nLab
FFRS-formalism

Idea

The FFRS-formalism (after Fröhlich, Fuchs, Runkel, Schweigert) is a state sum model? description of the topological part of rational 2-dimensional conformal field theory.

It proceeds by constructing a kind of boundary field theory to the Reshetikhin-Turaev construction: given a modular tensor category $C$ the a 2-dimensional TQFT which is such that

• if the MTC $C$ is equivalent to a representation category of a vertex operator algebra, one can construct an identification of the linear map assigned by the construction to a surface to an element in the space of conformal blocks of that surface.

  • A conformal block is a function that depends onthe conformal structure on the surface, so that after this identification the construction yields an assignment of data to conformal surfaces. It is well known that a CFT assigns to a surface a conformal block in this way. What the FFRS prescription achieves is that it provides a way to pick all these conformal blocks in such a way that they actually satisfy the sewing law, i.e. that they actually conspire to yields a functor on conformal cobordisms.

References

A brief summary on a few pages is in

• Jürgen Fuchs?, Ingo Runke?, Christoph Schweigert, Open strings and 3d topological field theory (pdf)

For a list of references see

• FRS reviews

A survey of the central theorem that the FRS construction solves the sewing constraints is at

• The FRS theorem on CFT

and a discussion of the converse, that every rarional 2-d CT is obtained this way is at

• FFRS on uniqueness of conformal field theory

An indication of how the FRS formalism follows from 3-dimensional Chern-Simons theory regarded as an extended topological quantum field theory with defects and the holographic principle is in

• Anton Kapustin, Natalia Saulina, _ Surface operators in 3d TFT and 2d Rational CFT_ in Hisham Sati, Urs Schreiber, Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings in Symposia in Pure Mathematics, AMS, (2011)