Relating the archetypes of logarithmic conformal field theory. (English) Zbl 1282.81157
Summary: Logarithmic conformal field theory is a rich and vibrant area of modern mathematical physics with well-known applications to both condensed matter theory and string theory. Our limited understanding of these theories is based upon detailed studies of various examples that one may regard as archetypal. These include the \(c=-2\) triplet model, the Wess-Zumino-Witten model on \(\mathrm{SL}(2;\mathrm{R})\) at level \(k=-\frac{1}{2}\), and its supergroup analogue on \(\mathrm{GL}(1|1)\). Here, the latter model is studied algebraically through representation theory, fusion and modular invariance, facilitating a subsequent investigation of its cosets and extended algebras. The results show that the archetypes of logarithmic conformal field theory are in fact all very closely related, as are many other examples including, in particular, the \(\mathrm{SL}(2|1)\) models at levels 1 and \(-\frac{!}{2}\). The conclusion is then that the archetypal examples of logarithmic conformal field theory are practically all the same, so we should not expect that their features are in any way generic. Further archetypal examples must be sought.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
References:
| [1] | Maldacena, J., The large \(N\) limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys., 2, 231-252 (1998) · Zbl 0914.53047 |
| [2] | Zirnbauer, M., Conformal field theory of the integer quantum hall plateau transition · Zbl 1360.81339 |
| [3] | Guruswamy, S.; LeClair, A.; Ludwig, A., \(g l(N | N)\) super current algebras for disordered Dirac fermions in two-dimensions, Nucl. Phys. B, 583, 475-512 (2000) · Zbl 0984.82029 |
| [4] | Quella, T.; Schomerus, V.; Creutzig, T., Boundary spectra in superspace Sigma-models, JHEP, 0810, 024 (2008) · Zbl 1245.81247 |
| [5] | Ashok, S.; Benichou, R.; Troost, J., Conformal current algebra in two dimensions, JHEP, 0906, 017 (2009) |
| [6] | Creutzig, T., Yangian superalgebras in conformal field theory, Nucl. Phys. B, 849, 636-653 (2011) · Zbl 1215.81097 |
| [7] | Candu, C.; Schomerus, V., Exactly marginal parafermions, Phys. Rev. D, 84, 051704 (2011) |
| [8] | Quella, T.; Schomerus, V., Free fermion resolution of supergroup WZNW models, JHEP, 0709, 085 (2007) · Zbl 1023.81033 |
| [9] | Hikida, Y.; Schomerus, V., Structure constants of the \(OSP(1 | 2)\) WZNW model, JHEP, 0712, 100 (2007) · Zbl 1246.81260 |
| [10] | Creutzig, T.; Hikida, Y.; Rønne, P., Supergroup-extended super Liouville correspondence, JHEP, 1106, 063 (2011) · Zbl 1298.81259 |
| [11] | Creutzig, T.; Rønne, P., From world-sheet supersymmetry to super target spaces, JHEP, 1011, 021 (2010) · Zbl 1294.81187 |
| [12] | Creutzig, T., Geometry of branes on supergroups, Nucl. Phys. B, 812, 301-321 (2009) · Zbl 1194.81193 |
| [13] | Gotz, G.; Quella, T.; Schomerus, V., The WZNW model on \(PSU(1, 1 | 2)\), JHEP, 0703, 003 (2007) |
| [14] | Saleur, H.; Schomerus, V., On the \(SU(2 | 1)\) WZW model and its statistical mechanics applications, Nucl. Phys. B, 775, 312-340 (2007) · Zbl 1119.82014 |
| [15] | Creutzig, T.; Hikida, Y., Branes in the \(OSP(1 | 2)\) WZNW model, Nucl. Phys. B, 842, 172-224 (2011) · Zbl 1207.81109 |
