Contents

# Contents

## 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 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 $c=-2$-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).

## References

### General

Original articles:

Review:

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:

• 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

Review in:

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

### On fractional-level WZW models as logarithmic CFTs

with a good review in:

and interpreted as logarithmic CFTs:

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 quasi-characters at fractional level:

• Sachin Grover, Quasi-Characters in $\widehat{\mathfrak{su}}(2)$ Current Algebra at Fractional Levels $[$arXiv:2208.09037$]$

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

category: physics

Last revised on June 12, 2022 at 21:05:55. See the history of this page for a list of all contributions to it.