Abstract
We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the ’t Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.
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References
J.L. Cardy, Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 275 (1986) 200 [INSPIRE].
E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
J. Cardy, Boundary conditions in conformal field theory, Adv. Stud. Pure Math. 19 (1989) 127
J.L. Cardy, Boundary conformal field theory, Encycl. Math. Phys. (2006) 333, hep-th/0411189 [INSPIRE].
M. Oshikawa and I. Affleck, Defect lines in the Ising model and boundary states on orbifolds, Phys. Rev. Lett. 77 (1996) 2604 [hep-th/9606177] [INSPIRE].
M. Oshikawa and I. Affleck, Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line, Nucl. Phys. B 495 (1997) 533 [cond-mat/9612187] [INSPIRE].
V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators 1. Partition functions, Nucl. Phys. B 646 (2002) 353 [hep-th/0204148] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators. 2. Unoriented world sheets, Nucl. Phys. B 678 (2004) 511 [hep-th/0306164] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators. 3. Simple currents, Nucl. Phys. B 694 (2004) 277 [hep-th/0403157] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators IV: Structure constants and correlation functions, Nucl. Phys. B 715 (2005) 539 [hep-th/0412290] [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93 (2004) 070601 [cond-mat/0404051] [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [INSPIRE].
T. Quella, I. Runkel and G.M.T. Watts, Reflection and transmission for conformal defects, JHEP 04 (2007) 095 [hep-th/0611296] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, The fusion algebra of bimodule categories, Appl. Categ. Struct. 16 (2008) 123 [math/0701223] [INSPIRE].
J. Fuchs, M.R. Gaberdiel, I. Runkel and C. Schweigert, Topological defects for the free boson CFT, J. Phys. A 40 (2007) 11403 [arXiv:0705.3129] [INSPIRE].
C. Bachas and I. Brunner, Fusion of conformal interfaces, JHEP 02 (2008) 085 [arXiv:0712.0076] [INSPIRE].
L. Kong and I. Runkel, Cardy algebras and sewing constraints. I., Commun. Math. Phys. 292 (2009) 871 [arXiv:0807.3356] [INSPIRE].
V.B. Petkova, On the crossing relation in the presence of defects, JHEP 04 (2010) 061 [arXiv:0912.5535] [INSPIRE].
Communications in Mathematical Physics 313 (2012) 351 [arXiv:1104.5047].
N. Carqueville and I. Runkel, Orbifold completion of defect bicategories, Quantum Topol. 7 (2016) 203 [arXiv:1210.6363] [INSPIRE].
I. Brunner, N. Carqueville and D. Plencner, A quick guide to defect orbifolds, Proc. Symp. Pure Math. 88 (2014) 231 [arXiv:1310.0062] [INSPIRE].
A. Davydov, L. Kong and I. Runkel, Functoriality of the center of an algebra, Adv. Math. 285 (2015) 811 [arXiv:1307.5956] [INSPIRE].
L. Kong, Q. Li and I. Runkel, Cardy algebras and sewing constraints, II, Adv. Math. 262 (2014) 604 [arXiv:1310.1875] [INSPIRE].
V.B. Petkova, Topological defects in CFT, Phys. Atom. Nucl. 76 (2013) 1268 [INSPIRE].
M. Bischoff, R. Longo, Y. Kawahigashi and K.-H. Rehren, Tensor categories of endomorphisms and inclusions of von Neumann algebras, arXiv:1407.4793 [INSPIRE].
L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [arXiv:1704.02330] [INSPIRE].
M. Hauru, G. Evenbly, W.W. Ho, D. Gaiotto and G. Vidal, Topological conformal defects with tensor networks, Phys. Rev. B 94 (2016) 115125 [arXiv:1512.03846] [INSPIRE].
D. Aasen, R.S.K. Mong and P. Fendley, Topological defects on the lattice I: the Ising model, J. Phys. A 49 (2016) 354001 [arXiv:1601.07185] [INSPIRE].
J.C. Bridgeman and D.J. Williamson, Anomalies and entanglement renormalization, Phys. Rev. B 96 (2017) 125104 [arXiv:1703.07782] [INSPIRE].
R. Vanhove et al., Mapping topological to conformal field theories through strange correlators, Phys. Rev. Lett. 121 (2018) 177203 [arXiv:1801.05959] [INSPIRE].
