Abstract
We study four-dimensional gauge theories with arbitrary simple gauge group with 1-form global center symmetry and 0-form parity or discrete chiral symmetry. We canonically quantize on 𝕋3, in a fixed background field gauging the 1-form symmetry. We show that the mixed 0-form/1-form ’t Hooft anomaly results in a central extension of the global-symmetry operator algebra. We determine this algebra in each case and show that the anomaly implies degeneracies in the spectrum of the Hamiltonian at any finite- size torus. We discuss the consistency of these constraints with both older and recent semiclassical calculations in SU(N) theories, with or without adjoint fermions, as well as with their conjectured infrared phases.
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References
G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, NATO Sci. Ser. B 59 (1980) 135 [INSPIRE].
J. L. Rosner, Explorations of compositeness, Comments Mod. Phys. A 1 (1999) 11 [hep-ph/9812537] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
D. Gaiotto, Z. Komargodski and N. Seiberg, Time-reversal breaking in QCD4, walls, and dualities in 2 + 1 dimensions, JHEP 01 (2018) 110 [arXiv:1708.06806] [INSPIRE].
H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev. D 97 (2018) 105011 [arXiv:1706.06104] [INSPIRE].
Z. Komargodski, T. Sulejmanpasic and M. Ünsal, Walls, anomalies, and deconfinement in quantum antiferromagnets, Phys. Rev. B 97 (2018) 054418 [arXiv:1706.05731] [INSPIRE].
M. M. Anber and E. Poppitz, On the baryon-color-flavor (BCF) anomaly in vector-like theories, JHEP 11 (2019) 063 [arXiv:1909.09027] [INSPIRE].
M. M. Anber and E. Poppitz, Generalized ’t Hooft anomalies on non-spin manifolds, JHEP 04 (2020) 097 [arXiv:2002.02037] [INSPIRE].
M. M. Anber, Condensates and anomaly cascade in vector-like theories, JHEP 03 (2021) 191 [arXiv:2101.04132] [INSPIRE].
D. Delmastro, D. Gaiotto and J. Gomis, Global Anomalies on the Hilbert Space, arXiv:2101.02218 [INSPIRE].
M. M. Anber and E. Poppitz, Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls, JHEP 09 (2018) 076 [arXiv:1807.00093] [INSPIRE].
A. Armoni and S. Sugimoto, Vacuum structure of charge k two-dimensional QED and dynamics of an anti D-string near an O1− -plane, JHEP 03 (2019) 175 [arXiv:1812.10064] [INSPIRE].
T. Misumi, Y. Tanizaki and M. Ünsal, Fractional θ angle, ’t Hooft anomaly, and quantum instantons in charge-q multi-flavor Schwinger model, JHEP 07 (2019) 018 [arXiv:1905.05781] [INSPIRE].
Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, Symmetries and strings of adjoint QCD2 , JHEP 03 (2021) 103 [arXiv:2008.07567] [INSPIRE].
A. Cherman and T. Jacobson, Lifetimes of near eternal false vacua, Phys. Rev. D 103 (2021) 105012 [arXiv:2012.10555] [INSPIRE].
M. M. Anber and E. Poppitz, Domain walls in high-T SU(N) super Yang-Mills theory and QCD(adj), JHEP 05 (2019) 151 [arXiv:1811.10642] [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].
Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, Comments on Abelian Higgs Models and Persistent Order, SciPost Phys. 6 (2019) 003 [arXiv:1705.04786] [INSPIRE].
T. Sulejmanpasic and Y. Tanizaki, C-P-T anomaly matching in bosonic quantum field theory and spin chains, Phys. Rev. B 97 (2018) 144201 [arXiv:1802.02153] [INSPIRE].
Y. Tanizaki and T. Sulejmanpasic, Anomaly and global inconsistency matching: θ-angles, SU(3)/U(1)2 nonlinear sigma model, SU(3) chains and its generalizations, Phys. Rev. B 98 (2018) 115126 [arXiv:1805.11423] [INSPIRE].
