Bi-Hamiltonian recursion, Liu-Pandharipande relations, and vanishing terms of the second Dubrovin-Zhang bracket

TL;DR

This paper investigates the second Poisson bracket in the Dubrovin-Zhang hierarchy, proving rationality of its coefficients and the vanishing of certain non-polynomial terms, thus supporting a key conjecture in integrable systems.

Contribution

It provides a new proof of rationality of the second bracket's coefficients and confirms the vanishing of non-polynomial terms, advancing understanding of bi-Hamiltonian structures.

Findings

Coefficients of the second bracket are rational functions with prescribed singularities.

Terms with negative degree in the dispersion expansion vanish.

Supports the conjecture that the second bracket's coefficients are polynomial in the dispersion parameter.

Abstract

The Dubrovin-Zhang hierarchy is a Hamiltonian infinite-dimensional integrable system associated to a semi-simple cohomological field theory or, alternatively, to a semi-simple Dubrovin-Frobenius manifold. Under an extra assumption of homogeneity, Dubrovin and Zhang conjectured that there exists a second Poisson bracket that endows their hierarchy with a bi-Hamiltonian structure. More precisely, they gave a construction for the second bracket, but the polynomiality of its coefficients in the dispersion parameter expansion is yet to be proved. In this paper we use the bi-Hamiltonian recursion and a set of relations in the tautological rings of the moduli spaces of curves derived by Liu and Pandharipande in order to analyze the second Poisson bracket of Dubrovin and Zhang. We give a new proof of a theorem of Dubrovin and Zhang that the coefficients of the dispersion parameter expansion of the second bracket are rational functions with prescribed singularities. We also prove that all terms in the expansion of the second bracket in the dispersion parameter that cannot be realized by polynomials because they have negative degree do vanish, thus partly confirming the conjecture of Dubrovin and Zhang.