Cohomological Field Theory — TQFT

Partition functions, cobordism categories, and topological invariants on 2D surfaces

Mode

Z: 2-Cob → Vect_k
Z(M) ∈ V, Z(W): V→W

A 2D TQFT is a symmetric monoidal functor from the cobordism category 2-Cob to vector spaces. Atiyah's axioms (1988) formalize Witten's topological field theories.

TQFT Axioms

1. Z(∅) = k (ground field)
2. Z(M⊔N) = Z(M)⊗Z(N)
3. Z(M̄) = Z(M)* (dual)
4. Z(M∘W) = Z(M)∘Z(W)
5. Diffeomorphism invariant

These axioms are equivalent to a commutative Frobenius algebra structure on Z(S¹).