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Alexander Varchenko

Publications and source records attributed to Alexander Varchenko.

At least 19 recordsLinked to original sources
arXiv2026

Quaternities, correspondences, and tetrahedron equations (Summa tetralogiae)

The aim of this note is: (a) to propose a generalization of tetrahedron equations from \cite{S} and of their solutions. Due to appearance of a larger number of parameters the $R$-matrices from \cite{S} will be replaced by "$R$-correspondences". (b) To rephrase these equations in terms of Wronskian evolutions in the spirit of \cite{SV}. (c) To discuss some elementary structures of cohomological flavour lying behind our considerations. We call them "quaternities", or "bibitorsors"; they might be not without an independent interest.

math.RA↗
arXiv2025

$p$-curvature operators and Satake-type phenomenon for $\frak{sl}_2$ KZ equations with $\kappa=\pm 2$

The $\frak{sl}_2$ KZ differential equations with values in the tensor power of the fundamental representation with parameter $\kappa=\pm 2$ are considered. A Satake-type correspondence is established over complex numbers and subsequently reduced to finite characteristic. This correspondence enables the study of the KZ equations on the lower weight subspaces of the tensor power in terms of the wedge powers of the weight subspace of the weight just below the highest weight. We apply this approach to analyze the $p$-curvature operators associated with our KZ equations, evaluate the dimension of the solution space in characteristic $p$, and determine whether all solutions are generated by the so-called $p$-hypergeometric solutions. In particular, we show that not all solutions of the KZ equations with $\kappa=2$ in characteristic $p$ are generated by $p$-hypergeometric solutions. Previously, no such examples were known.

math.NT↗
arXiv2025

Knizhnik-Zamolodchikov equations in Deligne categories

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 case and compute monodromy in both cases.

math.RT↗
arXiv2025

Finding All Solutions of qKZ Equations in Characteristic $p$

In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages, arXiv:2208.09721], the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper, we study in detail the first family of nontrivial examples of the qKZ equations in characteristic $p$. We describe all solutions of these qKZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. We also prove a Lagrangian property (called the orthogonality property) of the subbundle of the qKZ bundle spanned by the $p$-hypergeometric sections. This paper extends the results of [arXiv:2405.05159] on the differential KZ equations to the difference qKZ equations.

math-ph↗
arXiv2024

On the Satake correspondence for the equivariant quantum differential equations and qKZ difference equations of Grassmannians

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 between this joint system for $G(k,n)$ and the corresponding system for the projective space $\mathbb{P}^{n-1}$. Specifically, we show that, under suitable \textit{Satake identifications} of the equivariant cohomologies of $G(k,n)$ and $\mathbb{P}^{n-1}$, the joint system for $G(k,n)$ is gauge equivalent to a differential-difference system on the $k$-th exterior power of the cohomology of $\mathbb{P}^{n-1}$. Secondly, we demonstrate that the \textcyr{B}-theorem for Grassmannians, as stated in arXiv:1909.06582, arXiv:2203.03039, is compatible with the Satake identification. This implies that the \textcyr{B}-theorem for $\mathbb{P}^{n-1}$ extends to $G(k,n)$ through the Satake identification. As a consequence, we derive determinantal formulas and new integral representations for multi-dimensional hypergeometric solutions of the joint qDE and qKZ system for $G(k,n)$. Finally, we analyze the Stokes phenomenon for the joint system of qDE and qKZ equations associated with $G(k,n)$. We prove that the Stokes bases of solutions correspond to explicit $K$-theoretical classes of full exceptional collections in the derived category of equivariant coherent sheaves on $G(k,n)$. Furthermore, we show that the Stokes matrices equal the Gram matrices of the equivariant Euler-Poincar\'e-Grothendieck pairing with respect to these exceptional $K$-theoretical bases.

math.AG↗
arXiv2024

Notes on $2D$ $\mathbb F_p$-Selberg integrals

We prove a two-dimensional $\mathbb F_p$-Selberg integral formula, in which the two-dimensional $\mathbb F_p$-Selberg integral $\bar S(a,b,c;l_1,l_2)$ depends on positive integer parameters $a,b,c$, $l_1,l_2$ and is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.

math.AG↗
arXiv2024

On $p$-adic solutions to KZ equations, ordinary crystals, and $p^s$-hypergeometric solutions

We consider the KZ connection associated with a family of hyperelliptic curves of genus $g$ over the ring of $p$-adic integers $\mathbb{Z}_p$. Then the dual connection is the Gauss-Manin connection of that family. We observe that the Gauss-Manin connection has an ordinary $F$-crystal structure and its unit root subcrystal is of rank $g$. We prove that all local flat sections of the KZ connection annihilate the unite root subcrystal, and the space of all local flat sections of the KZ connection is a free $\mathbb{Z}_p$-module of rank $g$. We also consider the reduction modulo $p^s$ of the unit root subcrystal for any $s\geq 1$. We prove that its annihilator is generated by the so-called $p^s$-hypergeometric flat sections of the KZ connection. In particular, that means that the reduction modulo $p^s$ of an arbitrary local flat section of the KZ connection over $\mathbb{Z}_p$ is a linear combination of the $p^s$-hypergeometric flat sections.

