Tau-structure for the Double Ramification Hierarchies

TL;DR

This paper advances the understanding of double ramification hierarchies by establishing tau-structure properties, relating them to Dubrovin-Zhang hierarchies through Miura transformations, and proving conjectures for various CohFTs and genus levels.

Contribution

It proves the tau-structure of the DR hierarchy, relates it to Dubrovin-Zhang hierarchies via Miura transformations, and verifies the conjecture for multiple CohFTs and genus cases.

Findings

DR hierarchy satisfies tau-symmetry.

Partition functions obey string, dilaton, and divisor equations.

Conjecture proven for several CohFTs and genus 1.

Abstract

In this paper we continue the study of the double ramification hierarchy of [Bur15]. After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton and divisor equations plus some important degree constraints). We then formulate a stronger version of the conjecture from [Bur15]: for any semisimple cohomological field theory, the Dubrovin-Zhang and double ramification hierarchies are related by a normal (i.e. preserving the tau-structure [DLYZ14]) Miura transformation which we completely identify in terms of the partition function of the CohFT. In fact, using only the partition functions, the conjecture can be formulated even in the non-semisimple case (where the Dubrovin-Zhang hierarchy is not defined). We then prove this conjecture for various CohFTs (trivial CohFT, Hodge class, Gromov-Witten theory of $\mathbb{CP}^1$, $3$-, $4$- and $5$-spin classes) and in genus $1$ for any semisimple CohFT. Finally we prove that the higher genus part of the DR hierarchy is basically trivial for the Gromov-Witten theory of smooth varieties with non-positive first Chern class and their analogue in Fan-Jarvis-Ruan-Witten quantum singularity theory [FJRW].