TQFT

Topological quantum field theory (TQFT) is a framework in which the physical observables of a quantum field theory depend only on the topology of the underlying spacetime, not on its geometric details such as distances or curvature. In a TQFT, one aims to produce topological invariants of manifolds and of embedded structures like knots or links.

Mathematically, a TQFT is formulated as a functor from a bordism category to a linear category. In Atiyah’s formulation, an n-dimensional TQFT assigns to every closed (n−1)-manifold a finite-dimensional vector space and to every n-dimensional cobordism a linear map between the corresponding vector spaces; gluing cobordisms corresponds to the composition of maps. This framework encodes how topology controls quantum amplitudes.

Two-dimensional TQFTs are classified by commutative Frobenius algebras. In higher dimensions, concrete models include Chern–Simons theory in three dimensions, which produces knot invariants and 3-manifold invariants such as the Witten–Reshetikhin–Turaev invariants. In four dimensions, theories associated with Donaldson and Seiberg–Witten invariants provide topological information about smooth structures on 4-manifolds. These examples illustrate how TQFTs connect physics with profound mathematical invariants.

Historically, TQFTs gained prominence with Edward Witten’s 1989 work linking Chern–Simons theory to the Jones polynomial, offering a physical derivation of knot invariants. At the level of foundations, Atiyah’s axioms helped formalize the subject and unify diverse constructions under a common language.

Extensions of the basic framework, known as extended TQFTs, assign data to lower-dimensional manifolds and are described using higher category theory. The Cobordism Hypothesis, formulated by Baez and Dolan and developed by Lurie, provides a broad structural principle for these theories. TQFTs also appear in condensed matter as models of topological phases with potential applications to fault-tolerant quantum computation.