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Topological defect lines and renormalization group flows in two dimensions

  • Chi-Ming Chang
  • Ying-Hsuan Lin
  • Shu-Heng ShaoEmail author
  • Yifan Wang
  • Xi Yin
Open Access
Regular Article - Theoretical Physics
  • 45 Downloads

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.

Keywords

Anomalies in Field and String Theories Conformal Field Theory Global Symmetries 

Notes

Open Access

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References

  1. [1]
    J.L. Cardy, Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 275 (1986) 200 [INSPIRE].
  2. [2]
    E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
  3. [3]
    J. Cardy, Boundary conditions in conformal field theory, Adv. Stud. Pure Math. 19 (1989) 127Google Scholar
  4. [4]
    J.L. Cardy, Boundary conformal field theory, Encycl. Math. Phys. (2006) 333, hep-th/0411189 [INSPIRE].
  5. [5]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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].
  7. [7]
    V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [INSPIRE].
  8. [8]
    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].
  9. [9]
    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].
  10. [10]
    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].
  11. [11]
    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].
  12. [12]
    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].
  13. [13]
    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].
  14. [14]
    T. Quella, I. Runkel and G.M.T. Watts, Reflection and transmission for conformal defects, JHEP 04 (2007) 095 [hep-th/0611296] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. Fuchs, I. Runkel and C. Schweigert, The fusion algebra of bimodule categories, Appl. Categ. Struct. 16 (2008) 123 [math/0701223] [INSPIRE].
  16. [16]
    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].
  17. [17]
    C. Bachas and I. Brunner, Fusion of conformal interfaces, JHEP 02 (2008) 085 [arXiv:0712.0076] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    L. Kong and I. Runkel, Cardy algebras and sewing constraints. I., Commun. Math. Phys. 292 (2009) 871 [arXiv:0807.3356] [INSPIRE].
  19. [19]
    V.B. Petkova, On the crossing relation in the presence of defects, JHEP 04 (2010) 061 [arXiv:0912.5535] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    Communications in Mathematical Physics 313 (2012) 351 [arXiv:1104.5047].
  21. [21]
    N. Carqueville and I. Runkel, Orbifold completion of defect bicategories, Quantum Topol. 7 (2016) 203 [arXiv:1210.6363] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    I. Brunner, N. Carqueville and D. Plencner, A quick guide to defect orbifolds, Proc. Symp. Pure Math. 88 (2014) 231 [arXiv:1310.0062] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    A. Davydov, L. Kong and I. Runkel, Functoriality of the center of an algebra, Adv. Math. 285 (2015) 811 [arXiv:1307.5956] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    L. Kong, Q. Li and I. Runkel, Cardy algebras and sewing constraints, II, Adv. Math. 262 (2014) 604 [arXiv:1310.1875] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    V.B. Petkova, Topological defects in CFT, Phys. Atom. Nucl. 76 (2013) 1268 [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    M. Bischoff, R. Longo, Y. Kawahigashi and K.-H. Rehren, Tensor categories of endomorphisms and inclusions of von Neumann algebras, arXiv:1407.4793 [INSPIRE].
  27. [27]
    L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [arXiv:1704.02330] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    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].
  29. [29]
    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].
  30. [30]
    J.C. Bridgeman and D.J. Williamson, Anomalies and entanglement renormalization, Phys. Rev. B 96 (2017) 125104 [arXiv:1703.07782] [INSPIRE].
  31. [31]
    R. Vanhove et al., Mapping topological to conformal field theories through strange correlators, Phys. Rev. Lett. 121 (2018) 177203 [arXiv:1801.05959] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    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].
  36. [36]
    D. Gaiotto, Open Verlinde line operators, arXiv:1404.0332 [INSPIRE].
  37. [37]
    G.W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    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).Google Scholar
  39. [39]
    V.G. Turaev, Quantum invariants of knots and 3-manifolds, 2nd edition, De Gruyter studies in mathematics 18, Walter De Gruyter, Germany (2010).Google Scholar
  40. [40]
    A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321 (2006) 2.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories, Mathematical Surveys and Monographs, American Mathematical Society, U.S.A. (2016).Google Scholar
  42. [42]
    P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, Ann. Math. 162 (2005) 581.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    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].
  44. [44]
    G. ’t Hooft et al., Recent developments in gauge theories, NATO Sci. Ser. B 59 (1980) 1.Google Scholar
  45. [45]
    A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in various dimensions and group cohomology, arXiv:1404.3230 [INSPIRE].
  46. [46]
    C. Vafa, Modular invariance and discrete torsion on orbifolds, Nucl. Phys. B 273 (1986) 592 [INSPIRE].
  47. [47]
    I. Brunner, N. Carqueville and D. Plencner, Discrete torsion defects, Commun. Math. Phys. 337 (2015) 429 [arXiv:1404.7497] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    C. Bachas and M. Gaberdiel, Loop operators and the Kondo problem, JHEP 11 (2004) 065 [hep-th/0411067] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    T.J. Hagge and S.-M. Hong, Some non-braided fusion categories of rank 3, arXiv:0704.0208.
