Abstract
We study four-dimensional gauge theories on oriented and non-spin spacetime manifolds. On such manifolds, each line operator arises only either as a boson or a fermion. Based on physical arguments, a method of systematically assigning spin labels to line operators is proposed, and several consistency checks are performed. This is used to classify all possible sets of allowed line operators — including their spins — for gauge theories with simple Lie algebras. The Lagrangian descriptions of the theories with these sets of allowed line operators are given. Finally, the one-form symmetries of these theories are studied by coupling to background gauge fields, and their ’t Hooft anomalies are computed.
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References
K.G. Wilson, Confinement of quarks, Phys. Rev.D 10 (1974) 2445 [INSPIRE].
G. ’t Hooft, On the phase transition towards permanent quark confinement, Nucl. Phys.B 138 (1978) 1 [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys.B 125 (1977) 1 [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev.D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
P.A.M. Dirac, The theory of magnetic poles, Phys. Rev.74 (1948) 817 [INSPIRE].
J.S. Schwinger, Magnetic charge and quantum field theory, Phys. Rev.144 (1966) 1087 [INSPIRE].
D. Zwanziger, Quantum field theory of particles with both electric and magnetic charges, Phys. Rev.176 (1968) 1489 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS states, Adv. Theor. Math. Phys.17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
R. Jackiw and C. Rebbi, Spin from isospin in a gauge theory, Phys. Rev. Lett.36 (1976) 1116 [INSPIRE].
P. Hasenfratz and G. ’t Hooft, A fermion-boson puzzle in a gauge theory, Phys. Rev. Lett.36 (1976) 1119 [INSPIRE].
A.S. Goldhaber, Spin and statistics connection for charge-monopole composites, Phys. Rev. Lett.36 (1976) 1122 [INSPIRE].
N. Seiberg and E. Witten, Gapped boundary phases of topological insulators via weak coupling, PTEP2016 (2016) 12C101 [arXiv:1602.04251] [INSPIRE].
M.A. Metlitski, S-duality of u(1) gauge theory with θ = π on non-orientable manifolds: applications to topological insulators and superconductors, arXiv:1510.05663 [INSPIRE].
D.S. Freed and M.J. Hopkins, Reflection positivity and invertible topological phases, arXiv:1604.06527 [INSPIRE].
C. Córdova and T.T. Dumitrescu, Candidate phases for SU(2) adjoint QCD4with two flavors from \( \mathcal{N} \) = 2 supersymmetric Yang-Mills theory, arXiv:1806.09592 [INSPIRE].
J. Wang, X.-G. Wen and E. Witten, A new SU(2) anomaly, J. Math. Phys.60 (2019) 052301 [arXiv:1810.00844] [INSPIRE].
Z. Wan, J. Wang and Y. Zheng, Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory, Phys. Rev.D 100 (2019) 085012 [arXiv:1904.00994] [INSPIRE].
P.-S. Hsin and S.-H. Shao, Lorentz symmetry fractionalization and dualities in (2 + 1)d, SciPost Phys.8 (2020) 018 [arXiv:1909.07383] [INSPIRE].
M. Guo, P. Putrov and J. Wang, Time reversal, SU(N ) Yang–Mills and cobordisms: interacting topological superconductors/insulators and quantum spin liquids in 3 + 1D, Annals Phys.394 (2018) 244 [arXiv:1711.11587] [INSPIRE].
Z. Wan, J. Wang and Y. Zheng, New higher anomalies, SU(N ) Yang-Mills gauge theory and ℂℙN −1sigma model, Ann. Phys.414 (2020) 168074 [arXiv:1812.11968].
Z. Wan and J. Wang, Beyond standard models and grand unifications: anomalies, topological terms and dynamical constraints via cobordisms, arXiv:1910.14668 [INSPIRE].
J. Wang, Y.-Z. You and Y. Zheng, Gauge enhanced quantum criticality and time reversal domain wall: SU(2) Yang-Mills dynamics with topological terms, Phys. Rev. Research.2 (2020) 013189 [arXiv:1910.14664] [INSPIRE].
Z. Wan and J. Wang, Adjoint QCD4 , deconfined critical phenomena, symmetry-enriched topological quantum field theory and higher symmetry-extension, Phys. Rev.D 99 (2019) 065013 [arXiv:1812.11955] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
C. Wang and T. Senthil, Time-reversal symmetric U(1) quantum spin liquids, Phys. Rev.X 6 (2016) 011034 [arXiv:1505.03520] [INSPIRE].
