ADE surfaces and their moduli

Riemann Surfaces and Their Moduli

The purpose of these notes is to give an overview of Riemann surfaces, and their moduli spaces. In general, theorems will be presented without proof, although references will be provided. These notes were originally prepared for the summer 2004 session of the UIUC Graduate String Seminar1. Proofs of nearly all the theorems may be found, unless otherwise noted, in [FK80]. CONTENTS 1. Mathematica...

Moduli spaces of hyperbolic surfaces and their Weil–Petersson volumes

Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil–Petersson form. Mirzakhani proved that Weil–Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.

ADE-bundle over rational surfaces, configuration of lines and rulings

To each del Pezzo surface Xn (resp. ruled surface, ruled surface with a section), we describe a natural Lie algebra bundle of type En (resp. Dn−1, An−2) over it . Using lines and rulings on any such surface, we describe various representation bundles corresponding to fundamental representations of the corresponding Lie algebra. When we specify a geometric structure on the surface to reduce the ...

ADE-bundle over rational surfaces, configuration of lines and rulings

Unitons and Their Moduli

We sketch the proof that unitons (harmonic spheres in U(N)) correspond to holomorphic ‘uniton bundles’, and that these admit monad representations analogous to Donaldson’s representation of instanton bundles. We also give a closed-form expression for the unitons involving only matrix operations, a finite-gap result (two-unitons have energy ≥ 4), computations of fundamental groups of energy ≤ 4 ...