nLab conformal block

Contents

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

Relation to braiding of field insertions

In the case of 2d rational CFT the representation category of the corresponding vertex operator algebra is a modular tensor category (see there), hence a braided monoidal category with extra properties.

In this case the monoidal- and the braided structure (hence the modular tensor structure) on the underlying representation category is entirely fixed by the space of conformal blocks of the 2d CFT on the Riemann sphere (the “genus zero conformal blocks”).

This may be found highlighted in EGNO 15, p. 266, Runkel, Sec. 4.3. The essentially equivalent fact that the genus=0 conformal blocks already determine the modular functor of the CFT is proven in Andersen & Ueno 2012.

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

  • 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

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)

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

Braid representations via twisted cohomology of configuration spaces

The “hypergeometric integral” construction of conformal blocks for affine Lie algebra/WZW model-2d CFTs and of more general solutions to the Knizhnik-Zamolodchikov equation, via twisted de Rham cohomology of configuration spaces of points, originates with:

following precursor observations due to:

The proof that for rational levels this construction indeed yields conformal blocks is due to:

Review:

See also:

This “hypergeometric” construction uses results on the twisted de Rham cohomology of configuration spaces of points due to:

also:

reviewed in:

  • Yukihito Kawahara, The twisted de Rham cohomology for basic constructionsof hyperplane arrangements and its applications, Hokkaido Math. J. 34 2 (2005) 489-505 [[doi:10.14492/hokmj/1285766233]]

Discussion for the special case of level=0=0 (cf. at logarithmic CFT – Examples):

Interpretation of the hypergeometric construction as happening in twisted equivariant differential K-theory, showing that the K-theory classification of D-brane charge and the K-theory classification of topological phases of matter both reflect braid group representations as expected for defect branes and for anyons/topological order, respectively:

Laughlin wavefunctions as conformal blocks

Relating anyonic topologically ordered Laughlin wavefunctions to conformal blocks:

Specifically for logarithmic CFT:

Specifically for su(2)-anyons:

Last revised on June 10, 2022 at 11:03:39. See the history of this page for a list of all contributions to it.