Line operators from M-branes on compact Riemann surfaces

Notes on Compact Riemann Surfaces

Non-abelian vortices on compact Riemann surfaces

We consider the vortex equations for a U(n) gauge field A coupled to a Higgs field φ with values on the n × n matrices. It is known that when these equations are defined on a compact Riemann surface Σ, their moduli space of solutions is closely related to a moduli space of τ -stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix φ, we show t...

The resultant on compact Riemann surfaces

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the ...

Loop operators and S-duality from curves on Riemann surfaces

We study Wilson-’t Hooft loop operators in a class of N = 2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We then show that this precisely matches Dehn’s classification of homotopy clas...

Introduction to Compact Riemann Surfaces

The theory of Riemann surfaces is a classical field of mathematics where geometry and analysis play equally important roles. The purpose of these notes is to present some basic facts of this theory to make this book more self contained. In particular we will deal with classical descriptions of Riemann surfaces, Abelian differentials, periods on Riemann surfaces, meromorphic functions, theta fun...