Towards a description of the double ramification hierarchy for Witten's r-spin class

Buryak, Alexandr; Guéré, Jérémy

doi:10.1016/j.matpur.2016.03.013

Cited by 14 publications

(29 citation statements)

References 44 publications

(113 reference statements)

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“…As an application of the genus-1 computation of the previous section we compute the quantum DR hierarchy of Witten's r-spin class, for r = 3, 4. In light of the results of [BG15,BDGR16], which establish that the DR hierarchy in these cases coincides with the DZ hierarchy once we pass to the normal coordinates u α = η αµ δg µ,0 δu 1 (which, for r = 4 also changes the form of the Hamiltonian operator, see [BDGR16]), and the fact that the DZ hierarchies for the 3-and 4-spin theories correspond in turn to the 3-and 4-KdV Gelfand-Dickey hierarchies [DZ05,Dic03], we obtain this way a quantization for such two well-known integrable systems.…”

Section: Quantum Double Ramification Hierarchy In Genusmentioning

confidence: 95%

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: Quantum Double Ramification Hierarchy In Genusmentioning

confidence: 95%

Integrable Systems of Double Ramification Type

Buryak

Dubrovin

Guéré

et al. 2019

International Mathematics Research Notices

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“…, a n+g ) = g!a 2 n+1 · · · a 2 n+g [M g,n ]. (5) It is also useful to remember that (see e.g. [19])…”

Section: Double Ramification Cycle and The Definition Of The A-classmentioning

confidence: 99%

“…The argument from the proof of Proposition 5.2 in [5] shows that using this system together with the string and the dilaton equations for F one can uniquely reconstruct the whole solution (w sol ) α starting from the dispersionless part (w sol ) α | ε=0 . After that using the string and the dilaton equations it is easy to reconstruct the whole function F .…”

Section: Restricted Set Of Relationsmentioning

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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“…Let r ≥ 3 and consider the cohomological field theory formed by Witten's r-spin classes (see Section 3.2 or, e.g., [4]). In this case we have V = e i i=1,...,r−1 and the metric is given by η αβ = δ α+β,r .…”

Section: Gelfand-dickeymentioning

confidence: 99%

“…For the 5-spin theory we content ourselves to write the classical Hamiltonian (see [4]), The relation with the Gelfand-Dickey hierarchies is described as follows. First let us recall the definition of the Gelfand-Dickey hierarchies.…”

Section: Gelfand-dickeymentioning

confidence: 99%

Integrability, Quantization and Moduli Spaces of Curves

Rossi¹

2017

SIGMA

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Abstract. This paper has the purpose of presenting in an organic way a new approach to integrable (1 + 1)-dimensional field systems and their systematic quantization emerging from intersection theory of the moduli space of stable algebraic curves and, in particular, cohomological field theories, Hodge classes and double ramification cycles. This methods are alternative to the traditional Witten-Kontsevich framework and its generalizations by Dubrovin and Zhang and, among other advantages, have the merit of encompassing quantum integrable systems. Most of this material originates from an ongoing collaboration with A. Buryak, B. Dubrovin and J. Guéré.

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Towards a description of the double ramification hierarchy for Witten's r-spin class

Cited by 14 publications

References 44 publications

Integrable Systems of Double Ramification Type

Integrable Systems of Double Ramification Type

DR/DZ equivalence conjecture and tautological relations

Integrability, Quantization and Moduli Spaces of Curves

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