A. Davydov, L. Kong and I. Runkel, Invertible defects and isomorphisms of rational CFTs, Adv. Theor. Math. Phys. 15 (2011) 43 [arXiv:1004.4725] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
N. Drukker, D. Gaiotto and J. Gomis, The virtue of defects in 4D gauge theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [INSPIRE].
D. Gaiotto, Open Verlinde line operators, arXiv:1404.0332 [INSPIRE].
G.W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].
G. W. Moore and N. Seiberg, Lectures on RCFT, in the proceedings of the 1989 Banff NATO ASI: Physics, Geometry and Topology, August 14-25, Banff, Canada (1989).
V.G. Turaev, Quantum invariants of knots and 3-manifolds, 2nd edition, De Gruyter studies in mathematics 18, Walter De Gruyter, Germany (2010).
A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321 (2006) 2.
P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories, Mathematical Surveys and Monographs, American Mathematical Society, U.S.A. (2016).
P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, Ann. Math. 162 (2005) 581.
M.A. Levin and X.-G. Wen, String net condensation: a physical mechanism for topological phases, Phys. Rev. B 71 (2005) 045110 [cond-mat/0404617] [INSPIRE].
G. ’t Hooft et al., Recent developments in gauge theories, NATO Sci. Ser. B 59 (1980) 1.
A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in various dimensions and group cohomology, arXiv:1404.3230 [INSPIRE].
C. Vafa, Modular invariance and discrete torsion on orbifolds, Nucl. Phys. B 273 (1986) 592 [INSPIRE].
I. Brunner, N. Carqueville and D. Plencner, Discrete torsion defects, Commun. Math. Phys. 337 (2015) 429 [arXiv:1404.7497] [INSPIRE].
C. Bachas and M. Gaberdiel, Loop operators and the Kondo problem, JHEP 11 (2004) 065 [hep-th/0411067] [INSPIRE].
T.J. Hagge and S.-M. Hong, Some non-braided fusion categories of rank 3, arXiv:0704.0208.
V. Ostrik, Pivotal fusion categories of rank 3, Mosc. Math. J. 15 (2015) 373 [arXiv:1309.4822].
G.W. Moore and G. Segal, D-branes and k-theory in 2D topological field theory, hep-th/0609042 [INSPIRE].
J. C. Baez and J. Dolan, Higher dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995) 6073.
J. Lurie, On the classification of topological field theories, Curr. Devel. Math. 2008 (2009) 129.
D.S. Freed, M.J. Hopkins, J. Lurie and C. Teleman, Topological quantum field theories from compact Lie groups, in A celebration of Raoul Bott’s legacy in mathematics montreal, Canada, June 9-13, 2008, arXiv:0905.0731 [INSPIRE].
C.J. Schommer-Pries, The classification of two-dimensional extended topological field theories, arXiv:1112.1000 [INSPIRE].
E. Witten, The “parity” anomaly on an unorientable manifold, Phys. Rev. B 94 (2016) 195150 [arXiv:1605.02391] [INSPIRE].
J. Wang, X.-G. Wen and E. Witten, Symmetric gapped interfaces of SPT and SET states: systematic constructions, Phys. Rev. X 8 (2018) 031048 [arXiv:1705.06728] [INSPIRE].
Y. Tachikawa, On gauging finite subgroups, arXiv:1712.09542 [INSPIRE].
M. Mueger, Tensor categories: a selective guided tour, arXiv:0804.3587.
A. Davydov, L. Kong and I. Runkel, Field theories with defects and the centre functor, arXiv:1107.0495 [INSPIRE].
N. Carqueville, Lecture notes on 2-dimensional defect TQFT, 2016, arXiv:1607.05747 [INSPIRE].
H. Sonoda, Sewing conformal field theories, Nucl. Phys. B 311 (1988) 401 [INSPIRE].
H. Sonoda, Sewing conformal field theories. 2, Nucl. Phys. B 311 (1988) 417 [INSPIRE].
B. Bakalov and A. Kirillov, On the Lego-Teichmüller game, Transf. Groups 5 (2000) 207.
L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65 (1944) 117.
C.N. Yang, The spontaneous magnetization of a two-dimensional Ising model, Phys. Rev. 85 (1952) 808.
