Notes on Gauge Theory
These notes stemmed from a reading course we (Matei and Nilay) took with Prof. Peter Woit at Columbia University in the fall semester of 2013. They are an attempt to write down clearly and concisely the basic structures and techniques used in gauge theory, largely motivated by a general lack of readable/comprehensive/available sources. We recommend a working knowledge of basic differential geometry (including differential forms, vector bundles, integration, de Rham cohomology, and Lie groups/algebras) and a minimal exposure to representation theory (Lie algebras and adjoint actions). For the later chapters (that have not yet been started), there will likely be a significant amount of algebraic topology. A background in physics (electromagnetism, quantum mechanics, and the basics of quantum field theory) is not required, but may be useful.
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The following outlines the material we intend to cover. Much of the material is, at the moment, rather formal and presented without much motivation or explanation (especially in regards to physics). This will be addressed at a later date. Note: topics marked with a (?) are highly tentative.
- Principal bundles
- Connections and curvature
- Yang-Mills theory and instantons
- Chern-Simons theory and link invariants
- Chern-Weil theory (?)
- An introduction to Donaldson theory (?)
- An introduction to Seiberg-Witten theory (?)