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
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,
(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+
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,
(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. + ei)"y Substituting (2.24) and (2.25) into (2.17) we obtain the first order equations [26]:
d u = -u'" + 6uu( - 12^ €7 - W&fJ - 24( + i)Ay + (3! - u)ay + (20 + 0')ft - 6^y = a_ (2.29) + 60'0" + 200'" + 12^V - 60V -
fa - 120'£ - 40"6 + 6u0£ - 2£ 0a (2.27) 2 2 2 for j = 0,1,..., n — 1. For j = n, we obtain, the evolution equations (2.19). In this case,
?+6 = -4£2" + 3£2«' • the first Poisson bracket structure is given by [33]
(2.30) 3+0 = 0
Using the recursion relations (2.17), (2.18) and the functions (2.25) and (2.26) one can while others vanishing. The second Poisson bracket structure is still given by (2.21). obtain j42, a?, 02, B2 which when substituted into the dynamical equations (2.19), give If we now set, AQ = |, ao = /3o — 0; -Bo = 0, then the zeroth order evolution equations
10 11 from (2.26) are The evolution equations in this case would be
2 d+U = d_ ( - tt" + 3u - 3^1 -
(2.31) (2.35)
d+
on = -f2 4, - i \ -u + 4? £! + 1 <*+|A 0 (2.36) A = 6 ( 0 -fi + 6 2 Therefore, the evolution equations (2.19) in this case would have the form in the first case and
1 * - 5 A 0 (2.37) 0 -6 + 6 24> A
(2.33) in the second case. Also, the first biHamiltonian system (2.27) is not supersymmetric under the transformations
Using (2.32) in the recursion relations (2.29} we obtain <-*) (2.38) = u +
(2.34)
which leave the equations (2.15) invariant. In fact, the shift u — £ A2 does not respect this superBymmetry whereas the shift 4> — ^ A respects it. So, the second biHamiltonian system
12 13
T" I , j,. It 1™'
(2.35) is in fact supersymmetric. The other N = 2 supersymmetric extensions of the KdV from (2.19) as equation can also be obtained in this method. Since the functions A+, A+\ A^ , Al - 2(a2 have to satisfy the supersymmetry relations given in (2.38), all the modes of these functions 12 in the A-expanskm will also satisfy (2.38). Making use of the scaling property the most - (2a2 - l)^
general forms of At, <*i, 0i, Bi will be ' - 2(1 (2.42)
= 3_ [^' + 3u£2
(2.39) It has been noted in ref. 26, that for a = 2, -1 and £ the system (2.42) has Lax pair Pi = *»£[ s
representation which is an indication that for these values of a2, the system (2.42) would
be integrable. We notice that for a2 = 2, the equations (2.42) match exactly with one of the biHamiltonian system (2.35) obtained earlier. We would like to close this section by Since they have to satisfy the supersyraraetry relations (2.38) and the consistency condi- pointing out that the light-cone component of our gauge field A- is related with Polyakov- tions following from (2.20) are Bershadsky (P-B) gauge field by a gauge transformation of the form
6AX SBi
5£2 Du dtp (2.40) (2.42) OQLI oB\ Sp\ Hi 0 -
In their W^ bosonic algebra (£t + £2) and (-£1 + £2) are treated as conformal spin 3/2 all the parameters in (2.39) can be solved in terms of a2 and therefore the functions in (2.39) can be obtained as bosonic fields, but in our case they are fermionic fields. So, the algebra obtained in P-B gauge fixing is exactly the same as N = 2 superconformal algebra when the bosonic fields
2 Ai = u + a24> are replaced by the fermionic fields and the commutator is replaced by the anticommutator. This then clarifies the reason behind the similarities between W3 algebra and N = 2 (2.41) superconformal algebra.
