study of this theory. -2- Majorana spinors which obey

further Yang-Mills gauging of a hidden S0(6). Thus in l+-dimensions, the theory Ta abe may altogether exhibit a Yang-Mills gauging of an S0(5) X S0(6) symmetry plus x = C (2) the gauging of three U(l)'s, corresponding to a total of 28 vector fields in the theory when reduced to d = h. Such gaugings may be important for if = [\\i\i ) T ,, and similarly for x • The ten vectors and five where li pa u ptienomenologic al prospects of the theory. The variety of these gaugings antisymmetric tensor fields have the following symmetries and and the eompactification patterns (including those relevant to Kaluaa-KLein) properties; enhance the value of this theory as a theoretical laboratory. (3a) P uy

n II. One way to construct the d = 7 theory is to perform ordinary dimensional reduction of the d = 11 theory. This would lead to a fairly where ( ) denotes (unit strength) symmetriaation and [ ]| denotes (unit complicated theory in 7-dimensiona with global SL(1*,R) (as a result of strength) sympletic tracelesa antisymmetrization. The reason for choosing d • 11 general co-ordinate) and a local.SO(l*) (as a result of local d = 11 A and B „ to transform as cogradient 10 and contragradient 5, Lorentz) invariances. According to Cremmer and Julia's conjecture, these respectively, as well as their opposite pseudo-reality properties is a symmetries can be enlarged to global SL(5,H) and local S0(5) by field re- consequence of the symmetries of the Lagrangian (see discussion after Eq.(8)). definitions and duality transformations. Guided "by this and the structure "/T- is SL(5jR) valued 5x5 matrix. Its symmetries and reality properties of the known d = h and 5, N = 8 theories, we start the construction directly in 7-dimensions, thereby avoiding the highly elaborate field re- definitions. Due to the presence of the antisymmetric tensor fields, some

terms that do not arise in It- or 5r-dimensions will be present. To deal with ab (3c) them we shall be guided by the results of the ordinary dimensional reduction from 11-dimensions. With, these considerations in mind, we assign the fields y is parametrized by 2k scalar fields. Only lit of them are physical of d=7 theory to the irreducible representations of global SL{5,E) and due to the local USp(lt) invariance. It is an essential feature (apparently local S0(5) (which is isomorphic to USp(lt)). The field content and common to all extended supergravity theories) that scalar fields are described representation assignments sire given "below. by a non-linear a-model- type Lagrangian. Accordingly, V"^ transforms under * Ofp global SL(5,E) from the left and local USp(M from the right. The infinitesimal r aoc V X B V c yv afi of SL(5,R) can be described as follows:

Global SL(5,H) 1 i 1 10 5 5 [Y ab

Local USp(it) 1 h 16 i 1 5

where Z are the lU a$ Y Y 8 * La Y* ^B The SL(5,E) indices a,g,..,, and the U8p(U) indices a,b,... run from 1 & generators of SL(5,E)/USp(lt) coset, and •A, -= ,>A ... are the ten to h. The l6-spinor~fields x ° have mixed symmetry and they are symplectic Cip CL Op ^ »K > traceless generators of the USp(lt) subgroup of SL(5,R). To verify this, we write (It)

abc ab (1) convention ). We can choose all F-matrices •) In our nrs = ( + Here antidiagd, and are 8-dimensional sympleetic to be real except rQ which is imaginary. In this case C = 1 and T lh] {IT ,rv)= 2 n • r» V -3- G =3B *2F fB „ +2 perms •, (8a) (5) livpa ii vpa yv paB vhere k 3 B + 12 P B - [w vpa] [nv po]a

(6a)

Here B^^ and B^ = B , , are related to the well-known third rank anti- 1 1 r ^ © i + a. © r symmetric of the d = 11 theory while F a = (S A* - 3 Aa) where A" is essentially the off-diagonal component of the d = 11 metric. The index i runs from 1 to 5. The matrices are the usual U * It Dirac These field strengths are under a set of generalized antisymmetric matrices including y : Pa = yay (a = 1,...,10, F y and gauge transformations. These transformations result from the reduction (to 1 It is clear from (6) that E ^ ana AN otiey the algebra of SL(5,E), Cartan seven dimensions) of the simple d = 11 law: + 2 perms lAip p decomposed with respect to Its maximal subalgebra USp(U): [1,1] Z3 A , (w = l,...,ll) and are given by [£,A]r>E and [A,A] O A.

