Christian Mercat

Discrete Riemann Surfaces and the Ising Model

Abstract: We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyls lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality.

Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function

Abstract Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable (in the sense of 3D consistency, which yields also the existense of zero curvature representations, Bäcklund transformations etc.). We demonstrate that in some precise sense the linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions which take real (imaginary) values on two sublattices. We extend solutions of the basic equations of both theories to ℤ d , where d is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, we give an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions. We introduce the d-dimensional discrete logarithmic function which is a generalization of Kenyon’s discrete Green’s function, and uncover several new properties of this function. We prove that it is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle patterns along the family of isomonodromic discrete power functions.

Discrete Riemann surfaces

We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincare dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we define the discrete analogs of period matrices, Riemanns bilinear relations, exponential of constant argument and series. We present the notion of criticality and its relationship with integrability.In this survey paper we give a proof of hyperbolicity of the complex of curves for a non-exceptional surface S of finite type combining ideas of Masur/Minsky and Bowditch. We also shortly discuss the relation between the geometry of the complex of curves and the geometry of Teichmueller space.This survey article considers moduli of algebraic curves using techniques from the complex analytic Teichmuller theory of deformations for the underlying Riemann surfaces and combinatorial topology of surfaces. The aim is to provide a readable narrative, suitable for people with a little background in complex analysis, hyperbolic plane geometry and discrete groups, who wish to understand the interplay of combinatorial, geometric and topological processes in this area. We explore in some detail a natural relationship with Grothendieck dessins, which provides both an appropriate setting in which to describe Veech curves (a special type of Teichmuller disc) and also a framework for relating complex moduli to arithmetic data involving a field of definition for the associated algebraic curves.We study the boundary of Teichmueller disks in a partial compactification of Teichmueller space, and their image in Schottky space. We give a broad introduction to Teichmueller disks and explain the relation between Teichmueller curves and Veech groups. Furthermore, we describe Braungardts construction of this partial compactification and compare it with the Abikoff augmented Teichmueller space. Following Masur, we give a description of Strebel rays that makes it easy to understand their end points on the boundary. This prepares the description of boundary points that a Teichmueller disk has, with a particular emphasis to the case that it leads to a Teichmueller curve. Further on we turn to Schottky space and describe two different approaches to obtain a partial compactification. We give an overview how the boundaries of Schottky space, Teichmueller space and moduli space match together and how the actions of the diverse groups on them are linked. Finally we consider the image of Teichmueller disks in Schottky space and show that one can choose the projection from Teichmueller space to Schottky space in such a manner that the image of the Teichmueller disk is a quotient by an infinite group.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.We survey explicit coordinate descriptions for two (A and X) versions of Teichmuller and lamination spaces for open 2D surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the boundary. Main features, such as mapping class group action, Poisson and symplectic structures and others, are described in these terms. The lamination spaces are interpreted as the tropical limits of the Teichmuller ones. Canonical pairings between lamination and Teichmuller spaces are constructed. The paper could serve as an introduction to higher Teichmuller theory developed by the authors in math.AG/0311149, math.AG/0311245.This paper has been withdrawn by the author(s). The material contained in the paper will be published in a subtantially reorganized form, part of it is now included in math.QA/0510174

Integrable lattice realizations of conformal twisted boundary conditions

We construct integrable lattice realizations of conformal twisted boundary conditions for s�( 2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical A–D–E lattice models with positive spectral parameter. The integrable seam boundary conditions are labelled by (r, s,ζ) ∈ (Ag−2 ,A g−1 ,Γ )where Γ is the group of automorphisms of the graph G and g is the Coxeter number of G = A, D, E. Taking symmetries into account, these are identified with conformal twisted boundary conditions of Petkova and Zuber labelled by (a, b, γ) ∈ (Ag−2 ⊗ G, Ag−2 ⊗ G, Z2) and associated with nodes of the minimal analog of the Ocneanu quantum graph. Our results are illustrated using the Ising (A2 ,A 3) and 3-state Potts (A4 ,D 4) models.  2001 Elsevier Science B.V. All rights reserved.

Discrete complex structure on surfel surfaces

This paper defines a theory of conformal parametrization of digital surfaces made of surfels equipped with a normal vector. The main idea is to locally project each surfel to the tangent plane, therefore deforming its aspect-ratio. It is a generalization of the theory known for polyhedral surfaces. The main difference is that the conformal ratios that appear are no longer real in general. It yields a generalization of the standard Laplacian on weighted graphs.

Integrable Boundaries and Universal TBA Functional Equations

We derive the fusion hierarchy of functional equations for critical A-D-E lattice models related to the sl (2) unitary minimal models, the parafermionic models and the supersymmetric models of conformal field theory and deduce the related TBA functional equations. The derivation uses fusion projectors and applies in the presence of all known integrable boundary conditions on the torus and cylinder. The resulting TBA functional equations are universal in the sense that they depend only on the Coxeter number of the A-D-E graph and are independent of the particular integrable boundary conditions. We conjecture generally that TBA functional equations are universal for all integrable lattice models associated with rational CFTs and their integrable perturbations.

Cross-Curriculum Search for Intergeo

Intergeo is a European project dedicated to the sharing of interactive geometry constructions. This project is setting up an annotation and search web platform which will offer and provide access to thousands of interactive geometry constructions and resources using them. The search platform should cross the boundaries of the curriculum standards of Europe. A topics and competency based approach to retrieval for interactive geometry with designation of the semantic entities has been adopted: it requests the contributor of an interactive geometry resource to input the competencies and topics involved in a construction, and allows the searcher to find it by the input of competencies and topics close to them; both rely on plain-text-input. This paper describes the current prototypes, the input-methods, the workflows used, and the integration into the Intergeo platform.

Integrable and conformal twisted boundary conditions for sl(2) A-D-E lattice models

We study integrable realizations of conformal twisted boundary conditions for s�( 2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A, D, E lattice models with positive spectral parameter u> 0a nd Coxete rnumber g .I ntegrable seams are constructed by fusing blocks of elementary local face weights. The usual A-type fusions are labelled by the Kac labels (r, s) and are associated with the Ve rlinde fusion algebra. We introduce a new type of fusion in the two braid limits u →± i∞ associated with the graph fusion algebra, and labelled by nodes a, b ∈ G respectively. When combined with automorphisms, they lead to general integrable seams labelled by x = (r, a ,b , κ)∈ (Ag−2 ,H , H,Z2) where H is the graph G for type I theories and its parent for type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with th ec onformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for |G| 6. We also show, in the case of D2� ,t hat the non-commutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.

Mesh Parameterization with Generalized Discrete Conformal Maps

We introduce a new method to compute conformal parameterizations using a recent definition of discrete conformity, and establish a discrete version of the Riemann mapping theorem. Our algorithm can parameterize triangular, quadrangular and digital meshes. It can also be adapted to preserve metric properties. To demonstrate the efficiency of our method, many examples are shown in the experiment section.

Curvature estimation for discrete curves based on auto-adaptive masks of convolution

We propose a method that we call auto-adaptive convolution which extends the classical notion of convolution in pictures analysis to function analysis on a discrete set. We define an averaging kernel which takes into account the local geometry of a discrete shape and adapts itself to the curvature. Its defining property is to be local and to follow a normal law on discrete lines of any slope. We used it together with classical differentiation masks to estimate first and second derivatives and give a curvature estimator of discrete functions.