Pointed Drinfeld center functor. (English) Zbl 1460.81084
Summary: In this work, using the functoriality of the Drinfeld center of fusion categories, we generalize the functoriality of the full center of simple separable algebras in a fixed fusion category to all fusion categories. This generalization produces a new center functor, which involves both Drinfeld center and full center and is called pointed Drinfeld center functor. We prove that this pointed Drinfeld center functor is a symmetric monoidal equivalence. It turns out that this functor provides a precise and rather complete mathematical formulation of the boundary-bulk relation of 1 + 1D rational conformal field theories (RCFT). In this process, we also solve an old problem of computing the fusion of two 0D (or 1D) wall CFT’s along a non-trivial 1 + 1D bulk RCFT. At the end of this work, we explain the mysterious relation between the boundary-bulk relation in 2 + 1D topological orders and that in 1 + 1D RCFTs via the so-called topological Wick rotation.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81T45 | Topological field theories in quantum mechanics |
| 81T05 | Axiomatic quantum field theory; operator algebras |
| 11F52 | Modular forms associated to Drinfel’d modules |
| 18M20 | Fusion categories, modular tensor categories, modular functors |
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 46M18 | Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.) |
References:
| [1] | Ai, Y.; Kong, L.; Zheng, H., Topological orders and factorization homology, Adv. Theor. Math. Phys., 21, 8, 1845-1894 (2017) · Zbl 1387.81296 · doi:10.4310/ATMP.2017.v21.n8.a1 |
| [2] | Ayala, D., Francis, J.: A factorization homology primer, Chapter 2 in the ebook: Handbook of Homotopy Theory, edited by Haynes Miller, doi:10.1201/9781351251624arXiv:1903.10961 |
| [3] | Bénabou, J.: Introduction to bicategories, 1967 Reports of the Midwest Category Seminar, pp. 1-77. Springer, Berlin · Zbl 1375.18001 |
| [4] | Ben-Zvi, D.; Brochier, A.; Jordan, D., Quantum character varieties and braided module categories, Selecta Mathematica, 24, 4711-4748 (2018) · Zbl 1456.17010 · doi:10.1007/s00029-018-0426-y |
| [5] | Cardy, JL, Conformal invariance and surface critical behavior, Nucl. Phys. B, 240, 514-532 (1984) · doi:10.1016/0550-3213(84)90241-4 |
| [6] | Chang, C-M; Lin, Y-H; Shao, S-H; Wang, Y.; Yin, X., Topological defect lines and renormalization group flows in two dimensions, J. High Energ. Phys., 2019, 26 (2019) · Zbl 1409.81134 · doi:10.1007/JHEP01(2019)026 |
| [7] | Davydov, A., Centre of an algebra, Adv. Math., 225, 319-348 (2010) · Zbl 1227.18003 · doi:10.1016/j.aim.2010.02.018 |
| [8] | Davydov, A.; Kong, L.; Runkel, I., Invertible defects and isomorphisms of rational CFTs, Adv. Theor. Math. Phys., 15, 43-69 (2011) · Zbl 1246.81319 · doi:10.4310/ATMP.2011.v15.n1.a2 |
| [9] | Davydov, A.; Kong, L.; Runkel, I., Functoriality of the center of an algebra, Adv. Math., 285, 811-876 (2015) · Zbl 1345.18004 · doi:10.1016/j.aim.2015.06.023 |
| [10] | Davydov, A.; Müger, M.; Nikshych, D.; Ostrik, V., The Witt group of non-degenerate braided fusion categories, J. Reine Angew. Math., 677, 135-177 (2013) · Zbl 1271.18008 |
| [11] | Davydov, A.; Nikshych, D., The Picard corossed module of a braided tensor category, Algebra Number Theory, 7, 1365-1403 (2013) · Zbl 1284.18015 · doi:10.2140/ant.2013.7.1365 |
| [12] | Douglas, CL; Schommer-Pries, C.; Snyder, N., The balanced tensor product of module categories, Kyoto J. Math, 59, 1, 167-179 (2019) · Zbl 1478.18006 · doi:10.1215/21562261-2018-0006 |
| [13] | Douglas, C.L., Schommer-Pries, C., Snyder, N.: Dualizable tensor categories, arXiv:1312.7188 · Zbl 1478.18006 |
| [14] | Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories, Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI (2015) · Zbl 1365.18001 |
| [15] | Etingof, P.; Nikshych, D.; Ostrik, V., On fusion categories, Ann. Math., 162, 581-642 (2005) · Zbl 1125.16025 · doi:10.4007/annals.2005.162.581 |
| [16] | Etingof, P.; Nikshych, D.; Ostrik, V., Fusion categories and homotopy theory, Quantum Topol., 1, 3, 209-273 (2010) · Zbl 1214.18007 · doi:10.4171/QT/6 |
| [17] | Etingof, P.; Ostrik, V., Finite tensor categories, Mosc. Math. J., 4, 3, 627-654 (2004) · Zbl 1077.18005 · doi:10.17323/1609-4514-2004-4-3-627-654 |
| [18] | Fjelstad, J.; Fuchs, J.; Runkel, I.; Schweigert, C., Uniqueness of open/closed rational CFT with given algebra of open states, Adv. Theor. Math. Phys., 12, 1283-1375 (2008) · Zbl 1151.81034 · doi:10.4310/ATMP.2008.v12.n6.a4 |