V. Ostrik, Fusion categories of rank 2, Math. Res. Lett. 10 (2003) 177 [math/0203255].
D. Tambara and S. Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, J. Algebra 209 (1998) 692.
P. Etingof, S. Gelaki and V. Ostrik, Classification of fusion categories of dimension pq, math/0304194.
V. Ostrik, Pre-modular categories of rank 3, Mosc. Math. J. 8 (2008) 111 [math/0503564].
R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Defect lines, dualities and generalised orbifolds, in the proceedings of the 16th International Congress on Mathematical Physics (ICMP09), August 3-8, Prague, Czech Republic, (2009), arXiv:0909.5013 [INSPIRE].
H.A. Kramers and G.H. Wannier, Statistics of the two-dimensional ferromagnet. Part I, Phys. Rev. 60 (1941) 252.
G.W. Moore and N. Seiberg, Taming the conformal zoo, Phys. Lett. B 220 (1989) 422 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer, Germany (1997).
D.A. Huse, Exact exponents for infinitely many new multicritical points, Phys. Rev. B 30 (1984) 3908.
A. Alekseev and S. Monnier, Quantization of Wilson loops in Wess-Zumino-Witten models, JHEP 08 (2007) 039 [hep-th/0702174] [INSPIRE].
C. Bachas and S. Monnier, Defect loops in gauged Wess-Zumino-Witten models, JHEP 02 (2010) 003 [arXiv:0911.1562] [INSPIRE].
V.A. Fateev and A.B. Zamolodchikov, Integrable perturbations of Z(N) parafermion models and O(3) σ-model, Phys. Lett. B 271 (1991) 91 [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
A.B. Zamolodchikov, S matrix of the subleading magnetic perturbation of the tricritical Ising model, PUPT-1195 (1990).
R.M. Ellem and V.V. Bazhanov, Thermodynamic Bethe ansatz for the subleading magnetic perturbation of the tricritical Ising model, Nucl. Phys. B 512 (1998) 563 [hep-th/9703026] [INSPIRE].
V.P. Yurov and A.B. Zamolodchikov, Truncated conformal space approach to scaling Lee-Yang model, Int. J. Mod. Phys. A 5 (1990) 3221 [INSPIRE].
A.B. Zamolodchikov, Renormalization group and perturbation theory near fixed points in two-dimensional field theory, Sov. J. Nucl. Phys. 46 (1987) 1090 [Yad. Fiz. 46 (1987) 1819] [INSPIRE].
A.W.W. Ludwig and J.L. Cardy, Perturbative evaluation of the conformal anomaly at new critical points with applications to random systems, Nucl. Phys. B 285 (1987) 687 [INSPIRE].
A.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models. scaling three state Potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
A.B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B 358 (1991) 497 [INSPIRE].
M.J. Martins, The off critical behavior of the multicritical Ising models, Int. J. Mod. Phys. A 7 (1992) 7753 [INSPIRE].
G. Feverati, E. Quattrini and F. Ravanini, Infrared behavior of massless integrable flows entering the minimal models from \( \phi \)(31), Phys. Lett. B 374 (1996) 64 [hep-th/9512104] [INSPIRE].
D. Gaiotto, Domain walls for two-dimensional renormalization group flows, JHEP 12 (2012) 103 [arXiv:1201.0767] [INSPIRE].
V. Dotsenko, J.L. Jacobsen, M.-A. Lewis and M. Picco, Coupled Potts models: self-duality and fixed point structure, Nucl. Phys. B 546 (1999) 505 [INSPIRE].
G. Sarkissian, Defects and Permutation branes in the Liouville field theory, Nucl. Phys. B 821 (2009) 607 [arXiv:0903.4422] [INSPIRE].
C. Bachas, I. Brunner and D. Roggenkamp, Fusion of critical defect lines in the 2D Ising model, J. Stat. Mech. 1308 (2013) P08008 [arXiv:1303.3616] [INSPIRE].
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Chang, CM., Lin, YH., Shao, SH. et al. Topological defect lines and renormalization group flows in two dimensions. J. High Energ. Phys. 2019, 26 (2019). https://doi.org/10.1007/JHEP01(2019)026
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DOI: https://doi.org/10.1007/JHEP01(2019)026