E. Sharpe, Undoing decomposition, Int. J. Mod. Phys. A 34 (2020) 1950233 [arXiv:1911.05080] [INSPIRE].
A. Cherman, T. Jacobson, Y. Tanizaki and M. Ünsal, Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD, SciPost Phys. 8 (2020) 072 [arXiv:1908.09858] [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].
A. Smilga, A comment on instantons and their fermion zero modes in adjoint QCD_2, SciPost Phys. 10 (2021) 152 [arXiv:2104.06266] [INSPIRE].
M. Ünsal and L. G. Yaffe, Center-stabilized Yang-Mills theory: Confinement and large N volume independence, Phys. Rev. D 78 (2008) 065035 [arXiv:0803.0344] [INSPIRE].
K. Aitken, A. Cherman and M. Ünsal, Dihedral symmetry in SU(N) Yang-Mills theory, Phys. Rev. D 100 (2019) 085004 [arXiv:1804.05845] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
J. Greensite, An introduction to the confinement problem, vol. 821 (2011), 10.1007/978-3-642-14382-3 [INSPIRE].
G. ’t Hooft, Aspects of Quark Confinement, Phys. Scripta 24 (1981) 841 [INSPIRE].
P. van Baal, Some Results for SU(N) Gauge Fields on the Hypertorus, Commun. Math. Phys. 85 (1982) 529 [INSPIRE].
A. Gonzalez-Arroyo, Yang-Mills fields on the four-dimensional torus. Part 1.: Classical theory, in Advanced Summer School on Nonperturbative Quantum Field Physics, pp. 57–91, 6, 1997 [hep-th/9807108] [INSPIRE].
G. ’t Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl. Phys. B 153 (1979) 141 [INSPIRE].
P. van Baal, Twisted Boundary Conditions: A Nonperturbative Probe for Pure Nonabelian Gauge Theories, Ph.D. Thesis, , Utrecht University, Utrecht The Netherlands (1984).
A. Gonzalez Arroyo and C. P. Korthals Altes, The Spectrum of Yang-Mills Theory in a Small Twisted Box, Nucl. Phys. B 311 (1988) 433 [INSPIRE].
P. Ramond, Group theory: A physicist’s survey, Cambridge University Press, Cambridge U.K. (2010).
E. Witten, Large N Chiral Dynamics, Annals Phys. 128 (1980) 363 [INSPIRE].
G. Gabadadze and M. Shifman, QCD vacuum and axions: What’s happening?, Int. J. Mod. Phys. A 17 (2002) 3689 [hep-ph/0206123] [INSPIRE].
E. Poppitz and F. D. Wandler, Topological terms and anomaly matching in effective field theories on ℝ3 × 𝕊1. Part I. Abelian symmetries and intermediate scales, JHEP 01 (2021) 091 [arXiv:2009.14667] [INSPIRE].
M. Ünsal, Theta dependence, sign problems and topological interference, Phys. Rev. D 86 (2012) 105012 [arXiv:1201.6426] [INSPIRE].
E. Poppitz, T. Schäfer and M. Ünsal, Universal mechanism of (semi-classical) deconfinement and theta-dependence for all simple groups, JHEP 03 (2013) 087 [arXiv:1212.1238] [INSPIRE].
M. M. Anber, Θ dependence of the deconfining phase transition in pure SU (Nc) Yang-Mills theories, Phys. Rev. D 88 (2013) 085003 [arXiv:1302.2641] [INSPIRE].
A. Bhoonah, E. Thomas and A. R. Zhitnitsky, Metastable vacuum decay and θ dependence in gauge theory. Deformed QCD as a toy model, Nucl. Phys. B 890 (2014) 30 [arXiv:1407.5121] [INSPIRE].
M. M. Anber and A. R. Zhitnitsky, Oblique Confinement at θ ≠ 0 in weakly coupled gauge theories with deformations, Phys. Rev. D 96 (2017) 074022 [arXiv:1708.07520] [INSPIRE].
K. Aitken, A. Cherman and M. Ünsal, Vacuum structure of Yang-Mills theory as a function of θ, JHEP 09 (2018) 030 [arXiv:1804.06848] [INSPIRE].
