Deformations of the Riemann hierarchy and the geometry of $\overline{\mathcal{M}}_{g,n}$
Alexandr Buryak, Paolo Rossi
TL;DR
This work investigates deformations of the Riemann hierarchy through the lens of double ramification (DR) hierarchies derived from rank-one (partial) CohFTs and F-CohFTs. It formalizes how tau-symmetric deformations and conservation-law deformations can be encoded as DR hierarchies, and demonstrates that, under conjectures by DLYZ and ALM, DR hierarchies are universal representatives of these deformation spaces. The paper proves a weaker form of DLYZ, shows that ALM implies the main part of DLYZ, and provides explicit DR-constructions realizing standard and normal forms; it also characterizes when DR hierarchies are Hamiltonian. Overall, the results connect deformation theory of the Riemann hierarchy with the geometry of moduli spaces via DR hierarchies, offering a pathway to geometric origins for ALM/DLYZ parameters and suggesting directions for proving the conjectures in full generality.
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.