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Ecole Normale Supérieure Laboratoire d'Anneey-le-Vieux de Lyon de Physique des Particules

SUPERSYMMETRIC BLACK HOLES FROM TODA THEORIES1

François DELDUC2 Laboratoire de Physique Théorique ENSLAPP1, Ecole Normale Supérieure de Lyon 46 Allée d'Italie, 69364 Lyon CEDEX 07, France. Jean-Loup GERVAIS Laboratoire de Physique Théorique de l'École Normale Supérieure1, 24 rue Lhomond, 75231 Paris CEDEX 05, France. and Mikhail V. SAVELIEV' Laboratoire de Physique Théorique ENSLAPP, Ecole Normale Supérieure de Lyon 46 Allée d'Italie, 69364 Lyon CEDEX 07, France.

Abstract By the example of nonabelian Toda type theory associated with the Lie ojp(2|4) we show that this integrable dynamical system is relevant to a background metric in the corresponding target space. In the even sector the model under consideration reduces to the exactly solvable conformai theory (nonabelian B> Toda system) in the presence of a black hole recently proposed in [I].

hepth@xxx/9207027 LPTENS-92/20 ENS[LAPP-L-393/92 July 1992

1UTgX file available from [email protected] (# 9207027) -On leave of absence from the LPTHE, Université Paris 7, F-75251 Paris Cedex 05. 'URA 14-36 du CNRS, associée à TENS de Lyon et au L.A.P.P. d'Annecy-le-Vieux. 'Unité Propre du Centre National de la Recherche Scientifique, associée k l'École Normale Supérieure et à l'Université de Paris-Sud. 'On leave of absence from the Institute for High Energy Physics, 142284, Protvino, Moscow region, Russia. , H ' +*•'

1. It was shown in paper [Ij that exactly solvable nonabelian Toda systems [2] are rel- evant to two dimensional conformai field theories in the presence of a black hole. Here the fields arising as the potential terms in the corresponding Lagrangian, play the role of the 2d cosmological terms. Moreover, the corresponding i?n-WZNW model is gauged by an appropriate nilpotent group, in distinction with those of Witten's scheme, see [3] and ref- erences therein. The simplest nontrivial example containing all main features characteristic of the black holes in the framework of this approach, is based on the JS2 supplied with a noncanonical gradation. In the present note we discuss a supersymmetric black hole by the example of N = 1 ï supersymmetric extension of nonabeliaa Toda system associated with the classical finite dimensional Lie superalgebra oap(2\i). Here we use such a choice of its simple roots that in the corresponding Dynkin diagram the first simple root is odd (it corresponds to a/(l|l) superalgebra) and other two simple roots are even. With an appropriate gradation, this superalgebra provides, in the even sector, the reduction of the system under consideration to the case of the Lie algebra B2. 2. Consider the classical Lie superalgebra Q with the defining relations, see e.g., [4],

[A11A,] = 0, [h,,X±j}=±kiiX±J, [X+^X-J] = SnHi, (1) where k is the Cartan matrix of Q\ h, and X±,, 1 < i < rank G, are its Cartan and Chevalley generators. Here the numerical indices of the elements of the superalgebra Q correspond to its roots, in particular, subscripts of the generators denote the simple roots. For the case under consideration k is defined in accordance with the above mentioned Dynkin scheme. Realize the gradation of the oap(2|4) by the Cartan element H — A2 + 2A-* of its o«p(l|2) subsuperalgebra with the odd elements Y± = X^ 7 Xtm, and even elements (generators of the sl(2,C) subalgebra of ojp(l|2)): H = [Y+1Y-)+,X± = Y£. Then the subspaces G,n,m = 0, ±1, ±2, of oap(2\4) = ®mQm, are the following:

(2) ,1 Q±\ = (3) G±2 — (4) Start with the N = I superextension [5, 6] of the Toda systems (Abelian as well as their nonabelian versions) associated with finite dimensional classical Lie Q, : -lD-g) = (5) where g is a regular element of the Grassmann span of the Lie group Gt> with Lie algebra Gu, and depends on the coordinates of the 2|2 C2^; D± are the supersymmetric covariant derivatives in this superspace. Then, in the case under consideration, parametrize 1 the function g as + = e\p{a X+2} exp{a (6)

with the unknown functions (superfields) depending on the points in We come to the following system of equations:

- e a~ D-a+} =

U+[e [Lf^a — (a ) Lf-a )\ = —e ' o . (7)

Now, it is the very time to note that there is such an element Ht> — hi — h:i which commutes with Y±, and, consequently, with X±, while hi commutes only with X±, and just '.• due to this reason the r.h.s. of (5) in our case contains only hi and H but not hi (as it ' should be in the even case). Then, making use the gauge properties of our system, equations '" (7) can be transformed, in terms of the functions

AL/ + -W/2 •* I «-I + a3 . .TC p = -Arsh(a+a )" - Î-; V = ^— + *2' and + + + D+W = --T- —-\a~D+a - (1 + 2a a~)a D+a~ 2(1 + a+a-y v ' + + —2o a"(l + a a~)D+(at + 2a2 — 03)], 1 r[-a+D_a" + (1 + 2a+a~)a"D_o+] "-" 2(l + a+a--)) x ' to the form sh/j D+D-p = chpe^ - —j- D+Ui D.w, . ; D+(thV^-w) - D_(thVD+u;) = 0,

The Lagrangian density C corresponding to the system (8) is the following: i C, = -DJ-O D-O + -DjL-1U) D xb — —th oD±.w D ui + sh

Therefore, the target-space metric G, \ t G.-,-= diag(l,l,-th2/»), contains the hyperbolic-tangent-square function (superfield) which, in even case, is charac- ^t* ' teristic for the black hole in the spirit of Witten. The potential term in (9) plays the role of ;] ' the cosmological term in C2|2- I We are not going to discuss here more details concerning the properties of supersymmetric • black holes in the framework of the given approach; only note once more that they are K i . described by exactly solvable system of equations. A relation with the corresponding gauged 1 |r supersymmetric WZNW model is quite evident. , . ; ' Acknowledgements. One of the authors (M.S.) would like to thank A. Izergin, A. 1 Sevrin, and P. Sorba for useful discussions; and ENSLAPP in Lyon for kind hospitality. ft* ' References

v [1] J.-L.Gervais, M.V.Saveliev: Black holes from non-abelian Todatheori.es, » Preprint LPTENS-92/07, EENSLAPP-L-370/92, 1992; to appear in Phys. Lett. B. I j [2] A.N.Leznov, M.V.Saveliev: Lett. Math. Phys. 6, 505 (1982); Comm. Math. Phys. 89, 59 (1983). [3] E.Witten: Phys. Rev. D 44, 314 (1991); Two-dimensional theory and black holes, - in "Topics in quantum gravity", Univ.of Cincinnati, April 1992. [4] V.G.Kac: Comm. Math. Phys. 53, 31 (1977). (5) M.V.Saveliev: Comm. Math. Phys. 95, 199 (1984); A.N.Leznov, M.V.Saveliev: Sov. J. Theor. Math. Phys. 61, 150 (1984); D.A.Leites, M.V.Saveliev, V.V.Serganova: - in "Group Theoretical Methods in Physics" V.I.Manko, M.A.Markov, eds., VNU Sci. Press 1, 255 (1986). [6] F. Delduc, E. Ragoucy and P. Sorba: Comm. Math. Phys. 146 403 (1992).

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