Integrable Systems of Double Ramification Type

Abstract

In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the first author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus 1 quantum correction and, as an application, compute completely the quantization of the 3-and 4-KdV hierarchies (the DR hierarchies for Witten's 3-and 4-spin theories). We then focus on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometric origin, we use it to characterize and construct a new family of quantum and classical integrable systems which we call of double ramification type, as they satisfy all of the main properties of the DR hierarchy. In the second part, we obtain new insight towards the Miura equivalence conjecture between the DR and Dubrovin-Zhang hierarchies, via a geometric interpretation of the correlators forming the double ramification tau-function. We then show that the candidate Miura transformation between the DR and DZ hierarchies (which we uniquely identified in our previous paper) indeed turns the Dubrovin-Zhang Poisson structure into the standard form. Eventually, we focus on integrable hierarchies associated with rank-1 cohomological field theories and their deformations, and we prove the DR/DZ equivalence conjecture up to genus 5 in this context.

Simple Lax Description of the ILW Hierarchy

“…Here and in what follows we will use the formal loop space formalism in the notations of [3]. The Hamiltonian structure of the ILW equation (1.1) is given by the Hamiltonian…”

“…· dξ represents the principal value integral. This equation can be rewritten in the formal loop space formalism (see for instance [3], from which we borrow notations for the rest of the paper) as Hamiltonian structure, the conserved densities generate an infinite number of commuting flows. However the explicit relation between the Lax representation of the ILW equation, its higher flows and the Hamiltonian structure was never, to the best of our knowledge, clarified in the literature.…”

DR/DZ equivalence conjecture and tautological relations

“…The motivation for this conjecture comes from the study of the double ramification (DR) hierarchy, an integrable system of Hamiltonian PDEs associated to a CohFT and involving the geometry of the DR cycle, introduced by the first author in [1] and further studied in [7,8,5,3,4] (see also [2,23] for a review). In [3], sharpening a conjecture from [1], it was conjectured that (the logarithm of) the tau-function of (a particular solution of) the DR hierarchy coincides with the reduced potential of the CohFT.…”

Integrability, Quantization and Moduli Spaces of Curves

“…Its properties, quantization and relation with the DZ hierarchy were studied and clarified in the series of joint papers [2,3,5,8,9], partly guided by our previous investigations of the classical and quantum integrable systems arising in SFT [19,29,30,31,32,33].…”

“…After a self contained introduction to the language of integrable systems in the formal loop space and the needed notions from the geometry of the moduli space of stable curves we will explain the double ramification hierarchy construction and present its main features, with an accent on the quantization procedure, concluding with a list of examples worked out in detail. This paper does not contain new results with respect to the series of papers [2,3,5,8,9]. It is however a complete reorganization and, in part, a rephrasing of those results with the aim of showcasing the power of our methods and making them more accessible to the mathematical physics community.…”