Related papers: Regularized Integrals on Riemann Surfaces and Modu…
In this article, we deduce a series of integral formulas for a foliated sub-Riemannian manifold, which is a new geometric concept denoting a Riemannian manifold equipped with a distribution ${\mathcal D}$ and a foliation ${\mathcal F}$,…
We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point, and additional analyticity properties. This situation occurs…
We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.
We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schr\"odinger…
We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space…
Meromorphic differentials on Riemann surfaces are said to be real-normalized if all their periods are real. Moduli spaces of real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles and residues…
We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential…
In this paper we develop a general approach to dimer models analogous to Krichever's scheme in the theory of integrable systems. We start with a Riemann surface and the simplest generic meromorphic functions on it and demonstrate how to…
In this article we provide a classification and description of compact Riemann surfaces admitting a triangular action of a group of order $2p^2,$ where $p$ is an odd prime number. We obtain that all such Riemann surfaces are isomorphic to…
By analogy with work of Hitchin on integrable systems, we construct natural relaxations of several kinds of moduli spaces of difference equations, with special attention to a particular class of difference equations on an elliptic curve…
This article is a result of the AIM workshop on Moment Maps and Surjectivity in Various Geometries (August 9 - 13, 2004) organized by T.Holm, E.Lerman and S.Tolman. At that workshop I was introduced to the work of T.Hausel and N.Proudfoot…
The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.
In this paper, we elaborate on the connection between leading singularities and canonical bases of Feynman integrals beyond polylogarithms. We start by discussing a notion of leading singularities in dimensional regularization, which can be…
This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations.…
We review the following subjects: 1. Basic theory on algebraic curves and their moduli space, 2. Schottky uniformization theory of Riemann surfaces, and its extension called arithmetic uniformization theory, 3. Application to these theories…
We study the gluing theory of Riemann surfaces using formal algebraic geometry, and give computable relations between the associated parameters for different gluing processes. As its application to the Liouville conformal field theory, we…
We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve…
Let S be a surface obtained from a plane polygon by identifying infinitely many pairs of segments along its boundary. A condition is given under which the complex structure in the interior of the polygon extends uniquely across the quotient…
We study the compactification of the moduli space of a certain class of rank-two irregular connections on the Riemann sphere, presenting one double pole and two simple poles. To construct the compactification explicitly, we identify a class…
We study the evolution equations for a regularized version of Dirac-harmonic maps from closed Riemannian surfaces. We establish the existence of a global weak solution for the regularized problem, which is smooth away from finitely many…