math.NT↗
arXiv2024

Finding all solutions to the KZ equations in characteristic $p$

The KZ equations are differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the $n$-point functions of affine primary fields. In [SV1] the KZ equations were identified with equations for flat sections of suitable Gauss-Manin connections, and solutions of the KZ equations were constructed in the form of multidimensional hypergeometric integrals. In [SV2] the KZ equations were considered modulo a prime number $p$, and, for rational levels, polynomial solutions of the KZ equations modulo $p$ were constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper we study in detail the first nontrivial example of the KZ equations in characteristic $p$. In particular, if the level is irrational, we prove a version of the steepest descent result that relates the KZ local system to the space of functions on the critical locus of the master function. We use this result to prove the generic irreducibility of the KZ local system at any irrational level. If the level is rational, we describe all solutions of the KZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. Finally, we prove a Lagrangian property of the subbundle of the KZ bundle spanned by the $p$-hypergeometric sections.

math-ph↗
arXiv2024

$p$-curvature of periodic pencils of flat connections

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 $\lbrace x_i\rbrace$ are coordinates on $X$ and $B_{ij}: X\to {\rm Mat}_N$ are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts $s_j\mapsto s_j+1$ up to isomorphism. In this paper we show that in characteristic $p>0$, the $p$-curvature operators $\lbrace C_i,1\le i\le r\rbrace$ of a periodic pencil $\nabla$ are isospectral to the commuting endomorphisms $C_i^*:=\sum_{j=1}^n (s_j-s_j^p)B_{ij}^{(1)}$, where $B_{ij}^{(1)}$ is the Frobenius twist of $B_{ij}$. Using the results of arXiv:2401.00636, this allows us to compute the eigenvalues of the $p$-curvature for many important examples of pencils of flat connections, including Knizhnik-Zamolodchikov (KZ), Casimir, and Dunkl connections, their confluent limits, and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. In particular, for rational values of parameters these eigenvalues are zero, so the connections are globally nilpotent. We also show that every periodic pencil has regular singularites and its residues have rational eigenvalues for rational values of parameters. In particular, this holds for the aforementioned quantum connections if they have rational coefficients. Also we generalize these results to irregular pencils (KZ, Casimir, Dunkl, and Toda), and relate them in the Dunkl case to representations of rational Cherednik algebras. Finally, we extend our main result to pseudo-pencils and discuss the generalization to difference equations.

math.AG↗
arXiv2024

Periodic and quasi-motivic pencils of flat connections

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 $s_1,...,s_n$ and generically invariant, up to isomorphism, under the shifts $s_i\mapsto s_i+1$ for all $i$. If in addition $\nabla$ has regular singularities, we call it a quasi-motivic pencil. We use tools from complex analysis to establish various remarkable properties of such pencils over $\mathbb C$. For example, we show that the monodromy of a quasi-motivic pencil is defined over the field of algebraic functions in $e^{2\pi is_j}$, and that its singularities are constrained to an arrangement of hyperplanes with integer normal vectors. Then we show that many important examples of families of flat connections, such as Knizhnik-Zamolodchikov, Dunkl, and Casimir connections, are quasi-motivic and thus periodic pencils. Besides being interesting in its own right, the periodic property of a pencil of flat connections turns out to be very useful in computing the eigenvalues of the $p$-curvature of its reduction to positive characteristic. This will be done in our forthcoming paper.

math.AG↗
arXiv2023

Calogero-Moser eigenfunctions modulo $p^s$

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 constant. The truncation procedure uses the integral representation for the hypergeometric solutions to KZ equations. The $s\rightarrow \infty$ limit to the pure $p$-adic case has been analyzed in the $n=2$ case

hep-th↗
arXiv2023

Polynomial superpotential for Grassmannian $Gr(k,n)$ from a limit of vertex function

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 function, where $\hbar$ denotes the equivariant parameter of a torus acting on $X$ by dilating the cotangent fibers. We show that in this limit the integral turns into the standard mirror integral representation for the $A$-series of the Grassmannian $Gr(k,n)$ with the Laurent polynomial Landau-Ginzburg superpotential of Eguchi, Hori and Xiong. We also observe some Dwork type congruences for the coefficients of the $A$-series.

math-ph↗
arXiv2023

The p-adic approximations of vertex functions via 3D-mirror symmetry

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 sequence $T_s(z)$ converges in the $p$-adic norm to the Okounkov's vertex function of $X$ as $s\to \infty$. We prove that $T_s(z)$ satisfy Dwork-type congruences which lead to a new infinite product presentation of the vertex function modulo $p^s$.

math-ph↗
arXiv2022

Monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian

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 integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant K-theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant K-theory algebra of the cotangent bundle are Laurent polynomials with integer coefficients in the exponentiated equivariant parameters.

math-ph↗
arXiv2022

De Rham - Witt KZ equations

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.

math-ph↗
arXiv2022

Solutions of the $sl_2$ qKZ equations modulo an integer

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 $\kappa$. For any integer $N$ relatively prime to the step $\kappa$, we construct a family of polynomials $f_r(z)$ in variables $z_1,\dots,z_n$ with values in $V^{\otimes n}$ such that the coordinates of these polynomials with respect to the standard basis of $V^{\otimes n}$ are polynomials with integer coefficients. We show that the polynomials $f_r(z)$ satisfy the qKZ equations modulo $N$. Polynomials $f_r(z)$ are modulo $N$ analogs of the hypergeometric solutions of the \qKZ/ equations given in the form of multidimensional Barnes integrals.

math.QA↗
arXiv2022

Dynamical and qKZ equations modulo $p^s$, an example

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 prime integer $p$. We show that the $p$-adic limit of these solutions as $n\to\infty$ determines a sequence of line bundles, each of which is invariant with respect to the corresponding dynamical connection, and that sequence of line bundles is invariant with respect to the corresponding qKZ difference connection.

math.NT↗