Abstract
Motivated by Wakimoto free field realisations, the bosonic ghost system of central charge c = 2 is studied using a recently proposed formalism for logarithmic conformal field theories. This formalism addresses the modular properties of the theory with the aim being to determine the (Grothendieck) fusion coefficients from a variant of the Verlinde formula. The key insight, in the case of bosonic ghosts, is to introduce a family of parabolic Verma modules which dominate the spectrum of the theory. The results include S-transformation formulae for characters, non-negative integer Verlinde coefficients, and a family of modular invariant partition functions. The logarithmic nature of the corresponding ghost theories is explicitly verified using the Nahm–Gaberdiel–Kausch fusion algorithm.
Similar content being viewed by others
References
Friedan D., Martinec E., Shenker S.: Conformal invariance: supersymmetry and string theory. Nucl. Phys. B 271, 93–165 (1986)
Wakimoto M.: Fock representation of the algebra \({A_1^{(1)}}\). Comm. Math. Phys. 104, 605–609 (1986)
Feigin B., Frenkel E.: Quantization of the Drinfeld–Sokolov reduction. Phys. Lett. B 246, 75–81 (1990)
Gurarie V.: Logarithmic operators in conformal field theory. Nucl. Phys. B 410, 535–549 (1993) arXiv:hep-th/9303160
Kausch, H.: Curiosities at c = −2. arXiv:hep-th/9510149
Kausch H.: Symplectic fermions. Nucl. Phys. B 583, 513–541 (2000) arXiv:hep-th/0003029
Lesage F., Mathieu P., Rasmussen J., Saleur H.: Logarithmic lift of the \({\widehat{su} \left(2 \right)_{-1/2}}\) model. Nucl. Phys. B 686, 313–346 (2004) arXiv:hep-th/0311039
Creutzig T., Ridout D.: Logarithmic conformal field theory: beyond an introduction. J. Phys. A 46, 494006 (2013) arXiv:1303.0847 [hep-th]
Huang Y.-Z., Lepowsky J.: Tensor categories and the mathematics of rational and logarithmic conformal field theory. J. Phys. A 46, 494009 (2013) arXiv:1304.7556 [hep-th]
Creutzig T., Ridout D.: Modular data and Verlinde formulae for fractional level WZW models I. Nucl. Phys. B 865, 83–114 (2012) arXiv:1205.6513 [hep-th]
Ridout D.: Fusion in fractional level \({\widehat{\mathfrak{sl}} \left(2 \right)}\) -theories with \({k=-\frac{1}{2}}\) . Nucl. Phys. B 848, 216–250 (2011) arXiv:1012.2905 [hep-th]
Schellekens A., Yankielowicz S.: Simple currents, modular invariants and fixed points. Int. J. Mod. Phys. A 5, 2903–2952 (1990)
Ridout, D., Wood, S.: The Verlinde formula in logarithmic CFT. arXiv:1409.0670 [hep-th]
Creutzig T., Ridout D.: Relating the archetypes of logarithmic conformal field theory. Nucl. Phys. B 872, 348–391 (2013) arXiv:1107.2135 [hep-th]
Lesage F., Mathieu P., RasmussenJ. Saleur H.: The \({\widehat{su} \left(2 \right)_{-1/2}}\) WZW model and the βγ system. Nucl. Phys. B 647, 363–403 (2002) arXiv:hep-th/0207201
Ridout D.: \({\widehat{\mathfrak{sl}}\left(2 \right)_{-1/2}}\) and the triplet model. Nucl. Phys. B 835, 314–342 (2010) arXiv:1001.3960 [hep-th]
Humphreys, J.: Representations of semisimple Lie algebras in the BGG category \({\mathcal{O}}\) , In: Graduate Studies in Mathematics, vol. 94. American Mathematical Society, Providence (2008)
Ridout D.: \({\widehat{\mathfrak{sl}} \left(2 \right)_{-1/2}}\) : a case study. Nucl. Phys. B 814, 485–521 (2009) arXiv:0810.3532 [hep-th]
Nahm W.: Quasirational fusion products. Int. J. Mod. Phys. B 8, 3693–3702 (1994) arXiv:hep-th/9402039
Gaberdiel M., Kausch H.: Indecomposable fusion products. Nucl. Phys. B 477, 293–318 (1996) arXiv:hep-th/9604026
Kytöllä K., Ridout D.: On staggered indecomposable Virasoro modules. J. Math. Phys. 50, 123503 (2009) arXiv:0905.0108 [math-ph]
Block R.: The irreducible representations of the Weyl algebra A 1. Lect. Notes Math. 740, 69–79 (1979)
Mazorchuk, V.: Lectures on \({\mathfrak{sl}_2 \left( \mathbb{C} \right)}\) -Modules. Imperial College Press, London (2010)
Feigin B., Semikhatov A., Tipunin Yu. I.: Equivalence between chain categories of representations of affine sl (2 ) and N = 2 superconformal algebras. J. Math. Phys. 39, 3865–3905 (1998) arXiv:hep-th/9701043
Kac V., Peterson D.: Infinite-dimensional Lie algebras, theta functions and modular forms. Adv. Math. 53, 125–264 (1984)
Creutzig T., Ridout D.: Modular data and Verlinde formulae for fractional level WZW models II. Nucl. Phys. B 875, 423–458 (2013) arXiv:1306.4388 [hep-th]
Huang Y.Z.: Vertex operator algebras, the Verlinde conjecture, and modular tensor conjectures. Proc. Natl. Acad. Sci. USA 102, 5352–5356 (2005) arXiv:math/0412261 [math.QA]
Gaberdiel M.: Fusion rules of chiral algebras. Nucl. Phys. B 417, 130–150 (1994) arXiv:hep-th/9309105
Mathieu P., Ridout D.: From percolation to logarithmic conformal field theory. Phys. Lett. B 657, 120–129 (2007) arXiv:0708.0802 [hep-th]
Gotz G., Quella T., Schomerus V.: Representation theory of \({{\mathfrak{sl}} (2\vert 1)}\) . J. Alg. 312, 829–848 (2007) arXiv:hep-th/0504234
Quella T., Schomerus V.: Free fermion resolution of supergroup WZNW models. JHEP 0709, 085 (2007) arXiv:0706.0744 [hep-th]
Gaberdiel M., Kausch H.: A local logarithmic conformal field theory. Nucl. Phys. B 538, 631–658 (1999) arXiv:hep-th/9807091
Fuchs J., Schweigert C., Stigner C.: From non-semisimple Hopf algebras to correlation functions for logarithmic CFT. J. Phys. A 46, 494012 (2013) arXiv:1302.4683 [hep-th]
Ridout D., Wood S.: Modular transformations and Verlinde formulae for logarithmic (p +, p −)-models. Nucl. Phys. B 880, 175–202 (2014) arXiv:1310.6479 [hep-th]
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ridout, D., Wood, S. Bosonic Ghosts at c = 2 as a Logarithmic CFT. Lett Math Phys 105, 279–307 (2015). https://doi.org/10.1007/s11005-014-0740-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-014-0740-z