Searcharxiv⌕ Search

arXiv subjects

Alexander Varchenko

Publications and source records attributed to Alexander Varchenko.

At least 37 records · Page 2Linked to original sources
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 $κ$. For any integer $N$ relatively prime to the step $κ$, 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↗
arXiv2022

Dwork-type congruences and $p$-adic KZ connection

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 the irreducibility of its monodromy representation. The construction of the invariant subbundle is based on new Dwork--type congruences for associated Hasse--Witt matrices.

math.NT↗
arXiv2022

On the number of $p$-hypergeometric solutions of KZ equations

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, and solutions of the KZ equations modulo a prime number $p$ were constructed. These solutions modulo $p$, called the $p$-hypergeometric solutions, are polynomials with integer coefficients. A general problem is to determine the number of independent $p$-hypergeometric solutions and understand the meaning of that number. In this paper we consider the KZ equations associated with the space of singular vectors of weight $n-2r$ in the tensor power $W^{\otimes n}$ of the vector representation of $\frak{sl}_2$. In this case, the hypergeometric solutions of the KZ equations are given by $r$-dimensional hypergeometric integrals. We consider the module of the corresponding $p$-hypergeometric solutions, determine its rank, and show that the rank equals the dimension of the space of suitable square integrable differential $r$-forms.

math-ph↗
arXiv2022

Frobenius-like structure in Gaudin model

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 and the action of Gaudin Hamiltonians in terms of derivatives of the potential of the second kind.

math.QA↗
arXiv2021

Ghosts and congruences for $p^s$-approximations of hypergeometric periods

We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and $p$-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and KZ equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the simplest example of a $p$-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of the monodromy group.

math.NT↗
arXiv2021

Congruences for Hasse--Witt matrices and solutions of $p$-adic KZ equations

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 modulo $p^s$ of Knizhnik--Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the $p$-adic KZ connection associated with the family of hyperelliptic curves $y^2=(t-z_1)\dots (t-z_{2g+1})$ has an invariant subbundle of rank $g$. Notice that the corresponding complex KZ connection has no nontrivial subbundles due to the irreducibility of its monodromy representation.

math.NT↗
arXiv2021

Notes on solutions of KZ equations modulo $p^s$ and $p$-adic limit $s\to\infty$

We consider the KZ equations over $\mathbb C$ in the case, when the hypergeometric solutions are hyperelliptic integrals of genus $g$. Then the space of solutions is a $2g$-dimensional complex vector space. We also consider the same equations modulo $p^s$, where $p$ is an odd prime and $s$ is a positive integer, and over the field $\mathbb Q_p$ of $p$-adic numbers. We construct polynomial solutions of the KZ equations modulo $p^s$ and study the space $\mathcal M_{p^s}$ of all constructed solutions. We show that the $p$-adic limit of $\mathcal M_{p^s}$ as $s\to\infty$ gives us a $g$-dimensional vector space of solutions of the KZ equations over $\mathbb Q_p$. The solutions over $\mathbb Q_p$ are power series at a certain asymptotic zone of the KZ equations. In the appendix written jointly with Steven Sperber we consider all asymptotic zones of the KZ equations in the case $g=1$ of elliptic integrals. The $p$-adic limit of $\mathcal M_{p^s}$ as $s\to \infty$ gives us a one-dimensional space of solutions over $\mathbb Q_p$ at every asymptotic zone. We apply Dwork's theory and show that our germs of solutions over $\mathbb Q_p$ defined at different asymptotic zones analytically continue into a single global invariant line subbundle of the associated KZ connection. Notice that the corresponding KZ connection over $\mathbb C$ does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over $\mathbb Q_p$. We describe the Frobenius transformations of solutions of the KZ equations for $g =1$ and then recover the unit roots of the zeta functions of the elliptic curves defined by the equations $y^2= β\,x(x-1)(x-α)$ over the finite field $\mathbb F_p$. Here $α,β\in\mathbb F_p^\times, α\ne 1$.

math.AG↗
arXiv2020

Derived KZ equations

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".

math.AG↗
arXiv2020

Determinant of $\mathbb F_p$-hypergeometric solutions under ample reduction

We consider the KZ differential equations over $\mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work joint with V.\,Schechtman and called the $\mathbb F_p$-hypergeometric solutions. The dimension of the space of $\mathbb F_p$-hypergeometric solutions depends on the prime number $p$. We say that the KZ equations have ample reduction for a prime $p$, if the dimension of the space of $\mathbb F_p$-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over $\mathbb C$. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis $\mathbb F_p$-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials $(z_i-z_j)^{M_i+M_j}$ are replaced with $(z_i-z_j)^{M_i+M_j-p}$ and the Euler gamma function $Γ(x)$ is replaced with a suitable $\mathbb F_p$-analog $Γ_{\mathbb F_p}(x)$ defined on $\mathbb F_p$.

math.AG↗
arXiv2020

The $\mathbb F_p$-Selberg Integral

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 multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.

math.AG↗
arXiv2020

The $\mathbb F_p$-Selberg integral of type $A_n$

We prove an $\mathbb F_p$-Selberg integral formula of type $A_n$, 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 multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$. For the type $A_1$ the formula was proved in a previous paper by the authors.

math.AG↗
arXiv2020

The $*$-Markov equation for Laurent polynomials

We consider the $*$-Markov equation for the symmetric Laurent polynomials in three variables with integer coefficients, which is an equivariant analog of the classical Markov equation for integers. We study how the properties of the Markov equation and its solutions are reflected in the properties of the $*$-Markov equation and its solutions.

math.AG↗
arXiv2020

3d Mirror Symmetry and Elliptic Stable Envelopes

We consider a pair of quiver varieties (X;X') related by 3d mirror symmetry, where X =T*Gr(k,n) is the cotangent bundle of the Grassmannian of k-planes of n-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on X $\times$ X' (the Mother function) whose restrictions to X and X' are the elliptic stable envelopes of those varieties. This implies, that the restriction matrices of the elliptic stable envelopes for X and X' are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.

math.AG↗
arXiv2020

Equivariant quantum differential equation and $qKZ$ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

In arXiv:1901.02990v1 the equivariant quantum differential equation ($qDE$) for a projective space was considered and a compatible system of difference $qKZ$ equations was introduced; the space of solutions to the joint system of the $qDE$ and $qKZ$ equations was identified with the space of the equivariant $K$-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant $K$-theory algebra. This paper is a continuation of arXiv:1901.02990v1. We describe the relation between solutions to the joint system of the $qDE$ and $qKZ$ equations and the topological-enumerative solution to the $qDE$ only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant $K$-theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant $K$-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincaré pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani.

math.AG↗
arXiv2020

Hypergeometric Integrals Modulo $p$ and Hasse--Witt Matrices

We consider the KZ differential equations over $\mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the space of polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work by V. Schechtman and the second author. Using Hasse-Witt matrices we identify the space of these polynomial solutions over $\mathbb F_p$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over $\mathbb F_p$ and the hypergeometric solutions over $\mathbb C$.

math.AG↗