IC/8U/217
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
CONSTRUCTION OF THE SUPERALGEBRAS FOR N a 1 SUPERGRAVITY
E.A. Ivanov
and
J. Niederle
INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL SCIENTIFIC AND CULTURAL ORGANIZATION 1984 MIRAMARE-TRIESTE iiittili' rri lift >'•«,.« «>l TC/8U/21T
International Atomic Energy Agency 1. INTRODUCTION and United H.. :ons Educational Scientific and Cultural Organization It is well known /I' •""/' that the action of the infinite-parameter general covarianee algebra of the Einstein gravitation theory can INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS be reduced to the alternating actions of its two finite-parameter eubalgebrae, the conformal eubalgebra and the special linear sub- algebra. This not only essentially simplifies the study of the transformation properties of any quantity under the action of the CONSTRUCTION OF THE SUPERALGEBRAS FOR H = 1 SUPERGRAVITY • general covariant algebra but also gives the possibility to treat gravitons ae Goldstone bosone in the frajnework of non-linear E.A. IVftnov Joint Institute for luclear Research, Dubna, USSR, realization of the conformal and linear (reap, affine) subalgebras .
In the paper we shall diacuss the various supergravity models and in the same spirit. Tor that purpose first we briefly recall J. Niederle •* the geometric formulation of the N = 1 supergravity models £for International Centre for Theoretical Physics, Trieste, Italy, and details see /'-J»4/$. It is based on the (4 + 4)-dimensional International School for Advanced Studies, Trieste, Italy. complex superspace 4 *} ABSTRACT c" - l It is shown that the infinite parameter gauge superalgetjras of the conforms! and of the N = 1 Einstein supergravities can he obtained as the J f 6 closures of various two finite-parameter superalgebras. In the conformal with (»".•£ *f > and C»5. H , A - ^ t .*£>* case the standard, minimal and Einsteinian closures are studied. In the being its left- or right-banded parametrizations. In superspace case of the N = 1 Einstein supergravities the minimal and non-minimal closures are discussed. C the "triangular" gauge supergroup is introduced which is infinitesimal?1 defined by MIRAMAEE - TRIESTE (1.2) November * To be submitted for publication. */ The vector (spinor) coordinates are labaled by l,i&,n,... •* Permanent address: Institute of Physics, Czechoslovak Academy of Sciences, Prague, Czechoslovakia. -2- where A , •* , p are arbitral^ superfunction-parameters. Using the relations between transformations on the super- It leaves invariant the "chiral" superspace -4/2 volume elements in I and i.e. (1.3) (1.6 where Ber denote superdeterminants (Berezians) '*' and n 4/4 The real, physical superspace is a hypersurface in one of the values of the Gates - Siegel parameter ' ' we can select different subeupergroups of supergroup (1.2) as invariance groups of (1.6). They correspond to the various M = 1 Einstein supergravities " " . Depending on the value of n one can the shape of which is determined by the superfunctions H™, H" distinguish three types of the Einstein supergravities: 1 and S " defined by: 1 i/ non-minimal in which n , 0 ii/ minimal for which n = - -i and iii/ new minimal for n - 0. The non-minimal and minimal cases of the Einstein supergravity Recall that after imposing' a dynamical postulete (action principal) are studied in detail in Sees. 3.1 and S.2. respectively. the superfunctions Hs become the supergravity superfielde Cprepo- 2. CONFORMAL SUFERGRAVITi tentials). The infinitesimal general coordinate transformations of the The complex supergroup (1.2) in (4 • 2)-dimensional chiral x conformal supergravity in C ' = {( lm > "t ) { are defined by euperspaces C plays the role of the gauge supergroup of the conformal