Abstract
We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c = 1 CFTs and uncover a rich structure. We find that even those c = 1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Gukov and A. Kapustin, Topological quantum field theory, nonlocal operators, and gapped phases of gauge theories, arXiv:1307.4793 [INSPIRE].
A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, Prog. Math. 324 (2017) 177 [arXiv:1309.4721] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, Prog. Math. 324 (2017) 177 [arXiv:1309.4721] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal, and temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [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].
A. Kapustin and M. Tikhonov, Abelian duality, walls and boundary conditions in diverse dimensions, JHEP 11 (2009) 006 [arXiv:0904.0840] [INSPIRE].
A. Kapustin and N. Saulina, Surface operators in 3d topological field theory and 2d rational conformal field theory, arXiv:1012.0911 [INSPIRE].
J. Fuchs, C. Schweigert and A. Valentino, Bicategories for boundary conditions and for surface defects in 3d TFT, Commun. Math. Phys. 321 (2013) 543 [arXiv:1203.4568] [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].
L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [arXiv:1704.02330] [INSPIRE].
R. Vanhove et al., Mapping topological to conformal field theories through strange correlators, Phys. Rev. Lett. 121 (2018) 177203 [arXiv:1801.05959] [INSPIRE].
C.-M. Chang et al., Topological defect lines and renormalization group flows in two dimensions, JHEP 01 (2019) 026 [arXiv:1802.04445] [INSPIRE].
Y.-H. Lin and S.-H. Shao, Duality defect of the monster CFT, J. Phys. A 54 (2021) 065201 [arXiv:1911.00042] [INSPIRE].
R. Thorngren and Y. Wang, Fusion category symmetry. Part I. Anomaly in-flow and gapped phases, JHEP 04 (2024) 132 [arXiv:1912.02817] [INSPIRE].
T. Lichtman et al., Bulk anyons as edge symmetries: boundary phase diagrams of topologically ordered states, Phys. Rev. B 104 (2021) 075141 [arXiv:2003.04328] [INSPIRE].
D. Aasen, P. Fendley and R.S.K. Mong, Topological defects on the lattice: dualities and degeneracies, arXiv:2008.08598 [INSPIRE].
Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, Symmetries and strings of adjoint QCD2, JHEP 03 (2021) 103 [arXiv:2008.07567] [INSPIRE].
C.-M. Chang and Y.-H. Lin, Lorentzian dynamics and factorization beyond rationality, JHEP 10 (2021) 125 [arXiv:2012.01429] [INSPIRE].
T.-C. Huang and Y.-H. Lin, Topological field theory with Haagerup symmetry, J. Math. Phys. 63 (2022) 042306 [arXiv:2102.05664] [INSPIRE].
W. Ji and X.-G. Wen, Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions, Phys. Rev. Res. 2 (2020) 033417 [arXiv:1912.13492] [INSPIRE].
L. Kong et al., Algebraic higher symmetry and categorical symmetry — a holographic and entanglement view of symmetry, Phys. Rev. Res. 2 (2020) 043086 [arXiv:2005.14178] [INSPIRE].
A. Kapustin, Topological field theory, higher categories, and their applications, in the proceedings of the International Congress of Mathematicians, (2010) [arXiv:1004.2307] [INSPIRE].
T. Johnson-Freyd, On the classification of topological orders, Commun. Math. Phys. 393 (2022) 989 [arXiv:2003.06663] [INSPIRE].
A. Davydov, L. Kong and I. Runkel, Field theories with defects and the centre functor, arXiv:1107.0495 [INSPIRE].
P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories, Mathematical Surveys and Monographs. American Mathematical Society, U.S.A. (2015).
K. Graham and G.M.T. Watts, Defect lines and boundary flows, JHEP 04 (2004) 019 [hep-th/0306167] [INSPIRE].
E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
P.H. Ginsparg, Curiosities at c = 1, Nucl. Phys. B 295 (1988) 153 [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, (1989) [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. 4. Structure constants and correlation functions, Nucl. Phys. B 715 (2005) 539 [hep-th/0412290] [INSPIRE].
J. Fjelstad, J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators. 5. Proof of modular invariance and factorisation, Theor. Appl. Categor. 16 (2006) 342 [hep-th/0503194] [INSPIRE].
C. Bachas and S. Monnier, Defect loops in gauged Wess-Zumino-Witten models, JHEP 02 (2010) 003 [arXiv:0911.1562] [INSPIRE].
D. Tambara and S. Yamagami, Tensor categories with fusion rules of self-duality for finite Abelian groups, J. Algebra 209 (1998) 692.
A. Recknagel and V. Schomerus, Boundary conformal field theory and the worldsheet approach to D-branes, Cambridge University Press, Cambridge, U.K. (2013) [https://doi.org/10.1017/CBO9780511806476] [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].
