Quantum field theory
In a conformal field theory the conditions on correlators can be divided into two steps
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);
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 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.
holographic principle in quantum field theory
For 2d CFT
A review is around p. 30 of
- A. Tsuchiya, K. 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
A. Beauville, Y. 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 -dimensions (pdf)
Revised on November 6, 2013 03:34:13
by Urs Schreiber