New Gauge Supergravity in Seven and Eleven Dimensions

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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 (ωµ ), two gravitini (ψµ)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 | a ab highly nontrivial issue in general and has often remained pergravity containing, apart from eµ and ωµ , one gravitino an unsolvable problem [7,8]. ψµ, 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-| 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 ωb 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 ω in terms of e. This implies that AdS a even classically, ω and e should be assumed as dynami- LT 3 = L3∗(ω)+2eaT , (4) cally independent fields and torsion necessarily contains a a n where dL2∗n 1(ωb )=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 := ωab ωab(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∗(ω) (R bR a +2[T Ta R eaeb])L3∗(ω) render the action dimensionless allowing the interpreta- − 4 − 3 a b a ab a a b c tion of ω 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 ωab 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. The Lagrangian (1) is an AdS- variant homogeneous polynomial of degree n in the cur- CS form in the sense that its exterior derivative is the vature, that is, a characteristic class [28]. In the examples Euler class, above, (3) is the CS form for the Euler characteristic class

AdS A1A2 A2n 1A2n 2n-form, while the exotic lagrangians are related to dif- dLG 2n 1 = κA1 A2n R R − , (3) − ··· ··· ferent combinations of Chern characters. Thus, a generic AB CS action in 2n 1 dimensions for a Lie algebra g can where R is the AdS curvature and κ is quantized [23] − (in the following we will set κ = l =1). be written as Although torsion in general appears in the field equa- g n dL2n 1 =< F >, (6) tions, it has not been necessary to introduce it explicitly − in the action until now. As shown in [24], the LL actions where <>stands for a multilinear function in the Lie can be extended to allow for torsion so that for each di- algebra g, invariant under cyclic permutations such as Tr mension there is a unique set of possible additional terms or STr. The problem of finding all possible CS actions for to be considered. Like in the pure LL theories, the most a given group is equivalent to finding all possible invari- general action contains a large number of arbitrary con- ant tensors of rank n in the algebra. This is in general stants, and again as in the LL case, their number can be an open problem, and for the groups relevant for super- reduced to two if the Lorentz invariance is enlarged to gravity discussed below (e.g., OSp(32 1)) the number of AdS symmetry. invariant tensors can be rather large.| Most of these in- variants, however, give rise to bizarre lagrangians and

2 ¯i T ji ¯i ji the real problem is to find the appropriate invariant that In (8) ψ = ψj Cu (ψ = ψj†Cu for D =4k+1), describes a sensible theory. where C and u are given in the table above. These al- The R.H.S. of (5) is a particular form of (6) in which gebras admit (m + N) (m + N) matrix representations <>is the ordinary trace over spinor indices. Other [32], where the J and Z×have entries in the m m block, n p p × possibilities of the form , are not used in the Mij ’s in the N N block, while the fermionic gener- our construction as they would not lead to the minimal ators Q have entries× in the complementary off-diagonal supersymmetric extensions of AdS containing the Hilbert blocks. action. In the supergravity theories discussed below, the Under a gauge transformation, A transforms by δA= 1 AdS 1 AdS gravitational sector is given by 2n LG 2n 1 2 LT 2n 1 λ,where is the covariant derivative for the connec- [29]. ± − − − tion∇ A. In particular,∇ under a supersymmetry transfor- i i Gauge Supergravity.