auprgravity. In the next section we Bhow that this (2.1) infinite-parameter supergroup can be obtained as the closure of its two finite-parameter subeupergroupe in three different ways - where are arbitrary infinitesimal chiral auperfunctione- Standard, minimal and Eineteinian. The standard closure was parameters. Expanding J- }X into infinite series of powers of discussed in and we can find that the infinite number of generators -3- of (2.1) split into g types, namely m 2 k'1 lx*-—L > 7> = X r "n - (2.3) (2.2) V (n) n m (0) Here the index in the bracket denotes dimension of the correspond- ing generator and (eL ©Ll = ©L ©L^ . In the next subsections Z) —. = / we shall see that all generators (2.2) can be obtained by means (O), of the generators of two finite-parameter supergroups. We shall distinguish three cases. These 24 generators satisfy relations of the conformal super- 2.1 Standard closure algebra. We shall distinguish two Weyt subeuperalgebras of the In this case two aboTe-mentioned finite-parameter supergroups conformal superalgebra: are the conforms! and the affine supergroups. They are generated Weyl superalgebra I with the generators :|Q , Q - , P , M , by the following generators of (2.2): Dlt S^ ] and a) Conformal superalgebra ( for notations see Appendix and /I/ Weyl superalgebra II generated by : { Q^ , Q^ , P^ b) Affine superalgebra - X 1- -5- -6- I Jj ^ 1 t. f\ Jl U ™" (2.4) in Section 2.1 and in 2.3 Sinsteinian closure In this case we first complete the affine superalgebra to and P and Q „ as in the previous case. the general linear auperalgebra by adding generator I^2"'m . Theorem; AH.generators (2.2) can be obtained by repeatedly applying From the conformal superalgebra we take only the Weyl subsuper- commutation and anticommutation r«lations satisfied by {.he algebra I . It is clear that the closure of these two supar- generators of the conformal and the affine superalgebra. algebrae yields all generators (2.2) . Indeed by commuting S from the general linear superalgebra with the generators from This result can be proved applying the induction method in several the Weyl superalgebra I we get all generators of the conformal ways. For instance it is sufficient first to produce all generators superalgebra and thus the situation discussed in section 2.1. linear in x? and the quadratic generator x^ xm^ —- . Then by repeatedly applying the quadratic generator to the linear Remark: We cannot take only the generators QM , P , Q^ and ones we obtain all generators (2.2). In the second proof we first aa in the minimal closure since we cannot produce generator i product all generators mn . by commuting E with . We K (this is possible due to /1//). Then we commute '111^ with The situation in conformal eupergravity is summarized in 1 11 n) Q<" and obtain T' "!* and s' *Th e commutator of R mn with Fig. 1. S,; gives ity^j . Then commuting with jC n • Standard closure we obtain all generators P ~"t •*'* .Finally, commuting D-^ D...1 0 . 0 . li . P, . I* II */•< * ,ti ' Tin* k* inn1 /-i** ' jfi <"" with we get the remaining two types of gene- conformal auperalgebra affine superalgebra rators from (2.2). Minimal closure 2.2 Minimal closure of the In fact it is sufficient to take instead'whole affine snper- conformal superalgebra algebra its aubsuperalgebra generated by Q^ , ?m , K^, 0^ . Since anticommuXator { Q^ , S; } gives generators B B ., Einsteinian closure P T and H - T V E aince in the conforms! super- R l d { mn 4 L mn p DI» Q^ ' Q^ ' Mmn' Pk > mn> Vv ' ^ p algebra there are x£ ^r and ^ ^.^ ^ our disposal w Weyl superalgebra 1 general linear superalgebra separately we have in fact all generatore of the .affine and of the conformal BUperalgebraa and thus the previous Situation treated Fig. 1 -T- -8- 3. K = 1 EINSTEIN SUPERGRAVITY (3.2) 3_. 