Abstract
A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda equation, is shown to have a Lax formalism and an infinite hierarchy of higher flows. The Lax formalism is very similar to the case of the self-dual vacuum Einstein equation and its hyper-Kähler version, however now based upon a symplectic structure on a cylinderS 1×R. An analogue of the Toda lattice tau function is introduced. The existence of hidden SDiff(2) symmetries are derived from a Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function turn out to have commutator anomalies, hence give a representation of a central extension of the SDiff(2) algebra.
Similar content being viewed by others
References
Saveliev, M. V. and Vershik, A. M., Continual analogues of contragredient Lie algebras,Comm. Math. Phys. 126, 367–378 (1989).
Bakas, I., The structure of theW ∞ algebra.Comm. Math. Phys. 134, 487–508 (1990).
Park, Q-Han, Extended conformal symmetries in real heavens,Phys. Lett. 236B, 429–432 (1990).
Boyer, C. and Finley, J. D., Killing vectors in self-dual, Euclidean Einstein spaces,J. Math. Phys. 23, 1126–1128 (1982).
Gegenberg, J. D. and Das, A., Stationary Riemannian space-times with self-dual curvature,Gen. Relativity Gravitation 16, 817–829 (1984).
Hitchin, N. J., Complex manifolds and Einstein's equations, in H. D.Doebner and T.Weber (eds.),Twistor Geometry and Non-linear Systems, Lecture Notes in Mathematics vol. 970, Springer-Verlag, New York, 1982.
Jones, P. E. and Tod, K. P., Minitwistor spaces and Einstein-Weyl spaces,Classical Quantum Gravity 2, 565–577 (1985).
Ward, R. S., Einstein-Weyl spaces and SU(∞) Toda fieldsClassical Quantum Gravity 7, L95-L98 (1990).
LeBrun, C., Explicit self-dual metrics onCP 2 # ⋯ #CP 2,J. Diff. Geom. (to appear).
Sato, M. and Sato, Y., Soliton equations as dynamical systems in an infinite dimension Grassmann manifold, in P. D.Lax, H.Fujita and G.Strang (eds.),Nonlinear Partial Differential Equations in Applied Sciences, North-Holland, Amsterdam, and Kinokuniya, Tokyo, 1982.
Date, E., Kashiwara, M., Jimbo, M. and Miwa, T., Transformation groups for soliton equations, in M.Jimbo and T.Miwa (eds.),Nonlinear Integrble Systems-Classical Theory and Quantum Theory, World Scientific, Singapore, 1983.
Ueno, K. and Takasaki, K., Toda lattice hierarchy, inGroup Representations and Systems of Differential Equations, Advanced Studies in Pure Mathematics vol. 4, Kinokuniya, Tokyo, 1984.
Takebe, T., Toda lattice hierarchy and conservation laws,Comm. Math. Phys. 129, 281–318 (1990).
Takasaki, K., Symmetries of hyper-Kähler (or Poisson gauge field) hierarchy,J. Math. Phys. 31, 1877–1888 (1990).
Kashaev, R. M., Saveliev, M. V., Savelieva, S. A., and Vershik, A. M., On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds, Institute for High Energy Physics preprint 90-I (1990).
Golenisheva-Kutuzova, M. I. and Reiman, A. G., Integrable equations related to Poisson algebras,Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 169, 44 (1988) (in Russian).
Krichever I. M., The dispersionless Lax equations and topological minimal models, preprint (1991).
Penrose, R., Nonlinear gravitons and curved twistor theory,Gen. Relativity Gravitation 7, 31–52 (1976).
Park, Q-Han, Self-dual Yang-Mills (+ gravity) as a 2D sigma model,Phys. Lett. B257, 105–110 (1991).
Takasaki, K., Differential algebras andD-modules in super Toda lattice hierarchy,Lett. Math. Phys. 19, 229–236 (1990).
Arakelyan, T. A. and Savvidy, G. K., Cocycles of area-preserving diffeomorphisms and anomalies in the theory of relativistic surfaces,Phys. Lett. 214B, 350–356 (1988).
Bars, I., Pope, C. N., and Sezgin, E., Central extensions of area preserving membrane algebras,Phys. Lett. 210B, 85–91 (1988).
Floratos, F. G. and Iliopoulos, J., A note on the classical symmetries of the closed bosonic membranes,Phys. Lett. 201B, 237–240 (1988).
Hoppe, J., Diff A T' and the curvature of some infinite dimensional manifolds,Phys. Lett. 215B, 706–710 (1988).