- The supersymmetric exten- mation, λ =¯Qi Q¯i,and sion of a theory invariant under AdS requires new bosonic − k¯ k generators to close the superalgebra [14]. In standard  ψk ψ ¯k Dj δA= − i i i , (9) D¯ ¯ ψj ψ¯ j supergravities, Lorentz tensors of rank higher than two  − −  were usually excluded from the superalgebra on the where D is the covariant derivative on the bosonic con- grounds that elementary particle states of spin higher nection, D =(d+1[eaΓ +1ωabΓ + 1 b[r]Γ ]) ai  . than 2 are inconsistent [30]. However, this does not rule j 2 a 2 ab r! [r] j j i D=7 The smallest AdS superalgebra in seven dimen-− out the relevance of those tensor generators in theories 1 ab sions is osp(2 8). The connection (8) is A = ω Jab + of extended objects [31]. | 2 eaJ + Q¯iψ + 1 a M ij ,whereMij are the generators of In [12], we discussed family of theories in odd dimen- a i 2 ij sions whose algebra contains the Poincar´e generators . sp(2). In the representation given above, the bracket < The anticommutator of the supersymmetry generators is > is the supertrace and, in terms of the component fields a combination of a translation plus a tensorial “central” appearing in the connection, the CS form is extension, osp(2 8) 4 AdS 1 AdS L7 | (A)=2− LG 7 (ω,e) LT 7 (ω,e) α ¯ a α abcde α − 2 Q , Qβ = i(Γ )β Pa i(Γ )β Zabcde. (7) Sp(2) { } − − L∗ (a)+LF(ψ,ω,e,a). (10) − 7 This algebra gives rise to supergravity theories with off- Here the fermionic Lagrangian is shell Poincar´e superalgebra. The existence of these the- ories suggests that there should be similar supergravities ¯j 2 i i 2 i LF =4ψ (R δj +Rfj +(f )j)Dψi based on the AdS symmetry. It is our purpose here to i j k j k k +4(ψ¯ ψj)[(ψ¯ ψk)(ψ¯ Dψi) ψ¯ (Rδ + f )Dψk] present these theories. − i i ¯i ¯j k k ¯j Superalgebra and Connection.- The smallest su- 2(ψ Dψj)[ψ (Rδi + fi )ψk + Dψ Dψi], peralgebra containing the AdS algebra in the bosonic − i i i k 1 ab a b 1 a sector is found following the same approach as in [14], where fj = daj +akaj ,andR= 4(R +e e )Γab+ 2 T Γa but lifting the restriction of N = 1 [27]. The result, for are the sp(2) and so(8) curvatures, respectively. The D>3is: supersymmetry transformations (9) read a 1 i a ab 1 i ab δe = 2 ¯ Γ ψi δω = 2 ¯ Γ ψi i i− ¯i δψi = Di δaj =¯ψj ψj. D S-Algebra Conjugation Matrix Internal Metric D=11 In this case, the smallest AdS− superalgebra is 8k 1 osp(N m) CT = C uT = u 1 ab a osp(32 1) and the connection is A = ω Jab + e Ja + − | T T − | 2 8k +3 osp(m N) C = C u = u 1 babcdeJ + Qψ¯ ,wherebis a totally antisymmetric | − 5! abcde 4k +1 su(m N) C† = C u† = u | fifth-rank Lorentz tensor one-form. Now, in terms of the [D/2] elementary bosonic and fermionic fields, the CS form in In each of these cases, m =2 and the connection (6) reads takes the form osp(32 1) sp(32) L | (A)=L (Ω) + LF (Ω,ψ), (11) 1 ab a 1 [r] 11 11 A = ω Jab + e Ja + b Z[r] + 2 r! 1 a 1 ab 1 abcde where Ω 2 (e Γa+ 2 ω Γab+ 5! b Γabcde)isansp(32) 1 i i 1 ij ≡ (ψ¯ Qi Q¯ ψi)+ aij M . (8) connection. The bosonic part of (11) can be written as 2 − 2 sp(32) 6 AdS 1 AdS b i L (Ω) = 2− LG (ω,e) LT (ω,e)+L (b, ω, e). The generators Jab,Ja span the AdS algebra, Qα gen- 11 11 − 2 11 11 erate (extended) supersymmetry transformations, and [r] denotes a set of r antisymmetrized Lorentz indices. The The fermionic Lagrangian is Q0s transform as vectors under the action of Mij and as 4 2 LF =6(ψR¯ Dψ) 3 (DψDψ¯ )+(ψRψ¯ ) (ψR¯ Dψ) spinors under the Lorentz group. Finally, the Z’s com- − 3 (ψR¯ 3ψ)+(DψR¯ 2Dψ) (ψDψ¯ )+ plete the extension of AdS into the larger algebras so(m), −   sp(m)orsu(m). ¯ 2 ¯ 2 ¯ ¯ ¯ 2 (DψDψ) +(ψRψ) +(ψRψ)(DψDψ) (ψDψ),   3 where R = dΩ+Ω2 is the sp(32) curvature. The super- a region where the rank of the symplectic form is maximal symmetry transformations (9) read the theory behaves as a normal gauge system, and this a 1 a ab 1 ab condition is stable under perturbations. As it is shown δe = 8 ¯Γ ψδω = 8 ¯Γ ψ abcde − 1 abcde in [38], there exists a nontrivial extension of the AdS δψ = D δb = 8 ¯Γ ψ. Discussion.- The supergravities presented here have superalgebra with one abelian generator for which anti- two distinctive features: The fundamental field is always de Sitter space without matter fields is a background of the connection A and, in their simplest form, these are maximal rank, and the gauge superalgebra is realized in pure CS systems (matter couplings are discussed below). the Dirac brackets. For example, for D =11andN= 32, As a result, these theories possess a larger gravitational the only nonvanishing anticommutator reads sector, including propagating spin connection. Contrary i j 1 ij a ab abcde Q , Q¯ = δ CΓ Ja + CΓ Jab + CΓ Zabcde to what one could expect, the geometrical interpretation { α β} 8 αβ is quite clear, the field structure is simple and, in con- ij M Cαβ ,  trast to the standard cases, the supersymmetry transfor- − mations close off shell without auxiliary fields. where M ij are the generators of SO(32) internal group. A. Torsion. It can be observed that the torsion la- On this background the D = 11 theory has 212 fermionic 12 grangians (LT )are odd while the torsion-free terms (LG) and 2 1 bosonic degrees of freedom. The (su- are even under spacetime reflections. The minimal super- per)charges− obey the same algebra with a central exten- symmetric extension of the AdS group in 4k 1 dimen- sion. This fact ensures a lower bound for the mass as a sions requires using chiral spinors of SO(4k) [33].