R. Kitano, R. Matsudo, N. Yamada and M. Yamazaki, Peeking into the θ vacuum, Phys. Lett. B 822 (2021) 136657 [arXiv:2102.08784] [INSPIRE].
P. van Baal, QCD in a finite volume, hep-ph/0008206 [INSPIRE].
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
E. Witten, Supersymmetric index in four-dimensional gauge theories, Adv. Theor. Math. Phys. 5 (2002) 841 [hep-th/0006010] [INSPIRE].
G. ’t Hooft, Some Twisted Selfdual Solutions for the Yang-Mills Equations on a Hypertorus, Commun. Math. Phys. 81 (1981) 267 [INSPIRE].
M. Lüscher, Some Analytic Results Concerning the Mass Spectrum of Yang-Mills Gauge Theories on a Torus, Nucl. Phys. B 219 (1983) 233 [INSPIRE].
J. B. Kogut and L. Susskind, Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys. Rev. D 11 (1975) 395 [INSPIRE].
H. Reinhardt, On ’t Hooft’s loop operator, Phys. Lett. B 557 (2003) 317 [hep-th/0212264] [INSPIRE].
M. M. Anber and E. Poppitz, On the global structure of deformed Yang-Mills theory and QCD(adj) on ℝ3 × 𝕊1, JHEP 10 (2015) 051 [arXiv:1508.00910] [INSPIRE].
Y. Kikuchi and Y. Tanizaki, Global inconsistency, ’t Hooft anomaly, and level crossing in quantum mechanics, PTEP 2017 (2017) 113B05 [arXiv:1708.01962] [INSPIRE].
A. Behtash, T. Sulejmanpasic, T. Schäfer and M. Ünsal, Hidden topological angles and Lefschetz thimbles, Phys. Rev. Lett. 115 (2015) 041601 [arXiv:1502.06624] [INSPIRE].
A. González-Arroyo, Constructing SU(N) fractional instantons, JHEP 02 (2020) 137 [arXiv:1910.12565] [INSPIRE].
M. Ünsal, Strongly coupled QFT dynamics via TQFT coupling, arXiv:2007.03880 [INSPIRE].
Z. Wan, J. Wang and Y. Zheng, New higher anomalies, SU(N) Yang-Mills gauge theory and ℂℙN−1 sigma model, Annals Phys. 414 (2020) 168074 [arXiv:1812.11968] [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].
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].
C. Córdova and K. Ohmori, Anomaly Constraints on Gapped Phases with Discrete Chiral Symmetry, Phys. Rev. D 102 (2020) 025011 [arXiv:1912.13069] [INSPIRE].
C. Córdova and K. Ohmori, Anomaly Obstructions to Symmetry Preserving Gapped Phases, arXiv:1910.04962 [INSPIRE].
J. C. Myers and M. C. Ogilvie, New phases of SU(3) and SU(4) at finite temperature, Phys. Rev. D 77 (2008) 125030 [arXiv:0707.1869] [INSPIRE].
J. C. Myers and M. C. Ogilvie, Phase diagrams of SU(N) gauge theories with fermions in various representations, JHEP 07 (2009) 095 [arXiv:0903.4638] [INSPIRE].
C. Bonati, M. Cardinali and M. D’Elia, θ dependence in trace deformed SU(3) Yang-Mills theory: a lattice study, Phys. Rev. D 98 (2018) 054508 [arXiv:1807.06558] [INSPIRE].
C. Bonati, M. Cardinali, M. D’Elia and F. Mazziotti, θ-dependence and center symmetry in Yang-Mills theories, Phys. Rev. D 101 (2020) 034508 [arXiv:1912.02662] [INSPIRE].
M. Ünsal, TQFT at work for IR-renormalons, resurgence and Lefschetz decomposition, arXiv:2106.14971 [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992).
S. L. Adler and D. G. Boulware, Anomalous commutators and the triangle diagram, Phys. Rev. 184 (1969) 1740 [INSPIRE].
M. Ünsal, Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions, Phys. Rev. D 80 (2009) 065001 [arXiv:0709.3269] [INSPIRE].
