nLab logarithmic CFT

Idea

A 2d CFT is called logarithmic (LCFT) if it looks like a rational conformal field theory except that some of the 2-point functions may have a logarithmic dependence on the field insertion, in addition to the usual power law (see, e.g. Fuchs & Schweigert 2019, (8) for a clear account).

Examples

A classical example of a logarithmic CFT it the “triple algebra of the $c=-2$ Virasoso model”, see the references below.

More recently it was understood (Gaberdiel 2001, review in Ridout 2010, 2020) that the chiral WZW models at “admissible” fractional levels are examples of logarithmic CFTs, see the references further below. (The WZW model at non-integer-level does not have a geometric sigma-model-description but but it still does make sense as a current algebra/vertex operator algebra/modular functor of conformal blocks).

In fact, the triple algebra $c=-2$-model turns out to be essentially equivalent to the $\mathfrak{su}(2)$-WZW model at admissible level $k = 0$ (Nichols 2002, cf. at KZ-equation, here).

Related concepts

• rational CFT

• boundary CFT

References

General

Original articles:

• John Cardy, Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications, J. Phys. A: Math. Theor. 46 (2013) 494001 $[$arXiv:1302.4279, doi:10.1088/1751-8113/46/49/494001$]$

Review:

• Matthias R. Gaberdiel, An algebraic approach to logarithmic conformal field theory, Int. J. Mod. Phys. A 18 (2003) 4593-4638 $[$arXiv:hep-th/0111260, doi:10.1142/S0217751X03016860$]$

• Michael Flohr, 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$]$

• Thomas Creutzig, David Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A: Math. Theor. 46 (2013) 494006 (doi:10.1088/1751-8113/46/49/494006, arXiv:1303.0847)

• Matthijs Hogervorst, Miguel Paulos, Alessandro Vichi, The ABC (in any D) of logarithmic CFT, J. High Energ. Phys. 2017 201 (2017) $[$doi:10.1007/JHEP10(2017)201201), arXiv:1605.03959$]$

On the Verlinde formula for logarithmic CFTs:

• David Ridout, Simon Wood, The Verlinde formula in logarithmic CFT, J. Phys.: Conf. Ser. 597 012065 $[$arXiv:1409.0670, doi:10.1088/1742-6596/597/1/012065$]$

• Thomas Creutzig, Terry Gannon, Logarithmic conformal field theory, log-modular tensor categories and modular forms, J. Phys. A 50, 404004 (2017) (doi:10.1088/1751-8121/aa8538, arXiv:1605.04630)

• Thomas Creutzig: Resolving Verlinde’s formula of logarithmic CFT [arXiv:2411.11383]

On the braided tensor categories corresponding to logarithmic CFTs:

• Thomas Creutzig, Simon Lentner, Matthew Rupert, Characterizing braided tensor categories associated to logarithmic vertex operator algebras $[$arXiv:2104.13262$]$

On generalizing the FRS-theorem on rational 2d CFT to logarithmic 2dCFTs:

• Jürgen Fuchs, Christoph Schweigert, Full Logarithmic Conformal Field theory - an Attempt at a Status Report, Proc. of Higher Structures in M-Theory 2018, Fortschr. Phys. 67 8-9 (2019) $[$arXiv:1903.02838, doi:10.1002/prop.201910018$]$

Logarithmic CFT describing generalized Laughlin wavefunctions with topological order:

• Victor Gurarie, Michael Flohr, Chetan Nayak, The Haldane-Rezayi Quantum Hall State and Conformal Field Theory, Nucl. Phys. B 498 (1997) 513-538 $[$arXiv:cond-mat/9701212, doi:10.1016/S0550-3213%2897%2900351-9$]$

  (review in Flohr 2003, §5.4)

See also:

• D. Adamović, A. Milas, Lattice construction of logarithmic modules for certain vertex algebras, Selecta Math. New Ser. 15 (2009), 535–561 arxiv/0902.3417; An analogue of modular BPZ equation in logarithmic (super)conformal field theory, to appear in Contemporary Mathematics 497; An explicit realization of logarithmic modules for the vertex operator algebra $W_{p,p'}$, arxiv/1202.6667; $C_2$-cofinite W-algebras and their logarithmic representations_, arxiv/1212.6771; Logarithmic intertwining operators and $W(2,2p-1)$-algebras, J.Math.Phys.48:073503, 2007, arxiv/math.RT/0702081