1 Non-Minimal ease r'frl'*t.,*ihft, mt.. In this case the gauge superalgebra is obtained by imposing constraint(1.6) with n ¥ - 3 , 0 on transformations (1.2). The infinitesimal form of relation (1.6) is given by (3.1) We have still to add to them . the generators of type 2/ and 3/: Due to (3.1) all generators split into three types: H>•*••••> , /~H , (3.3) 21/ / A*7jr 8)— + iff* 6 )'i 'fe jzz . (3.4) and the generators obtained by multiplying them with the monomials 6L and (6L 9, ) . Consequently generators (2.2) are modified in the following way: Let us now discuss the modification of the generators from the 1/ conformal and affine supergroups. We shall begin with the Weyl sub- •7- k group I from the conformal supergroup. Ihe generators Q , Q^; , P are not modified. The generators of the Lorentz group are modified in am obvious way , _ 7- #•••?+ Since the generators and Dj belong to type 1/ they are also modified 8» (3.7) -10- ia guarantee by the fact that the Keneratora of the modified general It ia easy to check that the generators nun linear and Weyl supergroups satisfy condition (3.1) which form the same s1 -eralgebra as the generators without tilde. preserves the algebraic structure (commutation relation^. At the same time it is not possible to modify generator S • of How to obtain generators of type 2/ and 3/ ? It is clear the Weyl supergroup II in such a way that the new generator by that we have to take from/very beginning the generators commuting with Q- gives the new Lorentz generators M . Thus the only possibility to get the extra term on the right-hand aide of = l8 the commutator /si -j ^ • [O"mn ) ^•a~ i> to change Q • -? Cfe -^^^^ and to add the term -v f< (w —= ) to the generator S-. However, Since we have no way to obtain the generators thia term is quadratic with respect to u> and those parts are A *" ~ 'i -~ V (li^ t4 excluded by the group constraint (3.1) . t -11- -ia- .a1 st- $ ft = f C3.14) r r H; F W ? . o> a f T' . Wfclf 1 ft r yf p ft) It is clear that by commuting generators of type 1/ with the generators from superalgebra F we can obtain an arbitrary This euperalgebra has the following structure: power of x? for generators from J and thus all generators T['}\ F'?' ff F of type 2/ and 3/. Consequently t }' -parameter^ Theorem : The infinite^'Buperalgebra of the non-minimal Einstein subauperalgebra its ideal subeuperalgebra FIX (ideal in superalgebra) supergravitj- can be obtaintttas a closure of two finite-parameter superalgebras: Remark: T^ and FJJ can be obtained by gauging generators l T !~t) I on a pure Grassmannian base { dfl . It appears that F can be joined, to the modified Weyl euper- general linear auperalgebra and algebra I. The correeponding relations are of the form : Q, Pn' w ^ Weyl superalgebra I I Their intersection io the superalgebra also ,1\ Remark: We may/add F 5' to Aj . Then the generator F .« -13- belongs to the intersection. The same ia true for Dj . Note that The generators of type 1/ are of the fern.: F forms an ideal in kjy 3.2 Minimal Einatein supergravity In thio case the gauge transformatione of the infinite - parameter supergroup The generators of type 2/ UL> $I . f\> $i (mtHt are restricted by the condition (1.6) with n which The generators of type 3/ (3.19) the infinitesimal form (3.17) (3.20) The structure of infinitesimal transformations fulfilling (3.17) The generators of type 4/ leads to the generators of five types ' .m,, (3.21) The generators of type 5/ 3/ \* U) 2_ (3-22) / * h+3 The only generators of the conformal and affine euperalgebrae which satisfy (3.17) are the generators of the «eyl superalgebra II: 5/ Q» . Q.