M. M. Anber and E. Poppitz, Two-flavor adjoint QCD, Phys. Rev. D 98 (2018) 034026 [arXiv:1805.12290] [INSPIRE].
E. Poppitz and T. A. Ryttov, Possible new phase for adjoint QCD, Phys. Rev. D 100 (2019) 091901 [arXiv:1904.11640] [INSPIRE].
C. Córdova and T. T. Dumitrescu, Candidate Phases for SU(2) Adjoint QCD4 with Two Flavors from \( \mathcal{N} \) = 2 Supersymmetric Yang-Mills Theory, arXiv:1806.09592 [INSPIRE].
S. Catterall, J. Giedt, F. Sannino and J. Schneible, Phase diagram of SU(2) with 2 flavors of dynamical adjoint quarks, JHEP 11 (2008) 009 [arXiv:0807.0792] [INSPIRE].
A. J. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, Spectrum of SU(2) lattice gauge theory with two adjoint Dirac flavours, JHEP 05 (2009) 025 [arXiv:0812.1467] [INSPIRE].
L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, Conformal versus confining scenario in SU(2) with adjoint fermions, Phys. Rev. D 80 (2009) 074507 [arXiv:0907.3896] [INSPIRE].
A. Athenodorou, Bennett, G. Bergner and B. Lucini, Investigating the conformal behaviour of SU(2) with one adjoint Dirac flavor, arXiv:2103.10485 [INSPIRE].
E. Poppitz and M. Ünsal, Conformality or confinement: (IR)relevance of topological excitations, JHEP 09 (2009) 050 [arXiv:0906.5156] [INSPIRE].
T. Banks and A. Zaks, On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions, Nucl. Phys. B 196 (1982) 189 [INSPIRE].
M. M. Anber, E. Poppitz and B. Teeple, Deconfinement and continuity between thermal and (super) Yang-Mills theory for all gauge groups, JHEP 09 (2014) 040 [arXiv:1406.1199] [INSPIRE].
S. Chen, K. Fukushima, H. Nishimura and Y. Tanizaki, Deconfinement and \( \mathcal{CP} \) breaking at θ = π in Yang-Mills theories and a novel phase for SU(2), Phys. Rev. D 102 (2020) 034020 [arXiv:2006.01487] [INSPIRE].
N. M. Davies, T. J. Hollowood and V. V. Khoze, Monopoles, affine algebras and the gluino condensate, J. Math. Phys. 44 (2003) 3640 [hep-th/0006011] [INSPIRE].
M. Nguyen, Y. Tanizaki and M. Ünsal, Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond N -ality, JHEP 03 (2021) 238 [arXiv:2101.02227] [INSPIRE].
B. Hall, Lie groups, lie algebras, and representations: an elementary introduction, Springer, Heidelberg Germany (2015).
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].
S. Vandoren and P. van Nieuwenhuizen, Lectures on instantons, arXiv:0802.1862 [INSPIRE].
G. W. Gibbons and C. N. Pope, CP2 as a gravitational instanton, Commun. Math. Phys. 61 (1978) 239 [INSPIRE].
T. Eguchi, P. B. Gilkey and A. J. Hanson, Gravitation, Gauge Theories and Differential Geometry, Phys. Rept. 66 (1980) 213.
R. P. Geroch, Spinor structure of space-times in general relativity. I, J. Math. Phys. 9 (1968) 1739 [INSPIRE].
R. P. Geroch, Spinor structure of space-times in general relativity. II, J. Math. Phys. 11 (1970) 343 [INSPIRE].
S. W. Hawking and C. N. Pope, Generalized Spin Structures in Quantum Gravity, Phys. Lett. B 73 (1978) 42 [INSPIRE].
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Cox, A.A., Poppitz, E. & Wandler, F.D. The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles. J. High Energ. Phys. 2021, 69 (2021). https://doi.org/10.1007/JHEP10(2021)069
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DOI: https://doi.org/10.1007/JHEP10(2021)069