On a class of integrable deformations of the integrable hierarchy of topological type associated to a semisimple Frobenius manifold
Si-Qi Liu, Paolo Rossi, Di Yang, Youjin Zhang
TL;DR
The paper develops a general framework for integrable deformations of the Dubrovin–Zhang hierarchy (the hierarchy of topological type) associated to semisimple Frobenius manifolds, and proves these deformations can possess polynomial tau-structures. It introduces the $L$-deformed partition function $Z_L=e^{sL}Z$ and establishes a genus expansion $Z_L=\exp(\sum_{g\ge0} \epsilon^{2g-2}{\mathcal H}_g)$ with ${\mathcal H}_0={\mathcal F}_0$ and jet dependence up to ${\bf v}_{3g-2}$, providing recursive equations that uniquely determine ${\mathcal H}_g$. It then proves that these deformations admit a tau-structure (via two-point functions $\Omega$) and that the deformation is preserved under suitable Miura-type changes (Theorem ab). The paper further develops a Virasoro-like deformation program based on LYZZ's operators, showing universality in the one-dimensional case and giving explicit Sawada–Kotera-type examples, including a normalization that fixes coefficients for a standard form. Overall, the work identifies a universal deformation object in the 1D setting and connects the deformation theory to classical integrable hierarchies and topological field theory data.
Abstract
Given a semisimple Frobenius manifold, we construct a class of integrable deformations of its hierarchy of topological type. We show that these integrable deformations have polynomial tau-structures, and conjecture that for the one-dimensional Frobenius manifold they give a universal object for integrable deformations of the Riemann--Hopf hierarchy having a tau-structure.