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Boris Dubrovin

Publications and source records attributed to Boris Dubrovin.

At least 19 recordsLinked to original sources
arXiv2023

Simple Lie algebras, Drinfeld--Sokolov hierarchies, and multi-point correlation functions

For a simple Lie algebra $\mathfrak{g}$, we derive a simple algorithm for computing logarithmic derivatives of tau-functions of Drinfeld--Sokolov hierarchy of $\mathfrak{g}$-type in terms of $\mathfrak{g}$-valued resolvents. We show, for the topological solution to the lowest-weight-gauge Drinfeld--Sokolov hierarchy of $\mathfrak{g}$-type, the resolvents evaluated at zero satisfy the $\textit{topological ODE}$.

math-ph↗
arXiv2022

Affine Kac-Moody Algebras and Tau-Functions for the Drinfeld-Sokolov Hierarchies: the Matrix-Resolvent Method

For each affine Kac-Moody algebra $X_n^{(r)}$ of rank $\ell$, $r=1,2$, or $3$, and for every choice of a vertex $c_m$, $m=0,\dots,\ell$, of the corresponding Dynkin diagram, by using the matrix-resolvent method we define a gauge-invariant tau-structure for the associated Drinfeld-Sokolov hierarchy and give explicit formulas for generating series of logarithmic derivatives of the tau-function in terms of matrix resolvents, extending the results of [Mosc. Math. J. 21 (2021), 233-270, arXiv:1610.07534] with $r=1$ and $m=0$. For the case $r=1$ and $m=0$, we verify that the above-defined tau-structure agrees with the axioms of Hamiltonian tau-symmetry in the sense of [Adv. Math. 293 (2016), 382-435, arXiv:1409.4616] and [arXiv:math.DG/0108160].

math-ph↗
arXiv2022

Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality

For each of the simple Lie algebras $\mathfrak{g}=A_l$, $D_l$ or $E_6$, we show that the all-genera one-point FJRW invariants of $\mathfrak{g}$-type, after multiplication by suitable products of Pochhammer symbols, are the coefficients of an algebraic generating function and hence are integral. Moreover, we find that the all-genera invariants themselves coincide with the coefficients of the unique calibration of the Frobenius manifold of $\mathfrak{g}$-type evaluated at a special point. For the $A_4$ (5-spin) case we also find two other normalizations of the sequence that are again integral and of at most exponential growth, and hence conjecturally are the Taylor coefficients of some period functions.

math.AG↗
arXiv2021

On tau-functions for the KdV hierarchy

For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in this paper two new formulae for the generating series by introducing a pair of wave functions of the solution. Applications to the Witten--Kontsevich tau-function, to the generalized Brézin--Gross--Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lamé tau-function are also given.

math-ph↗
arXiv2020

Integrable systems of double ramification type

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.

math-ph↗
arXiv2020

Local Moduli of Semisimple Frobenius Coalescent Structures

We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities and mutual constraints are allowed in the definition of monodromy data, in view of their importance for conjectural relationships between Frobenius manifolds and derived categories. Detailed examples and applications are taken from singularity and quantum cohomology theories. We explicitly compute the monodromy data at points of the Maxwell Stratum of the A3-Frobenius manifold, as well as at the small quantum cohomology of the Grassmannian G(2,4). In the latter case, we analyse in details the action of the braid group on the monodromy data. This proves that these data can be expressed in terms of characteristic classes of mutations of Kapranov's exceptional 5-block collection, as conjectured by one of the authors.

math.DG↗
arXiv2020

Matrix resolvent and the discrete KdV hierarchy

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are then obtained, and applications to enumeration of ribbon graphs with even valencies and to the special cubic Hodge integrals are considered.

math-ph↗
arXiv2019

Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials

For the root systems of type $B_l, C_l$ and $D_l$, we generalize the result of \cite{DZ1998} by showing the existence of Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups that correspond to any vertex of the Dynkin diagram instead of a particular choice made in \cite{DZ1998}. It also depends on certain additional data. We also construct LG superpotentials for these Frobenius manifold structures.

