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Toda Equation Daniel Finley And For Publisher's use GENERALIZED SYMMETRIES FOR THE SDIFF(2) TODA EQUATION DANIEL FINLEY AND JOHN K. MCIVER University of New Mexico, Albuquerque, NM 87131, USA E-mail: ¯[email protected] Self-dual Einstein metrics which admit one (rotational) symmetry vector are determined by solutions of the sDi®(2) Toda equation, which has also been studied in a variety of other physical contexts. Non- trivial solutions are di±cult to obtain, with considerable e®ort in that direction recently. Therefore much e®ort has been involved with determining solutions with symmetries, and also with a speci¯c lack of symmetries. The contact symmetries have been known for some time, and form an in¯nite- dimensional Lie algebra over the jet bundle of the equation. Generalizations of those symmetries to include derivatives of arbitrary order are often referred to as higher- or generalized-symmetries. Those symmetries are described, with the unexpected result that their existence also requires prolongations to \potentials" for the original dependent variables for the equation: potentials which are generalizations of those already usually introduced for this equation. Those prolongations are described, and the prolongations of the commutators for the symmetry generators are created. The generators so created form an in¯nite-dimensional, Abelian, Lie algebra, de¯ned over these prolongations. 1 The sDiff(2) Toda equation, formulation for an h-space, a heaven, which and the standard Toda lattice is a 4-dimensional, complex manifold with a self-dual curvature tensor. He of course All self-dual vacuum solutions of the Einstein showed that such a space is determined ¯eld equations that admit (at least) one rota- by a single function of 4 variables, ­ = tional Killing vector are determined by solu- ­(p; p;¹ q; q), which must satisfy one con- tions of the sDiff(2) Toda equation, which straining pde, and then determines the met- may be written in various equivalent forms: ric via its second derivatives, as follows: u;ss r g = 2(­;pp¹ dp dp¹ + ­;pq dp dq u;qq + e = 0 () r;qq + (e;s);s = 0 +­ dq dp¹ + ­ dq dq) ; v ;qp¹ ;qq () v;qq + (e );ss = 0 ; v ´ r;s ´ u;ss : (1) ­;pp¹­;qq ¡ ­;pq­;qp¹ = 1 : (2) How this comes about was shown by Charles Boyer and myself1 in 1982. [In Restricting attention to those complex spaces fact Pleba¶nskiand I ¯rst wrote the equation that allow real metrics of Euclidean signa- down, for complex-valued, self-dual spaces in ture, there are only two possible \sorts" 1979.2] Since that time there has been con- of Killing vectors, \translations" and \ro- siderable interest in this equation, in general tations." Noting that the covariant deriva- relativity, and also in some other ¯elds of tive of any Killing tensor must be skew- physics and mathematics. Nonetheless, most symmetric, by virtue of Killing's equations, currently known solutions describe metrics we may make this division more technical that also allow a translational Killing vec- by dividing the class of Killing vectors based tor. Such solutions do not provide much new, on this skew-symmetric tensor's anti-self-dual real understanding of this equation since they part, which must be constant. For \trans- were susceptible to discovery by a much sim- lational" Killing vectors, this anti-self-dual pler route, as solutions of the 3-dimensional part vanishes, while it does not for the \rota- Laplace equation. tional" ones. The self-dual case|where the To understand how this process occurs, anti-self-dual part vanishes|has been com- we may begin with the standard Pleba¶nski3 pletely resolved.4;5 (In this case the con- PlebTalkLatex: submitted to World Scienti¯c on January 16, 2009 1 For Publisher's use straining equation for ­ reduces simply to the to the hierarchy for the KP equation, which 3-dimensional Laplace equation.) contained operator realizations of sDiff(2). We continue by insisting that the space The name also emphasized its relationship to under study admit a rotational Killing vector, certain limits of the 2-dimensional Toda lat- »e, we re-de¯ne the variables so that they are tice equations: adapted to it: a b b a K bu a a v u;xy = e or v;xy = K b e ; e e » = i(p@p ¡ p@¹ p¹) ´ @Á ; »(­) = 0 ; a a b p p (v ´ K b u ) ; a; b = 1; 2; : : : ; n ; (7) p ´ r eiÁ ; p¹ ´ r e¡iÁ ; (3) a where K b is the Cartan matrix for the Lie al- which changes the constraining equation as gebra which is also the generator of the sym- follows, construing ­ to now depend on the metries of these same Toda equations . variables fr; Á; q; qg: The hoped-for virtue