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.
Edward Witten, Introduction to cohomological field theory, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 (pdf)
Maxim Kontsevich, Yuri 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
Shuhan Jiang: Mathematical structures of cohomological field theories, Journal of Geometry and Physics 185 (2023) 104744 [arXiv:2202.12425, doi:10.1016/j.geomphys.2022.104744]
Particularly on generalized Seiberg-Witten theory etc.:
Shuhan Jiang, Jürgen Jost: Cohomological field theories and generalized Seiberg-Witten equations [arXiv:2407.04019]
Shuhan Jiang: Cohomological Field Theories and Generalized Seiberg-Witten Equations, talk at CQTS (Feb 2025) [slides:pdf]
Last revised on February 14, 2025 at 11:46:47. See the history of this page for a list of all contributions to it.