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A. Buryak

Publications and source records attributed to A. Buryak.

At least 19 recordsLinked to original sources
arXiv2020

Open intersection numbers and the wave function of the KdV hierarchy

Recently R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of the intersection numbers is a specific solution of a system of PDEs, that they called the open KdV equations. In this paper we show that the open KdV equations are closely related to the equations for the wave function of the KdV hierarchy. This allows us to give an explicit formula for the specific solution in terms of Witten's generating series of the intersection numbers on the moduli space of stable curves.

math-ph↗
arXiv2018

Tau-structure for the Double Ramification Hierarchies

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 [DLYZ14]) 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 $\mathbb{CP}^1$, $3$-, $4$- 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 [FJRW].

math-ph↗
arXiv2016

ILW equation for the Hodge integrals revisited

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 formula for the one-point linear Hodge integrals that was found independently by Faber, Pandharipande and Ekedahl, Lando, Shapiro, Vainshtein.

math-ph↗
arXiv2015

Double ramification cycles and quantum integrable systems

In this paper we define a quantization of the Double Ramification Hierarchies of [Bur15b] and [BR14], 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 the first descendant of the unit of the cohomological field theory only. We study various examples which provide, in very explicit form, new $(1+1)$-dimensional integrable quantum field theories whose classical limits are well-known integrable hierarchies such as KdV, Intermediate Long Wave, Extended Toda, etc. Finally we prove polynomiality in the ramification multiplicities of the integral of any tautological class over the double ramification cycle.

math-ph↗
arXiv2015

Recursion relations for Double Ramification Hierarchies

In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced in [Bur15] 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).

math-ph↗
arXiv2015

Dubrovin-Zhang hierarchy for the Hodge integrals

In this paper we prove that the generating series of the Hodge integrals over the moduli space of stable curves is a solution of a certain deformation of the KdV hierarchy. This hierarchy is constructed in the framework of the Dubrovin-Zhang theory of the hierarchies of the topological type. It occurs that our deformation of the KdV hierarchy is closely related to the hierarchy of the Intermediate Long Wave equation.

math-ph↗
arXiv2015

Equivalence of the open KdV and the open Virasoro equations for the moduli space of Riemann surfaces with boundary

In a recent paper R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. The authors conjectured KdV and Virasoro type equations that completely determine all intersection numbers. In this paper we study these equations in detail. In particular, we prove that the KdV and the Virasoro type equations for the intersection numbers on the moduli space of Riemann surfaces with boundary are equivalent.

math.AG↗
arXiv2015

The hypergeometric functions of the Faber-Zagier and Pixton relations

The relations in the tautological ring of the moduli space M_g of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space of stable pointed curves by Pixton in 2012 are based upon two hypergeometric series A and B. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius structures associated to 3-spin curves and to CP1). The series A and B also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of A and B starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the 19th century). Several open questions are proposed.

math.AG↗
arXiv2015

Double ramification cycles and integrable hierarchies

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.

math-ph↗
arXiv2014

The classes of the quasihomogeneous Hilbert schemes of points on the plane

In this paper we give a formula for the classes (in the Grothendieck ring of complex quasi-projective varieties) of irreducible components of $(1,k)$-quasi-homogeneous Hilbert schemes of points on the plane. We find a new simple geometric interpretation of the $q,t$-Catalan numbers. Finally, we investigate a connection between $(1,k)$-quasi-homogeneous Hilbert schemes and homogeneous nested Hilbert schemes.

math.AG↗
arXiv2014

Generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes

In this paper we prove that the generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We also compute the generating series of the numbers of quasihomogeneous components in a moduli space of sheaves on the projective plane. The answer is given in terms of characters of the affine Lie algebra $\hat{sl}_m$.

math.AG↗
arXiv2014

Homogeneous components in the moduli space of sheaves and Virasoro characters

The moduli space $\mathcal M(r,n)$ of framed torsion free sheaves on the projective plane with rank $r$ and second Chern class equal to $n$ has the natural action of the $(r+2)$-dimensional torus. In this paper, we look at the fixed point set of different one-dimensional subtori in this torus. We prove that in the homogeneous case the generating series of the numbers of the irreducible components has a beautiful decomposition into an infinite product. In the case of odd $r$ these infinite products coincide with certain Virasoro characters. We also propose a conjecture in a general quasihomogeneous case.

math.AG↗
arXiv2012

Integrals of psi-classes over double ramification cycles

DR-cycles are certain cycles on the moduli space of curves. Intuitively, they parametrize curves that allow a map to \mathbb{P}^1 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, we compute the intersection numbers of those DR-cycles with any monomials in psi-classes when this intersection is zero-dimensional.

math.AG↗
arXiv2011

On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket

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 involved computation of a deformation formula for the bracket with respect to the Givental-Y.-P. Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we reprove it using a deformation formula for weak quasi-Miura transformation that relates our hierarchy of PDE's with its dispersionless limit.

math-ph↗
arXiv2010

A polynomial bracket for the Dubrovin--Zhang hierarchies

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 case of a conformal (homogeneous) Frobenius structure, our hierarchy coincides with the Dubrovin-Zhang hierarchy that is canonically associated to the underlying Frobenius structure. Therefore, our approach allows to prove the polynomiality of the equations, Hamiltonians and one of the Poisson brackets of these hierarchies, as conjectured by Dubrovin and Zhang.

math-ph↗