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We consider the KZ equations over 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 ps, where p is an odd prime and s is a positive integer, and over the field Q_p of p-adic numbers. We construct polynomial solutions of the KZ equations modulo ps and study the space Mps of all constructed solutions. We show that the p-adic limit of Mps as s→∞ gives us a g-dimensional vector space of solutions of the KZ equations over Qp. The solutions over Qp 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 Mps as s→∞ gives us a one-dimensional space of solutions over Qp at every asymptotic zone. We apply Dwork's theory and show that our germs of solutions over Qp 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 C does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over Qp. 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 y2=βx(x−1)(x−α) over the finite field Fp. Here α,β∈F×p,α≠1 

We consider the KZ differential equations over C in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field F_p. We study the polynomial solutions of these differential equations over F_p, constructed in a previous work joint with V. Schechtman and called the F_p-hypergeometric solutions. The dimension of the space of 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 F_p-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over C. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis 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)^{Mi+Mj−p} and the Euler gamma function Γ(x) is replaced with a suitable F_p-analog defined on F_p 

We consider the space of solutions of the Bethe ansatz equations of the sl_N XXX quantum integrable model, associated with the trivial representation of sl_N. We construct a family of commuting flows on this space and identify the flows with the flows of coherent rational Ruijesenaars-Schneider systems. For that we develop in full generality the spectral transform for the rational Ruijesenaars-Schneider system. 

In [TV19a] 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 [TV19a]. 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. 

It is considered the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra A(2)2n. The population is naturally partitioned into an infinite collection of complex cells C^m, where m are some positive integers. For each cell it is defined an injective rational map C^m→M(A^{(2)}_{2n}) of the cell to the space M(A^{(2)}_{2n}) of Miura opers of type A^{(2)}_{2n}. It is shown that the image of the map is invariant with respect to all mKdV flows on M(A^{(2)}_{2n}) and the image is point-wise fixed by all mKdV flows ∂/∂t_r with index r greater than 4m.