II. Zero Curvature as an Anomaly Equation In this section we will interpret the zero curvature condition as an anomaly equation, So, the one parameter family of JV — 2 supersynunetric KdV equations can be written following our earlier proposal [30], In other words, we shall
14 15 regard Eqs. (2.9) as the Ward identities for the supergravity associated with OSp(2/2) We recognize these to be the O(2) superconformal transformations of the stress-tensor, group after some suitable identifications. the Bupercurrents and the spin-1 current of a O(2) supersymmetric conformal field theory. Let us note first that the zero curvature condition is invariant under an infinitesimal Therefore, we identify transformation
SA±=d±t + \A±,i\ (3.1) (3.5) where, for the OSp(2/2) case, we may write
= 5_ -(e-i +«("*)) (3.2)
Similarly, we can calculate the residual gauge variations of A+, A+ , BB*+ * and JS^. The Since we have partially fixed the gauge with the choice in E
t' = - Bl (3.3)
The residual gauge transformations are easily seen to be If we identify
A+ = h++
*(°) (3.7) Hi = (| i-B- (3.4)
{a} + u) then Eqs. (3.6) are recognized as the residual transformation of the metric, the graviti- nos, and the U(l) gauge field in the light-cone gauge [34]. With these identifications the
16 17
¥ dynamical equations of the N = 2 sKdV hierarchy is nothing but the Ward identities of r ,j*Jl,S-] can be thought as an effective action resulting from a theory in the (2,0) induced 2D supergravity. background of a stress-tensor, two supercurrents and a spin-1 current. An appropriate As seen previously [30], it seems more appropriate to identify action for the (2,0) supergravity can be defined through the Legendre transform
(3.1S) 0 (3.8) / + 2B+S-) 2S_ If this is taken to be the dynamical action of the supergravity theory, it follows now that and the classical equations will be ST (3.9) = T__ = 0
With this identification, the zero curvature condition takes the same form as the gauge anomaly equation for OSp(2/2), namely, (3.16) ST
L J+ = 3_J+ (3.10) ST = 2S_ = 0 which is invariant under 6B+ 8 A- =d-e Imposing Eqs. (3.16) on Eqs. (2.13) and Eqs. (3.4) we obtain respectively (3.11) dih++ =o 6J+=d+c+[J+,i\ with f(T ,jl,jl,S~) given in Eq. (3.2). (3.17) To determine the current algebra associated with this supergravity theory, let us look 3_B+ = 0 at, the effective action which gives and din- =0 6W\A.\ (3.12) SA. (3.18)
Infinitesimally, therefore, we can write
2 Thus, we can write 6W[A-] = f d xati(J+SA.) (3.13)
Here we use supertrace instead of the ordinary trace because both J+ and A- are super- \ (J*(x (3.19) matrices. Using Eqs. (3.8), (3.9) and (2.8) we get
! x ( - fe++£T__ + 2x+6jL - 2x%Sjl - 2B+6S-) (3.14)
18 19 Therefore, we have from Eqs. (3.9) and (2.8) It now follows from the transformation in Eq. (3.11) that the J°'s generate an OSp(2/2)
current algebra. J+ (r__ = o,ji = ojl = o,s_ = o) = J + h{x+)x- (3.20) where By introducing the rescaling J" -» | J°, A++ -v \ k++, x+ -» f X+i X+ ~* f X+ and B^. —• j B+, the operator product expansion of the currents takes the form (3.21)
and
- (r__ = O.jl = Ojl = 0,£_ = 0)] n = 2,3 (3.22) (3.27)
Similarly, we have
t3"23) where JJ"* is the inverse of the Killing metric. In this case, the only nonzero values of t}ab are and
T__ =O,JA=O,J! (3.24) where n(0)(0) _ _£ ,°° = (3.28)
v+- = ,-+ = ,- = ,(-*)* = -,*<-*) = -, (3.25)
and
i- (T_- = Qjl =O,jl = O,fl_ =0)] n = 2,3 (3.26) For completeness, we write down these operator product expanaions explicitly.
20 21
TT - I B+U±*f.+ 1»T£toT+pegular
J*(i+)J±*(y+) - regular
J--*(i+)J±*(y+)~ regular (3.29) ( ] J±i(a;+)J<±* V) - ^ ^ + regular
regular -)- regular
{)~ regular Using Eqs. (3.19) and (3.29) we can now write down the operator product expansions regular involving the fields as regular
regular
regular regular
regular -^ + regular
~ regular V)
regular
22 23 Acknowledgement: One of us (A.D.) would like to thank Prof. A. Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics. This work was supported in part by the U.S. Department of Energy Contract No. regular (3.30) DE-AC02-76ER13065.
- 4f*++-^ + 4^-V) d"-X+{y) + r6gUlar
B+[x)B+{v) ~ -
Finally, let us remark here that our results Eqs. (3.19) and (3.29) agree with those of (2,0) induced 2D supergravity in the superchiral gauge, obtained by using superspace approach [34].
IV. Conclusion: We have derived N — 2 superconformal algebra from the zero curvature condition on OSp(2/2) group. The associated TV = 2 fermionic extensions of KdV equation are also studied. We have obtained two biHamiltonian extensions by shifting appropriate bosonic variables by a constant. One of them is manifestly supersymmetric but the other is not. We have also constructed a one-parameter family of N = 2 supersymmetric KdV equation. The similarity between N = 2 superconformal algebra without a biHamiltonian structure and the origin of the W^ ' bosonic algebra is clarified by comparing the explicit forms of the corresponding gauge fixings. As we have proposed earlier, we have regarded the zero curvature condition as a gauge anomaly equation. This immediately enables us to obtain the current algebra for the (2,0) 2D supergravity theory. The operator product expansions of currents and appropriate fields are also written down explicitly. These results agree with those obtained from the superspace approach. Therefore, this viewpoint provides a simpler way of understanding the Kac-Moody symmetry in the supergravity theory.
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