Double Ramification Cycles and Integrable Hierarchies

Buryak, Alexandr

doi:10.1007/s00220-014-2235-2

Cited by 49 publications

(149 citation statements)

References 43 publications

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“…In particular, g α,d = g α,d dx, expressed in terms of the p-variables, coincides with the definition given in [Bur15a]. The system of local functionals g α,d , for α = 1, .…”

Section: The Double Ramification Hierarchysupporting

confidence: 56%

“…In this section we briefly recall the main definitions in [Bur15a]. 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: The Double Ramification Hierarchymentioning

confidence: 99%

“…In a recent paper, [Bur15a], one of the authors, inspired by ideas from symplectic field theory [EGH00], has introduced a new integrable hierarchy of PDEs associated to a given cohomological field theory. The construction makes use of the intersection numbers of the given cohomological field theory with the double ramification cycle, the top Chern class of the Hodge bundle and psi-classes on the moduli space of stable Riemann surfaces M g,n .…”

mentioning

confidence: 99%

“…In [Bur15a] the author further conjectures, guided by the examples of the trivial and the Hodge cohomological field theories (which give the KdV and the ILW hierarchies, respectively) that the double ramification hierarchy is Miura equivalent to the Dubrovin-Zhang hierarchy associated to the same cohomological field theory via the construction described, for instance, in [DZ05].…”

mentioning

confidence: 99%

“…In the second part we focus instead on the string solution of the double ramification hierarchy (see [Bur15a]) and prove that the divisor equation for the Hamiltonians implies the divisor equation for the string solution. Our main application of this fact is a proof of the Miura equivalence described above in the case of the Gromov-Witten theory of the complex projective line.…”

mentioning

confidence: 99%

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Recursion Relations for Double Ramification Hierarchies

Buryak

Rossi

2015

Commun. Math. Phys.

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Abstract. In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced in [Bur15a] using intersection theory of the double ramification cycle in the moduli space of stable curves. In particular, we prove a recursion formula that recovers the full hierarchy starting from just one of the Hamiltonians, the one associated to the first descendant of the unit of a cohomological field theory. Moreover, we introduce analogues of the topological recursion relations and the divisor equation both for the hamiltonian densities and for the string solution of the double ramification hierarchy. This machinery is very efficient and we apply it to various computations for the trivial and Hodge cohomological field theories, and for the r-spin Witten's classes. Moreover we prove the Miura equivalence between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for the Gromov-Witten theory of the complex projective line (extended Toda hierarchy).

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“…In particular, g α,d = g α,d dx, expressed in terms of the p-variables, coincides with the definition given in [Bur15a]. The system of local functionals g α,d , for α = 1, .…”

Section: The Double Ramification Hierarchysupporting

confidence: 56%

Section: The Double Ramification Hierarchymentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Recursion Relations for Double Ramification Hierarchies

Buryak

Rossi

2015

Commun. Math. Phys.

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Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

Gisonni

Грава

Ruzza

2020

Ann. Henri Poincaré

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We consider the Laguerre partition function and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven in Cunden et al. (Ann. Inst. Henri Poincaré D, to appear) that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers that can be explicitly computed from our formulae. As a second result, we identify the Laguerre partition function with only positive couplings and a special value of the parameter $$\alpha =-1/2$$ α = - 1 / 2 with the modified GUE partition function, which has recently been introduced in Dubrovin et al. (Hodge-GUE correspondence and the discrete KdV equation. arXiv:1612.02333) as a generating function for Hodge integrals. This identification provides a direct and new link between monotone Hurwitz numbers and Hodge integrals.

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Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case

Liu

Zhang

2013

Commun. Math. Phys.

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We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless KdV hierarchy starting from any of its infinitesimal deformations.

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Double Ramification Cycles and Integrable Hierarchies

Abstract: Abstract. It this paper we present a new construction of a hamiltonian hierarchy associated to a cohomological field theory. We conjecture that in the semisimple case our hierarchy is related to the Dubrovin-Zhang hierarchy by a Miura transformation and check it in several examples.

Cited by 49 publications

References 43 publications

Recursion Relations for Double Ramification Hierarchies

Recursion Relations for Double Ramification Hierarchies

Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case

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