− This in function of the other bosonic charges [39]. turn implies that the gravitational action has no definite D. Classical solutions. The field equations for these parity, but requires the combination of LT and LG as de- theories in terms of the Lorentz components (ω, e, b, n 1 scribed above. In D =4k+ 1 this issue doesn’t arise due a, ψ) are spread-out expressions for =0, to the vanishing of the torsion invariants, allowing con- where G(a) are the generators of the superalgebra. It is structing a supergravity theory based on LG only, as in rather easy to verify that in all these theories the anti-de [10]. If one tries to exclude torsion terms in 4k 1 dimen- Sitter space is a classical solution , and that for ψ = b = sions, one is forced to allow both chiralities for− SO(4k) a = 0 there exist spherically symmetric, asymptotically duplicating the field content, and the resulting theory has AdS standard [22], as well as topological [40] black holes. two copies of the same system [34]. In the extreme case these black holes can be shown to be B. Field content and extensions with N>1.The BPS states. field content compares with that of the standard super- E. Matter couplings. It is possible to introduce a gravities in D = 7, 11 as follows: minimal couplings to matter of the form A J.ForD= 11, the matter content is that of a theory with· (super-) 0, D Standard supergravity New supergravity 2, and 5–branes, whose respective worldhistories couple 7 ea A ψαi ai λα φ ea ωab ψαi ai to the spin connection and the b fields. µ [3] µ µj µ µ µ µj F. Standard SUGRA. Some sector of these the- 11 ea A ψα ea ωab ψα babcde µ [3] µ µ µ µ µ ories might be related to the standard supergravities Standard seven-dimensional supergravity is an N =2 if one identifies the totally antisymmetric part of the theory (its maximal extension is N=4), whose gravita- contorsion tensor in a coordinate basis, kµνλ,withthe tional sector is given by Einstein-Hilbert gravity with abelian 3-form, A[3]. In 11 dimensions one could also cosmological constant and with a background invariant identify the antisymmetric part of b with an abelian 6- under OSp(2 8) [35,36]. Standard eleven-dimensional su- form A , whose exterior derivative, dA , is the dual of | [6] [6] pergravity [6] is an N=1 supersymmetric extension of F[4] = dA[3]. Hence, in D =11theCStheorypossibly Einstein-Hilbert gravity that cannot accomodate a cos- contains the standard supergravity as well as some kind mological constant [37]. An N>1 extension of this of dual version of it. theory is not known. In the case presented here, the extensions to larger N The authors are grateful to S. Deser, G. Gibbons, M. are straighforward in any dimension. In D = 7, the index Henneaux, C. Teitelboim and the CECS research group i is allowed to run from 2 to 2s, and the Lagrangian is a for many enlightening discussions, and to E. Witten for CS form for osp(2s 8). In D = 11, one must include an helpful comments. This work was supported in part internal so(N) field| and the Lagrangian is an osp(32 N) by grants 1960229, 1970151, 1980788 and 3960009 from CS form [27]. The cosmological constant is necessarily| FONDECYT (Chile), and 27-953/ZI-DICYT (USACH). nonzero in all cases. Institutional support to CECS from Fuerza A´erea de C. Spectrum. The stability and positivity of the en- Chile and a group of Chilean private companies (Business ergy for the solutions of these theories is a highly nontriv- Design Associates, CGE, CODELCO, COPEC, Empre- ial problem. As shown in Ref. [20], the number of degrees sas CMPC, Minera Collahuasi, Minera Escondida, NO- of freedom of bosonic CS systems for D 5isnotcon- VAGAS and XEROX-Chile) is also acknowledged. J. Z. stant throughout phase space and different≥ regions can thanks the John Simon Guggenheim Foundation for sup- have radically different dynamical content. However, in port.

