nLab
conformal block

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 $G$-WZW model the assignment of spaces of conformal blocks, hence by the above equivalently 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.

Related concepts

• FFRS-formalism

holographic principle in quantum field theory

bulk field theory	boundary field theory
dimension $n+1$	dimension $n$
field	source
wave function	correlation function
space of quantum states	conformal blocks

References

For 2d CFT

A review is around p. 30 of

• Krzysztof Gawędzki, Conformal field theory: a case study (arXiv:hep-th/9904145)

Detailed discussion in terms of conformal nets is in

• Arthur Bartels, Christopher Douglas, André Henriques, Conformal nets II: conformal blocks (arXiv:1409.8672)

See also

• A. Tsuchiya, Kenji Ueno, Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Studies in Pure Math. 19, 459–566, Academic Press (1989) MR92a:81191

• Kenji Ueno, Conformal field theory with gauge symmetry, Fields Institute Monographs 2008 book page

Relation to theta functions

Relation to theta functions:

• Arnaud Beauville, Yves Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385 - 419, euclid, alg-geom/9309003, MR1289330

• Arnaud Beauville, Conformal blocks, fusion rings and the Verlinde formula, Proc. of the Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc. 9, 75-96 (1996) pdf

• Krzysztof Gawędzki, Lectures on CFT (from Park City, published in QFT and strings for mathematicians, Dijkgraaf at al editors, site, source, dvi, ps

• A.A. Beilinson, Yu.I. Manin, V.V. Schechtman, Sheaves of Virasoro and Neveu-Schwarz algebras, Lecture Notes in Math. 1289, Springer 1987, 52–66

• A.Mironov, A.Morozov, Sh.Shakirov, Conformal blocks as Dotsenko-Fateev integral discriminants, arxiv/1001.0563

For higher dimensional CFT

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

• Kiyonori Gomi, An analogue of the space of conformal blocks in $(4k+2)$-dimensions (pdf)