nLab fractional-level WZW model -- references

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$]$