International Centre for Theoretical Physics V

IC/91/125 I C UR-1215 - wU ER-13065-667 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS V ZERO CURVATURE CONDITION OF OSp(2/2) AND THE ASSOCIATED SUPERGRAVITY THEORY Ashok Das Wen-Jui Huang INTERNATIONAL ATOMIC ENERGY and AGENCY Shibaji Roy UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL MIRAMARE - TRIESTE ORGANIZATION T " I 1091/125 UR-1215 ER-13065-667 Abstract The N = 2 fermionic extensions of the KdV equations are derived from the zero International Atomic Energy Agency curvature condition associated with the graded Lie algebra of OSp(2/2). These equations and United Nations Educational Scientific and Cultural Organization lead to two bi-Hamiltonian systems one of which is supersymmetric. We also derive the INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS one-parameter family of JV = 2 supersymmetric KdV equation without a bi-Hamiltonian structure in this approach. Following our earlier proposal, we interpret the zero curvature condition as a gauge anomaly equation which brings out the underlying current algebra for ZERO CURVATURE CONDITION OF OSp(2/2) AND THE ASSOCIATED SUPERGRAVTTY THEORY the corresponding 2D-supergravity theory. This current algebra is, then, used to obtain the operator product expansions of various fields of this theory. Ashok Das * International Centre for Theoretical Physics, Trieste, Italy Wen-Jui Huang and Shibaji Roy *" Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA. MIRAMARE-TRIESTE June 1991 Permanent address: Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA. Address after 1 October 1991: International Centre for Theoretical Physics, Trieste, Italy. T * 9 I. Introduction with the appropriate field variables of a supergravity theory, the dynamical equations of the Integrable models have been studied quite extensively in the recent past [1-8]. Apart sKdV hierarchy take the form of the Ward identity of the supergravity theory in the light- from being interesting on their own, these models are closely related with two dimensional cone gauge. Following our earlier proposal [30], we treat the zero curvature condition as a (2-D) conformal field theories (9-14) as well as 2-D gravity theories [15-20]. Integrable gauge anomaly equation which can then be integrated following Polyakov and Wiegrnann models are Hamiltonian systems with the right number (i.e. n if In is the phase-space [31] to give the WZNW action and the associated Kac-Moody algebra associated with the dimension and infinite for continuum systems) of functionally independent conserved quan- appropriate theory. tities which are in involution. Each of these conserved quantities determines its own flow. Our paper is organized as follows. In section II, we describe how the N — 2 fermionic In other words, for every non-linear integrable equation there is a hierarchy of equations extensions of the KdV hierarchy equation can be obtained from the zero curvature condi- associated with it which share the same set of conserved quantities and therefore they tion of OSp(2/2). We also show how the one parameter family of JV = 2 supersymmetric are also integrable. This hierarchial structure comes out naturally in the zero curvature KdV equation can be obtained in this formulation. The reason behind the similarites be- formulation. For example, the KdV equation and its hierarchy can be obtained from the tween the JV = 2 superconformal algebra and the bosonic H^2) algebra of Polyakov and zero curvature condition associated with the group SL(2,R) [21]. Similarly, the Boussinesq Bershadsky is also explained. In section III we interpret the zero curvature condition as a hierarchy naturally comes out from the zero curvature condition of SL(3,R) [22] and so on gauge anomaly equation. We show how the current algebra structure comes out naturally [23]. for the corresponding supergravity theory. The operator product expansions of various Bosonic integrable models are known to possess a biHamiltonian structure, although fields are also calculated. We present our conclusions in section IV. for fermionic ones it is not necessarily true. In a zero curvature formulation associated II. Zero Curvature Condition of OSp(2/2) and sKdV Equation: with the graded Lie algebra OSp(2/l) [24,25], we have shown that there are two integrable The OSp(2/2) Lie superalgebra consists of eight generators, four bosonic and four JV = 1 fermionic extensions of KdV equation - one of which is biHamiltonian but not su- fermionic. Let us denote them by persymmetric while the other one