| [16] | Giribet, G.; Hikida, Y.; Takayanagi, T., Topological string on \(OSP(1 | 2) / U(1)\), JHEP, 0909, 001 (2009) |
| [17] | Creutzig, T.; Rønne, P.; Schomerus, V., \(N = 2\) superconformal symmetry in super coset models, Phys. Rev. D, 80, 066010 (2009) |
| [18] | Rozansky, L.; Saleur, H., S and T matrices for the super \(U(1, 1)\) WZW model: Application to surgery and three manifolds invariants based on the Alexander-Conway polynomial, Nucl. Phys. B, 389, 365-423 (1993) |
| [19] | Rozansky, L.; Saleur, H., Quantum field theory for the multivariable Alexander-Conway polynomial, Nucl. Phys. B, 376, 461-509 (1992) |
| [20] | Saleur, H.; Schomerus, V., The \(GL(1 | 1)\) WZW model: From supergeometry to logarithmic CFT, Nucl. Phys. B, 734, 221-245 (2006) · Zbl 1192.81185 |
| [21] | Creutzig, T.; Schomerus, V., Boundary correlators in supergroup WZNW models, Nucl. Phys. B, 807, 471-494 (2009) · Zbl 1192.81260 |
| [22] | Creutzig, T.; Quella, T.; Schomerus, V., Branes in the \(GL(1 | 1)\) WZNW-model, Nucl. Phys. B, 792, 257-283 (2008) · Zbl 1146.81037 |
| [23] | Creutzig, T.; Rønne, P., The \(GL(1 | 1)\)-symplectic Fermion correspondence, Nucl. Phys. B, 815, 95-124 (2009) · Zbl 1194.81194 |
| [24] | Gurarie, V., Logarithmic operators in conformal field theory, Nucl. Phys. B, 410, 535-549 (1993) · Zbl 0990.81686 |
| [25] | Cardy, J., Logarithmic correlations in quenched random magnets and polymers |
| [26] | Piroux, G.; Ruelle, P., Pre-logarithmic and logarithmic fields in a sandpile model, J. Stat. Mech., 0410, P005 (2004) · Zbl 1073.82524 |
| [27] | Pearce, P.; Rasmussen, J.; Zuber, J.-B., Logarithmic minimal models, J. Stat. Mech., 0611, 017 (2006) |
| [28] | Read, N.; Saleur, H., Associative-algebraic approach to logarithmic conformal field theories, Nucl. Phys. B, 777, 316-351 (2007) · Zbl 1200.81136 |
| [29] | Mathieu, P.; Ridout, D., From percolation to logarithmic conformal field theory, Phys. Lett. B, 657, 120-129 (2007) · Zbl 1246.81181 |
| [30] | Ridout, D., On the percolation BCFT and the crossing probability of Watts, Nucl. Phys. B, 810, 503-526 (2009) · Zbl 1192.81304 |
| [31] | Vasseur, R.; Jacobsen, J.; Saleur, H., Indecomposability parameters in chiral logarithmic conformal field theory, Nucl. Phys. B, 851, 314-345 (2011) · Zbl 1229.81271 |
| [32] | Kytölä, K., SLE local martingales in logarithmic representations, J. Stat. Mech., 0908, P08005 (2009) · Zbl 1456.60215 |
| [33] | Grumiller, D.; Johansson, N., Instability in cosmological topologically massive gravity at the chiral point, JHEP, 0807, 134 (2008) · Zbl 1173.83315 |
| [34] | Flohr, M., Bits and pieces in logarithmic conformal field theory, Int. J. Mod. Phys. A, 18, 4497-4592 (2003) · Zbl 1062.81125 |
| [35] | Gaberdiel, M.; Kausch, H., Indecomposable fusion products, Nucl. Phys. B, 477, 293-318 (1996) · Zbl 1485.81053 |
| [36] | Gaberdiel, M.; Kausch, H., A rational logarithmic conformal field theory, Phys. Lett. B, 386, 131-137 (1996) |
| [37] | Feigin, B.; Gainutdinov, A.; Semikhatov, A.; Tipunin, I. Yu., Logarithmic extensions of minimal models: Characters and modular transformations, Nucl. Phys. B, 757, 303-343 (2006) · Zbl 1116.81059 |
| [38] | Eberle, H.; Flohr, M., Virasoro representations and fusion for general augmented minimal models, J. Phys. A, 39, 15245-15286 (2006) · Zbl 1106.81065 |
