LMS online mini-lecture series

Lecture 1 - Homogenous spaces: Introduction, properties, and applications
- Motivation, definition and properties of homogeneous spaces
- Examples to build intuition
- Applications (in physics)

Lecture 2 - Homogenous space and maximally symmetric spacetimes
- Introduction to spacetimes
- Sketch of classification of maximally symmetric spacetimes
- Discussion of properties and invariants
- Relevance in physics

The lectures will highlight some recent work on solvable models of topological solitons. The first involves generalisations of the U(1) Abelian-Higgs model whose integrability is intimately related to the geometry of constant curvature Riemann surfaces. The second piece of work is a study of magnetic skyrmions in chiral magnets. Recently a family of solvable models for magnetic skyrmions in chiral magnets was introduced. The energy functional for these models is bounded below by the topological charge, configurations which attain this bound solve first order equations. The explicit solutions of these first order equations are given in terms of arbitrary holomorphic functions. Finally I will explain how this model can be interpreted as a gauged non-linear sigma model.

Lecture 1: A primer on solitons.
I will introduce the concept of a topological soliton through two prototypical examples, the phi^4 and Sine-Gordon models in 1+1 dimensions. Next, we will meet Derrick's theorem and learn why solitons are hard to construct in higher dimensions. Finally, we will meet some examples of higher dimensional models possessing soliton solutions.

Lecture 2: Solitons in chiral magnets.
We will meet a specific model of two dimensional chiral magnetic systems which admits soliton solutions. For a special potential term exact, degree 1, skyrmion solutions can be constructed. This leads up to meeting a critically coupled version of the model where there is a whole zoo of analyitic skyrmion solutions.

In these lectures we will discuss elegant constructions of 4d N = 4, 3d N = 4 and 2d N = (0,4) suspersymmetric gauge theories in string theory. We will learn several aspects of the relations between the dynamics of branes in string theory and the classical and quantum physics of supersymmetric gauge theories in various dimensions with different amounts of supersymmetry. In particular we will discuss the dualities in supersymmetric gauge theories predicted from string theory via branes, which involve more recent topics, dualities of boundary and corner configurations in supersymmetric field theories.

Lecture 1 - Homogenous spaces: Introduction, properties, and applications
- Motivation, definition and properties of homogeneous spaces
- Examples to build intuition
- Applications (in physics)

Lecture 2 - Homogenous space and maximally symmetric spacetimes
- Introduction to spacetimes
- Sketch of classification of maximally symmetric spacetimes
- Discussion of properties and invariants
- Relevance in physics

In these lectures we will discuss elegant constructions of 4d N = 4, 3d N = 4 and 2d N = (0,4) suspersymmetric gauge theories in string theory. We will learn several aspects of the relations between the dynamics of branes in string theory and the classical and quantum physics of supersymmetric gauge theories in various dimensions with different amounts of supersymmetry. In particular we will discuss the dualities in supersymmetric gauge theories predicted from string theory via branes, which involve more recent topics, dualities of boundary and corner configurations in supersymmetric field theories.

The lectures will highlight some recent work on solvable models of topological solitons. The first involves generalisations of the U(1) Abelian-Higgs model whose integrability is intimately related to the geometry of constant curvature Riemann surfaces. The second piece of work is a study of magnetic skyrmions in chiral magnets. Recently a family of solvable models for magnetic skyrmions in chiral magnets was introduced. The energy functional for these models is bounded below by the topological charge, configurations which attain this bound solve first order equations. The explicit solutions of these first order equations are given in terms of arbitrary holomorphic functions. Finally I will explain how this model can be interpreted as a gauged non-linear sigma model.

Lecture 1: A primer on solitons.
I will introduce the concept of a topological soliton through two prototypical examples, the phi^4 and Sine-Gordon models in 1+1 dimensions. Next, we will meet Derrick's theorem and learn why solitons are hard to construct in higher dimensions. Finally, we will meet some examples of higher dimensional models possessing soliton solutions.

Lecture 2: Solitons in chiral magnets.
We will meet a specific model of two dimensional chiral magnetic systems which admits soliton solutions. For a special potential term exact, degree 1, skyrmion solutions can be constructed. This leads up to meeting a critically coupled version of the model where there is a whole zoo of analyitic skyrmion solutions.