- H*~ . K . S- D. (3.23) -16- r^> ^x"^2- + i r*, A 9 and the sneratore from the affine supergroup: (3.24) Thus in comparison with the conformal supergravity JT whioh and X*L— do not enter separately but intl/e in: DTT . The closure of these two supergroups cannot generate ell subsupergroups fulfilling (3.17) since jione > of these super- groups contains generators bilinear with respect to coordinates. Thus we have to add this generator either to (3.23) or to (3.24) Theorem. The infinite-parameter superalgebra fulfilling (3-17) the can be obtained by talcing the closure of two finite-parameter They can beobtained from the generators of above specified super- subsuperalgebras; A algebra ^>y using the relations: i/ Weyl superalgebra II with generators Q«, Q^, Pffi, S^, Dn and ii/ the aubsuperalgebra the generators of which are the operator P , Q ., , Q£ , MJJ^ from the affine superalgebra and the quadratic rt „ -3 D'fe) generator &L The closure is illustrated in theFig. 2 } Fig. 2 Remark. We can include DJJ also to the first superalgebra D th*n the intersection contains {Q., , Q^ , PB, HL,, TT 1 • Proof: It will be done by induction method. First we shall write all lowest generators which are not included in the initial superalgebras. -17- -18- Thus we have obtained all lowest generators and we have to show how to get the higher/ We aaaume that we have at our dieposal APPENDIX the generator ^-1)^ ^ Basic notations and identities uaed in the paper: 7,,, The generators of other types can be obtained in the same way w ueing commutator* with f Aftf J For eia]1ipie assuming f 1 that we have 't our aispooal generator j^-j) "*'" ">n ^^ higher generator! l^+Tt)^ ••• ">n+^ follow from j = J t 1 $ In the same vay we get the remaining generators. ACKNOWLEDGMENTS One of the authors (J.H.) would like to thank Professor Atidus Salam and Professor P. Budinich, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, and the International School for Advanced Studies, Trieste. -19- -20- ? ••• • '1. Jiff-11 - 20 - REFERENCES REFERENCES Akulov, V., Volkov, D. and Soroka, V. (1977) Teor. Mat. Piz. 33^, 12- Borisov, A.B. and Ogievetsky, V.I. (197M , Teor. Mat. Fiz. 21, 329. [l] Ogieveteky V.; Lettere al Nuovo Cimento £ (1973), 968. Breitenlohner, P. (1977), Nucl. Phys, B12]*_, 500. Borisov A.B., Ogievetsky V.I.: Teor. Mat. Fix. gl (1974). 329 (in Russian). Ferrara, S. and van Wieuwenhuizen, P. (1976), Phys. Lett, TUB, 333. f3j Galperin, A., Ogievetsky, V. and.Sokatchev, E. (1985), J. Phys. A 1£, 3765. Galperin A., Ogievetaky V.f Sokatchev 1.: JrPbye.A: Math.Gen.ll (1982), 3785. Lukierski, J. and Nowotnik, H. (1983), Phys. Lett. 12 5B, 1*52- W Ogievetsky V., Sokatchev E.: Hiys.Lett. B^2 (1978), 222; Ogievetsky, V. (1973), Lettere al Huovo Cimento B_, 988. Ttadernaya Fiz. 2i (1980), 264, 821; 22 (1980), 862; 870; 1192; Ogievetsky, V. ana Sokatchev, E. (1978), Phys, Lett. BT9., 222. Sokatchev E.: in "Superspace and Supergravity" eds. Ogievetsky, V. and Sokatchev, E. (1980), Yadernaya Fiz. 31., 26h, 821. S.Hawking and M.Rocek, Cambridge Univ. Press, 1981, p.197. Ogievetsky, V. and Sokatchev, E. (i960), Yadernaya Fiz. 32., 862, 870, 1192. [5] Lukierski J.Nowotnik H.: Hiys.Letters 12?B (1983), 452. Siegel, V, and Gates, J. (1979), Nucl. Phys. BIU7, 77. [&] Siegel W., Gates J.: Nucl.Phye. B147 (1979), 77. Sohnius, M.F. and West, P. (1981), Phys. Lett. 105B, 353. [7] Stelle K.S., West P.C: Phys.Lett. J4JJ (1978), 330; Sokatehev, K. (I98l), in Superspace and Supergravity, Eds. S. Hawking Ferrara S., van Nieuwenhuioen P.: Phys.Lett. 24.B (1978), 333. and M, Eocek (Cambridge University Press), p.197. [B] Breitenlohner P.: Nucl.Phys. B124 (1977), 500. Stelle, K.S. and West, P.C. (1976), Phys. Lett. T^B, 330. [97 Siegel W., Gates J.: Nucl.Phys. B147 (1979), 77. fioj Akulov V., Volkov D., Soroka V.: Theor.Math.Phys. ^1(1977), 12; Sohnius M.F., West P.: Phys.Lett. 105B (1981), 353. -21- -n-