### ?

## Double ramification cycles and integrable hierarchies

Communications in Mathematical Physics. 2015. Vol. 336. No. 3. P. 1085-1107.

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.

Buryak A., Dubrovin B., Guere J. et al., Communications in Mathematical Physics 2018 Vol. 363 No. 1 P. 191-260

In this paper we continue the study of the double ramification hierarchy introduced by the first author. 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 ...

Added: September 27, 2020

Buryak A., Posthuma H., Shadrin S., Journal of Differential Geometry 2012 Vol. 92 No. 1 P. 153-185

We define a hierarchy of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the equations, the Hamiltonians, and the bracket are weighted-homogeneous polynomials in the derivatives of the dependent variables with respect to the space variable. In the particular ...

Added: September 30, 2020

Buryak A., Rossi P., Letters in Mathematical Physics 2016 Vol. 106 No. 3 P. 289-317

In this paper we define a quantization of the Double Ramification Hierarchies using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with ...

Added: September 28, 2020

Buryak A., Rossi P., Communications in Mathematical Physics 2016 Vol. 342 No. 2 P. 533-568

In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced by the first author 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 ...

Added: September 28, 2020

Buryak A., Posthuma H., Shadrin S., Journal of Geometry and Physics 2012 Vol. 62 No. 7 P. 1639-1651

In our recent paper we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of PDE's associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the first Poisson structure of their hierarchy. Our approach was based on a very ...

Added: September 30, 2020

Buryak A., Rossi P., Advances in Mathematics 2021 Vol. 386 No. 6 Article 107794

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's r-spin classes. They are parameterized by a phase space which has one extra dimension and in genus 0 they correspond to the extended r-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while ...

Added: October 29, 2021

Arsie A., Buryak A., Lorenzoni P. et al., Communications in Mathematical Physics 2021 Vol. 388 P. 291-328

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple ...

Added: October 29, 2021

Kruglinskaya O., Marshakov A., Journal of Physics A: Mathematical and Theoretical 2015 P. 1-30

We extend the construction of the relativistic Toda chains as integrable systems on the Poisson submanifolds in Lie groups beyond the case of A-series. For the simply-laced case this is just a direct generalization of the well-known relativistic Toda chains, and we construct explicitly the set of the Ad-invariant integrals of motion on symplectic leaves, ...

Added: December 15, 2014

Marshakov A., Fock V., / Cornell University. Series math "arxiv.org". 2014.

We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups PGLˆ(N), which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups (Wˆ×Wˆ)♯. Their phase spaces admit cluster coordinates, whereas the integrals of motion are cluster functions. We show, that this class of integrable systems coincides with the ...

Added: October 29, 2014

Buryak A., Clader E., Tessler R., Journal of Geometry and Physics 2019 Vol. 137 P. 132-153

We study a generalization of genus-zero r-spin theory in which exactly one insertion has a negative-one twist, which we refer to as the "closed extended" theory, and which is closely related to the open r-spin theory of Riemann surfaces with boundary. We prove that the generating function of genus-zero closed extended intersection numbers coincides with the ...

Added: September 27, 2020

Buryak A., Mathematical Research Letters 2016 Vol. 23 No. 3 P. 675-683

In a previous paper we proved that after a simple transformation the generating series of the linear Hodge integrals on the moduli space of stable curves satisfies the hierarchy of the Intermediate Long Wave equation. In this paper we present a much shorter proof of this fact. Our new proof is based on an explicit ...

Added: September 28, 2020

Buryak A., Успехи математических наук 2017 Т. 72 № 5(437) С. 63-112

Обзор посвящён обширному классу систем уравнений в частных производных, которые, с одной стороны, возникают в классических задачах математической физики, а с другой стороны, являются эффективным инструментом для описания перечислительных инвариантов в алгебраической геометрии. Особое внимание уделено новым подходам к этим системам, в частности подходу, предложенному в недавней работе автора. ...

Added: September 27, 2020

Marshall I., International Mathematics Research Notices 2015 Vol. 18 P. 8925-8958

A Poisson structure is defined on the space {\mathcal {W}} of twisted polygons in {\mathbb {R}}^{\nu }. Poisson reductions with respect to two Poisson group actions on {\mathcal {W}} are described. The \nu =2 and \nu =3 cases are discussed in detail. Amongst the Poisson structures arising in examples are to be found the lattice ...

Added: November 28, 2014

Buryak A., Shadrin S., Spitz L. et al., American Journal of Mathematics 2015 Vol. 137 No. 3 P. 699-737

DR-cycles are certain cycles on the moduli space of curves. Intuitively, they parametrize curves that allow a map to the complex projective line with some specified ramification profile over two points. They are known to be tautological classes, but in general there is no known expression in terms of standard tautological classes. In this paper, ...

Added: September 30, 2020

Nirov Khazret S., Razumov A. V., Journal of Geometry and Physics 2017 Vol. 112 P. 1-28

A detailed construction of the universal integrability objects related to the integrable
systems associated with the quantum loop algebra Uq(L(sl2)) is given. The full proof of the
functional relations in the form independent of the representation of the quantum loop
algebra on the quantum space is presented. The case of the general gradation and general
twisting is treated. The ...

Added: January 29, 2018

Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2019 No. 074003 P. 1-22

We establish the exact laws of large numbers for two time additive quantities in the raise and peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related stationary state correlation functions then follow. The proof is based on the ...

Added: October 8, 2019

Buryak A., Dubrovin B., Guere J. et al., International Mathematics Research Notices 2020 Vol. 2020 No. 24 P. 10381-10446

In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the 1st 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 ...

Added: April 21, 2020

Kazaryan M., Zvonkine D., Lando S., International Mathematics Research Notices 2018 No. 22 P. 6817-6843

We consider families of curve-to-curve maps that have no singularities except those of genus 0 stable maps and that satisfy a versality condition at each singularity. We provide a universal expression for the cohomology class Poincaré dual to the locus of any given singularity. Our expressions hold for any family of curve-to-curve maps satisfying the ...

Added: July 10, 2017

Khoroshkin S. M., Tsuboi Z., Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1-11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

Dunin-Barkowski P., Norbury P., Orantin N. et al., Journal of the Institute of Mathematics of Jussieu 2019 Vol. 18 No. 3 P. 449-497

We apply the spectral curve topological recursion to Dubrovin's universal Landau-Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold. ...

Added: December 22, 2016

Buryak A., Moscow Mathematical Journal 2017 Vol. 17 No. 1 P. 1-13

In this paper, using the formula for the integrals of the psi-classes over the double ramification cycles found by S. Shadrin, L. Spitz, D. Zvonkine and the author, we derive a new explicit formula for the n-point function of the intersection numbers on the moduli space of curves. ...

Added: September 27, 2020

Derbyshev A. E., Povolotsky A. M., Priezzhev V. B., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2015 Vol. 91 P. 022125

The generalized totally asymmetric exclusion process (TASEP) [J. Stat. Mech. (2012) P05014] is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as ...

Added: February 19, 2015

Kharchev S., Levin A., Olshanetsky M. et al., Journal of Mathematical Physics 2018 Vol. 59 No. 103509 P. 1-36

We define the quasi-compact Higgs G -bundles over singular curves introduced in our previous paper for the Lie group SL(N). The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of G at marked points of the curves. We demonstrate that in particular cases, this construction leads ...

Added: October 20, 2018

Krichever I. M., Функциональный анализ и его приложения 2012 Т. 46 № 2 С. 37-51

Using meromorphic differentials with real periods, we prove Arbarello's conjecture that any compact complex cycle of dimension g−n in the moduli space M_g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n. ...

Added: April 17, 2014