| [19] | Fröhlich, J.; Fuchs, J.; Runkel, I.; Schweigert, C., Duality and defects in rational conformal field theory, Nucl. Phys. B, 763, 354-430 (2007) · Zbl 1116.81060 · doi:10.1016/j.nuclphysb.2006.11.017 |
| [20] | Fuchs, J.; Runkel, I.; Schweigert, C., TFT construction of RCFT correlators. I: partition functions, Nucl. Phys. B, 646, 353-497 (2002) · Zbl 0999.81079 · doi:10.1016/S0550-3213(02)00744-7 |
| [21] | Fuchs, J.; Runkel, I.; Schweigert, C., The fusion algebra of bimodule categories, Appl. Cat. Str., 16, 123-140 (2008) · Zbl 1154.18003 · doi:10.1007/s10485-007-9102-7 |
| [22] | Fuchs, J.; Schweigert, C., Category theory for conformal boundary conditions, Fields Institute Commun., 39, 25-71 (2003) · Zbl 1084.17012 |
| [23] | Fuchs, J.; Schweigert, C.; Valentino, A., Bicategories for boundary conditions and for surface defects in 3-d TFT, Commun. Math. Phys., 321, 2, 543-575 (2013) · Zbl 1269.81169 · doi:10.1007/s00220-013-1723-0 |
| [24] | Greenough, J., Monoidal 2-structure of bimodule categories, J. Algebra, 324, 8, 1818-1859 (2010) · Zbl 1242.18009 · doi:10.1016/j.jalgebra.2010.06.018 |
| [25] | Huang, Y-Z; Kong, L., Full field algebras, Commun. Math. Phys., 272, 345-396 (2007) · Zbl 1153.17012 · doi:10.1007/s00220-007-0224-4 |
| [26] | Huang, Y-Z, Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math., 10, 871 (2008) · Zbl 1169.17019 · doi:10.1142/S0219199708003083 |
| [27] | Joyal, A.; Street, R., Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Alg., 71, 1, 43-51 (1991) · Zbl 0726.18004 · doi:10.1016/0022-4049(91)90039-5 |
| [28] | Kitaev, A.; Kong, L., Models for gapped boundaries and domain walls, Commun. Math. Phys., 313, 351-373 (2012) · Zbl 1250.81141 · doi:10.1007/s00220-012-1500-5 |
| [29] | Kong, L., Open-closed field algebras, Comm. Math. Phys., 280, 207-261 (2008) · Zbl 1173.17021 · doi:10.1007/s00220-008-0446-0 |
| [30] | Kong, L.: Conformal field theory and a new geometry. In: Sati, Hisham, Schreiber, Urs (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory. Proceedings of Symposia in Pure Mathematics, AMS, Vol. 83, 199-244 (2011) · Zbl 1248.81199 |
| [31] | Kong, L.; Runkel, I., Morita classes of algebras in modular tensor categories, Adv. Math., 219, 1548-1576 (2008) · Zbl 1156.18003 · doi:10.1016/j.aim.2008.07.004 |
| [32] | Kong, L.; Runkel, I., Cardy Algebras and Sewing Constraints, I, Commun. Math. Phys., 292, 871-912 (2009) · Zbl 1214.81251 · doi:10.1007/s00220-009-0901-6 |
| [33] | Kong, L., Wen, X.-G., Zheng, H.: Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers, arXiv:1502.01690 |
| [34] | Kong, L.; Zheng, H., The center functor is fully faithful, Adv. Math., 339, 749-779 (2018) · Zbl 1419.18013 · doi:10.1016/j.aim.2018.09.031 |
| [35] | Kong, L.; Zheng, H., Drinfeld center of enriched monoidal categories, Adv. Math., 323, 411-426 (2018) · Zbl 1387.18018 · doi:10.1016/j.aim.2017.10.038 |
| [36] | Kong, L., Zheng, H.: Semisimple and separable algebras in multi-fusion categories, arXiv:1706.06904 |
| [37] | Kong, L.; Zheng, H., A mathematical theory of gapless edges of 2d topological orders. Part I, J. High Energ. Phys., 2020, 150 (2020) · Zbl 1435.81184 · doi:10.1007/JHEP02(2020)150 |
| [38] | Kong, L., Zheng, H.: A mathematical theory of gapless edges of 2d topological orders. Part II, arXiv:1912.01760 · Zbl 1435.81184 |
| [39] | Lurie, J.: Higher algebras, a book available online · Zbl 1175.18001 |
| [40] | Majid, S., Representations, duals and quantum doubles of monoidal categories, Rend. Circ. Math. Palermo (2) Suppl., 26, 197-206 (1991) · Zbl 0762.18005 |
| [41] | Ostrik, V., Module categories, weak Hopf algebras and modular invariants, Transform. Groups, 8, 177-206 (2003) · Zbl 1044.18004 · doi:10.1007/s00031-003-0515-6 |
| [42] | Savit, R., Duality in field theory and statistical systems, Rev. Mod. Phys., 52, 453 (1980) · doi:10.1103/RevModPhys.52.453 |
| [43] | Street, R., Monoidal categories in, and linking, geometry and algebra, Bull. Belg. Math. Soc. Simon Stevin, 19, 5, 769-821 (2012) · Zbl 1267.18006 · doi:10.36045/bbms/1354031551 |
| [44] | Tambara, D., A duality for modules over monoidal categoires of representations of semisimple Hopf algebras, J. Algebra, 241, 515-547 (2001) · Zbl 0987.16034 · doi:10.1006/jabr.2001.8771 |
| [45] | Zheng, H.: Extended TQFT arising from enriched multi-fusion categories. arXiv:1704.05956 |
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.