of the relationship (r­;r);r­;qq ¡ r ­;qr­;qr = 1 : (4) with the Toda lattice lay in the fact that the Toda lattice equations, when based on any It is however often more convenient to rewrite ¯nite-dimensional, semi-simple algebra, has the constraining equation, and the metric, in symmetries which allow the determination terms of a new set of coordinates, obtained of BÄacklund transformations which generate from the original ones via a Legendre trans- new solutions from old ones. We ¯rst use form based on variables r and s ´ r­;r. Tak- the gauge freedom in the original equations ing fs; q; qg, along with Á, as the new coor- to divide into two parts the unknown func- dinates, and v ´ ln r as the function of these tions va ´ aa + ba. Then for the case when coordinates that will generate the metric, we the Lie algebra is sl(n + 1), one ¯nds that ¯nd the following new presentation, which the explicit ¯rst-order pde's for the BÄacklund show the agreement with the sDiff(2) Toda transformation as the following, where the equation: wa = wa(x; y) are the \other" set of depen- ¡1 2 dent variables, i.e., the \pseudopotentials" g = V γ + V (dÁ + »!) ; γ ´ ds2 + 4 ev dq ^ dq ; (5) involved in the transformation: 1 ! i a a wa+ba wa¡1+ba¡1 V ´ 2 v;s ; » ´ 2 fv;qdq ¡ v;qdqg ; fw ¡ a g;x = ¡e + e ; a a ¡wa+aa ¡wa+1+aa+1 and we still require that fw + b g;y = e ¡ e : (8) v v;qq + (e );ss = 0 ; (6) The zero-curvature conditions, for the 2 di®erence of the two cross-derivatives, then and ¤ (d»!) = ¡iV d(2s ¡ 1=V ) : γ generate exactly the original Toda equations, The name I have used for Eq. (1) was ¯rst Eqs. (7), in the variables va, as desired, and used by Mikhail Saveliev,6 and also Kanehisa expected. Moreover, if one adds the two 7 a Takasaki and T. Takebe, emphasizing their cross-derivatives, inserts the form for b;xy understanding of its relationship to the alge- from the Toda equations, and changes to bra of all area-preserving di®eomorphisms of the new, translated pseudopotentials, `k ´ a 2-surface. Saveliev and Vershik used8 this wk ¡ wk+1 + ak+1 + bk, then these new de- equation as a non-trivial example of the use of pendent variables are also required to satisfy their development of continuum Lie algebras, the Toda equations, although for sl(n), since it having the symmetry group which was a there are only n ¡ 1 of them: limit of An as n went to +1. Takasaki and a a¡1 a+1 ¡ ¢a b `a = 2e` ¡e` ¡ e` = e` ; Takebe created a (double) hierarchy of equa- ;xy Kn¡1 b tions connected with this equation, analogous a; b = 1; : : : ; n ¡ 1 : (9) PlebTalkLatex: submitted to World Scienti¯c on January 16, 2009 2 For Publisher's use 2 The continuous limit of the Toda pseudopotentials wa, we acquire the following lattice equations limiting forms for the \proposed" BÄacklund transformations: Following the straight-forward existence of (W ¡ A) = ¡@ e+(W +B) ; both soliton-type solutions and BÄacklund ;z s ¡(W ¡A) transformations of the Toda lattice equa- (W + B);z = ¡@se : (13) tions, we, earlier, studied limits of these equa- However, the integrability conditions of tions to the continuous case, with the intent these equations are not what was desired: of course that these limits would carry over V = ¡@ eV @ V = ¡@2eV ; (14) to the existence of BÄacklund transformations ;zz s s n s o V for our equation with three independent vari- and L;zz = ¡@s e @sL ; (15) ables. To accomplish the change from dis- L ´ 2W + B ¡ A: crete indices to functions of a (new) contin- uous variable, we begin with a new function, The ¯rst of these equations is of course V = V (z; z; s), that depends on a third con- what we expect; but the second is not. This tinuous variable, s, which varies, say, from 0 particular pair of equations is just the system 10 to ¯. We then superpose on these values for s of equations that LeBrun requires to de- a lattice of n points, a distance ± apart, ¯ll in termine his \weak heavens," which have only the space between the lattice points by tak- self-dual conformal curvature, and therefore ing the limit as n ! 1, with ¯ ¯xed, which a possibly non-zero matter tensor. There is is the same as taking the limit as ± ! 0, and quite a lot of interesting work on the complete following earlier work of Park,9 re-scale the resolution of this pair of equations; perhaps other continuous variables so that appropri- we ought to look at it as a system and \try ate di®erences of the exponentials of the va's again"? Nonetheless, it certainly does not will create second derivatives with respect to create a BÄacklund equation for the original s: problem.
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