Recursion relations for Double Ramification Hierarchies

TL;DR

This paper develops recursion formulas and relations for the double ramification hierarchy, an integrable system in algebraic geometry, enabling efficient computations and establishing equivalences with known hierarchies.

Contribution

It introduces a recursion formula for the hierarchy, analogues of topological recursion and divisor equations, and proves Miura equivalence with the Dubrovin-Zhang hierarchy.

Findings

Recursion formula recovers the full hierarchy from a single Hamiltonian.

Analogues of topological recursion and divisor equations are established.

Proved Miura equivalence with the Dubrovin-Zhang hierarchy for Gromov-Witten theory.

Abstract

In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced in [Bur15] using intersection theory of the double ramification cycle in the moduli space of stable curves. In particular, we prove a recursion formula that recovers the full hierarchy starting from just one of the Hamiltonians, the one associated to the first descendant of the unit of a cohomological field theory. Moreover, we introduce analogues of the topological recursion relations and the divisor equation both for the hamiltonian densities and for the string solution of the double ramification hierarchy. This machinery is very efficient and we apply it to various computations for the trivial and Hodge cohomological field theories, and for the $r$-spin Witten's classes. Moreover we prove the Miura equivalence between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for the Gromov-Witten theory of the complex projective line (extended Toda hierarchy).