| [39] | Gaberdiel, M.; Runkel, I., The logarithmic triplet theory with boundary, J. Phys. A, 39, 14745-14780 (2006) · Zbl 1107.81045 |
| [40] | Adamović, D.; Milas, A., On the triplet vertex algebra \(W(p)\), Adv. Math., 217, 2664-2699 (2008) · Zbl 1177.17017 |
| [41] | Mathieu, P.; Ridout, D., Logarithmic \(M(2, p)\) minimal models, their logarithmic couplings, and duality, Nucl. Phys. B, 801, 268-295 (2008) · Zbl 1189.82053 |
| [42] | Pearce, P.; Rasmussen, J.; Ruelle, P., Integrable boundary conditions and \(W\)-extended fusion in the logarithmic minimal models \(L M(1, p)\), J. Phys. A, 41, 295201 (2008) · Zbl 1149.81019 |
| [43] | Gaberdiel, M., Fusion rules and logarithmic representations of a WZW model at fractional level, Nucl. Phys. B, 618, 407-436 (2001) · Zbl 0989.81063 |
| [44] | Lesage, F.; Mathieu, P.; Rasmussen, J.; Saleur, H., Logarithmic lift of the \(\hat{s u}(2)_{- 1 / 2}\) model, Nucl. Phys. B, 686, 313-346 (2004) · Zbl 1107.81328 |
| [45] | Adamović, D., A construction of admissible \(A_1^{(1)}\)-modules of level \(- \frac{4}{3} \), J. Pure Appl. Alg., 196, 119-134 (2005) · Zbl 1066.17016 |
| [46] | Ridout, D., Fusion in fractional level \(\hat{sl}(2)\)-theories with \(k = - \frac{1}{2} \), Nucl. Phys. B, 848, 216-250 (2011) · Zbl 1215.81102 |
| [47] | Lesage, F.; Mathieu, P.; Rasmussen, J.; Saleur, H., The \(\hat{s u}(2)_{- 1 / 2}\) WZW model and the βγ, System. Nucl. Phys. B, 647, 363-403 (2002) · Zbl 1001.81050 |
| [48] | Fuchs, J.; Hwang, S.; Semikhatov, A.; Tipunin, I. Yu., Nonsemisimple fusion algebras and the Verlinde formula, Comm. Math. Phys., 247, 713-742 (2004) · Zbl 1063.81062 |
| [49] | Ridout, D., \( \hat{sl}(2)_{- 1 / 2} \): A case study, Nucl. Phys. B, 814, 485-521 (2009) · Zbl 1194.81223 |
| [50] | Fjelstad, J., On duality and extended chiral symmetry in the \(SL(2, R)\) WZW model, J. Phys. A, 44, 235404 (2011) · Zbl 1219.81212 |
| [51] | Rohsiepe, F., On reducible but indecomposable representations of the Virasoro algebra |
| [52] | Fjelstad, J.; Fuchs, J.; Hwang, S.; Semikhatov, A.; Tipunin, I. Yu., Logarithmic conformal field theories via logarithmic deformations, Nucl. Phys. B, 633, 379-413 (2002) · Zbl 0995.81129 |
| [53] | Feigin, B.; Gainutdinov, A.; Semikhatov, A.; Tipunin, I. Yu., Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center, Comm. Math. Phys., 065, 47-93 (2006) · Zbl 1107.81044 |
| [54] | Feigin, B.; Gainutdinov, A.; Semikhatov, A.; Tipunin, I. Yu., Kazhdan-Lusztig correspondence for the representation category of the triplet \(W\)-algebra in logarithmic CFT, Theo. Math. Phys., 2006, 1210-1235 (2006) · Zbl 1177.17012 |
| [55] | Huang, Y.-Z.; Lepowsky, J.; Zhang, L., Logarithmic tensor product theory for generalized modules for a conformal vertex algebra |
| [56] | Kytölä, K.; Ridout, D., On staggered indecomposable Virasoro modules, J. Math. Phys., 50, 123503 (2009) · Zbl 1373.81234 |
| [57] | Adamovic, D.; Milas, A., The structure of Zhuʼs algebras for certain \(W\)-algebras, Adv. Math., 227, 2425-2456 (2011) · Zbl 1302.17041 |
| [58] | Kac, V., Lie superalgebras, Adv. Math., 26, 8-96 (1977) · Zbl 0366.17012 |
| [59] | Kac, V., A sketch of Lie superalgebra theory, Comm. Math. Phys., 53, 31-64 (1977) · Zbl 0359.17009 |