Y. Tachikawa, On gauging finite subgroups, SciPost Phys. 8 (2020) 015 [arXiv:1712.09542] [INSPIRE].
Y.-H. Lin and S.-H. Shao, Anomalies and bounds on charged operators, Phys. Rev. D 100 (2019) 025013 [arXiv:1904.04833] [INSPIRE].
B. Heidenreich et al., Non-invertible global symmetries and completeness of the spectrum, JHEP 09 (2021) 203 [arXiv:2104.07036] [INSPIRE].
R. Thorngren, Anomalies and bosonization, Commun. Math. Phys. 378 (2020) 1775 [arXiv:1810.04414] [INSPIRE].
A. Karch, D. Tong and C. Turner, A web of 2d dualities: Z2 gauge fields and Arf invariants, SciPost Phys. 7 (2019) 007 [arXiv:1902.05550] [INSPIRE].
W. Ji, S.-H. Shao and X.-G. Wen, Topological transition on the conformal manifold, Phys. Rev. Res. 2 (2020) 033317 [arXiv:1909.01425] [INSPIRE].
R. Thorngren and C. von Keyserlingk, Higher SPT’s and a generalization of anomaly in-flow, arXiv:1511.02929 [INSPIRE].
L. Fidkowski and A. Kitaev, The effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81 (2010) 134509 [arXiv:0904.2197] [INSPIRE].
S. Ryu and S.-C. Zhang, Interacting topological phases and modular invariance, Phys. Rev. B 85 (2012) 245132 [arXiv:1202.4484] [INSPIRE].
H. Yao and S. Ryu, Interaction effect on topological classification of superconductors in two dimensions, Phys. Rev. B 88 (2013) 064507 [arXiv:1202.5805] [INSPIRE].
X.-L. Qi, A new class of (2 + 1)-dimensional topological superconductors with Z topological classification, New J. Phys. 15 (2013) 065002 [arXiv:1202.3983] [INSPIRE].
A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, JHEP 12 (2015) 052 [arXiv:1406.7329] [INSPIRE].
T. Numasawa and S. Yamaguchi, Mixed global anomalies and boundary conformal field theories, JHEP 11 (2018) 202 [arXiv:1712.09361] [INSPIRE].
I. Runkel and L. Szegedy, Topological field theory on r-spin surfaces and the Arf-invariant, J. Math. Phys. 62 (2021) 102302 [arXiv:1802.09978] [INSPIRE].
D. Radicevic, Spin structures and exact dualities in low dimensions, arXiv:1809.07757 [INSPIRE].
Y. Yao and A. Furusaki, Parafermionization, bosonization, and critical parafermionic theories, JHEP 04 (2021) 285 [arXiv:2012.07529] [INSPIRE].
E.H. Fradkin and L.P. Kadanoff, Disorder variables and parafermions in two-dimensional statistical mechanics, Nucl. Phys. B 170 (1980) 1 [INSPIRE].
V.A. Fateev and A.B. Zamolodchikov, Parafermionic currents in the two-dimensional conformal quantum field theory and selfdual critical points in Zn invariant statistical systems, Sov. Phys. JETP 62 (1985) 215 [INSPIRE].
J.A. Harvey and G.W. Moore, An uplifting discussion of T-duality, JHEP 05 (2018) 145 [arXiv:1707.08888] [INSPIRE].
N. Carqueville, I. Runkel and G. Schaumann, Line and surface defects in Reshetikhin-Turaev TQFT, Quantum Topol. 10 (2018) 399 [arXiv:1710.10214] [INSPIRE].
R. Longo and K.-H. Rehren, Nets of subfactors, Rev. Math. Phys. 7 (1995) 567 [hep-th/9411077] [INSPIRE].
J. Bockenhauer and D.E. Evans, Modular invariants, graphs and alpha induction for nets of subfactors. 1, Commun. Math. Phys. 197 (1998) 361 [hep-th/9801171] [INSPIRE].
J. Bockenhauer and D.E. Evans, Modular invariants, graphs and alpha induction for nets of subfactors. 2, Commun. Math. Phys. 200 (1999) 57 [hep-th/9805023] [INSPIRE].
J. Bockenhauer and D.E. Evans, Modular invariants, graphs and alpha induction for nets of subfactors. 3, Commun. Math. Phys. 205 (1999) 183 [hep-th/9812110] [INSPIRE].
J. Bockenhauer, D.E. Evans and Y. Kawahigashi, Chiral structure of modular invariants for subfactors, Commun. Math. Phys. 210 (2000) 733 [math/9907149] [INSPIRE].
V. Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003) 177 [math/0111139] [INSPIRE].
J. Milnor and D. Husemöller, Symmetric bilinear forms, Springer, Berlin, Heidelberg, Germany (1973) [https://doi.org/10.1007/978-3-642-88330-9].
C.T.C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963) 281.
