Studying topological quantum field theories (TQFTs) requires a solid foundation in several mathematical and physical concepts. Here are some key prerequisites to consider:
Quantum Mechanics: Understanding the principles and mathematical formalism of quantum mechanics is essential. This includes concepts such as wave functions, observables, operators, and the Schrödinger equation.
Quantum Field Theory: Familiarity with quantum field theory is crucial since TQFTs build upon its framework. You should be comfortable with concepts like Lagrangian and Hamiltonian formalism, symmetries, Feynman diagrams, path integrals, and quantization of fields.
Differential Geometry: TQFTs heavily rely on differential geometry, particularly the study of manifolds and their properties. Knowledge of differential forms, tensors, vector bundles, connections, curvature, and Lie groups will be helpful.
Algebraic Topology: An understanding of algebraic topology is fundamental to TQFTs. Topics such as homotopy theory, fundamental groups, homology and cohomology, characteristic classes, and the Mayer-Vietoris sequence are relevant.
Category Theory: TQFTs are formulated in the language of category theory, so familiarity with this branch of mathematics is beneficial. Concepts like functors, natural transformations, limits, colimits, and adjunctions play a crucial role.
Symplectic Geometry: Some TQFTs, such as the ones related to Chern-Simons theory, are deeply connected to symplectic geometry. Knowledge of symplectic manifolds, symplectic forms, and Hamiltonian dynamics will be useful.
Mathematical Physics: Having a broad understanding of mathematical physics topics can be beneficial. This includes topics such as group theory, representation theory, Lie algebras, and mathematical aspects of quantum field theory.
Mathematical Rigor: TQFTs involve rigorous mathematical reasoning, so a strong background in mathematical rigor and proof techniques is essential. This includes familiarity with logic, set theory, and mathematical analysis.
It's important to note that studying TQFTs is a highly advanced topic that requires considerable mathematical maturity and a solid foundation in physics. It is recommended to follow a structured curriculum or consult textbooks and research papers on the subject to gain a deeper understanding.