Abstract
In many gauge theories, the existence of particles in every representation of the gauge group (also known as completeness of the spectrum) is equivalent to the absence of one-form global symmetries. However, this relation does not hold, for example, in the gauge theory of non-abelian finite groups. We refine this statement by considering topological operators that are not necessarily associated with any global symmetry. For discrete gauge theory in three spacetime dimensions, we show that completeness of the spectrum is equivalent to the absence of certain Gukov-Witten topological operators. We further extend our analysis to four and higher spacetime dimensions. Since topological operators are natural generalizations of global symmetries, we discuss evidence for their absence in a consistent theory of quantum gravity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
T. Pantev and E. Sharpe, GLSM’s for Gerbes (and other toric stacks), Adv. Theor. Math. Phys. 10 (2006) 77 [hep-th/0502053] [INSPIRE].
T. Pantev and E. Sharpe, Notes on gauging noneffective group actions, hep-th/0502027 [INSPIRE].
S. Hellerman, A. Henriques, T. Pantev, E. Sharpe and M. Ando, Cluster decomposition, T-duality, and gerby CFT’s, Adv. Theor. Math. Phys. 11 (2007) 751 [hep-th/0606034] [INSPIRE].
A. Caldararu, J. Distler, S. Hellerman, T. Pantev and E. Sharpe, Non-birational twisted derived equivalences in abelian GLSMs, Commun. Math. Phys. 294 (2010) 605 [arXiv:0709.3855] [INSPIRE].
C.W. Misner and J.A. Wheeler, Classical physics as geometry: Gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space, Annals Phys. 2 (1957) 525 [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
L. Susskind, Trouble for remnants, hep-th/9501106 [INSPIRE].
T. Banks and L.J. Dixon, Constraints on String Vacua with Space-Time Supersymmetry, Nucl. Phys. B 307 (1988) 93 [INSPIRE].
D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, arXiv:1810.05338 [INSPIRE].
J. Polchinski, Monopoles, duality, and string theory, Int. J. Mod. Phys. A 19S1 (2004) 145 [hep-th/0304042] [INSPIRE].
N. Craig, I. Garcia Garcia and S. Koren, Discrete Gauge Symmetries and the Weak Gravity Conjecture, JHEP 05 (2019) 140 [arXiv:1812.08181] [INSPIRE].
B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius and I. Valenzuela, Non-Invertible Global Symmetries and Completeness of the Spectrum, to appear.
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
S. Gukov and E. Witten, Rigid Surface Operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [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].
A. Feiguin et al., Interacting anyons in topological quantum liquids: The golden chain, Phys. Rev. Lett. 98 (2007) 160409 [cond-mat/0612341] [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Defect lines, dualities, and generalised orbifolds, in 16th International Congress on Mathematical Physics, 9, 2009, DOI [arXiv:0909.5013] [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].
M. Buican and A. Gromov, Anyonic Chains, Topological Defects, and Conformal Field Theory, Commun. Math. Phys. 356 (2017) 1017 [arXiv:1701.02800] [INSPIRE].
L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [arXiv:1704.02330] [INSPIRE].
C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, Topological Defect Lines and Renormalization Group Flows in Two Dimensions, JHEP 01 (2019) 026 [arXiv:1802.04445] [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].
Y.-H. Lin and S.-H. Shao, Duality Defect of the Monster CFT, arXiv:1911.00042 [INSPIRE].
R. Thorngren and Y. Wang, Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases, arXiv:1912.02817 [INSPIRE].
S. Pal and Z. Sun, High Energy Modular Bootstrap, Global Symmetries and Defects, JHEP 08 (2020) 064 [arXiv:2004.12557] [INSPIRE].
J.M. Maldacena, G.W. Moore and N. Seiberg, D-brane charges in five-brane backgrounds, JHEP 10 (2001) 005 [hep-th/0108152] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, Symmetries and Strings of Adjoint QCD2, arXiv:2008.07567 [INSPIRE].
A. Kapustin and N. Saulina, Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory, arXiv:1012.0911 [INSPIRE].
G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].
E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].
V.G. Drinfeld, Quantum groups, J. Sov. Math. 41 (1988) 898 [INSPIRE].
A. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].
M.D.F. de Wild Propitius, Topological interactions in broken gauge theories, Ph.D. Thesis, University of Amsterdam (1995) [hep-th/9511195] [INSPIRE].
Y. Hu, Y. Wan and Y.-S. Wu, Twisted quantum double model of topological phases in two dimensions, Phys. Rev. B 87 (2013) 125114 [arXiv:1211.3695] [INSPIRE].