  50. [50]
    V. Ostrik, Pivotal fusion categories of rank 3, Mosc. Math. J. 15 (2015) 373 [arXiv:1309.4822].MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    G.W. Moore and G. Segal, D-branes and k-theory in 2D topological field theory, hep-th/0609042 [INSPIRE].
  52. [52]
    J. C. Baez and J. Dolan, Higher dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995) 6073.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    J. Lurie, On the classification of topological field theories, Curr. Devel. Math. 2008 (2009) 129.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    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].
  55. [55]
    C.J. Schommer-Pries, The classification of two-dimensional extended topological field theories, arXiv:1112.1000 [INSPIRE].
  56. [56]
    E. Witten, The “parity” anomaly on an unorientable manifold, Phys. Rev. B 94 (2016) 195150 [arXiv:1605.02391] [INSPIRE].
  57. [57]
    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].
  58. [58]
    Y. Tachikawa, On gauging finite subgroups, arXiv:1712.09542 [INSPIRE].
  59. [59]
    M. Mueger, Tensor categories: a selective guided tour, arXiv:0804.3587.
  60. [60]
    A. Davydov, L. Kong and I. Runkel, Field theories with defects and the centre functor, arXiv:1107.0495 [INSPIRE].
  61. [61]
    N. Carqueville, Lecture notes on 2-dimensional defect TQFT, 2016, arXiv:1607.05747 [INSPIRE].
  62. [62]
    H. Sonoda, Sewing conformal field theories, Nucl. Phys. B 311 (1988) 401 [INSPIRE].
  63. [63]
    H. Sonoda, Sewing conformal field theories. 2, Nucl. Phys. B 311 (1988) 417 [INSPIRE].
  64. [64]
    B. Bakalov and A. Kirillov, On the Lego-Teichmüller game, Transf. Groups 5 (2000) 207.CrossRefzbMATHGoogle Scholar
  65. [65]
    L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65 (1944) 117.Google Scholar
  66. [66]
    C.N. Yang, The spontaneous magnetization of a two-dimensional Ising model, Phys. Rev. 85 (1952) 808.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    V. Ostrik, Fusion categories of rank 2, Math. Res. Lett. 10 (2003) 177 [math/0203255].
  68. [68]
    D. Tambara and S. Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, J. Algebra 209 (1998) 692.MathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    P. Etingof, S. Gelaki and V. Ostrik, Classification of fusion categories of dimension pq, math/0304194.
  70. [70]
    V. Ostrik, Pre-modular categories of rank 3, Mosc. Math. J. 8 (2008) 111 [math/0503564].
  71. [71]
    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    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].
  73. [73]
    H.A. Kramers and G.H. Wannier, Statistics of the two-dimensional ferromagnet. Part I, Phys. Rev. 60 (1941) 252.Google Scholar
  74. [74]
    G.W. Moore and N. Seiberg, Taming the conformal zoo, Phys. Lett. B 220 (1989) 422 [INSPIRE].
  75. [75]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer, Germany (1997).Google Scholar
  76. [76]
    D.A. Huse, Exact exponents for infinitely many new multicritical points, Phys. Rev. B 30 (1984) 3908.Google Scholar
  77. [77]
    A. Alekseev and S. Monnier, Quantization of Wilson loops in Wess-Zumino-Witten models, JHEP 08 (2007) 039 [hep-th/0702174] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    C. Bachas and S. Monnier, Defect loops in gauged Wess-Zumino-Witten models, JHEP 02 (2010) 003 [arXiv:0911.1562] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  79. [79]
    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].
  80. [80]
    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  81. [81]
    A.B. Zamolodchikov, S matrix of the subleading magnetic perturbation of the tricritical Ising model, PUPT-1195 (1990).Google Scholar
  82. [82]
    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].
  83. [83]
    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].
  84. [84]
    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].
  85. [85]
    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].
  86. [86]
    A.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models. scaling three state Potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
  87. [87]
    A.B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B 358 (1991) 497 [INSPIRE].
  88. [88]
    M.J. Martins, The off critical behavior of the multicritical Ising models, Int. J. Mod. Phys. A 7 (1992) 7753 [INSPIRE].
  89. [89]
    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].
  90. [90]
    D. Gaiotto, Domain walls for two-dimensional renormalization group flows, JHEP 12 (2012) 103 [arXiv:1201.0767] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  91. [91]
    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].
  92. [92]
    G. Sarkissian, Defects and Permutation branes in the Liouville field theory, Nucl. Phys. B 821 (2009) 607 [arXiv:0903.4422] [INSPIRE].
  93. [93]
    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].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Chi-Ming Chang
    • 1
  • Ying-Hsuan Lin
    • 2
  • Shu-Heng Shao
    • 3
    Email author
  • Yifan Wang
    • 4
  • Xi Yin
    • 5
  1. 1.Center for Quantum Mathematics and Physics (QMAP)University of CaliforniaDavisU.S.A.
  2. 2.Walter Burke Institute for Theoretical Physics, California Institute of TechnologyPasadenaU.S.A.
  3. 3.School of Natural Sciences, Institute for Advanced StudyPrincetonU.S.A.
  4. 4.Joseph Henry LaboratoriesPrinceton UniversityPrincetonU.S.A.
  5. 5.Jefferson Physical LaboratoryHarvard UniversityCambridgeU.S.A.

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