nLab
conformal block

Contents

Idea

In a conformal field theory the conditions on correlators can be divided into two steps

  1. for a fixed cobordism the correlators need to depend in a certain way on the choice of conformal structure, they need to satisfy the Ward identities (e.g. Gawedzki 99, around p. 30);

  2. the correlators need to glue correctly underly composition of cobordisms.

The spaces of functionals that satisfy the first of these conditions are called conformal blocks . The second condition is called the sewing constraint on conformal blocks.

So conformal blocks are something like “precorrelators” or “potential correlators” of a CFT.

The assignment of spaces of conformal blocks to surfaces and their isomorphisms under diffeomorphisms of these surfaces together constitutes the modular functor. Under CS/WZW holography this is essentially the data also given by the Hitchin connection, see at quantization of 3d Chern-Simons theory for more on this.

From a point of view closer to number theory and geometric Langlands correspondence elements of conformal blocks are naturally thought of (Beauville-Laszlo 93) as generalized theta functions (see there for more).

Properties

Holographic correspondence

The conformal blocks at least of the WZW model are by a holographic correspondence given by the space of quantum states of 3d Chern-Simons theory. See at AdS3-CFT2 and CS-WZW correspondence.

Relation to equivariant elliptic cohomology

For the GG-WZW model the assignment of spaces of conformal blocks, hence by the above equivalently 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.

holographic principle in quantum field theory

bulk field theoryboundary field theory
dimension n+1n+1dimension nn
fieldsource
wave functioncorrelation function
space of quantum statesconformal blocks

References

For 2d CFT

A review is around p. 30 of

On the Knizhnik-Zamolodchikov connection on configuration spaces of point? and conformal blocks:

Detailed discussion in terms of conformal nets is in

See also

Relation to theta functions

Relation to theta functions:

For higher dimensional CFT

Conformal blocks for self-dual higher gauge theory are discussed in