4 Int. J. Mod. Phys. A2(1987) 273. [31] J. A. de Azc´arraga, J. P. Gauntlet, J. M. Izquierdo and P. K. Townsend, Phys. Rev. Lett. 63 (1989) 2443. [32] P.Freund, Introduction to Supersymmetry Cambridge University Press (Cambridge, U.K., 1989). [1] E. Witten, Nucl. Phys. B443 (1995) 85. [33] M. Gunaydin, hep-th/9803133. [2] J. Schwarz, Phys. Lett. B367 (1996) 971. [34] P. Horava, hep-th/9712130. [3] P. K. Townsend, p-Brane Democracy, hep-th/9507048 [35] P. K. Townsend and P. van Nieuwenhuizen, Phys. Lett. [4] H. Nishino and S. J. Gates, Phys. Lett. B388 (1996) 504. 125B (1983) 41. [5] M.Green and P. Vanhove, Phys. Lett. B408, 122(1997). [36] A. Salam and E. Sezgin, Phys. Lett. 126B (1983) 295. [6] E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76B [37] K. Bautier, S. Deser, M. Henneaux and D.Seminara, (1978) 409. Phys. Lett. B406,(1997) 49. [7] P. van Nieuwenhuizen, Phys. Rep. 68, 4 (1981). [38] O. Chand´ıa, R. Troncoso and J. Zanelli, ”Dynamical [8] V. O. Rivelles and J. G. Taylor, Phys. Lett. 104B (1981) Structure of Chern-Simons Supergravities”, (in prepara- 131; 121 (1983) 38. tion). [9] A. Ach´ucarro and P. K. Townsend, Phys. Lett. B180 [39] G.W. Gibbons and C.M. Hull, Phys. Lett. 109B (1982) (1986) 89. 190. [10] A.Chamseddine, Nucl.Phys. B346 (1990) 213. [40] S. Aminneborg, I. Bengtsson, S. Holst and P. Peldan, [11] P. S. Howe, J. M. Izquierdo, G. Papadopoulos and P. K. Class. Quantum Grav.13 (1996) 2707. M. Ba˜nados,Phys. Townsend, Nucl. Phys. B467 (1996) 183. Rev. D57 (1998), 1068. R.B. Mann, gr-qc/9709039. [12] M. Ba˜nados, R. Troncoso and J. Zanelli, Phys. Rev. D54, (1996) 2605. [13] For simplicity, we will not distinguish between different signatures. Thus, the AdS algebra in D dimensions will be denoted as so(D +1). [14] J. W. van Holten and A. Van Proeyen, J. Phys. A15 (1982) 3763. [15] C. Lanczos, Ann. Math. 39 (1938) 842. [16] D. Lovelock, J. Math. Phys. 12 (1971) 498. [17] B. Zumino, Phys. Rep. 137 (1986) 109. [18] C. Teitelboim and J. Zanelli, Class. and Quantum Grav. 4(1987) L125. [19] B. Zwiebach, Phys. Lett. 156B (1985) 315. [20] M. Ba˜nados, L. J. Garay and M. Henneaux, Phys. Rev. D53 R593 (1996); Nucl. Phys. B476 611 (1996). [21] A. Chamseddine, Phys. Lett. B233 (1989) 113. [22] M. Ba˜nados, C. Teitelboim and J. Zanelli, in J. J. Gi- ambiagi Festschrift, H. Falomir et al.(eds.),(World Scien- tific, Singapore 1991); Phys. Rev. D49(1994) 975. [23] J. Zanelli, Phys. Rev. D51 (1995) 490. [24] A. Mardones and J. Zanelli, Class. Quantum Grav. 8 (1991) 1545. [25] E. Witten, Nucl. Phys. B311 (1988) 46. [26] M. Nakahara,Geometry, Topology and Physics (Adam Hilger, New York 1990). [27] R. Troncoso, Tesis Doctoral, Universidad de Chile (1996); and R. Troncoso and J. Zanelli, in the proceedings of the meeting on Quantum Gravity in the Siuthern Cone, Bariloche, Argentina, January 1998 (to appear). g [28] An explicit expression for the CS form in (6) is L2n 1 = − 1 2 2 n 1 dλ < A(λdA + λ A ) − >. If the manifold M has 0 no boundary, the action can be written as a topological theoryR in a 2n-dimensional space Ω, with ∂Ω=M. [29] The sign correspond to the two choices of inequivalent representations± of Γ’s, which in turn reflect the two chiral representations in D + 1. As mentioned earlier, the su- persymmetric extensions of LG or LT , require using both chiralities, thus doubling the algebras. Here we choose the + sign, which gives the minimal extension. [30] W. Nahm, Nucl. Phys. B135 (1978) 149. J. Strathdee,

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