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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.

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

See also

  • 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

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)(4k+2)-dimensions (pdf)

Revised on August 21, 2014 04:18:15 by Urs Schreiber (193.175.4.219)