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New Gauge Supergravity in Seven and Eleven Dimensions

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New Gauge Supergravity in Seven and Eleven Dimensions View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server New Gauge Supergravity in Seven and Eleven Dimensions Ricardo Troncoso and Jorge Zanelli Centro de Estudios Cient´ıficos de Santiago, Casilla 16443, Santiago 9, Chile Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile. fields neither belong to an irreducible representation of Locally supersymmetric systems in odd dimensions whose the supergroup nor do they transform as gauge connec- Lagrangians are Chern-Simons forms for supersymmetric ex- tions. This precludes a fiber-bundle interpretation of the tensions of anti-de Sitter gravity are discussed. The construc- theory, as is the case with standard Yang-Mills gauge tion is illustrated for = 7 and 11. In seven dimensions the D theories, where this interpretation is crucial in proving theory is an N = 2 supergravity whose fields are the vielbein a ab i renormalizability. In order to accomodate the supergrav- (eµ), the spin connection (!µ ), two gravitini ( µ)andan i ity multiplets in tensor representations, it is usually nec- sp(2) gauge connection (aµj ). These fields form a connection for osp(2 8). In eleven dimensions the theory is an N =1su- essary to introduce a host of auxiliary fields . This is a j a ab highly nontrivial issue in general and has often remained pergravity containing, apart from eµ and !µ , one gravitino an unsolvable problem [7,8]. µ, and a totally antisymmetric fifth rank Lorentz tensor one- abcde Still, there is a handful of supergravities whose super- form, bµ . These fields form a connection for osp(32 1). The actions are by construction invariant under local super-j algebras close off shell without requiring auxiliary fields: symmetry and the algebra closes off shell without requiring Anti-de Sitter (AdS)inD= 3 [9], and D = 5 [10]; auxiliary fields. The N =2[D=2]-theory can be shown to have Poincar´einD= 3 [11], and in general for D =2n 1 nonnegative energy around an AdS background, which is a [12]. These are genuine gauge systems for graded− Lie classical solution that saturates the Bogomolnyi bound ob- algebras and therefore make interesting candidates for tained from the superalgebra. renormalizable theories of gravity. PACS numbers: 04.50.+h, 04.65.+e, 11.10.Kk. Here, we present a family of supergravity theories in Introduction.- In recent years, M-Theory has become 2n 1 dimensions whose Lagrangians are Chern-Simons the preferred description for the underlying structure of (CS−) forms related to the n-th Chern character of a su- string theories [1,2]. However, although many features of pergroup in 2n dimensions [13]. As anticipated in the M-Theory have been identified, still no action principle pioneering work of Ref. [6], and underlined by other au- for it has been given. thors [14,3], D = 11 supergravity should be related to Some of the expected features of M-theory are: (i)Its a gauge system for the group OSp(32 1), a symmetry dynamics should somehow exhibit a superalgebra in which is not reflected in the CJS theory.| The theory pro- which the anticommutator of two supersymmetry gen- posed here, for D = 11 turns out to be naturally a gauge erators coincides with the AdS superalgebra in 11 di- system for this group. mensions osp(32 1) [3]; (ii)The low energy regime should Gauge Gravity.- Our aim is to construct a locally be described by| an eleven dimensional supergravity of supersymmetric theory whose generators form a closed new type which should stand on a firm geometric foun- off-shell algebra. One way to ensure this by construc- dation in order to have an off-shell local supersymmetry tion is by considering a gauge theory for a graded Lie [4]; (iii) The perturbation expansion for graviton scatter- algebra, that is, one where the structure constants are ing in M-theory has recently led to conjecture that the independent of the fields, and of the field equations. new supergravity lagrangian should contain higher pow- In order to achieve this, we relax two implicit assump- ers of curvature [5]. Since supersymmetry and geometry tions usually made about the purely gravitational sector: are two essential ingredients, most practitioners use the (i) gravitons are described by the Hilbert action, and, standard 11-dimensional Cremer-Julia-Scherk (CJS)su- (ii) torsion does not contain independently propagating pergravity [6] as a good approximation to M-Theory, in degrees of freedom. spite of the conflict with points (i) and (iii). In this let- The first assumption is historical and dictated