Stokes matrices for the quantum differential equations of some Fano varieties

The classical Stokes matrices for the quantum differential equation of projective n-space are computed, using multisummation and the so-called monodromy identity. Thus, we recover the results of D. Guzzetti that confirm Dubrovin's conjecture for projective spaces. The same method yields explicit formulas for the Stokes matrices of the quantum differential equations of smooth Fano hypersurfaces in projective n-space and for weighted projective spaces.Comment: 20 pages. Introduction has been changed. Small corrections in the tex

Lattice Gauge Fields Topology Uncovered by Quaternionic sigma-model Embedding

We investigate SU(2) gauge fields topology using new approach, which exploits the well known connection between SU(2) gauge theory and quaternionic projective sigma-models and allows to formulate the topological charge density entirely in terms of sigma-model fields. The method is studied in details and for thermalized vacuum configurations is shown to be compatible with overlap-based definition. We confirm that the topological charge is distributed in localized four dimensional regions which, however, are not compatible with instantons. Topological density bulk distribution is investigated at different lattice spacings and is shown to possess some universal properties.Comment: revtex4, 19 pages (24 ps figures included); replaced to match the published version, to appear in PRD; minor changes, references adde

On the numerical evaluation of algebro-geometric solutions to integrable equations

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solutions of the Davey-Stewartson and the multi-component nonlinear Schr\"odinger equations.Comment: 29 pages, 20 figure

Solution of the generalized periodic discrete Toda equation II; Theta function solution

We construct the theta function solution to the initial value problem for the generalized periodic discrete Toda equation.Comment: 11 page

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 Br\'ezin--Gross--Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lam\'e tau-function are also given

Classical hurwitz numbers and related combinatorics

We give a polynomial-time algorithm of computing the classical Hurwitz numbers Hg,d, which were defined by Hurwitz 125 years ago. We show that the generating series of Hg,d for any fixed g > 2 lives in a certain subring of the ring of formal power series that we call the Lambert ring. We then define some analogous numbers appearing in enumerations of graphs, ribbon graphs, and in the intersection theory on moduli spaces of algebraic curves, such that their generating series belong to the same Lambert ring. Several asymptotics of these numbers (for large g or for large d) are obtained

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 $\textit{topological ODE}$

Deformations with a resonant irregular singularity

I review topics of my talk in Alcal\ue1, inspired by the paper [1]. An isomonodromic system with irregular singularity at z= 1e (and Fuchsian at z=0) is considered, such that z= 1e becomes resonant for some values of the deformation parameters. Namely, the eigenvalues of the leading matrix at z= 1e coalesce along a locus in the space of deformation parameters. I give a complete extension of the isomonodromy deformation theory in this case