nLab cohomological field theory

Idea

The term “cohomological field theory” is mostly used for the “Witten style” topological quantum field theories which arise from a topological twist of a supersymmetric quantum field theory, notable example being the A-model and the B-model topological string.

Mathematically these topological field theories came to be known as TCFTs. With the proof of the cobordism hypothesis these were understood to be the “non-compact” 2-dimensional extended TQFTs with codomain an (infinity,1)-category of chain complexes, where “non-compact” refers to all cobordisms being required to have non-empty outgoing boundary.

For more on all this see at TCFT and at 2d TQFT.

References

• Edward Witten, Introduction to cohomological field theory, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 (pdf)
• M. Kontsevich, Yu. Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164.3 (1994) 525–562
• Rahul Pandharipande, Cohomological field theory calculations, Proc. ICM 2018 Rio de Janeiro, Vol. 1 (869–898) pdf