scholarly journals The topology of the moduli space of arc-length parametrised closed curves in Euclidean space

Topology ◽  
2000 ◽  
Vol 39 (3) ◽  
pp. 487-494
Author(s):  
Steven Lillywhite
Keyword(s):  
Euclidean Space ◽  
Moduli Space ◽  
Arc Length ◽  
Closed Curves

scholarly journals QUASISYMMETRIC SEWING IN RIGGED TEICHMÜLLER SPACE

2006 ◽  
Vol 08 (04) ◽  
pp. 481-534 ◽  
Author(s):  
DAVID RADNELL ◽  
ERIC SCHIPPERS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Operator Algebras ◽  
Complex Structure ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Closed Curves ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces

One of the basic geometric objects in conformal field theory (CFT) is the moduli space of Riemann surfaces whose n boundaries are "rigged" with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alternative model is the moduli space of n-punctured Riemann surfaces together with local biholomorphic coordinates at the punctures. We refer to both of these moduli spaces as the "rigged Riemann moduli space".By generalizing to quasisymmetric boundary parametrizations, and defining rigged Teichmüller spaces in both the border and puncture pictures, we prove the following results: (1) The Teichmüller space of a genus-g surface bordered by n closed curves covers the rigged Riemann and rigged Teichmüller moduli spaces of surfaces of the same type, and induces complex manifold structures on them; (2) With this complex structure, the sewing operation is holomorphic; (3) The border and puncture pictures of the rigged moduli and rigged Teichmüller spaces are biholomorphically equivalent.These results are necessary in rigorously defining CFT (in the sense of G. Segal), as well as for the construction of CFT from vertex operator algebras.

scholarly journals HOMOTOPY GROUPS OF MODULI SPACES OF STABLE QUIVER REPRESENTATIONS

2010 ◽  
Vol 21 (09) ◽  
pp. 1219-1238
Author(s):  
GRAEME WILKIN
Keyword(s):  
Euclidean Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Homotopy Groups ◽  
Quiver Representations ◽  
Dimension Vector ◽  
Fixed Dimension ◽  
Low Dimensional

The purpose of this paper is to describe a method for computing homotopy groups of the space of α-stable representations of a quiver with fixed dimension vector and stability parameter α. The main result is that the homotopy groups of this space are trivial up to a certain dimension, which depends on the quiver, the choice of dimension vector, and the choice of parameter. As a corollary we also compute low dimensional homotopy groups of the moduli space of α-stable representations of the quiver with fixed dimension vector, and apply the theory to the space of non-degenerate polygons in three-dimensional Euclidean space.

scholarly journals Dk Gravitational Instantons as Superpositions of Atiyah–Hitchin and Taub–NUT Geometries

2021 ◽  
Author(s):  
B J Schroers ◽  
M A Singer
Keyword(s):  
Euclidean Space ◽  
Moduli Space ◽  
Gravitational Instantons ◽  
Fifth Dimension ◽  
Finite Set ◽  
Set Of Points

Abstract We obtain Dk ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as nonlinear superpositions of the moduli space of centred SU(2) monopoles, equipped with the Atiyah–Hitchin metric, and k copies of the Taub–NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we show that the constituent Atiyah–Hitchin and Taub–NUT geometries arise as boundary components of the five-dimensional geometry as the adiabatic parameter is taken to zero.

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