Abstract
We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe ’t Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to “ungauge” the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Aasen, E. Lake and K. Walker, Fermion condensation and super pivotal categories, arXiv:1709.01941.
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].
M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, Symmetry fractionalization, defects, and gauging of topological phases, Phys. Rev. B 100 (2019) 115147.
L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, arXiv:1704.02330.
J.C. Bridgeman and D.J. Williamson, Anomalies and entanglement renormalization, Phys. Rev. B 96 (2017) 125104 [arXiv:1703.07782] [INSPIRE].
K.S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Springer New York (2012).
M. Buican and A. Gromov, Anyonic Chains, Topological Defects, and Conformal Field Theory, Commun. Math. Phys. 356 (2017) 1017 [arXiv:1701.02800] [INSPIRE].
N. Bultinck et al., Anyons and matrix product operator algebras, Annals Phys. 378 (2017) 183 [arXiv:1511.08090] [INSPIRE].
C.-M. Chang et al., Topological defect lines and renormalization group flows in two dimensions, JHEP 01 (2019) 026 [arXiv:1802.04445].
X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87 (2013) 155114.
J.I. Cirac, D. Pérez-García, N. Schuch and F. Verstraete, Matrix product density operators: Renormalization fixed points and boundary theories, Annals Phys. 378 (2017) 100.
C. Cordova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the Space of Coupling Constants and Their Dynamical Applications II, arXiv:1905.13361.
A. Davydov, M. Müger, D. Nikshych and V. Ostrik, The Witt group of non-degenerate braided fusion categories, J. Reine Angew. Math. (Crelles Journal) 2013 (2013).
A. Davydov, D. Nikshych and V. Ostrik, On the structure of the Witt group of braided fusion categories, arXiv:1109.5558.
R. Dijkgraaf, C. Vafa, E.P. Verlinde and H.L. Verlinde, The Operator Algebra of Orbifold Models, Commun. Math. Phys. 123 (1989) 485 [INSPIRE].
R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].
P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor Categories, American Mathematical Society (2015).
P. Etingof, D. Nikshych and V. Ostrik, Weakly group-theoretical and solvable fusion categories, Adv. Math. 226 (2011) 176.
P. Etingof, D. Nikshych and V. Ostrik, Fusion categories and homotopy theory, Quantum Topol. 1 (2010) 209.
A. Feiguin et al., Interacting Anyons in Topological Quantum Liquids: The Golden Chain, Phys. Rev. Lett. 98 (2007) 160409.
J. Fuchs, C. Schweigert and A. Valentino, Bicategories for boundary conditions and for surface defects in 3-d TFT, Commun. Math. Phys. 321 (2013) 543 [arXiv:1203.4568] [INSPIRE].
D. Gaiotto and E. Witten, S-Duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172.
S. Gukov and A. Kapustin, Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories, arXiv:1307.4793.
L.-Y. Hung and Y. Wan, Ground-State Degeneracy of Topological Phases on Open Surfaces, Phys. Rev. Lett. 114 (2015) 076401.
W. Ji and X.-G. Wen, Noninvertible anomalies and mapping-class-group transformation of anomalous partition functions, Phys. Rev. Res. 1 (2019) 033054.
A.K. Jr. and B. Balsam, Turaev-Viro invariants as an extended TQFT, arXiv:1004.1533.
A. Kapustin, Topological field theory, higher categories, and their applications, in proceedings of the International Congress of Mathematicians 2010 (ICM 2010), June 2011.
A. Kapustin and N. Saulina, Surface operators in 3d topological field theory and 2d rational conformal field theory, Mathematical Foundations of Quantum Field Theory and Perturbative String Theory 83 (2011) 175, DOI:https://doi.org/10.1090/pspum/083/2742429.
A. Kapustin and N. Saulina, Topological boundary conditions in abelian Chern-Simons theory, Nucl. Phys. B 845 (2011) 393 [arXiv:1008.0654] [INSPIRE].
A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in various dimensions and group cohomology, arXiv:1404.3230.
A. Kapustin and R. Thorngren, Higher Symmetry and Gapped Phases of Gauge Theories, Progr. Math. 324 (2017) 177.
A. Kapustin and M. Tikhonov, Abelian duality, walls and boundary conditions in diverse dimensions, JHEP 2009 (2009) 006.
G.I. Kats and V.G. Palyutkin, Finite ring groups, Trudy Moskovskogo Matematicheskogo Obshchestva 15 (1966) 224.
A. Kitaev and L. Kong, Models for Gapped Boundaries and Domain Walls, Commun. Math. Phys. 313 (2012) 351 [arXiv:1104.5047] [INSPIRE].
T. Lan, L. Kong and X.-G. Wen, Classification of (3+1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons, Phys. Rev. X 8 (2018) 021074.
T. Lan, J.C. Wang and X.-G. Wen, Gapped Domain Walls, Gapped Boundaries, and Topological Degeneracy, Phys. Rev. Lett. 114 (2015) 076402.
M. Levin, Protected Edge Modes without Symmetry, Phys. Rev. X 3 (2013) 021009.
M.A. Levin and X.-G. Wen, String-net condensation:A physical mechanism for topological phases, Phys. Rev. B 71 (2005) 045110.
K. Maya, A.M. Castaño and B. Uribe, Classification of Pointed Fusion Categories of dimension p3 up to weak Morita Equivalence, arXiv:1808.05139.
E. Meir and E. Musicantov, Module categories over graded fusion categories, J. Pure Appl. Algebra 216 (2012) 2449.
Á. Muñoz and B. Uribe, Classification of Pointed Fusion Categories of dimension 8 up to weak Morita equivalence, Commun. Algebra 46 (2018) 3873.
V. Ostrik, Module categories over the Drinfeld double of a finite group, Int. Math. Res. Not. 2003 (2003) 1507.
R.N.C. Pfeifer et al., Translation invariance, topology, and protection of criticality in chains of interacting anyons, Phys. Rev. B 86 (2012) 155111.
N.Y. Reshetikhin and V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127 (1990) 1 [INSPIRE].
T. Scaffidi, D.E. Parker and R. Vasseur, Gapless Symmetry-Protected Topological Order, Phys. Rev. X 7 (2017) 041048.
D. Tambara, Representations of tensor categories with fusion rules of self-duality for abelian groups, Israel J. Math. 118 (2000) 29.
D. Tambara and S. Yamagami, Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups, J. Algebra 209 (1998) 692.
R. Thorngren and Y. Wang, Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond, R. Thorngren and Y. Wang, Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond, arXiv:2106.12577 [INSPIRE].
V.G. Turaev and O.Y. Viro, State sum invariants of 3 manifolds and quantum 6j symbols, Topology 31 (1992) 865.
V.G. Turaev, Modular categories and 3-manifold invariants, Int. J. Mod. Phys. B 06 (1992) 1807.
R. Verresen, R. Thorngren, N.G. Jones and F. Pollmann, Gapless topological phases and symmetry-enriched quantum criticality, arXiv:1905.06969.
J. Vidal, R. Thomale, K.P. Schmidt and S. Dusuel, Self-duality and bound states of the toric code model in a transverse field, Phys. Rev. B 80 (2009) 081104.
D.J. Williamson et al., Matrix product operators for symmetry-protected topological phases: Gauging and edge theories, Phys. Rev. B 94 (2016) 205150.
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: 1912.02817
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 I. Anomaly in-flow and gapped phases. J. High Energ. Phys. 2024, 132 (2024). https://doi.org/10.1007/JHEP04(2024)132
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2024)132