Deformations of the Riemann hierarchy and the geometry of $\overline{\mathcal{M}}_{g,n}$

Abstract

The Riemann hierarchy is the simplest example of rank one, ($1$+$1$)-dimensional integrable system of nonlinear evolutionary PDEs. It corresponds to the dispersionless limit of the Korteweg-de Vries hierarchy. In the language of formal variational calculus, we address the classification problem for deformations of the Riemann hierarchy satisfying different extra requirements (general deformations, deformations as systems of conservation laws, Hamiltonian deformations, and tau-symmetric deformations), under the natural group of coordinate transformations preserving each of those requirements. We present several results linking previous conjectures of Dubrovin-Liu-Yang-Zhang (for the tau-symmetric case) and of Arsie-Lorenzoni-Moro (for systems of conservation laws) to the double ramification hierarchy construction of integrable hierarchies from partial CohFTs and F-CohFTs. We prove that, if the conjectures are true, DR hierarchies of rank one are universal objects in the space of deformations of the Riemann hierarchy. We also prove a weaker version of the DLYZ conjecture and that the ALM conjecture implies (the main part of) the DLYZ conjecture. Finally we characterize those rank one F-CohFTs which give rise to Hamiltonian deformations of the Riemann hierarchy.