| [60] | Kac, V.; van de Leur, J., Super Boson-Fermion correspondence, Ann. Inst. Fourier, 37, 99-137 (1987) · Zbl 0625.58041 |
| [61] | Ridout, D., \( \hat{sl}(2)_{- 1 / 2}\) and the triplet model, Nucl. Phys. B, 835, 314-342 (2010) · Zbl 1204.81157 |
| [62] | Kausch, H., Symplectic fermions, Nucl. Phys. B, 583, 513-541 (2000) · Zbl 0984.81141 |
| [63] | Creutzig, T., Branes in supergroups (2009), DESY Theory Group, PhD thesis · Zbl 1194.81193 |
| [64] | Humphreys, J., Representations of Semisimple Lie Algebras in the BGG Category \(O\), Graduate Studies in Mathematics, vol. 94 (2008), American Mathematical Society: American Mathematical Society Providence · Zbl 1177.17001 |
| [65] | Gotz, G.; Quella, T.; Schomerus, V., Representation theory of \(sl(2 | 1)\), J. Alg., 312, 829-848 (2007) · Zbl 1180.17009 |
| [66] | Nahm, W., Quasirational fusion products, Int. J. Mod. Phys. B, 8, 3693-3702 (1994) · Zbl 1264.81247 |
| [67] | Di Francesco, P.; Mathieu, P.; Sénéchal, D., Conformal Field Theory, Graduate Texts in Contemporary Physics (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0869.53052 |
| [68] | Semikhatov, A.; Sirota, V., Embedding diagrams of \(N = 2\) Verma modules and relaxed \(\hat{s l}(2)\) Verma modules |
| [69] | Gasper, G.; Rahman, M., Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, vol. 96 (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1129.33005 |
| [70] | Mathieu, P.; Ridout, D., The extended algebra of the \(SU(2)\) Wess-Zumino-Witten models, Nucl. Phys. B, 765, 201-239 (2007) · Zbl 1116.81061 |
| [71] | Mathieu, P.; Ridout, D., The extended algebra of the minimal models, Nucl. Phys. B, 776, 365-404 (2007) · Zbl 1200.81134 |
| [72] | Jacob, P.; Mathieu, P., A quasi-particle description of the \(M(3, p)\), Models Nucl. Phys. B, 733, 205-232 (2006) · Zbl 1119.81061 |
| [73] | Rasmussen, J.; Pearce, P., W-extended fusion algebra of critical percolation, J. Phys. A, 41, 295208 (2008) · Zbl 1141.82005 |
| [74] | Gaberdiel, M.; Runkel, I.; Wood, S., Fusion rules and boundary conditions in the \(c = 0\) triplet model, J. Phys. A, 42, 325403 (2009) · Zbl 1179.81145 |
| [75] | Adamovic, D.; Milas, A., On \(W\)-algebras associated to \((2, p)\) minimal models and their representations, Int. Math. Res. Not. IMRN, 3896-3934 (2010) · Zbl 1210.17031 |
| [76] | Feigin, B.; Semikhatov, A., \(W_n^{(2)}\) algebras, Nucl. Phys. B, 698, 409-449 (2004) · Zbl 1123.17302 |
| [77] | Creutzig, T.; Gao, P.; Linshaw, A., A commutant realization of \(W_n^{(2)}\) at critical level, Int. Math. Res. Not. IMRN (2013), in press |
| [78] | Odake, S., Extension of \(N = 2\) superconformal algebra and Calabi-Yau compactification, Mod. Phys. Lett. A, 4, 557 (1989) |
| [79] | Parker, M., Classification of real simple lie superalgebras of classical type, J. Math. Phys., 21, 689-697 (1980) · Zbl 0445.17002 |
| [80] | Berezin, F.; Kirillov, A.; Leites, D., Introduction to Superanalysis, Mathematical Physics and Applied Mathematics, vol. 9 (1987), D. Reidel: D. Reidel Dordrecht · Zbl 0659.58001 |
| [81] | Polyakov, A., Gauge transformations and diffeomorphisms, Int. J. Mod. Phys. A, 5, 833-842 (1990) · Zbl 0741.53057 |
| [82] | Bershadsky, M., Conformal field theories via Hamiltonian reduction, Comm. Math. Phys., 139, 71-82 (1991) · Zbl 0721.58046 |
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