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SDiff(2) Toda equation -- hierarchy, $τ$ function, and symmetries

Kanehisa Takasaki·Takashi Takebe
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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 and the group SDiff(2) of area preserving diffeomorphisms on a cylinder $S^1 \times \R$. 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.

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Kanehisa Takasaki, Takashi Takebe. 1991-12-17. SDiff(2) Toda equation -- hierarchy, $τ$ function, and symmetries. https://doi.org/10.1007/bf01885498

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