zbMATH — the first resource for mathematics

Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center. (English) Zbl 1107.81044
Summary: The \(\text{SL}(2,\mathbb Z)\)-representation \(\pi\) on the center of the restricted quantum group \(\overline{\mathcal U}s\ell(2)\) at the primitive \(2p\)th root of unity is shown to be equivalent to the \(\text{SL}(2,\mathbb Z)\)-representation on the extended characters of the logarithmic \((1,p)\) conformal field theory model. The multiplicative Jordan decomposition of the \(\overline{\mathcal U}s\ell(2)\) ribbon element determines the decomposition of \(\pi\) into a “pointwise” product of two commuting \(\text{SL}(2,\mathbb Z)\)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the \(\text{SL}(2,\mathbb Z)\)-representation on the \((1,p)\)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of \(\overline{\mathcal U}s\ell(2)\) at the primitive \(2 p\)th root of unity is shown to coincide with the fusion algebra of the \((1,p)\) logarithmic conformal field theory model. As a by-product, we derive \(q\)-binomial identities implied by the fusion algebra realized in the center of \(\overline{\mathcal U}s\ell(2)\).

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G05 Representation theory for linear algebraic groups
Full Text: DOI arXiv
[1] Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras. I. J. Amer. Math. Soc. 6, 905–947 (1993); II. J. Amer. Math. Soc. 6, 949–1011 (1993); III. J. Amer. Math. Soc. 7, 335–381 (1994); IV. J. Amer. Math. Soc. 7, 383–453 (1994) · Zbl 0786.17017 · doi:10.1090/S0894-0347-1993-99999-X
[2] Moore, G., Seiberg, N.: Lectures on RCFT. In: Physics, Geometry, and Topology (Trieste Spring School 1989), New York: Plenum, 1990, p. 263 · Zbl 0728.57012
[3] Finkelberg, M.: An equivalence of fusion categories. Geometric and Functional Analysis (GAFA) 6, 249–267 (1996) · Zbl 0860.17040 · doi:10.1007/BF02247887
[4] Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. Berlin–New York: Walter de Gruyter, 1994 · Zbl 0812.57003
[5] Lyubashenko, V.: Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity. Commun. Math. Phys. 172, 467–516 (1995); Modular properties of ribbon abelian categories. In: Symposia Gaussiana, Proc. of the 2nd Gauss Symposium, Munich, 1993, Conf. A , Berlin-New York: Walter de Gruyter, 1995, pp. 529–579; Modular Transformations for Tensor Categories. J. Pure Applied Algebra 98, 279–327 (1995) · Zbl 0844.57016 · doi:10.1007/BF02101805
[6] Lyubashenko, V., Majid, S.: Braided groups and quantum Fourier transform. J. Algebra 166, 506–528 (1994) · Zbl 0810.17006 · doi:10.1006/jabr.1994.1165
[7] Kausch, H.G.: Extended conformal algebras generated by a multiplet of primary fields. Phys. Lett. B 259, 448 (1991) · doi:10.1016/0370-2693(91)91655-F
[8] Gaberdiel, M.R., Kausch, H.G.: A rational logarithmic conformal field theory. Phys. Lett. B 386, 131 (1996) · Zbl 0948.81632 · doi:10.1016/0370-2693(96)00949-5
[9] Flohr, M.A.I.: On modular invariant partition functions of conformal field theories with logarithmic operators. Int. J. Mod. Phys. A11, 4147 (1996) · Zbl 1044.81713
[10] Flohr, M.: On Fusion Rules in Logarithmic Conformal Field Theories. Int. J. Mod. Phys. A12, 1943–1958 (1997) · Zbl 0985.81738
[11] Kerler, T.: Mapping class group action on quantum doubles. Commun. Math. Phys. 168, 353–388 (1995) · Zbl 0833.16039 · doi:10.1007/BF02101554
[12] Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge: Cambridge University Press, 1994 · Zbl 0839.17009
[13] Gaberdiel, M.R., Kausch, H.G.: Indecomposable fusion products. Nucl. Phys. B 477, 293 (1996) · doi:10.1016/0550-3213(96)00364-1
[14] Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360 (1988) · Zbl 1180.81120 · doi:10.1016/0550-3213(88)90603-7
[15] Fuchs, J., Hwang, S., Semikhatov, A.M., Tipunin, I.Yu.: Nonsemisimple fusion algebras and the Verlinde formula. Commun. Math. Phys. 247, 713–742 (2004) · Zbl 1063.81062 · doi:10.1007/s00220-004-1058-y