L. Wang and Z. Wang, In and around Abelian anyon models, J. Phys. A 53 (2020) 505203 [arXiv:2004.12048] [INSPIRE].
V. Turaev, Reciprocity for Gauss sums on finite Abelian groups, Math. Proc. Camb. Phil. Soc. 124 (1998) 205.
D. Tambara, Representations of tensor categories with fusion rules of self-duality for Abelian groups, Israel J. Math. 118 (2000) 29.
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, C = 1 conformal field theories on Riemann surfaces, Commun. Math. Phys. 115 (1988) 649 [INSPIRE].
D. Gepner and Z.-A. Qiu, Modular invariant partition functions for parafermionic field theories, Nucl. Phys. B 285 (1987) 423 [INSPIRE].
V.A. Fateev and A.B. Zamolodchikov, Integrable perturbations of ZN parafermion models and the O(3) sigma model, Phys. Lett. B 271 (1991) 91 [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].
R. Dijkgraaf, C. Vafa, E.P. Verlinde and H.L. Verlinde, The operator algebra of orbifold models, Commun. Math. Phys. 123 (1989) 485 [INSPIRE].
S.-K. Yang, Z4 × Z4 symmetry and parafermion operators in the selfdual critical Ashkin-Teller model, Nucl. Phys. B 285 (1987) 639 [INSPIRE].
E. Meir and E. Musicantov, Module categories over graded fusion categories, J. Pure Appl. Algebra 216 (2012) 2449.
D. Naidu, Categorical Morita equivalence for group-theoretical categories, math/0605530.
F.C. Alcaraz and R. Koberle, Duality and the phases of Zn spin systems, J. Phys. A 13 (1980) L153 [INSPIRE].
F.C. Alcaraz and R. Koberle, The phases of two-dimensional spin and four-dimensional gauge systems with ZN symmetry, J. Phys. A 14 (1981) 1169 [INSPIRE].
F.C. Alcaraz, The critical behavior of selfdual ZN spin systems: finite size scaling and conformal invariance, J. Phys. A 20 (1987) 2511 [INSPIRE].
P. Dorey, R. Tateo and K.E. Thompson, Massive and massless phases in selfdual ZN spin models: some exact results from the thermodynamic Bethe ansatz, Nucl. Phys. B 470 (1996) 317 [hep-th/9601123] [INSPIRE].
A. Cappelli and G. D’Appollonio, Boundary states of c = 1 and 3/2 rational conformal field theories, JHEP 02 (2002) 039 [hep-th/0201173] [INSPIRE].
M. Nguyen, Y. Tanizaki and M. Ünsal, Noninvertible 1-form symmetry and Casimir scaling in 2D Yang-Mills theory, Phys. Rev. D 104 (2021) 065003 [arXiv:2104.01824] [INSPIRE].
D. Baker, Differential characters and Borel cohomology, Topology 16 (1977) 441.
M.R. Gaberdiel, A. Recknagel and G.M.T. Watts, The conformal boundary states for SU(2) at level 1, Nucl. Phys. B 626 (2002) 344 [hep-th/0108102] [INSPIRE].
A.B. Zamolodchikov, Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys. 65 (1985) 1205 [INSPIRE].
R. Blumenhagen et al., W algebras with two and three generators, Nucl. Phys. B 361 (1991) 255 [INSPIRE].
K. Hornfeck, W algebras with set of primary fields of dimensions (3, 4, 5) and (3, 4, 5, 6), Nucl. Phys. B 407 (1993) 237 [hep-th/9212104] [INSPIRE].
J. de Boer, L. Feher and A. Honecker, A class of W algebras with infinitely generated classical limit, Nucl. Phys. B 420 (1994) 409 [hep-th/9312049] [INSPIRE].
R. Blumenhagen et al., Coset realization of unifying W algebras, Int. J. Mod. Phys. A 10 (1995) 2367 [hep-th/9406203] [INSPIRE].
C. Dong, C.H. Lam, Q. Wang and H. Yamada, The structure of parafermion vertex operator algebras, Commun. Math. Phys. 299 (2010) 783 [arXiv:0904.2758] [INSPIRE].
V.A. Fateev, Integrable deformations in ZN symmetrical models of conformal quantum field theory, Int. J. Mod. Phys. A 6 (1991) 2109 [INSPIRE].
Acknowledgements
We would like to thank Zohar Komargodski for collaborating on this work in its early stages. RT would also like to acknowledge Tsuf Lichtman, Erez Berg, Ady Stern, and Netanel Lindner for collaboration on a related project, as well as Dave Aasen and Dominic Williamson for many useful discussions. The work of YW is supported in part by the Center for Mathematical Sciences and Applications and the Center for the Fundamental Laws of Nature at Harvard University. YW would like to thank Ofer Aharony and Xi Yin for useful discussions. YW is also grateful to the Weizmann Institute of Science for hospitality where the project was initiated during his visit.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2106.12577
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Thorngren, R., Wang, Y. Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond. J. High Energ. Phys. 2024, 51 (2024). https://doi.org/10.1007/JHEP07(2024)051
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2024)051