nLab logarithmic CFT



Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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).


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

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



Original articles:


On the Verlinde formula for logarithmic CFTs:

On the braided tensor categories corresponding to logarithmic CFTs:

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

Logarithmic CFT describing generalized Laughlin wavefunctions with topological order:

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,pW_{p,p'}, arxiv/1202.6667; C 2C_2-cofinite W-algebras and their logarithmic representations_, arxiv/1212.6771; Logarithmic intertwining operators and W(2,2p1)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 sl^(2)\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 WW-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=2c = -2 model

Review in:

Relation to the SU(2) 0SU(2)_0 WZW-model:

On fractional-level WZW models as logarithmic CFTs

On WZW models at fractional level:

with a good review in:

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

On interpreting fractional level WZW models as logarithmic CFTs:

Reviewed in:

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

with a comprehensive account in:

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

and its lift to a logarithmic CFT:

  • F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, Logarithmic lift of the 𝔰𝔲(2) 1/2\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 𝔰𝔲^(2)\widehat{\mathfrak{su}}(2) Current Algebra at Fractional Levels [[arXiv:2208.09037]]

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

category: physics

Last revised on February 3, 2023 at 15:58:40. See the history of this page for a list of all contributions to it.