# Contents

## Idea

The local data for a CFT in dimension $d$ allows to assign to each $d$-dimensional cobordism $\Sigma$ a vector space of “possible correlator”s: those functions on the space of conformal structures on $\Sigma$ that have the correct behaviour to qualify as the (chiral) correlator of a CFT. This is called a space of conformal blocks $Bl(\Sigma)$. 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 $Bl(\Sigma)$ 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 $\Sigma$ is equivalently the space of quantum states of the TQFT on $\Sigma$. See at quantization of 3d Chern-Simons theory for more on this.

## Properties

### Relation to equivariant ellitpic cohomology / equivariant $tmf$

The modular functor for $G$-Chern-Simons theory restricted to genus-1 surfaces (elliptic curves) is essentially what is encoded in the universal $G$-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 April 16, 2014 08:20:04 by Urs Schreiber (145.116.129.4)