• 2011 workshop Logarithmic CFT and representation theory

• Matthias R Gaberdiel, Ingo Runkel, From boundary to bulk in logarithmic CFT, J. Phys. A: Math. Theor. 41 075402 (2008) doi

• A. M. Semikhatov, I. Yu. Tipunin, Logarithmic $\widehat{sl}(2)$ CFT models from Nichols algebras 1, arxiv/1301.2235

• A. M. Semikhatov, Factorizable ribbon quantum groups in logarithmic conformal field theories, Theor.Math.Phys.154:433–453, 2008 arxiv/0705.4267; A Heisenberg double addition to the logarithmic Kazhdan–Lusztig duality, Lett.Math.Phys.92:81–98, 2010 arxiv/0905.2215

• Boris Feigin, A. M. Gainutdinov, A. M. Semikhatov, I. Yu. Tipunin, Logarithmic extensions of minimal models: characters and modular transformations, Nucl.Phys. B757:303–343,2006 hep-th/0606196;

  • Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center, Commun. Math. Phys. 265 (2006) 47–93;

  • Kazhdan–Lusztig correspondence for the representation category of the triplet $W$-algebra in logarithmic CFT, Theor. Math. Phys. 148 (2006) 1210–1235;

  • Kazhdan–Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models, J. Math. Phys. 48 (2007) 032303

• Philippe Ruelle, Logarithmic conformal invariance in the Abelian sandpile model, arxiv/1303.4310

Example: The the $c = -2$ model

• Victor Gurarie, Logarithmic Operators in Conformal Field Theory, Nucl.Phys. B 410 (1993) 535-549 $[$arXiv:hep-th/9303160, doi:10.1016/0550-3213(93)90528-W$]$

• Matthias R. Gaberdiel, Horst G. Kausch, A Rational Logarithmic Conformal Field Theory, Phys.Lett. B 386 (1996) 131-137 $[$arXiv:hep-th/9606050, doi:10.1016/0370-2693(96)00949-5$]$

Review in:

• Gaberdiel 2001, §3

Relation to the $SU(2)_0$ WZW-model:

• Nichols 2002

On fractional-level WZW models as logarithmic CFTs

On WZW models at fractional level:

• Sunil Mukhi, Sudhakar Panda, Fractional-level current algebras and the classification of characters, Nuclear Physics B 338 1 (1990) 263-282 $[$doi:10.1016/0550-3213(90)90632-N$]$

• Gregory Moore, Nicholas Read, p. 389 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$]$

  (suggesting the fractional level $k = -5/4$ as related to Laughlin wavefunctions of anyons)

• Hidetoshi Awata, Yasuhiko Yamada, Fusion rules for the fractional level $\widehat{\mathfrak{sl}(2)}$ algebra, Mod. Phys. Lett. A 7 (1992) 1185-1196 $[$spire:332974, doi:10.1142/S0217732392003645$]$

• P. Furlan, A. Ch. Ganchev, R. Paunov, Valentina B. Petkova, Solutions of the Knizhnik-Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models, Nucl. Phys. B394 (1993) 665-706 (arXiv:hep-th/9201080, doi:10.1016/0550-3213(93)90227-G)

• J. L. Petersen, J. Rasmussen, M. Yu, Fusion, Crossing and Monodromy in Conformal Field Theory Based on $SL(2)$ Current Algebra with Fractional Level, Nucl. Phys. B481 (1996) 577-624 (arXiv:hep-th/9607129, doi:10.1016/S0550-3213(96)00506-8)

• Boris Feigin, Feodor Malikov, Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level, p. 357-405 in: Loday, Stasheff, Voronov (eds.) Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202, AMS (1997) $[$arXiv:q-alg/9511011, ams:conm-202$]$

  (an $\mathfrak{osp}(1\vert2)$-factor appears)

with a good review in:

• A. Ch. Ganchev, Valentina B. Petkova, G. M. T. Watts, A note on decoupling conditions for generic level $\widehat{s l}(3)_k$ and fusion rules, Nucl. Phys. B 571 (2000) 457-478 $[$doi:10.1016/S0550-3213(99)00745-2$]$

On braided fusion categories formed by affine Lie algebra-representations at admissible fractional level:

• Thomas Creutzig, Yi-Zhi Huang, Jinwei Yang, Braided tensor categories of admissible modules for affine Lie algebras, Commun. Math. Phys. 362 (2018) 827–854 $[$arXiv:1709.01865$]$

