Towards a description of the double ramification hierarchy for Witten's $r$-spin class

TL;DR

This paper investigates the double ramification hierarchy linked to Witten's r-spin classes, providing explicit computations and proving its Miura equivalence to Dubrovin--Zhang hierarchies, leading to quantizations of Gelfand--Dickey hierarchies for r=3,4,5.

Contribution

It introduces an effective method for computing the double ramification hierarchy for Witten's r-spin classes and establishes its equivalence to known hierarchies, enabling hierarchy quantization.

Findings

Explicit computations for r=3,4,5 cases.

Proof of Miura equivalence to Dubrovin--Zhang hierarchy.

Quantization of Gelfand--Dickey hierarchy for r=3,4,5.

Abstract

The double ramification hierarchy is a new integrable hierarchy of hamiltonian PDEs introduced recently by the first author. It is associated to an arbitrary given cohomological field theory. In this paper we study the double ramification hierarchy associated to the cohomological field theory formed by Witten's $r$-spin classes. Using the formula for the product of the top Chern class of the Hodge bundle with Witten's class, found by the second author, we present an effective method for a computation of the double ramification hierarchy. We do explicit computations for $r=3,4,5$ and prove that the double ramification hierarchy is Miura equivalent to the corresponding Dubrovin--Zhang hierarchy. As an application, this result together with a recent work of the first author with Paolo Rossi gives a quantization of the $r$-th Gelfand--Dickey hierarchy for $r=3,4,5$.