math.DG↗
arXiv2019

Helix Structures in Quantum Cohomology of Fano Varieties

In this paper we consider a conjecture formulated by the second author in occasion of the 1998 ICM in Berlin (arXiv:math/9807034v2). This conjecture states the equivalence, for a Fano variety $X$, of the semisimplicity condition for the quantum cohomology $QH^\bullet(X)$ with the existence condition of full exceptional collections in the derived category of coherent sheaves $\mathcal D^b(X)$. Furthermore, in its quantitative formulation, the conjecture also prescribes an explicit relationship between the monodromy data of $QH^\bullet(X)$ and characteristic classes of both $X$ and objects of the exceptional collections. In this paper we reformulate a refinement of (arXiv:math/9807034v2), which corrects a previous ansatz (lecture of the second author at Strasbourg) for what concerns the conjectural expression of the central connection matrix. We clarify the precise relationship between the refined conjecture presented in this paper and $Γ$-conjecture II of S. Galkin, V. Golyshev and H. Iritani (arXiv:1404.6407v4, arXiv:1508.00719v3). Through an explicit computation of the monodromy data and a detailed analysis of the action of the braid group on both the monodromy data and the set of exceptional collections, we prove the validity of our refined conjecture for all complex Grassmannians $\mathbb G(r,k)$. From these results, it is outlined an explicit description of the "geography" of the exceptional collections realizable at points of the small quantum cohomology of Grassmannians, i.e. corresponding to the monodromy data at these points. In particular, it is proved that Kapranov's exceptional collection appears at points of the small quantum cohomology only for Grassmannians of small dimension (namely, less or equal than 2). Finally, a property of quasi-periodicity of the Stokes matrices of complex Grassmannians, along the locus of the small quantum cohomology, is described.

math.AG↗
arXiv2018

Hodge-GUE correspondence and the discrete KdV equation

We prove the conjectural relationship recently proposed in [9] between certain special cubic Hodge integrals of the Gopakumar--Mariño--Vafa type [17, 28] and GUE correlators, and the conjecture proposed in [7] that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy.

math-ph↗
arXiv2018

A Riemann-Hilbert Approach to the Heun Equation

We describe the close connection between the linear system for the sixth Painlevé equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the Riemann-Hilbert formalism can be used to construct explicit polynomial solutions of the Heun equation.

math.CA↗
arXiv2018

Algebraic spectral curves over $\mathbb Q$ and their tau-functions

Let $W(z)$ be a $n\times n$ matrix polynomial with rational coefficients. Denote $C$ the spectral curve $\det \left( w\cdot{\bf 1}-W(z)\right) =0$. Under some natural assumptions about the structure of $W(z)$ we prove that certain combinations of logarithmic derivatives of the Riemann theta-function of $C$ of an arbitrary order starting from the third one all take rational values at the point of the Jacobi variety $J(C)$ specified by the line bundle of eigenvectors of $W(z)$.

math.AG↗
arXiv2018

Approximating tau-functions by theta-functions

We prove that the logarithm of an arbitrary tau-function of the KdV hierarchy can be approximated, in the topology of graded formal series by the logarithmic expansions of hyperelliptic theta-functions of finite genus, up to at most quadratic terms. As an example we consider theta-functional approximations of the Witten--Kontsevich tau-function.

math-ph↗
arXiv2018

Gromov--Witten invariants of the Riemann sphere

A conjectural formula for the $k$-point generating function of Gromov--Witten invariants of the Riemann sphere for all genera and all degrees was proposed in \cite{DY2}. In this paper, we give a proof of this formula together with an explicit analytic (as opposed to formal) expression for the corresponding matrix resolvent. We also give a formula for the $k$-point function as a sum of $(k-1)!$ products of hypergeometric functions of one variable. We show that the $k$-point generating function coincides with the $ε\rightarrow 0$ asymptotics of the analytic $k$-point function, and also compute three more asymptotics of the analytic function for $ε\rightarrow \infty$, $q\rightarrow 0$, $q\rightarrow\infty$, thus defining new invariants for the Riemann sphere.

math.AG↗
arXiv2017

Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincaré rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $Δ$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in Δ$. We analyse the case when the leading matrix remains diagonalisable at $Δ$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending to points of $Δ$. When the deformation is isomonodromic away from $Δ$, it is well known that a fundamental matrix solution has singularities at $Δ$. When the system also has a Fuchsian singularity at $z=0$, we show under minimal vanishing conditions on the residue matrix at $z=0$ that isomonodromic deformations can be extended to the whole polydisc, including $Δ$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $Δ$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $Δ$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlevé equations is discussed.

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