Alexander Varchenko
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View article: $p$-curvature operators and Satake-type phenomenon for $\frak{sl}_2$ KZ equations with $κ=\pm 2$
$p$-curvature operators and Satake-type phenomenon for $\frak{sl}_2$ KZ equations with $κ=\pm 2$ Open
The $\frak{sl}_2$ KZ differential equations with values in the tensor power of the fundamental representation with parameter $κ=\pm 2$ are considered. A Satake-type correspondence is established over complex numbers and subsequently reduce…
View article: Knizhnik-Zamolodchikov equations in Deligne categories
Knizhnik-Zamolodchikov equations in Deligne categories Open
We consider the Knizhnik-Zamolodchikov equations in Deligne Categories in the context of $(\mathfrak{gl}_m,\mathfrak{gl}_{n})$ and $(\mathfrak{so}_m,\mathfrak{so}_{2n})$ dualities. We derive integral formulas for the solutions in the first…
View article: Congruences for Hasse-Witt matrices and solutions of p-adic KZ equations
Congruences for Hasse-Witt matrices and solutions of p-adic KZ equations Open
We prove general Dwork-type congruences for Hasse–Witt matrices attached to tuples of Laurent polynomials.We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions mod…
View article: On the Satake correspondence for the equivariant quantum differential equations and qKZ difference equations of Grassmannians
On the Satake correspondence for the equivariant quantum differential equations and qKZ difference equations of Grassmannians Open
We consider the joint system of equivariant quantum differential equations (qDE) and qKZ difference equations for the Grassmannian $G(k,n)$, which parametrizes $k$-dimensional subspaces of $\mathbb{C}^n$. First, we establish a connection b…
View article: $p$-curvature of periodic pencils of flat connections
$p$-curvature of periodic pencils of flat connections Open
In arXiv:2401.00636 we introduced the notion of a periodic pencil of flat connections on a smooth variety $X$. Namely, a pencil is a linear family of flat connections $\nabla(s_1,...,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i,$ where $\l…
View article: Periodic and quasi-motivic pencils of flat connections
Periodic and quasi-motivic pencils of flat connections Open
We introduce a new notion of a periodic pencil of flat connections on a smooth algebraic variety $X$. This is a family $\nabla(s_1,...,s_n)$ of flat connections on a trivial vector bundle on $X$ depending linearly on parameters $s_1,...,s_…
View article: Congruences for Hasse-Witt matrices and solutions of $p$-adic KZ equations
Congruences for Hasse-Witt matrices and solutions of $p$-adic KZ equations Open
We prove general Dwork-type congruences for Hasse--Witt matrices attached to\ntuples of Laurent polynomials. We apply this result to establishing arithmetic\nand $p$-adic analytic properties of functions originating from polynomial\nsoluti…
View article: Calogero-Moser eigenfunctions modulo $p^s$
Calogero-Moser eigenfunctions modulo $p^s$ Open
In this note we use the Matsuo-Cherednik duality between the solutions to KZ equations and eigenfunctions of Calogero-Moser Hamiltonians to get the polynomial $p^s$-truncation of the Calogero-Moser eigenfunctions at a rational coupling con…
View article: Polynomial Solutions Modulo $p^s$ of Differential KZ and Dynamical Equations
Polynomial Solutions Modulo $p^s$ of Differential KZ and Dynamical Equations Open
We construct polynomial solutions modulo $p^s$ of the differential KZ and dynamical equations where $p$ is an odd prime number.
View article: Hypergeometric integrals, hook formulas and Whittaker vectors
Hypergeometric integrals, hook formulas and Whittaker vectors Open
We determine the coefficient of proportionality between two multidimensional hypergeometric integrals. One of them is a solution of the dynamical difference equations associated with a Young diagram and the other is the vertex integral ass…
View article: Polynomial superpotential for Grassmannian $Gr(k,n)$ from a limit of vertex function
Polynomial superpotential for Grassmannian $Gr(k,n)$ from a limit of vertex function Open
In this note we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, $X=T^{*} Gr(k,n)$. This integral representation can be used to compute the $\hbar\to \infty$ limit of the vertex func…
View article: Polynomial Solutions Modulo $p^s$ of Differential KZ and Dynamical Equations
Polynomial Solutions Modulo $p^s$ of Differential KZ and Dynamical Equations Open
We construct polynomial solutions modulo $p^s$ of the differential KZ and dynamical equations where $p$ is an odd prime number.
