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.
Highlights
Dubrovin–Frobenius manifolds were first introduced in [Dub96] as a way to study homogeneous solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) associativity equations [Wit90,DVV91] in a coordinate-free way
In this work the relation between Dubrovin–Frobenius manifolds, topological field theories (TFTs) and integrable systems is first explored: namely, one can construct two flat metrics related to a Dubrovin– Frobenius manifold, which give a pair of compatible Poisson brackets (P, K ) of hydrodynamic type
In [DZ01], Dubrovin and Zhang further explore the relationship between Dubrovin– Frobenius manifolds and integrable systems and deform this hierarchy via a quasi-Miura transformation wα = vα +
Summary
Dubrovin–Frobenius manifolds were first introduced in [Dub96] as a way to study homogeneous solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) associativity equations [Wit90,DVV91] in a coordinate-free way. G=1 given by weighted homogeneous differential rational functions Qαg to obtain the full dispersive hierarchy, which is known as the Dubrovin–Zhang (DZ) hierarchy They conjecture in [DZ01] that the transformed equations, Hamiltonians and brackets are differential polynomials in the coordinates wα. We start with a conformal semi-simple cohomological field theory, the construction of Dubrovin applied to the underlying Dubrovin–Frobenius manifold gives the second Poisson bracket in the dispersionless limit, and the quasi-Miura transformation (2) produces a possibly singular Poisson structure. As a source of suitable tautological relations we use the work of Liu and Pandharipande [LP11] The relations that they derive there appear to be exactly enough to prove the vanishing of all terms in the second Dubrovin–Zhang bracket whose standard degree is negative. The dimensional inequalities of the Liu–Pandharipande relations match exactly the standard degree count for the terms of the second Dubrovin– Zhang bracket in the equations that we derive from the bi-Hamiltonian recursion, so the Liu–Pandharipande relations say nothing about the non-negative standard degree terms of the second bracket
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