Recursion Relations for Double Ramification Hierarchies

Buryak, Alexandr; Rossi, Paolo

doi:10.1007/s00220-015-2535-1

Cited by 34 publications

(74 citation statements)

References 28 publications

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“…In this section we recall the main definitions and results from [Bur15,BR16a,BR16b]. The classical double ramification (DR) hierarchy is a system of commuting Hamiltonians on an infinite dimensional phase space that can be heuristically thought of as the loop space of a fixed vector space.…”

Section: Double Ramification Hierarchymentioning

confidence: 99%

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Integrable Systems of Double Ramification Type

Buryak

Dubrovin

Guéré

et al. 2019

International Mathematics Research Notices

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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.

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Section: Double Ramification Hierarchymentioning

confidence: 99%

“…Recursion for the qDR Hamiltonian densities. We recall some of the properties of the DR hierarchies, in particular a recursion equation, proven in [BR16a] for the classical case and in [BR16b] for the quantum case, allowing to recover all the Hamiltonian densities G α,p , α = 1, . .…”

Section: 2mentioning

confidence: 99%

Integrable Systems of Double Ramification Type

Buryak

Dubrovin

Guéré

et al. 2019

International Mathematics Research Notices

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“…In this section we briefly recall the main definitions from [Bur15] (see also [BR16a]). The double ramification hierarchy is a system of commuting Hamiltonians on an infinite dimensional phase space that can be heuristically thought of as the loop space of a fixed vector space.…”

Section: Double Ramification Hierarchymentioning

confidence: 99%

Tau-Structure for the Double Ramification Hierarchies

Buryak

Dubrovin

Guéré

et al. 2018

Commun. Math. Phys.

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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].

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“…We prove (7) by induction on m. It will be convenient for us to assume that the genus g(v) of a vertex v ∈ V(Γ) can be a rational number such that 2g(v) − 2 + n(v) > 0. So the total genus g = v∈V(Γ) g(v) can also be rational.…”

Section: Lemma 22mentioning

confidence: 99%

DR/DZ equivalence conjecture and tautological relations

2019

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In this paper we present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin-Zhang equivalence conjecture introduced in [3]. Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ramification cycle. We prove that both families behave the same way upon pullback and pushforward with respect to forgetting a marked point. We also prove that our conjectural relations are true in genus 0 and 1 and also when first pushed forward from M g,n+m to M g,n and then restricted to M g,n , for any g, n, m ≥ 0. Finally we show that, for semisimple CohFTs, the DR/DZ equivalence only depends on a subset of our relations, finite in each genus, which we prove for g ≤ 2. As an application we find a new formula for the class λ g as a linear combination of dual trees intersected with kappa and psi classes, and we check it for g ≤ 3.14H10; 37K10

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Recursion Relations for Double Ramification Hierarchies

Cited by 34 publications

References 28 publications

Integrable Systems of Double Ramification Type

Integrable Systems of Double Ramification Type

Tau-Structure for the Double Ramification Hierarchies

DR/DZ equivalence conjecture and tautological relations

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