6B . = 9 A In order to describe the kinetic term for the scalars and furthermore uaB u (9a) to construct a connection for local USp(U), we consider the derivative of 1 SB =31 - 9 A +2FPA (9b) the scalar matrix ^ . Since -\^~ 3 \?" lies in the algebra of SL(5,H) v va v pa uv g it has the following decomposition: (9c)

cd cd d] (7) ab 6b] To supplement this, we note that under the general co-ordinate transformations in the internal four directions, A™ transform as: P lies in the SL<5 ,R)/USp(.U) coset, and transforms homogeneously under local USpCt); it can therefore be used in the construction of the scalar kinetic (9d) term. Q lies in USp(lt), and it transforms inhomogeneously under local "I' \ \ USp(.U); it can be used as a USp(U) connection. P and Q satisfy identities which have been usea in the construction of the Lagrangian. All Sqa,(_Ba) and {.St>) suggest the following ansatz: these aspects are similar to those encountered already in the construction of the E(6) global ® USp(8) local theory in five dimensions and we shall 15) skip details Before we write the Lagrangian we note the following (10) points* With the transformation laws (9a) to (9a) combining aa; l) Besides supersymmetry, general co-ordinate, local Lorentz and SL(5,R) global ® USp(lt) local invariances, the theory is expected to have (lla) local U(l) invariances associated with the vector fields and the anti- symmetric tensor fields. These latter deserve special attention due to (lib) the fact that the coupling of the antisymmetric tensor fields to other fields may present consistency problems . To solve these problems, we generalise the work of Soherk and Schwarz , who have shown that in a The conjectured form of the FA term in (10) will be justified by the dimensional reduction from d = 11 to d = 7, there would arise the requirement of supersymmetry of the Lagrangian (see later), which also following field strengths: *) A similar FA term was found in the construction of Maxwell-Einstein supergravity theory in d = 10. -5- -6- specifies the coefficient in front of the FA term to be "-i". (This "-i" in turn dictates the pseudo-reality (3b) of B , from the hermicity The action is invariant under the following supersymmetry transformations requirements on the Lagrangian.) laws: 2) The contraction, as veil as the raising and lowering of indices in the FA and FA terms of (10) and (ll) are through S3 . No scalar matrix 6V1" = -i ea b OB 11 y ° is needed in the FA and FA terms because 10 S 10 of SL(5,B) already contains a 5 (i.e. G, -,. on the left-hand side). Actually, it ab = " \ is essential that the scalar matrix "^ does not appear in (10) and (ll)t G otherwise pvpr^wT I would not tie invariant under (ll). This also K B explains our choice of assigning B and A to 5 and 10 of SL(5,R). 3) We can now write all the kinetic as veil as Pauli and Yukawa SB terms. In addition to these terms there is need from supersymmetry,of bosonic v ...v cubic terms which may be traced to the well-known £ 1 -1-1 FU A

coupling of the d = 11 theory. From dimensional a p V-aV- £ _ —L~Y rP + 8 r reduction it is seen that a term of the form: fc * ' G G a ,„ r P« ab « P. could arise. However, this alone will not be invariant under (ll) and rP0T £ rpa ,T ^b 15 "POT ab t1 v 2 vj we find that one must add a term like &'" GFB with an appropriate coefficient,for invariance. 2 J2.

III. The final expressions for the d = 7 Lagrangian (up to the quartic fermion terms) and the supersymmetry transformations (up to quadratic fermion terms) are as follows (< = l):

(13)

IV-1 pvpay

uv r 1 r 2 ... r n with unit normalization. D is Lorentz and "U»Sp(M X rr r r , abc rS '[X T] covariant derivative. For instance, ^£a = |U + ^ u^ T l 6fl + Q^ J€

dc uu is a 10 x 10 matrix which is made of a product of i; r U 2 Xa ' ab x 1JVp T c pvp x + Ia" G (s r r r 4i + - ui r r x

