Prerequisites: SPA6308 Spacetime and Gravity, SPA6324 Mathematical Techniques 4 (or equivalent) 
Co-requisites: SPA7018 Relativistic Waves and Quantum Fields (or equivalent)

Course organiser: Dr Costis Papageorgakis
Course deputy: Prof Andreas Brandhuber

Synopsis:

The aim of this course is to provide the student with a number of advanced mathematical tools from differential geometry, essential for research in modern Theoretical Physics, and apply them to certain physical contexts.

More specifically, the notation of differential forms will be introduced and the geometric aspects of gauge theory will be explored. Gravity will be interpreted as a gauge theory in this geometric setting. Manifolds will be introduced and studied, leading to the definition of fibre bundles. Finally, we will explore the Dirac and 't Hooft-Polyakov monopoles, the Yang-Mills instanton and the gravitational instanton and their associated understanding in fibre-bundle language.
 

Syllabus:

Exterior algebra (Vector spaces - Dual basis - Alternating forms - Wedge product - Inner derivative - Pullback - Orientation - Vector-valued forms)

Differential forms on open subsets of R^n   (Tangent vectors - Frames - Differential forms - Tangent mapping - Pullback of differential forms - Exterior derivative - The Poincaré lemma and de Rham cohomology - Integration of n-forms - Integration of p-forms)

Metric structures (Metric on vector spaces - Induced metric on dual space - Hodge star - Isometries - Metric on open subset of R^n - Holonomic and orthonormal frames - Isometries for open subsets of  R^n - Coderivative)

Gauge theories (Maxwell's equations - Connection = potential - Curvature = field strength - Exterior covariant derivative - Yang-Mills theories)

Einstein-Cartan theory (Equivalence principle - Cartan's structure equations - Metric connections - Action and field equations - A farewell to the connection - Einstein gauge - Geodesics - Geometric interpretation of curvature and torsion)

Manifolds (Differential manifolds - Differentiable mappings - Cartesian product on manifolds - Submanifolds)

Fibre bundles (Notion of fibre bundles - Bundle maps - Examples - Associated bundle - Sections)

Monopoles, instantons and related fibre bundles (Dirac monopole - 't Hooft-Polyakov monopole - Yang-Mills instanton - Dirac monopole as a connection on a nontrivial bundle - Recovering the Dirac monopole from the 't Hooft-Polyakov monopole - Instanton bundle - Chern classes)

Recommended Books:

Göckeler and Schücker, "Differential geometry, gauge theories and gravity" (Cambridge Monographs on Mathematical Physics); the main textbook for this course.

Carroll, "Spacetime and Geometry: An introduction to General Relativity" (Pearson); the first chapters of this textbook also cover differential geometry from a more physical viewpoint.

Nakahara, "Geometry, topology, and physics" (IoP Publishing); for additional reading on mathematical concepts and applications.

Manton and Sutcliffe, "Topological Solitons" (Cambridge Monographs on Mathematical Physics); for additional reading on solitons and instantons.

Tong, "TASI Lectures on Solitons"; for additional reading on solitons and instantons