View article: The p-adic approximations of vertex functions via 3D-mirror symmetry
The p-adic approximations of vertex functions via 3D-mirror symmetry Open
Using the $3D$ mirror symmetry we construct a system of polynomials $T_s(z)$ with integral coefficients which solve the quantum differential equitation of $X=T^{*} Gr(k,n)$ modulo $p^s$, where $p$ is a prime number. We show that the sequen…
View article: Monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian
Monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian Open
We describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant K-theory algebra of the cotangent bundle. This description is based on the hypergeometric inte…
View article: De Rham - Witt KZ equations
De Rham - Witt KZ equations Open
We propose a de Rham - Witt version of the derived Knizhnik-Zamolodchikov equations, and of their hypergeometric realizations. We also propose de Rham - Witt versions of some classical theorems related to arbitrary hyperplane arrangements.
View article: Landau–Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety
Landau–Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety Open
We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau–Ginzburg mirror for that partia…
View article: Solutions of the $sl_2$ qKZ equations modulo an integer
Solutions of the $sl_2$ qKZ equations modulo an integer Open
We study the qKZ difference equations with values in the $n$-th tensor power of the vector $sl_2$ representation $V$, variables $z_1,\dots,z_n$ and integer step $κ$. For any integer $N$ relatively prime to the step $κ$, we construct a fami…
View article: Dynamical and qKZ equations modulo $p^s$, an example
Dynamical and qKZ equations modulo $p^s$, an example Open
We consider an example of the joint system of dynamical differential equations and qKZ difference equations with parameters corresponding to equations for elliptic integrals. We solve this system of equations modulo any power $p^n$ of a pr…
View article: Dwork-type congruences and $p$-adic KZ connection
Dwork-type congruences and $p$-adic KZ connection Open
We show that the $p$-adic KZ connection associated with the family of curves $y^q=(t-z_1)\dots (t-z_{qg+1})$ has an invariant subbundle of rank $g$, while the corresponding complex KZ connection has no nontrivial proper subbundles due to t…
View article: Landau-Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety
Landau-Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety Open
We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau-Ginzburg mirror for that partia…
View article: On the number of $p$-hypergeometric solutions of KZ equations
On the number of $p$-hypergeometric solutions of KZ equations Open
It is known that solutions of the KZ equations can be written in the form of multidimensional hypergeometric integrals. In 2017 in a joint paper of the author with V. Schechtman the construction of hypergeometric solutions was modified, an…
View article: The $${{\mathbb {F}}}_p$$-Selberg Integral
The $${{\mathbb {F}}}_p$$-Selberg Integral Open
We prove an $\mathbb F_p$-Selberg integral formula, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensi…
View article: Twisted de Rham Complex on Line and $\widehat{\frak{sl}_2}$ Singular Vectors
Twisted de Rham Complex on Line and $\widehat{\frak{sl}_2}$ Singular Vectors Open
This work studies the connection between the representation theory of affine Lie algebra $\widehat{\frak{sl}_2}$ and the relations between the cohomology classes of certain logarithmic differential forms. Following work of V. Schechtman an…
View article: Frobenius-like structure in Gaudin model
Frobenius-like structure in Gaudin model Open
We introduce a Frobenius-like structure for the $\frak{sl}_2$ Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form in terms of derivatives of the potential of the first kind a…
View article: Derived KZ Equations
Derived KZ Equations Open
In this paper we strengthen the results of [SV] by presenting their derived version. Namely, we define a "derived Knizhnik - Zamolodchikov connection"\ and identify it with a "derived Gauss - Manin connection".
View article: Path count asymptotics and Stirling numbers
Path count asymptotics and Stirling numbers Open
We obtain formulas for the growth rate of the numbers of certain paths in a multi-dimensional analogue of the Eulerian graph. Corollaries are new identities relating Stirling numbers of the first and second kinds.
View article: Reality Property of Discrete Wronski Map with Imaginary Step
Reality Property of Discrete Wronski Map with Imaginary Step Open
For a set of quasi-exponentials with real exponents, we consider the discrete Wronskian (also known as Casorati determinant) with pure imaginary step 2h. We prove that if the coefficients of the discrete Wronskian are real and for every it…