4- i -<=d pvp ) (12)- Clearly f[X_ transforms as 10 under SL(5,H) since ~~y transforms as 5 • ^e covariant derivative of H satisfies the following relation : 6) The 4uartic terms in the Lagrangian and quadratic fermion terms in the transformation lavs can, as usual, fce constructed lay requiring the closure = 2 (15) cd of the supersymmetry transformation laws ana supercovariance of the fermion field equations. One standard step 'in this direction is to supercovarianti2e This relation was crucial in proving supersymmetry. We now wish to make 5 , G ., and P the following comments on (12) and (13): the spin connection , the field strengths We leave this problem for the future. l) %i matrices in the F term, and 1^ matrices in the G2 term To conclude, in this note we have completed the construction of the are needed for SL(5,S) invariance (10 ® 10 ~p 1 and 5 ® 5 i 1), ungauged maximal extended supergravity theory in d = 7 (up to quartic In the G A and GBF terms there are no scalars and the contractions are with JJ , because 5 © j? 010 o 1, fermion terms). The problem of Yang-Mills gauging of this theory (again in d = 7) requires further study. 2) We emphasize that our Lagrangian is invariant under the generalized antisymmetric gauge transformations given in (ll) (actually, G and F are separately invariant under (ll)) which also determines the relative coefficient 'between the GGA and GBF terms. It is due to this gauge invariance that the supersymmetric variation of the cubic bosonic terms is greatly simplified. 3) The first term in "y&B in (13) is noteworthy. It is determined by requiring that the variation of G „ does not contain 3 [(SA )A ]. Such terms could not have been cancelled without losing supersymmetry. It is amusing that in Maxwell-Einstein supergravity theory in d = 10 similar terms have been discovered

k) lies in the SL(5,R)/USp(10 coset. This is similar to the situation encountered in d = U and 5 theories. It implies that 'Q is supercovariant (i.e. its supersymmetric variation does not contain 3 e).

5) Comparing our Lagrangian with the h- and 5-dimensional ones, we see that there are other similarities. For instance in the Pauli and Yukawa terms (and also in the transformation laws) similar combinations of T ***) matrices occur. *) • In proving this, we have used (7) (which can te read as

a at>Cd D ^' * = P "^gcd)and the fact that a third rank totally anti- symmetric, symplectic traceless USp(l4) tensor identically vanishes (Schouten identity). **) The action of SL(5,E) on F can "be described 'by defining

using the transformation .ab law which is contradgradient to that of Eq.{h). Note also that aB vS 1 Sv ctfi 1 cci S1 F = ^~ Si F + — fi F li\t 2 uv 2 u1

***) Some of these features can presumably be explained by the group -10- cd ab manifold approach ' . Mote also that x G x = 0 (this can be Xbcd cd proved by using the Schouten identity), while x Abcd Abcd 1_ cd ' 2 xx Xcdb' CURRENT ICl'P PUBLICATIONS AMD IHTERUAL REPORTS REFERENCES IC/82/23 SUN KUN OH - splitting between B and B mesons. IUT.REP.* 1) E. Cremmer and B. Julia, Nucl. Phys. B159, lUl (1979), IC/82/21* A, BREZIHI - Self-consistent study of localization near band edges. 2) See, for example, E. Cremmer, S. Ferrara, K. Stelle and P.C. West, INT.REr.* Phys. Letters .,1:^ - Infinity subtraction in a qi!:..'itarj flci.a V:::r.irj preprint IC/62/32.

10) P.G.O. Freund and M. Rubin, Phys. Letters 9J_B_, 233 (i960). IC/82/3'J H.W. L.-\[,'!:;-;ov,'ij;;i, iu i:tv/EJiYJ::;Ki ana L. $•/:•;i'j-zo'.-v.i - Or; u.^ i1 c.-..,:.ai'. i:. ISIT.iiZlV' 11) E. Cremmer in Superspace and Supergravity. Proc. of the Nuffield Workshop, Eds. S.W. Hawking and M. Roc"ek (C.U.P. London 1981), p.267; IC/3y/35 H.C, LEii, hi BTN'J-AH, SHH-J Ql-Xli:G, Z}].'\J!G I-iEI-M'M' :M'A^i\) TTof."G -+ IUT.REP,* J'Uectrovreali. interference effects in the hifjh energy e + e —» c + e B. Julia, ibid. p.331. + hadroiis process.

12) P.G.O. Freund, Institute, preprint EFI S2/8U. IC/82/36 G.A. CHRISTOS - Some aspects of the U(l) problem and the pat?udo:;calar- mass spectrum. 13) B. de Wit and H. Nicolai, CEEJ1 preprint (1981) TH.3183. IC/82/37 C. MUKKU - Guuce theories in hot enviroumwits: Fen^ion contributions 1*0 P. van Nieuwenhuizen, >"3tony Brook preprint TTO-GB-8I-67 (1981). to onj-loop.

