nLab
modular functor

Contents

Idea

The local data for a CFT in dimension dd allows to assign to each dd-dimensional cobordism Σ\Sigma a vector space of “possible correlators”: those functions on the space of conformal structures on Σ\Sigma that have the correct behaviour (satisfy the conformal Ward identities) to qualify as the (chiral) correlator of a CFT. This is called a space of conformal blocks Bl(Σ)Bl(\Sigma). This assignment is functorial under diffeomorphism. The corresponding functor is called a modular functor. (Segal 89, Segal 04, def. 5.1)

To get an actual collection of correlators one has to choose from each space of conformal blocks Bl(Σ)Bl(\Sigma) an element such that these choices glue under composition of cobordism: such that they solve the sewing constraints, see for instance at FRS-theorem on rational 2d CFT.

Dually, under a holographic principle such as CS3/WZW2 the space of conformal blocks on Σ\Sigma is equivalently the space of quantum states of the TQFT on Σ\Sigma. See at quantization of 3d Chern-Simons theory for more on this.

Properties

Relation to equivariant elliptic cohomology / equivariant tmftmf

The modular functor for GG-Chern-Simons theory restricted to genus-1 surfaces (elliptic curves) is essentially what is encoded in the universal GG-equivariant elliptic cohomology (equivariant tmf). In fact equivariant elliptic cohomology remembers also the pre-quantum incarnation of the modular functor as a systems of prequantum line bundles over Chern-Simons phase spaces (which are moduli stacks of flat connections) and remembers the quantization-process from there to the actual space of quantum states by forming holomorphic sections. See at equivariant elliptic cohomology – Idea – Interpretation in Quantum field theory for more on this.

References

Original formulations include

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

  • Graeme Segal, The definition of conformal field theory, Topology, geometry and quantum field theory London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, (2004), 421-577

  • Igor Kriz, On spin and modularity in conformal field theory, Ann. Sci. ANS (4) 36 (2003), no. 1, 57112

Lectures and reviews include

A nice review with a new concise construction is in

Discussion in the context of (2,1)-dimensional Euclidean field theories and tmf is in

A generalization of the modular functors is in

Revised on August 28, 2014 21:54:03 by David Corfield (129.12.18.145)