On interpreting fractional level WZW models as logarithmic CFTs:

• Matthias R. Gaberdiel, Fusion rules and logarithmic representations of a WZW model at fractional level, Nucl. Phys. B 618 (2001) 407-436 $[$arXiv:hep-th/0105046, doi:10.1016/S0550-3213(01)00490-4)$]$

• Matthias R. Gaberdiel, Section 5 of: An algebraic approach to logarithmic conformal field theory, Int. J. Mod. Phys. A 18 (2003) 4593-4638 $[$arXiv:hep-th/0111260, doi:10.1142/S0217751X03016860$]$

• David Ridout, $\widehat{\mathfrak{sl}}(2)_{-1/2}$: A Case Study, Nucl. Phys. B 814 (2009) 485-521 $[$arXiv:0810.3532, doi:10.1016/j.nuclphysb.2009.01.008$]$

• Thomas Creutzig, David Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models I, Nuclear Physics B 865 1 (2012) 83-114 $[$arXiv:1205.6513, doi:10.1016/j.nuclphysb.2012.07.018$]$

• Thomas Creutzig, David Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models II, Nuclear Physics B 875 2 (2013) 423-458 $[$arXiv:1306.4388, doi:10.1016/j.nuclphysb.2013.07.008$]$

• Thomas Creutzig, David Ridout, Section 4 of: Logarithmic conformal field theory: beyond an introduction, J. Phys. A: Math. Theor. 46 (2013) 494006 (doi:10.1088/1751-8113/46/49/494006, arXiv:1303.0847)

• Kazuya Kawasetsu, David Ridout, Relaxed highest-weight modules I: rank 1 cases, Commun. Math. Phys. 368 (2019) 627–663 $[$arXiv:1803.01989, doi:10.1007/s00220-019-03305-x$]$

• Kazuya Kawasetsu, David Ridout, Relaxed highest-weight modules II: classifications for affine vertex algebras, Communications in Contemporary Mathematics, 24 05 (2022) 2150037 $[$arXiv:1906.02935, doi:10.1142/S0219199721500371$]$

Reviewed in:

• David Ridout, Fractional Level WZW Models as Logarithmic CFTs (2010) $[$pdf, pdf$]$

• David Ridout, Fractional-level WZW models (2020) $[$pdf, pdf$]$

In particular, the logarithmic $c = -2$ model is essentially an admissible-level WZW model (namely at level $k = 0$):

• Alexander Nichols, Extended chiral algebras in the $SU(2)_0$ WZNW model, JHEP 04 (2002) $[$doi:10.1088/1126-6708/2002/04/056, arXiv:hep-th/0112094$]$

with a comprehensive account in:

• Alexander Nichols, $SU(2)_k$ Logarithmic Conformal Field Theories, PhD thesis, Oxford (2002) $[$arXiv:hep-th/0210070, spire:599081$]$

On the $c = -1$ model as the $\mathfrak{su}(2)$ WZW model at fractional level $-1/2$ and relation to the beta-gamma system:

• F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, The $\mathfrak{su}(2)_{-1/2}$ WZW model and the beta-gamma system, Nucl. Phys. B 647 (2002) 363-403 $[$arXiv:hep-th/0207201, doi:10.1016/S0550-3213(02)00905-7a$]$

and its lift to a logarithmic CFT:

• F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, Logarithmic lift of the $\mathfrak{su}(2)_{-1/2}$ model, Nuclear Physics B 686 3 (2004) 313-346 [doi:10.1016/j.nuclphysb.2004.02.039]

On quasi-characters at fractional level:

• Sachin Grover, Quasi-Characters in $\widehat{\mathfrak{su}}(2)$ Current Algebra at Fractional Levels, SciPost Phys. Core 6 068 (2023) $[$arXiv:2208.09037, doi:10.21468/SciPostPhysCore.6.4.068$]$

Identification of would-be fractional level $\mathfrak{su}(2)$ conformal blocks in twisted equivariant K-theory of configuration spaces of points:

• Hisham Sati, Urs Schreiber, Rem. 2.3 in: Anyonic defect branes in TED-K-theory, Rev. Math. Phys. 35 06 (2023) 2350009 $[$arXiv:2203.11838, doi:10.1142/S0129055X23500095$]$

category: physics