scholarly journals KNIZHNIK-ZAMOLODCHIKOV-BERNARD EQUATIONS ON RIEMANN SURFACES

1995 ◽  
Vol 10 (17) ◽  
pp. 2507-2536 ◽  
Author(s):  
D. IVANOV
Keyword(s):  
Spectral Parameter ◽  
Moduli Space ◽  
Correlation Functions ◽  
Riemann Surfaces ◽  
Projective Structure ◽  
Flat Connection ◽  
Conformal Blocks ◽  
Compact Riemann Surfaces ◽  
Marked Points

Knizhnik-Zamolodchikov-Bernard equations for twisted conformal blocks on compact Riemann surfaces with marked points are written explicitly in a general projective structure in terms of correlation functions in the theory of twisted b−c systems. It is checked that on the moduli space the equations provide a flat connection with the spectral parameter.

A Hodge theoretic projective structure on compact Riemann surfaces

2021 ◽  
Vol 149 ◽  
pp. 1-27
Author(s):  
Indranil Biswas ◽  
Elisabetta Colombo ◽  
Paola Frediani ◽  
Gian Pietro Pirola
Keyword(s):  
Riemann Surfaces ◽  
Projective Structure ◽  
Compact Riemann Surfaces

On complete curves in moduli space I

1991 ◽  
Vol 110 (3) ◽  
pp. 461-466 ◽  
Author(s):  
Gabino González Díez ◽  
William J. Harvey
Keyword(s):  
Moduli Space ◽  
General Position ◽  
Riemann Surfaces ◽  
Projective Variety ◽  
Dimension 2 ◽  
Compact Riemann Surfaces ◽  
Set Of Points

Letgdenote the moduli space of compact Riemann surfaces of genusg> 3. It is known thatgis a non-complete quasi-projective variety that contains many complete curves. This is because the Satake compactificationgofgis projective and the boundary\has co-dimension 2; thus by intersectingwith hypersurfaces in sufficiently general position one obtains a complete curve ingpassing through any given set of points [8].

A Hilbert manifold structure on the Weil–Petersson class Teichmüller space of bordered Riemann surfaces

2015 ◽  
Vol 17 (04) ◽  
pp. 1550016 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Banach Manifold ◽  
Holomorphic Maps ◽  
Discontinuous Group ◽  
Hilbert Manifold ◽  
Conformal Field ◽  
Compact Riemann Surfaces

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disk removed. We define a refined Teichmüller space of such Riemann surfaces (which we refer to as the WP-class Teichmüller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller space (which is a Banach manifold) is holomorphic. We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of non-overlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally, we show that the rigged moduli space is the quotient of the refined Teichmüller space by a properly discontinuous group of biholomorphisms.

scholarly journals REPRESENTATIONS OF SURFACE GROUPS IN THE PROJECTIVE GENERAL LINEAR GROUP

2011 ◽  
Vol 22 (02) ◽  
pp. 223-279 ◽  
Author(s):  
ANDRÉ GAMA OLIVEIRA
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Connected Components ◽  
Higgs Bundles ◽  
Semisimple Lie Group ◽  
Projective Bundles ◽  
Compact Riemann Surfaces ◽  
Hitchin Component ◽  
Projective General Linear Group ◽  
Closed Oriented Surface

Given a closed, oriented surface X of genus g ≥ 2, and a semisimple Lie group G, let [Formula: see text] be the moduli space of reductive representations of π1X in G. We determine the number of connected components of [Formula: see text], for n ≥ 4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in [Formula: see text] is homotopically equivalent to [Formula: see text].

scholarly journals On the existence of connected components of dimension one in the branch locus of moduli spaces of riemann surfaces

2012 ◽  
Vol 111 (1) ◽  
pp. 53 ◽  
Author(s):  
Antonio F. Costa ◽  
Milagros Izquierdo
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Fuchsian Groups ◽  
Connected Components ◽  
Branch Locus ◽  
Isolated Points ◽  
Compact Riemann Surfaces ◽  
Dimension One ◽  
Substantial Interest

Let $g$ be an integer $\geq3$ and let $B_{g}=\{X\in\mathcal{M}_{g}: \mathrm{Aut}(X)\neq Id\}$ be the branch locus of $M_{g}$, where $M_{g}$ denotes the moduli space of compact Riemann surfaces of genus $g$. The structure of $B_{g}$ is of substantial interest because $B_{g}$ corresponds to the singularities of the action of the modular group on the Teichmüller space of surfaces of genus $g$ (see [14]). Kulkarni ([15], see also [13]) proved the existence of isolated points in the branch loci of the moduli spaces of Riemann surfaces. In this work we study the isolated connected components of dimension 1 in such loci. These isolated components of dimension one appear if the genus is $g=p-1$ with $p$ prime $\geq11$. We use uniformization by Fuchsian groups and the equisymmetric stratification of the branch loci.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Sign in / Sign up

Export Citation Format

Share Document