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

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 $G$-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

• Laughlin wavefunction

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)

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

• Toshitake Kohno, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39, 575–598 (2014) [doi:10.1007/s40306-014-0088-6, pdf]

• Toshitake Kohno, §1.4 in: Conformal field theory and topology, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs 210. Iwanami Series in Modern Mathematics. Amer. Math. Soc. (2002) [AMS:mmono-210]

and via Bohr-Sommerfeld leaves in geometric quantization:

• Andrei Tyurin, On Bohr-Sommerfeld bases, Izvestiya: Mathematics 64 5 (2000) 1033–1064 [arXiv:math/9909084, doi:10.1070/IM2000v064n05ABEH000308, pdf]

Brief review:

• Jürgen Fuchs, Christoph Schweigert, Simon Wood, Yang Yang, pp. 8 of: Algebraic structures in two-dimensional conformal field theory, Encyclopedia of Mathematical Physics [arXiv:2305.02773]

Discussion in terms of conformal nets:

• 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

• Ratul Mahanta, Tanmoy Sengupta, Modular linear differential equations for four-point sphere conformal blocks [arXiv:2211.05158]

• Mikhail Pavlov, Global torus blocks in the necklace channel [arXiv:2302.10153]

• Vladimir Belavin, Juan Ramos Cabezas, Boris Runov: Shadow formalism for supersymmetric conformal blocks [arXiv:2408.07684]

Conformal blocks for self-dual higher gauge theory:

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

In particular, argument that the higher conformal blocks of the $D=6$ $\mathcal{N}=(2,0)$ SCFT reduce to confomal blocks of the ordinary 2d WZW model after KK-compactification on a Riemann surface:

• Edward Witten, p. 22 of: Geometric Langlands From Six Dimensions, in Peter Kotiuga (ed.) A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proceedings & Lecture Notes 50 AMS (2010) [arXiv:0905.2720, ISBN:978-0-8218-4777-0]

On irregular conformal blocks for Liouville theory

• Davide Gaiotto, Joel Lamy-Poirier: Irregular Singularities in the $H_3^+$ WZW Model [arXiv:1301.5342]

• Babak Haghighat, Yihua Liu, Nicolai Reshetikhin, Flat Connections from Irregular Conformal Blocks [arXiv:2311.07960]

• Xia Gu, Babak Haghighat, Kevin Loo, Irregular Fibonacci Conformal Blocks [arXiv:2311.13358]

• Babak Haghighat: Flat Connections from Irregular Conformal Blocks, talk at CQTS (Feb 2024) [video: zm, kt]

See also:

• Qianyu Hao, Andrew Neitzke: A new construction of $c=1$ Virasoro blocks [arXiv:2407.04483]

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

• Alexander Beilinson, Yuri Manin, Vadim 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:

• Vadim Schechtman, Alexander Varchenko, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$cds:1044951$]$

• Etsuro Date, Michio Jimbo, Atsushi Matsuo, Tetsuji Miwa, Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation, International Journal of Modern Physics B 04 05 (1990) 1049-1057 $[$doi:10.1142/S0217979290000528$]$

  (for $\mathfrak{sl}(2,\mathbb{C})$)

• Atsushi Matsuo, An application of Aomoto-Gelfand hypergeometric functions to the $SU(n)$ Knizhnik-Zamolodchikov equation, Communications in Mathematical Physics 134 (1990) 65–77 $[$doi:10.1007/BF02102089$]$

• Vadim Schechtman, Alexander Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 $[$doi:10.1007/BF00626523$]$

• Vadim Schechtman, Alexander Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 $[$dml:143938, pdf$]$

following precursor observations due to:

• Vladimir S. Dotsenko, Vladimir A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nuclear Physics B 240 3 (1984) 312-348 $[$doi:10.1016/0550-3213(84)90269-4$]$

• Philippe Christe, Rainald Flume, The four-point correlations of all primary operators of the $d = 2$ conformally invariant $SU(2)$ $\sigma$-model with Wess-Zumino term, Nuclear Physics B

  282 (1987) 466-494 $[$doi:10.1016/0550-3213(87)90693-6$]$

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

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by correlators in Wess-Zumino-Witten models, Lett Math Phys 20 (1990) 291–297 $[$doi:10.1007/BF00626525$]$

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun. Math. Phys. 163 (1994) 173–184 $[$doi:10.1007/BF02101739$]$

  (for $\mathfrak{sl}(2, \mathbb{C})$)

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 1 (1995) 219-247 [euclid:cmp/1104272957]

Review:

• Alexander Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics 21, World Scientific 1995 (doi:10.1142/2467)

• Ivan Cherednik, Section 8.2 of: Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Mathematical Society of Japan Memoirs 1998 (1998) 1-96 $[$doi:10.2969/msjmemoirs/00101C010$]$

• Pavel Etingof, Igor Frenkel, Alexander Kirillov, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998) $[$ISBN:978-1-4704-1285-2, review pdf$]$

• Toshitake Kohno, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 $[$doi:10.5427/jsing.2012.5g, pdf$]$

• Toshitake Kohno, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39 (2014) 575–598 $[$doi:10.1007%2Fs40306-014-0088-6, pdf$]$

• Toshitake Kohno, Introduction to representation theory of braid groups, Peking 2018 $[$pdf, pdf$]$

  (motivation from braid representations)

See also:

• Alexander Varchenko, Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys. 171 1 (1995) 99-137 $[$arXiv:hep-th/9403102, doi:10.1007/BF02103772$]$

• Edward Frenkel, David Ben-Zvi, Section 14.3 in: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, AMS 2004 $[$ISBN:978-1-4704-1315-6, web$]$

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

• Peter Orlik, Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent Math 56 (1980) 167–189 $[$doi:10.1007/BF01392549$]$

• Kazuhiko Aomoto, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan 39 2 (1987) 191-208 $[$doi:10.2969/jmsj/03920191$]$

• Hélène Esnault, Vadim Schechtman, Eckart Viehweg, Cohomology of local systems on the complement of hyperplanes, Inventiones mathematicae 109.1 (1992) 557-561 $[$pdf$]$

• Vadim Schechtman, H. Terao, Alexander Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, Journal of Pure and Applied Algebra 100 1–3 (1995) 93-102 $[$arXiv:hep-th/9411083, doi:10.1016/0022-4049(95)00014-N$]$

also:

• Peter Orlik, Hypergeometric integrals and arrangements, Journal of Computational and Applied Mathematics 105 (1999) 417–424 $[$doi:10.1016/S0377-0427(99)00036-9, pdf$]$

• Daniel C. Cohen, Peter Orlik, Arrangements and local systems, Math. Res. Lett. 7 (2000) 299-316 $[$arXiv:math/9907117, doi:10.4310/MRL.2000.v7.n3.a5$]$

reviewed in:

• Yukihito Kawahara, The twisted de Rham cohomology for basic constructions of 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$=0$ (cf. at logarithmic CFT – Examples):

• Fedor A. Smirnov, Remarks on deformed and undeformed Knizhnik-Zamolodchikov equations, $[$arXiv:hep-th/9210051$]$

• Fedor A. Smirnov, Form factors, deformed Knizhnik-Zamolodchikov equations and finite-gap integration, Communications in Mathematical Physics 155 (1993) 459–487 $[$doi:10.1007/BF02096723, arXiv:hep-th/9210052$]$

• S. Pakuliak, A. Perelomov, Relation Between Hyperelliptic Integrals, Mod. Phys. Lett. 9 19 (1994) 1791-1798 $[$doi:10.1142/S0217732394001647$]$

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:

• Hisham Sati, Urs Schreiber, Anyonic defect branes in TED-K-theory $[$arXiv:2203.11838$]$

Laughlin wavefunctions as conformal blocks

Relating anyonic topologically ordered Laughlin wavefunctions to conformal blocks:

• Gregory Moore, Nicholas Read, Section 2.2 of: Nonabelions in the fractional quantum hall effect, Nuclear Physics B 360 2–3 (1991) 362-396 [doi:10.1016/0550-3213(91)90407-O, pdf]

• Xiao-Gang Wen, Non-Abelian statistics in the fractional quantum Hall states, Phys. Rev. Lett. 66 (1991) 802 [doi:10.1103/PhysRevLett.66.802, pdf]

• B. Blok, Xiao-Gang Wen, Many-body systems with non-abelian statistics, Nuclear Physics B 374 3 (1992) 615-646 [doi:10.1016/0550-3213(92)90402-W]

• Xiao-Gang Wen, Yong-Shi Wu, Chiral operator product algebra hidden in certain fractional quantum Hall wave functions, Nucl. Phys. B 419 (1994) 455-479 [doi:10.1016/0550-3213(94)90340-9]

Review:

• Thors H. Hansson, M. Hermanns, Steven H. Simon, S. F. Viefers: Quantum Hall Physics – hierarchies and CFT techniques, Rev. Mod. Phys. 89 (2017) 025005 [doi:10.1103/RevModPhys.89.025005, arXiv:1601.01697]

Review in the broader context of the CS-WZW correspondence:

• Daniel C. Cabra, Gerardo L. Rossini, Explicit connection between conformal field theory and 2+1 Chern-Simons theory, Mod. Phys. Lett. A 12 (1997) 1687-1697 [doi:10.1142/S0217732397001722, arXiv:hep-th/9506054]

Specifically for logarithmic CFT:

• Victor Gurarie, Michael Flohr, Chetan Nayak, The Haldane-Rezayi Quantum Hall State and Conformal Field Theory, Nucl. Phys. B 498 (1997) 513-538 [doi:10.1016/S0550-3213(97)00351-9, arXiv:cond-mat/9701212]

• Michael Flohr, §5.4 in: Bits and pieces in logarithmic conformal field theory, International Journal of Modern Physics A, 18 25 (2003) 4497-4591 [doi:10.1142/S0217751X03016859, arXiv:hep-th/0111228]

Specifically for su(2)-anyons:

• Kazusumi Ino, Modular Invariants in the Fractional Quantum Hall Effect, Nucl. Phys. B 532 (1998) 783-806 [doi:10.1016/S0550-3213(98)00598-7, arXiv:cond-mat/9804198]

• Nicholas Read, Edward Rezayi, Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level, Phys. Rev. B 59 (1999) 8084 [doi:10.1103/PhysRevB.59.8084]

• Eddy Ardonne, Kareljan Schoutens: Wavefunctions for topological quantum registers, Annals Phys. 322 (2007) 201-235 [doi:10.1016/j.aop.2006.07.015, arXiv:cond-mat/0606217]

• Ludmil Hadjiivanov, Lachezar S. Georgiev, Braiding Fibonacci anyons [arxiv:2404.01778]