nLab D=2 CFT as functorial field theory -- references

$D=2$ CFT as functorial field theory

Discussion of D=2 conformal field theory as a functorial field theory, namely as a monoidal functor from a 2d conformal cobordism category to Hilbert spaces:

• Graeme Segal, The definition of conformal field theory, in: K. Bleuler, M. Werner (eds.), Differential geometrical methods in theoretical physics (Proceedings of Research Workshop, Como 1987), NATO Adv. Sci. Inst., Ser. C: Math. Phys. Sci. 250 Kluwer Acad. Publ., Dordrecht (1988) 165-171 $[$doi:10.1007/978-94-015-7809-7$]$

and including discussion of modular functors:

• Graeme Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol (1989) 22-37.

• Graeme Segal, The definition of conformal field theory, in: Ulrike Tillmann (ed.), Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser. 308, Cambridge University Press (2004) 421-577 $[$doi:10.1017/CBO9780511526398.019, pdf, pdf$]$

General construction for the case of rational 2d conformal field theory is given by the

• FRS-theorem on rational 2d CFT

See also:

• Greg Moore, Graeme Segal, D-branes and K-theory in 2D topological field theory (arXiv:hep-th/0609042)

• Richard Blute, Prakash Panangaden, Dorette Pronk, Conformal field theory as a nuclear functor, Electronic Notes in Theoretical Computer Science Volume 172, 1 April 2007, Pages 101-132 GDP Festschrift (pdf, doi:10.1016/j.entcs.2007.02.005)

A different but closely analogous development for chiral 2d CFT (vertex operator algebras, see there for more):

• Yi-Zhi Huang, Geometric interpretation of vertex operator algebras, Proc. Natl. Acad. Sci. USA 88 (1991) pp. 9964-9968 (doi:10.1073/pnas.88.22.9964)

Discussion of the case of Liouville theory:

• Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Segal’s axioms and bootstrap for Liouville Theory [arXiv:2112.14859]

Early suggestions to refine this to an extended 2-functorial construction:

• Stefan Stolz, Peter Teichner, What is an elliptic object?

A step towards generalization to 2d super-conformal field theory:

• Stephan Stolz, Peter Teichner, Supersymmetric field theories and generalized cohomology, in: H. Sati, U. Schreiber, Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics 83 (2011), 279–340 (arXiv:1108.0189, doi:10.1090/pspum/083/2742432)

Discussion of 2-functorial chiral 2d CFT:

• André Henriques, The complex cobordism 2-category, 2021 (video)