[16] Gurarie, V., Ludwig, A.W.W.: Conformal field theory at central charge c=0 and two-dimensional critical systems with quenched disorder. http://arxiv.org/list/hep-th/0409105, 2004 · Zbl 1082.81074
[17] Feigin, B.L., Gainutdinov, A.M., Semikhatov, A.M., Tipunin, I.Yu.: Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT, math.QA/0512621 · Zbl 1177.17012
[18] Reshetikhin, N.Yu., Turaev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys., 127, 1–26 (1990) · Zbl 0768.57003 · doi:10.1007/BF02096491
[19] Lachowska, A.: On the center of the small quantum group. http://arxiv.org/list/math.QA/0107098, 2001 · Zbl 1049.17011
[20] Ostrik, V.: Decomposition of the adjoint representation of the small quantum sl2. Commun. Math. Phys. 186, 253–264 (1997) · Zbl 0883.16028 · doi:10.1007/s002200050109
[21] Gluschenkov, D.V., Lyakhovskaya, A.V.: Regular representation of the quantum Heisenberg double {Uq (sl(2)), Funq(SL(2))} (q is a root of unity). http://arxiv.org/list/hep-th/9311075, 1993
[22] Jimbo, M., Miwa, T., Takeyama, Y.: Counting minimal form factors of the restricted sine-Gordon model http://arxiv.org/list/math-ph/0303059, 2003 · Zbl 1084.81066
[23] Gaberdiel, M.R.: An algebraic approach to logarithmic conformal field theory. Int. J. Mod. Phys. A18, 4593–4638 (2003) · Zbl 1055.81064
[24] Flohr, M.: Bits and Pieces in Logarithmic Conformal Field Theory. Int. J. Mod. Phys. A18, 4497–4592 (2003) · Zbl 1062.81125
[25] Gurarie, V.; Logarithmic operators in conformal field theory. Nucl. Phys. B410, 535 (1993) · Zbl 0990.81686
[26] Rohsiepe, F.: Nichtunitäre Darstellungen der Virasoro-Algebra mit nichttrivialen Jordanblöcken. Diploma Thesis, Bonn, (1996) [BONN-IB-96-19]
[27] Fjelstad, J., Fuchs, J., Hwang, S., Semikhatov, A.M., Tipunin, I.Yu.: Logarithmic conformal field theories via logarithmic deformations. Nucl. Phys. B633, 379 (2002) · Zbl 0995.81129
[28] Semikhatov, A.M., Taormina, A., Tipunin, I.Yu.: Higher-level Appell functions, modular transformations, and characters. http://arxiv.org/list/math.QA/0311314, 2003
[29] Kač, V.G.: Infinite Dimensional Lie Algebras. Cambridge: Cambridge University Press, 1990
[30] Fuchs, J.: Affine Lie algebras and quantum groups. Cambridge: Cambridge University Press, 1992 · Zbl 0925.17031
[31] Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: Quantum R-matrices and factorization problems. J. Geom. Phys. 5, 533–550 (1988) · Zbl 0711.17008 · doi:10.1016/0393-0440(88)90018-6
[32] Bakalov, B., Kirillov, A.A.: Lectures on Tensor Categories and Modular Functors. Providence, RI: AMS, 2001 · Zbl 0965.18002
[33] Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators I: Partition functions. Nucl. Phys. B 646, 353 (2002) · Zbl 0999.81079 · doi:10.1016/S0550-3213(02)00744-7
[34] Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators II: Unoriented world sheets. Nucl. Phys. B678, 511–637 (2004) · Zbl 1097.81736
[35] Kerler, T., Lyubashenko, V.V.: Non-Semisimple Topological Quantum Field Theories for 3- Manifolds with Corners. Springer Lecture Notes in Mathematics 1765, Berlin-Heidelberg-New York: Springer Verlag, 2001 · Zbl 0982.57013
[36] Larson, R.G., Sweedler, M.E.: An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91, 75–94 (1969) · Zbl 0179.05803 · doi:10.2307/2373270
[37] Radford, D.E.: The order of antipode of a finite-dimensional Hopf algebra is finite. Amer. J. Math 98, 333–335 (1976) · Zbl 0332.16007 · doi:10.2307/2373888
[38] Drinfeld, V.G.: On Almost Cocommutative Hopf Algebras. Leningrad Math. J. 1(2), 321–342 (1990) · Zbl 0718.16035
[39] Kassel, C.: Quantum Groups. New York: Springer-Verlag, 1995 · Zbl 0808.17003
[40] Sweedler, M.E.: Hopf Algebras. New York: Benjamin, 1969 · Zbl 0194.32901
[41] Radford, D.E.: The trace function and Hopf algebras. J. Alg. 163, 583–622 (1994) · Zbl 0801.16039 · doi:10.1006/jabr.1994.1033
[42] Gantmakher, F.R.: Teoriya Matrits [in Russian]. Moscow: Nauka, 1988
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.