is supersymmetric but not biHamiltonian. In this paper we study the N = 2 fermionic extensions of the KdV equation from the zero curvature t s (2.1) condition of OSp(2/2). There are several JV = 2 fermionic extensions of the KdV equation a as pointed out in ref. [26], With a suitable gauge fixing on OSp(2/2), all these equations where the Roman subscript T denotes the bosonic index and Greek subscript 'a' denotes as well as their hierarchy are obtained in this formulation. Our approach also clarifies the the fermionic index. The generators satisfy the OSp(2/2) algebra given by, reasons behind the similarities between the JV = 2 superconformal algebra [27,28] and the W^ bosonic algebra of Polyakov-Bershadsky [19,29]. Another motivation for studying integrable models in this approach is their connection (2.2) with 2-D gravity theories [19,30]. With the identification of the variables in sKdV equations 2 Here '+' denotes an anticommutator and the non-zero structure constants have the values In 1 + 1 dimension, we choose the light-cone components of the gauge fields associated r+ _ f- „ A _ r* _ M) _/("*) with OSp(2/2) to be [32] J+o Jo- /(o)} J{o)-i ;(i)(o)~ -" /0 _ rO •4 (-*) - M (2.3) 0 (2.7) 2<t> The simplest representation of the OSp(2/2) algebra can be given in terms of 3 x 3 matrices where the bosonic generators ij's take the form (2.8) J t+ = (-*) -A* J (2.4) where u and 4> are two real bosonic fields and ft are two real fermionic fields. A!{.'s to = 0 i 0 '(0) = 3 are, in general, functions of u, <j>, £j and fj. While A+, A+, A\.' are bosonic, A+ ' , J4+ 0 0 0, are fermionic functions of the fundamental variables. Similarly, the fermionic generators can be written, in this representation, as The zero curvature condition (2.9) (2.5) '0 0 0 \ ( (-i) = 0 0-1 associated with the potentials in (2.7) and (2.8) can be seen (with some work) to give four o ] constraint equations and four dynamical equations, namely, The matrices in (2.4) and (2.5) are supermatrices and have a generic block structure of the form = 0 \F B) (2'6) a- where Bi, B2 are two bosonic blocks of dim2 x 2 and lxl respectively and Fi, F$ are two (2.10) fermionic blocks of respective dimensionalities 2x1 and 1x2. We note that the matrices in (2.4) and (2.5) are supertraceless, i.e., tr Bi — trB2 = 0. - 2A+ = 0 4 and These relations can, in turn, be used to write the dynamical equations (2.11) in the form d+u = (i di + u' + 2U3_)A; + {Hid- + £i)^+* (2.13) * = 0 -A =0 -*-(4 £)A; + (31 + u}^- Here prime denotes g~. We now make changes in the variables to compare with other - (4 - ^') - (V - ^-*»)(-. = o A+ -> -2A+ u -+ -u (2.14) (2.11) The constraints (2.10) can be solved to express A%, A%, A\, A+* in terms of -A ;* Then the equations (2.13) would take the form d+u = -dtA~ -A -* a+<t> = -A Aj = a-A; - + a! A;* - UA;* (2.15) A+ = i a!A; + UA; - < 2 (2.12) * = To obtain the hierarchy of equations, let us shift u by it — ~ A2 where A is a constant [23] Let us remark here that the functions Aj, By, ay, j9y in the recursion relations (2.17) and and regard the functions A+, A+ , A+ , .A+ as functions of u, ^, 6> {2 and A. These (2.18) are related to the conserved quantities by functions can now be expanded as ;K *,£!,&, A) (2.20) tf/r,' J=0 (2.16) n ; » y=o So, one can construct infinite number of conserved quantities from them. The two Poisson y=o bracket structures associated with (2.17) and (2.18) are [26] Since u, <j>, 6> 6 d° n°t depend on A, we get the following recursion relations from (2.15) {u(x-},u(j/-)} = (-a_3 + 4u3_ + 2u')fi(x- - y~) (-dl+ iud.. + 2u')A; + (3^5- + el) ay + (360- + Qffj 3 (2.21) as well as a constraint equation -} Ay + f2ay - iipj + d^Bj = 0 (2.18) where £12 = 1 and (11 = £22 = 0, and for j = 0,1,..., n — 1. For j' — n, on the other hand, we have the dynamical equations d+u = ( - 3i + 4u9_ + 2u')An + (36 (2.22) 5+6 = (365- + 2£[)An + {dl ~ u)an and others are zero. Let us note that if we choose (2.19) Ao = - an = 0n = 0 Bo (2.23) •ir T I then the zeroth order equation follwing from (2.19) would have the form the 2nd order equations and in this way one recovers the whole hierarchy associated with this biHamiltonian system. The other biHamiltonian system can be obtained by shifting 0 by 0 - \ A. Again [ { i] (2.24) expanding the functions X+, A °\ A^, A ~ in powers of A as 3+0 = 0 y=o Using the recursion relations (2.17) we obtain y=o (2.28) (2.25) and furthermore (2.18) gives We obtain from (2.15), the following recursion relations (2.26) V + 2u')Aj + {3^3.

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