Winter School Gauge Theory
|Lecture 1-2: "Self-duality in higher dimensions"
Motivation from the division algebras.
Higher-dimensional instantons and special holonomy.
Concentrating on the cases of instanton on
Spin(7)-holonomy manifolds and the related notion of
monopoles on G_2-holonomy manifolds.
Lectures 3-4: "Self-duality and supersymmetry" 10-dimensional supersymmetric Yang--Mills theory. Dimensional reduction and higher-dimensional self-duality. Some consequences of supersymmetry: e.g., triholomorphic curves and octonionic instantons.
1-2 U(1) gauge theory in three complex
Donaldson--Thomas theory. Toric threefolds and the topological vertex.
Gauge-string duality: the MNOP conjecture.
Lecture 3 Calabi-Yau algebras and non-commutative U(1) gauge theory
Lecture 4 Higher-rank theory: bundles on threefolds. Fibred varieties:
elliptic and ALE fibrations
|Lecture 1: "The Arnold-Givental conjecture and Moment Floer homology"
The Arnold-Givental conjecture asks for a lower bound on the number of intersection points of two Hamiltonian isotopic Lagrangians which intersect transversally and are fixed points of an antisymplectic involution. We are proving this conjecture for a class of Lagrangians in Marsden-Weinstein quotients using Moment Floer homology. Moment Floer homology is an infinite dimensional analogon of Morse homology for a Lagrange multiplier action functional whose gradient flow lines are the symplectic vortex equations. These have better compactness properties than Floer's equations which enables us to define Moment Floer homology even in situations where the ordinary Floer homology cannot be defined via standard means.
Lecture 2: "Vortices on the cylinder"
We give a new proof of a result of Jaffe and Taubes on the structure of moduli spaces of vortices on the cylinder which also sheds new light on the connection between the moduli space of holomorphic curves in a toric variety and Givental's toric map spaces. Our proof is based on Floer theoretic methods, finite dimensional approximation, and a homotopy argument for Lagrange multiplier functionals which enables us to overcome an adiabatic limit analysis.
Lecture 3: "Rabinowitz action functional and obstructions for exact contact embeddings"
This is joint work with Kai Cieliebak. We define Floer homology for a Lagrange multiplier action functional generalizing the Moment action functional arising in the theory of Moment Floer homology. As an application of some computations we get obstructions for exact contact embeddings.
"Introduction to monopoles"
This will be an introduction to monopoles and their moduli spaces on
3-manifolds, in particular R3 and H3, together with some of their
Lecture 2 "Differential and algebraic geometry of monopole moduli spaces"
I shall discuss the deep interplay of differential and
algebro-geometric methods in the study of monopole moduli spaces and their
still mysterious geometries.
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