scholarly journals U(1)×U(1) Chern–Simons vortices in line bundles over a compact Riemann surface

2014 ◽  
Vol 75 ◽  
pp. 48-54
Author(s):  
Kwan Hui Nam
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Line Bundles ◽  
Chern Simons

scholarly journals A note on the theta characteristics of a compact Riemann surface

2004 ◽  
Vol 76 (3) ◽  
pp. 415-424
Author(s):  
Indranil Biswas
Keyword(s):  
Riemann Surface ◽  
Quadratic Form ◽  
Line Bundle ◽  
Vector Fields ◽  
Compact Riemann Surface ◽  
Line Bundles ◽  
Square Root ◽  
Topological Interpretation ◽  
Theta Characteristics

AbstractLet X be a compact connected Riemann surface and ξ a square root of the holomorphic contangent bundle of X. Sending any line bundle L over X of order two to the image of dim H0(X, ξ ⊗ L) − dim H0(X, ξ) in Z/2Z defines a quadratic form on the space of all order two line bundles. We give a topological interpretation of this quadratic form in terms of index of vector fields on X.

scholarly journals Uniqueness of bubbling solutions of mean field equations with non-quantized singularities

2020 ◽  
pp. 2050038
Author(s):  
Lina Wu ◽  
Lei Zhang
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Mean Field ◽  
Field Equations ◽  
Field Type ◽  
Dirac Mass ◽  
Mean Field Equations ◽  
The Mean ◽  
Chern Simons

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with the singularities of the Dirac data. If the strength of the Dirac mass at each blowup point is not a multiple of [Formula: see text], we prove that bubbling solutions are unique. This paper extends previous results of Lin-Yan [C. S. Lin and S. S. Yan, On the mean field type bubbling solutions for Chern–Simons–Higgs equation, Adv. Math. 338 (2018) 1141–1188] and Bartolucci et al. [D. Bartolucci, A. Jevnikar, Y. Lee and W. Yang, Uniqueness of bubbling solutions of mean field equations, J. Math. Pures Appl. (9) 123 (2019) 78–126].

THE SELF-DUAL CHERN–SIMONS HIGGS EQUATION ON A COMPACT RIEMANN SURFACE WITH BOUNDARY

2010 ◽  
Vol 21 (01) ◽  
pp. 67-76 ◽  
Author(s):  
MENG WANG
Keyword(s):  
Boundary Condition ◽  
Riemann Surface ◽  
Green Function ◽  
Neumann Boundary Condition ◽  
Compact Riemann Surface ◽  
Neumann Boundary ◽  
The Self ◽  
Almost Everywhere ◽  
Riemann Surface With Boundary ◽  
Chern Simons

We study the self-dual Chern–Simons Higgs equation on a compact Riemann surface with Neumann boundary condition. We show that the Chern–Simons Higgs equation with parameter λ > 0 has at least two solutions [Formula: see text] for λ sufficiently large, such that [Formula: see text] almost everywhere as λ → + ∞, and that [Formula: see text] almost everywhere as λ → ∞, where u0 is a (negative) Green function on M.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

scholarly journals ON CANONICAL QUANTIZATION OF THE GAUGED WZW MODEL WITH PERMUTATION BRANES

2011 ◽  
Vol 26 (26) ◽  
pp. 4647-4660
Author(s):  
GOR SARKISSIAN
Keyword(s):  
Riemann Surface ◽  
Phase Space ◽  
Boundary Conditions ◽  
Canonical Quantization ◽  
Simons Theory ◽  
Gauge Fields ◽  
Time Line ◽  
Chern Simons ◽  
Special Case ◽  
Chern Simons Theory

In this paper we perform canonical quantization of the product of the gauged WZW models on a strip with boundary conditions specified by permutation branes. We show that the phase space of the N-fold product of the gauged WZW model G/H on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of the double Chern–Simons theory on a sphere with N holes times the time-line with G and H gauge fields both coupled to two Wilson lines. For the special case of the topological coset G/G we arrive at the conclusion that the phase space of the N-fold product of the topological coset G/G on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of Chern–Simons theory on a Riemann surface of the genus N-1 times the time-line with four Wilson lines.

PARABOLIC GENERALIZED MODULAR FORMS AND THEIR CHARACTERS

2009 ◽  
Vol 05 (05) ◽  
pp. 845-857 ◽  
Author(s):  
MARVIN KNOPP ◽  
GEOFFREY MASON
Keyword(s):  
Riemann Surface ◽  
Modular Form ◽  
Finite Index ◽  
Modular Forms ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Eichler Cohomology

We make a detailed study of the generalized modular forms of weight zero and their associated multiplier systems (characters) on an arbitrary subgroup Γ of finite index in the modular group. Among other things, we show that every generalized divisor on the compact Riemann surface associated to Γ is the divisor of a modular form (with unitary character) which is unique up to scalars. This extends a result of Petersson, and has applications to the Eichler cohomology.

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