Topological quantum field theory

Summary

A topological quantum field theory assigns a vector space to each closed manifold of a fixed dimension $n$. It assigns a linear map to each $(n+1)$-manifold whose boundary joins two such manifolds. Gluing manifolds must correspond to composing linear maps, and disjoint union must correspond to the tensor product. A closed manifold of the top dimension then receives a number, which is an invariant of the manifold. Atiyah stated these axioms, drawing on work of Witten and Segal in physics. A main example is Chern–Simons theory, which produces quantum invariants of 3-manifolds and of knots.