nLab list of functorial field theories

This article is meant to give an exhaustive list of explicitly constructed nontopological functorial field theories in dimension 2 and higher. All currently known explicit constructions are nonextended, and with the exception of the Kandel construction, have dimension 2.

Free field theories

Massless boson

• Hessel Posthuma, The Heisenberg group and conformal field theory, arXiv.

Massive boson

• Santosh Kandel?, Functorial Quantum Field Theory in the Riemannian setting, arXiv.

Fermion

• James E. Tener, Construction of the unitary free fermion Segal CFT, arXiv.

Field theories with interaction

Pickrell

• Doug Pickrell: $P(\phi)_2$ Quantum Field Theories and Segal’s axioms, Commun. Math. Phys. 280 (2008) 403-425 [doi:10.1007/s00220-008-0467-8, arXiv:math-ph/0702077]

$$S(\Phi) \;=\; \int_\Sigma 2^{-1} \big(\|d\Phi\|^2+m^2\Phi^2\big) + P(\Phi)$$

Liouville field theory

See Liouville theory for more information.

• Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Segal’s axioms and bootstrap for Liouville Theory, arXiv.