Bi-Hamiltonian Recursion, Liu–Pandharipande Relations, and Vanishing Terms of the Second Dubrovin–Zhang Bracket

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