by sim- ter we present a family of supergravity theories which, plicity but in no way justified by need. In fact, for D>4 for D=11, exhibits all of the above features. the most general action for gravity –generally covariant In spite of its improved ultraviolet behavior, the renor- and with second order field equations for the metric– is malizability of standard supergravity beyond the first a polynomial of degree [D=2] in the curvature, first dis- loops has remained elusive. It is not completely absurd to cussed by Lanczos [15] for D = 5 and, in general, by speculate that this could be related to the fact that super- Lovelock [16,17]. This action contains the same degrees symmetry transformations of the dynamical fields form of freedom as the Hilbert action [18] and is the most gen- a closed algebra only on shell. An on-shell algebra might eral low-energy effective theory of gravity derived from seem satisfactory in the sense that the action is neverthe- string theory [19]. less invariant under local supersymmetry. It is unsatis- Assumption (ii) is also motivated by simplicity. It factory, however, because it means that the propagating means that the spin connection is not an independent 1 a field. Elimination of !b in favor of the remaining fields, The key point, however, is that torsion terms are nec- however, spoils the possibility of interpreting the local essary in general in order to further extend the AdS sym- translational invariance as a gauge symmetry of the ac- metry of the action into a supersymmetry for D>3. The tion. In other words, the spin connection and vielbein reason for this is analysed in the discussion. –the soldering between the base manifold and the tan- Let us now briefly examine how torsion appears in an gent space– cannot be identified as components of the AdS-invariant theory. The idea is best understood in connection for local Lorentz rotations and translations, 2+1 dimensions. For D = 3, apart from the standard respectively, as is the case in D = 3. Thus, the Einstein- action(3), there is a second CS form for the AdS group. Hilbert theory in D = 4 cannot be formulated purely The exterior derivative of this “exotic” lagrangian is the as a gauge theory on a≥ fiber bundle. Pontryagin form in 4 dimensions (2nd Chern character δS For a generic gravitational action in D>4, δω =0 for SO(4)). This alternative CS form is, [25] cannot be solved for ! in terms of e. This implies that AdS a even classically, ! and e should be assumed as dynami- LT 3 = L3∗(!)+2eaT ; (4) cally independent fields and torsion necessarily contains a a n where dL2∗n 1(!b )=Tr[(Rb ) ]isthen-th Chern char- many propagating degrees of freedom [20]. These de- − grees of freedom are described by the contorsion tensor, acter (see, e.g. [26]). Similar exotic actions, associ- kab := !ab !ab(e). Thus, the restriction to theories with ated to the Chern characters in 4k dimensions, exist µ µ − µ in D =4k 1. Since the (2n + 1)-th gravitational nonpropagating torsion would be a severe truncation in − general. Chern characters vanish, there are no exotic actions in D =4k+1. For D =4k 1, the number of possible The LL theory, which includes as a particular case the − EH system, is by construction invariant under Lorentz exotic forms grows as the partitions of k. As we shall see rotations in the tangent space. For D =2n 1, how- below, we will be interested in one particular combina- ever, there is a special choice of the coefficients− in the tion of these forms, which in the spinorial representation LL lagrangian which extends this invariance into an AdS of SO(4k) can be written as [27] AdS n 1 p symmetry [21]: LG 2n 1 = p=0− αpL ,whereαp = AdS 1 AB 2k 1n 1 2p D− dL = Tr[( R Γ ) ]: (5) κ(D 2p)−(−)l − ,and T 4k 1 AB − p P − − 4 p a1a2 a2p 1a2p a2p+1 aD It is important to note that in this Lagrangian, as well as LG =a1 aD R R − e e : (1) ··· ··· ··· in (4), torsion appears explicitly. For example, in seven Here wedge product is understood and the subscript “G” dimensions one finds denotes a Lagrangian for torsion-free gravity. The con- AdS 3 a b a ab stant l has dimensions of length and its purpose is to LT 7 = L7∗(!) (R bR a +2[T Ta R eaeb])L3∗(!) render the action dimensionless allowing the interpreta- − 4 − 3 a b a ab a a b c tion of ! and e as components of the AdS connection ( R bR a +2T Ta 4R eaeb)T ea +4TaR bR ae : [22] − 2 − The CS lagrangian (5) represents a particular choice !ab ea=l W AB = ; (2) of coefficients so that the local Lorentz symmetry is en- eb=l 0 − larged to AdS invariance. In general, a Chern-Simons D-form is defined so that its exterior derivative is an in- where A; B =1; :::D + 1.
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