15) E. Sezgin and P. van Hieuwenhuizen, Kucl. Phys. B195, 32!: (1982). IC/82/33 W. KOTAR:-;Ki and A. KOWALEWSKT - Optimal control of distributed parameter INT.REP.* system ifith incomplete Information about the initial condition. 16) H. Deser and E. Wit ten, Kucl. Phys. B178 , 1*91 (l98i). IC/82/39 H.I. YOUSEP' - Diffraction model analysis of polariv.ed triton t-:id He 17) I.. Castellani, P. Fro', F. Giani, K. Pilch and P. van Nieuwenhuizen, HIT.HEP.* elastic scattering. Stony Brook, preprint ITP-SB-82-19 (1982). IC/82/I+O S, SLLZK!1! and M. MAJLIS - Effects of surface exchange anisotropy INT.REP.* in Heisonberg ferromag;netic insulators.

IC/82/I1I H.R. HAROOU - Gubcritical assemblies, use and their feasibility INT.REP.* assessment. IC/82A2 W. AHDRHONI and M.P. T0S1 - Why is AgBr not a superionic conductor?

THKSE PR1TRIHTS ME AVAILABLE FROM THE PUBL1CATI0HS OFFICE, ICTP, P.O. BOX 586, I-3U100 TRIKSTE, ITALY.

-11- 'TC/82A3 U.S. CRAIGIE and J. STERN - What can we learn from sum rules for vertex functions in QCD?

IC/82/1A ERNEST C. HJAU - Distortions in power spectra of digitised signals - I: IHT.REP.* General formulations.

IC/82A5 ERHEST C. NJAU - Distortions in power spectra, of digitised signals - II: IHT.REP.* Suggested solution.

IC/82A6 H.R. DALAFI - A critique of nuclear 'behaviour at high angular . IHT.REP.*

IC/82A7 N.S. CRAIGIE and P. RATCLIFFE - Higher power QCD mechanisms for large p strange or charmed baryon production in polarised proton-proton collisions.

IC/82A8 B.R. BULKA - Electron density of states in a one-dimensional distorted IHT.REP.* system with impurities: Coherent potential approximation.

IC/82A9 J- GORECKI - On the resistivity of metal-tellurium alloys for low INT.EEP.* concentrations of tellurium.

IC/82/50 S. RANDJBAR-DAEMI and R. PERCACCI - Spontaneous compactification of a (l|+d)-dimensional Kaluza-Klein theory into l\ * G/H for arbitrary G and H. p JC/82/51 P,S, CHEE - On the extension of II -functions in polydiscs. IHT.EEP.*

IC/32/53 S. YOKSAH - Critical temperature of two-band superconductors containing IHT.REF.* Kondo impurities.

IC/82/5^ M, OZEB - More on generalized Gauge hierarchies.

IC/82/55 N.S. CRAIGIE and P. RATCLIFFE - A simple method for isolating order .ctg and ag corrections from polarized deep-inelastic scattering data,

IC/82/56 C. PANAGIOTAKOPOULOS - Renormalization of the QEMD of a dyon field.

IC/82/57 J.A. Mk&VASW - Towards an effective bilocal theory from QCD in a INT.REP.* background.

IC/82/58 S. CHAKEABARTI - On stability domain of stationary solitons in a many- charge field model with non-Abclian internal symmetry.

IC/82/59 J.K. ABOAYE and D.S. FAYIDA - High temperature internal friction in pure INT.REP.* aluminium.

IC/82/60 Y. FUJIMOTO and ZHAO:ZHIYOHG - M-S in SO(lO) and SU(6) supersymmetric groBd -unified models.

IC/82/61 J. GORECO and J. POPIELAWSKI - On the application of the long mean free path approximation to the theory of electron transport properties in liquid noble metals.

IC/82/62 I.M. REDA, J. HAFNER, P. PONGRATZ, A. WAGEKDRIETEL, H. EANGERT and P,K. BHAT - Amorphous Cu-Ag films with high stability. IC/PD/82/1 PHYSICS AND DEVELOPMENT (Winter College on and Reactors, INT.REP, 25 January - 19 March 1982),

-ii- - T