Towards a bihamiltonian structure for the double ramification hierarchy

Abstract

We propose a remarkably simple and explicit conjectural formula for a bihamiltonian structure of the double ramification hierarchy corresponding to an arbitrary homogeneous cohomological field theory. Various checks are presented to support the conjecture. ContentsIntroduction 1 Organization of the paper 3 Acknowledgements 3 1. Double ramification hierarchy and the main conjecture 3 1.1. Cohomological field theories 3 1.2. Double ramification hierarchy 5 1.3. Bihamiltonian structure for the double ramification hierarchy 9 2. Evidence I: checks for a general CohFT 10 2.1. Genus 0 10 2.2. The first equation of the bihamiltonian recursion 12 2.3. The Schouten-Nijenhuis bracket of two skew-symmetric operators 12 3. Evidence II: central invariants 15 3.1. Miura transformations 15 3.2. The central invariants of a pair of compatible Poisson operators 16 3.3. The central invariants of the pair (K 1 , K 2 ) 17 3.4. Relation with the Dubrovin-Zhang hierarchy 18 4. Evidence III: explicit examples 18 4.1. The trivial cohomological field theory 18 4.2. Higher r-spin theories 19 4.3. The Gromov-Witten theory of CP 1 24 References 26

Flat F-manifolds, F-CohFTs, and integrable hierarchies

“…Proof. For part (1), using the notations from paper [BRS20] let us note that P α β,d = η αµ δg β,d δu µ . Therefore, we have to check that η βµ K αµ 2 = R α β .…”

“…In [BRS20], the authors presented an explicit conjectural formula for a bihamiltonian structure of the DR hierarchy corresponding to a homogeneous CohFT. This in particular gives a recursion of certain type, called a bihamiltonian recursion, expressing the flows…”

“…Following [BRS20], we associate with a differential polynomial f ∈ A a sequence of differential operators indexed by α = 1, . .…”