Morita classes of algebras in modular tensor categories. (English) Zbl 1156.18003
The authors aim to study the relation between Morita classes of algebras and their centers in a modular tensor category for the sake of conformal field theory (CFT). It has recently become clear that there is a close relationship between rational CFT and non-degenerate algebras in modular tensor categories, both in the Euclidean and the Minkowski formulation of CFT. In the Euclidean setting, the modular tensor category arises as the category of representations of a vertex operator algebra with certain additional properties, in which the non-degenerate algebra is an algebra of boundary fields. It is well known that the non-degenerate algebra and the rational vertex operator algebra uniquely determine a CFT, though its existence is another matter. An important question is whether two non-Morita-equivalent open-string vertex operator algebras can give rise to the same full field algebra or not – in more physical terms – whether there may exist several incompatible sets of boundary conditions for a given bulk CFT or not. The main result of this paper, which claims that two simple algebras with non-degenerate trace pairing are Morita-equivalent if and only if their full centers are isomorphic as algebras, implies that for a rational CFT, this can not happen.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18D50 | Operads (MSC2010) |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
Keywords:
modular tensor category; conformal field theory; Morita equivalence; Frobenius algebra; vertex operator algebraReferences:
| [1] | Anderson, F. W.; Fuller, K. R., Rings and Categories of Modules, Grad. Texts in Math., vol. 13 (1973), Springer · Zbl 0242.16025 |
| [2] | Bichon, J., Cosovereign Hopf algebras, J. Pure Appl. Algebra, 157, 121-133 (2001) · Zbl 0976.16027 |
| [3] | Bakalov, B.; Kirillov, A. A., Lectures on Tensor Categories and Modular Functors (2001), American Mathematical Society: American Mathematical Society Providence · Zbl 0965.18002 |
| [4] | Etingof, P. I.; Nikshych, D.; Ostrik, V., On fusion categories, Ann. of Math., 162, 581-642 (2005) · Zbl 1125.16025 |
| [5] | Fjelstad, J.; Fuchs, J.; Runkel, I.; Schweigert, C., TFT construction of RCFT correlators. V: Proof of modular invariance and factorisation, Theory Appl. Categ., 16, 342-433 (2006) · Zbl 1151.81038 |
| [6] | Fjelstad, J.; Fuchs, J.; Runkel, I.; Schweigert, C., Topological and conformal field theory as Frobenius algebras, (Contemp. Math., vol. 431 (2007)), 225-247 · Zbl 1154.18006 |
| [7] | Fjelstad, J.; Fuchs, J.; Runkel, I.; Schweigert, C., Uniqueness of open/closed rational CFT with given algebra of open states · Zbl 1151.81034 |
| [8] | Fröhlich, J.; Fuchs, J.; Runkel, I.; Schweigert, C., Correspondences of ribbon categories, Adv. Math., 199, 192-329 (2006) · Zbl 1087.18006 |
| [9] | Fröhlich, J.; Fuchs, J.; Runkel, I.; Schweigert, C., Duality and defects in rational conformal field theory, Nuclear Phys. B, 763, 354-430 (2007) · Zbl 1116.81060 |
| [10] | Fuchs, J.; Runkel, I.; Schweigert, C., TFT construction of RCFT correlators. I: Partition functions, Nuclear Phys. B, 646, 353-497 (2002) · Zbl 0999.81079 |
| [11] | Fuchs, J.; Runkel, I.; Schweigert, C., TFT construction of RCFT correlators. II: Unoriented world sheets, Nuclear Phys. B, 678, 511-637 (2004) · Zbl 1097.81736 |
