Topological Gauge Theories and Symmetry TFT
A topological field theory assigns compatible amplitudes and state spaces to manifolds and their boundaries. This chapter develops that idea through Abelian Chern–Simons, BF, and finite gauge theories, then explains the bounded role of an auxiliary symmetry TFT in organizing symmetry and anomaly data.
Enter this chapter
Section titled “Enter this chapter”Begin with What Is a Topological Field Theory? before using an action or a degeneracy count as a diagnostic.
Helpful background. When Is a Topological Term Well Defined? and Fusion, Junctions, and Endpoints supply global action tests and defect junctions.
Topics
Section titled “Topics”- What Is a Topological Field Theory? — Distinguishes a topological theory from a topological term or response.
- State Spaces, Cobordisms, and Gluing — Relates cutting and gluing to composition of linear maps.
- Abelian Chern–Simons Theory — Computes finite line, braiding, and state-space data from an integral matrix.
- BF Theory as a Topological Gauge Theory — Shows how compact dynamical fields retain finite global sectors.
- Finite Gauge Theory and Dijkgraaf–Witten Twists — Constructs weighted finite-bundle sums and their possible twists.
- Operators, Boundaries, and Relative Topological Theories — Separates bulk operators from boundary and relative-theory data.
- Symmetry TFT: Encoding Symmetry, Anomaly, and Gauging — Explains conditional symmetry, anomaly, and gauging constructions using an auxiliary bulk.
How the ideas fit together
Section titled “How the ideas fit together”Gluing is the link between a local field-theory description and a global amplitude: a cut exposes a state space, and sewing contracts the matching boundary data. Knowing only its dimension loses the maps, operator products, and braiding that distinguish theories. The comparison of five nearby notions explains why a topological density or a finite state count is insufficient. See Atiyah 1988, § 2.
A symmetry TFT uses an auxiliary higher-dimensional theory with specified boundary conditions. Changing a symmetry boundary can implement gauging when the required interface exists; it is not a general identification of the auxiliary bulk with the original dynamical QFT.
Review the chapter
Section titled “Review the chapter”Can equal torus ground-state degeneracies prove that two topological theories are equivalent?
Solution
No. The count records only a state-space dimension. Theories can differ in line fusion, spins, braiding, mapping-class actions, or boundary data while sharing that dimension. Equivalence requires compatible identifications of the relevant assignments and operations.
Where to go next
Section titled “Where to go next”Continue to Operators, Boundaries, and Relative Topological Theories when a proposed symmetry construction depends on boundary choices.