P.-S. Hsin and A. Turzillo, Symmetry-enriched quantum spin liquids in (3 + 1)d, arXiv:1904.11550 [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
C. Córdova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications II, SciPost Phys.8 (2020) 002 [arXiv:1905.13361] [INSPIRE].
E. Witten, Dyons of charge e θ/2π, Phys. Lett.86B (1979) 283 [INSPIRE].
C. Wang, A.C. Potter and T. Senthil, Classification of interacting electronic topological insulators in three dimensions, Science343 (2014) 629 [arXiv:1306.3238] [INSPIRE].
R. Thorngren, Framed Wilson operators, fermionic strings and gravitational anomaly in 4d, JHEP02 (2015) 152 [arXiv:1404.4385] [INSPIRE].
S.M. Kravec, J. McGreevy and B. Swingle, All-fermion electrodynamics and fermion number anomaly inflow, Phys. Rev.D 92 (2015) 085024 [arXiv:1409.8339] [INSPIRE].
E. Witten, On S duality in Abelian gauge theory, Selecta Math.1 (1995) 383 [hep-th/9505186] [INSPIRE].
F. Benini, C. Córdova and P.-S. Hsin, On 2-group global symmetries and their anomalies, JHEP03 (2019) 118 [arXiv:1803.09336] [INSPIRE].
G. Hooft, Magnetic monopoles in unified gauge theories, Nucl. Phys.B 79 (1974) 276.
A.M. Polyakov, Particle spectrum in the quantum field theory, JETP Lett.20 (1974) 194 [INSPIRE].
H. Georgi and S.L. Glashow, Unified weak and electromagnetic interactions without neutral currents, Phys. Rev. Lett.28 (1972) 1494 [INSPIRE].
J.A. Harvey, Magnetic monopoles, duality and supersymmetry, in the proceedings of Fields, strings and duality. Theoretical Advanced Study Institute in Elementary Particle Physics (TASI’96), June 2–28, Boulder, U.S.A. (1996), hep-th/9603086 [INSPIRE].
M. Shifman, Advanced topics in quantum field theory, Cambridge University Press, Cambridge U.K. (2012).
A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Num. Theor. Phys.1 (2007) 1 [hep-th/0604151] [INSPIRE].
A. Kapustin, Holomorphic reduction of N = 2 gauge theories, Wilson-’t Hooft operators and S-duality, hep-th/0612119 [INSPIRE].
A. Kapustin and N. Saulina, The algebra of Wilson-’t Hooft operators, Nucl. Phys.B 814 (2009) 327 [arXiv:0710.2097] [INSPIRE].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Infrared computations of defect Schur indices, JHEP11 (2016) 106 [arXiv:1606.08429] [INSPIRE].
A. Kapustin and R. Thorngren, Topological field theory on a lattice, discrete theta-angles and confinement, Adv. Theor. Math. Phys.18 (2014) 1233 [arXiv:1308.2926] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal and temperature, JHEP05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
C. Córdova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications I, SciPost Phys.8 (2020) 001 [arXiv:1905.09315] [INSPIRE].
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].
C.-T. Hsieh, Y. Tachikawa and K. Yonekura, Anomaly of the electromagnetic duality of Maxwell theory, Phys. Rev. Lett.123 (2019) 161601 [arXiv:1905.08943] [INSPIRE].
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
Y. Tachikawa, On gauging finite subgroups, SciPost Phys.8 (2020) 015 [arXiv:1712.09542] [INSPIRE].
E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
C. Montonen and D.I. Olive, Magnetic monopoles as gauge particles?, Phys. Lett.B 72 (1977) 117.
H. Osborn, Topological charges for N = 4 supersymmetric gauge theories and monopoles of spin 1, Phys. Lett.B 83 (1979) 321.
C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys.B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
P.C. Argyres, A. Kapustin and N. Seiberg, On S-duality for non-simply-laced gauge groups, JHEP06 (2006) 043 [hep-th/0603048] [INSPIRE].
E. Witten, Supersymmetric index in four-dimensional gauge theories, Adv. Theor. Math. Phys.5 (2002) 841 [hep-th/0006010] [INSPIRE].
X. Gu, On the cohomology of the classifying spaces of projective unitary groups, arXiv:1612.00506.
E. Thomas, On the cohomology of the real grassmann complexes and the characteristic classes of n-plane bundles, Trans. Amer. Math. Soc.96 (1960) 67.
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Ang, J., Roumpedakis, K. & Seifnashri, S. Line operators of gauge theories on non-spin manifolds. J. High Energ. Phys. 2020, 87 (2020). https://doi.org/10.1007/JHEP04(2020)087
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DOI: https://doi.org/10.1007/JHEP04(2020)087