| [12] | Fuchs, J.; Runkel, I.; Schweigert, C., The fusion algebra of bimodule categories, Appl. Categ. Structures, 16, 123-140 (2008) · Zbl 1154.18003 |
| [13] | Fuchs, J.; Schweigert, C., Category theory for conformal boundary conditions, (Fields Inst. Commun., vol. 39 (2003)), 25-70 · Zbl 1084.17012 |
| [14] | Huang, Y.-Z., Vertex operator algebras and the Verlinde conjecture, Commun. Contemp. Math., 10, 103-154 (2008) · Zbl 1180.17008 |
| [15] | Huang, Y.-Z., Rigidity and modularity of vertex tensor categories · Zbl 1169.17019 |
| [16] | Huang, Y.-Z.; Kong, L., Open-string vertex algebras, tensor categories and operads, Comm. Math. Phys., 250, 433-471 (2004) · Zbl 1083.17010 |
| [17] | Huang, Y.-Z.; Kong, L., Full field algebras, Comm. Math. Phys., 272, 345-396 (2007) · Zbl 1153.17012 |
| [18] | Joyal, A.; Street, R., Braided tensor categories, Adv. Math., 102, 20-78 (1993) · Zbl 0817.18007 |
| [19] | Kong, L., Full field algebras, operads and tensor categories, Adv. Math., 213, 1, 271-340 (2007) · Zbl 1115.18002 |
| [20] | Kong, L., Open-closed field algebras, Comm. Math. Phys., 280, 207-261 (2008) · Zbl 1173.17021 |
| [21] | Kong, L., Cardy condition for open-closed field algebras · Zbl 1156.81023 |
| [22] | Kock, J., Frobenius Algebras and 2D Topological Quantum Field Theories (2003), Cambridge University Press: Cambridge University Press Cambridge |
| [23] | Kawahigashi, Y.; Longo, R.; Müger, M., Multi-interval subfactors and modularity of representations in conformal field theory, Comm. Math. Phys., 219, 631-669 (2001) · Zbl 1016.81031 |
| [24] | Kirillov, A. J.; Ostrik, V., On q-analog of McKay correspondence and ADE classification of \(\hat{s l}(2)\) conformal field theories, Adv. Math., 171, 183-227 (2002) · Zbl 1024.17013 |
| [25] | L. Kong, I. Runkel, Cardy algebras and sewing constrains, I, arXiv:0807.3356 [math.OA]; L. Kong, I. Runkel, Cardy algebras and sewing constrains, I, arXiv:0807.3356 [math.OA] · Zbl 1214.81251 |
| [26] | Lauda, A. D.; Pfeiffer, H., Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras, Topology Appl., 155, 623-666 (2008) · Zbl 1158.57038 |
| [27] | Longo, R.; Rehren, K. H., Local fields in boundary conformal QFT, Rev. Math. Phys., 16, 909-960 (2004) · Zbl 1078.81052 |
| [28] | Moore, G. W., Some comments on branes, G-flux, and K-theory, Internat. J. Modern Phys. A, 16, 936-944 (2001) |
| [29] | Moore, G. W.; Segal, G., D-branes and K-theory in 2D topological field theory |
| [30] | Müger, M., From subfactors to categories and topology II. The quantum double of tensor categories and subfactors, J. Pure Appl. Algebra, 180, 159-219 (2003) · Zbl 1033.18003 |
| [31] | M. Müger, Talk at workshop ‘Quantum Structures’, Leipzig, 28 June 2007, in preparation; M. Müger, Talk at workshop ‘Quantum Structures’, Leipzig, 28 June 2007, in preparation |
| [32] | Ostrik, V., Module categories, weak Hopf algebras and modular invariants, Transform. Groups, 8, 177-206 (2003) · Zbl 1044.18004 |
| [33] | Schellekens, A. N.; Stanev, Y. S., Trace formulas for annuli, J. High Energy Phys., 0112, 012 (2001) |
| [34] | Turaev, V. G., Quantum Invariants of Knots and 3-Manifolds (1994), de Gruyter: de Gruyter New York · Zbl 0812.57003 |
| [35] | Van Oystaeyen, F.; Zhang, Y. H., The Brauer group of a braided monoidal category, J. Algebra, 202, 96-128 (1998) · Zbl 0909.18005 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
