Tau-Structure for the Double Ramification Hierarchies

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 [DLYZ16]) 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 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 [FJR13].

DR/DZ equivalence conjecture and tautological relations

“…In this section we explain the relation between the above Conjecture 2.5 and the strong double ramification/Dubrovin-Zhang hierarchies equivalence conjecture from [3]. After recalling the main notions, we prove in particular how the first implies the second.…”

“…Therefore, τ di (e αi ) DR g = τ di (e αi ) red g , if d i ≤ 2g. Both potentials F DR and F red satisfy the assumptions of Lemma 5.1 (see [3]). Therefore, the lemma implies that F DR = F red .…”

“…In light of these two results the following conjecture was formulated in [3]. Conjecture 3.3 (Strong DR/DZ equivalence) Consider a semisimple cohomological field theory and the associated DZ and DR hierarchies.…”

Integrability, Quantization and Moduli Spaces of Curves

“…Finally, one of the main parts of this project is the proof of the conjecture (originally proposed in a weaker form by A. Buryak) that the DZ and DR hierarchies for a semisimple CohFT are in fact equivalent under a change of coordinates that preserves their tau-symmetry property (a normal Miura transformation), and which we completely identified in [2]. While the general proof of such conjecture is the object of an ongoing work, we managed to show its validity in a number of examples and classes of interesting special cases.…”

“…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].…”