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arXiv · 1104.2722

On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket

A. Buryak·H. Posthuma·S. Shadrin
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Abstract

In our recent paper we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of PDE's associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the first Poisson structure of their hierarchy. Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental-Y.-P. Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we reprove it using a deformation formula for weak quasi-Miura transformation that relates our hierarchy of PDE's with its dispersionless limit.

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A. Buryak, H. Posthuma, S. Shadrin. 2011-04-14. On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket. https://doi.org/10.1016/j.geomphys.2012.03.006

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