W. Burnside, Theorem of Groups of Finite Order, Dover Publications, Inc., New York, 2nd ed. (1955).
M. Müger, On the structure of modular categories, Proc. Lond. Math. Soc. 87 (2003) 291.
P.-S. Hsin, H.T. Lam and N. Seiberg, Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d, SciPost Phys. 6 (2019) 039 [arXiv:1812.04716] [INSPIRE].
S.X. Cui, S.-M. Hong and Z. Wang, Universal quantum computation with weakly integral anyons, Quant. Inf. Proc. 14 (2015) 2687.
R. Dijkgraaf, C. Vafa, E.P. Verlinde and H.L. Verlinde, The Operator Algebra of Orbifold Models, Commun. Math. Phys. 123 (1989) 485 [INSPIRE].
H. Moradi and X.-G. Wen, Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations, Phys. Rev. B 91 (2015) 075114 [arXiv:1404.4618] [INSPIRE].
T. Lan, L. Kong and X.-G. Wen, Classification of (3 + 1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are Al l Bosons, Phys. Rev. X 8 (2018) 021074 [INSPIRE].
T. Lan and X.-G. Wen, Classification of 3 + 1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions, Phys. Rev. X 9 (2019) 021005 [arXiv:1801.08530] [INSPIRE].
C. Wang and M. Levin, Braiding statistics of loop excitations in three dimensions, Phys. Rev. Lett. 113 (2014) 080403 [arXiv:1403.7437] [INSPIRE].
S. Jiang, A. Mesaros and Y. Ran, Generalized Modular Transformations in (3 + 1)D Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding, Phys. Rev. X 4 (2014) 031048 [arXiv:1404.1062] [INSPIRE].
J. Wang and X.-G. Wen, Non-Abelian string and particle braiding in topological order: Modular SL(3, ℤ) representation and (3 + 1)-dimensional twisted gauge theory, Phys. Rev. B 91 (2015) 035134 [arXiv:1404.7854] [INSPIRE].
C. Wang and M. Levin, Topological invariants for gauge theories and symmetry-protected topological phases, Phys. Rev. B 91 (2015) 165119 [arXiv:1412.1781] [INSPIRE].
P. Putrov, J. Wang and S.-T. Yau, Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2 + 1 and 3 + 1 dimensions, Annals Phys. 384 (2017) 254 [arXiv:1612.09298] [INSPIRE].
Q.-R. Wang, M. Cheng, C. Wang and Z.-C. Gu, Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in (3 + 1)-Dimensions, Phys. Rev. B 99 (2019) 235137 [arXiv:1810.13428] [INSPIRE].
P.-S. Hsin and A. Turzillo, Symmetry-enriched quantum spin liquids in (3 + 1)d, JHEP 09 (2020) 022 [arXiv:1904.11550] [INSPIRE].
D.V. Else and C. Nayak, Cheshire charge in (3 + 1)-dimensional topological phases, Phys. Rev. B 96 (2017) 045136 [arXiv:1702.02148] [INSPIRE].
J.S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Am. Math. Soc. 355 (2003) 3947.
C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
J. McNamara and C. Vafa, Baby Universes, Holography, and the Swampland, arXiv:2004.06738 [INSPIRE].
B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The Gravity Dual of a Density Matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
A.C. Wall, Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
Y.-H. Lin and S.-H. Shao, Anomalies and Bounds on Charged Operators, Phys. Rev. D 100 (2019) 025013 [arXiv:1904.04833] [INSPIRE].
C. Córdova, K. Ohmori, S.-H. Shao and F. Yan, Decorated ℤ2 symmetry defects and their time-reversal anomalies, Phys. Rev. D 102 (2020) 045019 [arXiv:1910.14046] [INSPIRE].
S. Fichet and P. Saraswat, Approximate Symmetries and Gravity, JHEP 01 (2020) 088 [arXiv:1909.02002] [INSPIRE].
Y. Nomura, Spacetime and Universal Soft Modes — Black Holes and Beyond, Phys. Rev. D 101 (2020) 066024 [arXiv:1908.05728] [INSPIRE].
T. Daus, A. Hebecker, S. Leonhardt and J. March-Russell, Towards a Swampland Global Symmetry Conjecture using weak gravity, Nucl. Phys. B 960 (2020) 115167 [arXiv:2002.02456] [INSPIRE].
C. Córdova, K. Ohmori and T. Rudelius, Symmetry Breaking Scales and Weak Gravity Conjectures, to appear.
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: 2006.10052
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
Rudelius, T., Shao, SH. Topological operators and completeness of spectrum in discrete gauge theories. J. High Energ. Phys. 2020, 172 (2020). https://doi.org/10.1007/JHEP12(2020)172
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)172