BOUNDARY REPARAMETRIZATIONS AS ADDITIONAL MODULI FOR THE STRING PROPAGATOR

1989 ◽  
Vol 04 (03) ◽  
pp. 283-291 ◽  
Author(s):  
R. KIRSCHNER
Keyword(s):  
Quadratic Differentials ◽  
Schottky Double

Analyzing the Polyakov integral on surfaces with boundaries, where the values of the string variables are fixed, we use the observation that there are more holomorphic quadratic differentials besides those obtained as restrictions from the Schottky double. They are naturally related to boundary reparametrizations. The corresponding additional moduli are used to express the integration over metrices. Some details are given for the vacuum functional and the propagator.

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

scholarly journals RECURSION FOR MASUR-VEECH VOLUMES OF MODULI SPACES OF QUADRATIC DIFFERENTIALS

2021 ◽  
pp. 1-6
Author(s):  
Maxim Kazarian
Keyword(s):  
Moduli Spaces ◽  
Recursion Relation ◽  
Quadratic Differentials ◽  
Hodge Integrals

Abstract We derive a quadratic recursion relation for the linear Hodge integrals of the form $\langle \tau _{2}^{n}\lambda _{k}\rangle $ . These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.

Stability conditions and braid group actions on affine An quivers

2021 ◽  
pp. 2250174
Author(s):  
Chien-Hsun Wang
Keyword(s):  
Moduli Space ◽  
Braid Group ◽  
Group Actions ◽  
Stability Conditions ◽  
Quadratic Differentials ◽  
Spherical Subgroup ◽  
Type Formula ◽  
Affine Type

We study stability conditions on the Calabi–Yau-[Formula: see text] categories associated to an affine type [Formula: see text] quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order [Formula: see text]. We follow Ikeda’s work to show that this moduli space of quadratic differentials is isomorphic to the space of stability conditions quotient by the spherical subgroup of the autoequivalence group. We show that the spherical subgroup is isomorphic to the braid group of affine type [Formula: see text] based on the Khovanov–Seidel–Thomas method.

PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS

2006 ◽  
Vol 08 (03) ◽  
pp. 381-399
Author(s):  
THOMAS KWOK-KEUNG AU ◽  
TOM YAU-HENG WAN
Keyword(s):  
Riemann Surface ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Sufficient Condition ◽  
Holomorphic Quadratic Differential ◽  
Simply Connected ◽  
The Given

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

scholarly journals The Prime Function, the Fay Trisecant Identity, and the van der Pauw Method

2021 ◽  
Author(s):  
Hiroyuki Miyoshi ◽  
Darren Crowdy ◽  
Rhodri Nelson
Keyword(s):  
Applied Sciences ◽  
Science Literature ◽  
Multiply Connected ◽  
Schottky Double ◽  
Van Der Pauw ◽  
Simply Connected ◽  
Genus One ◽  
Physical Quantities ◽  
Prime Function ◽  
Van Der Pauw Method

AbstractThe van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature.

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