Abstract
Higher-form symmetries are associated with transformations that only act on extended objects, not on point particles. Typically, higher-form symmetries live alongside ordinary, point-particle (0-form), symmetries and they can be jointly described in terms of a direct product symmetry group. However, when the actions of 0-form and higher-form symmetries become entangled, a more general mathematical structure is required, related to higher categorical groups. Systems with continuous higher-group symmetry were previously constructed in a top-down manner, descending from quantum field theories with a specific mixed ’t Hooft anomaly. I show that higher-group symmetry also naturally emerges from a bottom-up, low-energy perspective, when the physical system at hand contains at least two different given, spontaneously broken symmetries. This leads generically to a hierarchy of emergent higher-form symmetries, corresponding to the Grassmann algebra of topological currents of the theory, with an underlying higher-group structure. Examples of physical systems featuring such higher-group symmetry include superfluid mixtures and variants of axion electrodynamics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Y. Kosmann-Schwarzbach, The Noether theorems: invariance and conservation laws in the twentieth century, Springer, Germany (2011).
P.J. Olver, Applications of Lie groups to differential equations, Springer, Germany (1986).
G.W. Bluman and S.C. Anco, Symmetry and integration methods for differential equations, Springer, Germany (2002).
V.V. Zharinov, Differential algebras and low-dimensional conservation laws, Math. USSR Sborn. 71 (1992) 319.
R.L. Bryant and P.A. Griffiths, Characteristic cohomology of differential systems I: General Theory, J. Am. Math. Soc. 8 (1995) 507.
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in the antifield formalism. 1. General theorems, Commun. Math. Phys. 174 (1995) 57 [hep-th/9405109] [INSPIRE].
I.M. Anderson and C.G. Torre, Asymptotic conservation laws in field theory, Phys. Rev. Lett. 77 (1996) 4109 [hep-th/9608008] [INSPIRE].
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rep. 338 (2000) 439.
I.S. Krasilshchik and A.M. Vinogradov, Nonlocal symmetries and the theory of coverings: an addendum to A. M. Vinogradov’s ‘Local Symmetries and Conservation Laws’, Acta Appl. Math. 2 (1984) 79.
V.S. Vladimirov and I.V. Volovich, Construction of local and nonlocal conservation laws for nonlinear field equations, Annalen Phys. 47 (1990) 228 [INSPIRE].
I.S. Akhatov, R.K. Gazizov and N.K. Ibragimov, Nonlocal symmetries. Heuristic approach, J. Sov. Math. 55 (1991) 1401.
G.W. Bluman, A.F. Cheviakov and S.C. Anco, Applications of symmetry methods to partial differential equations, Springer, Germany (2010).
Z. Nussinov and G. Ortiz, A symmetry principle for topological quantum order, Annals Phys. 324 (2009) 977 [cond-mat/0702377] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
E. Sharpe, Notes on generalized global symmetries in QFT, Fortsch. Phys. 63 (2015) 659 [arXiv:1508.04770] [INSPIRE].
D.M. Hofman and N. Iqbal, Goldstone modes and photonization for higher form symmetries, SciPost Phys. 6 (2019) 006 [arXiv:1802.09512] [INSPIRE].
E. Lake, Higher-form symmetries and spontaneous symmetry breaking, arXiv:1802.07747 [INSPIRE].
L.V. Delacrétaz, D.M. Hofman and G. Mathys, Superfluids as higher-form anomalies, SciPost Phys. 8 (2020) 047 [arXiv:1908.06977] [INSPIRE].
S. Grozdanov, D.M. Hofman and N. Iqbal, Generalized global symmetries and dissipative magnetohydrodynamics, Phys. Rev. D 95 (2017) 096003 [arXiv:1610.07392] [INSPIRE].
J. Armas and A. Jain, Magnetohydrodynamics as superfluidity, Phys. Rev. Lett. 122 (2019) 141603 [arXiv:1808.01939] [INSPIRE].
P. Glorioso and D.T. Son, Effective field theory of magnetohydrodynamics from generalized global symmetries, arXiv:1811.04879 [INSPIRE].
J. Armas and A. Jain, One-form superfluids & magnetohydrodynamics, JHEP 01 (2020) 041 [arXiv:1811.04913] [INSPIRE].
S. Grozdanov and N. Poovuttikul, Generalized global symmetries in states with dynamical defects: The case of the transverse sound in field theory and holography, Phys. Rev. D 97 (2018) 106005 [arXiv:1801.03199] [INSPIRE].
J. Armas and A. Jain, Viscoelastic hydrodynamics and holography, JHEP 01 (2020) 126 [arXiv:1908.01175] [INSPIRE].
N. Sogabe and N. Yamamoto, Triangle anomalies and nonrelativistic Nambu-Goldstone modes of generalized global symmetries, Phys. Rev. D 99 (2019) 125003 [arXiv:1903.02846] [INSPIRE].
Y. Hidaka, Y. Hirono and R. Yokokura, Counting Nambu-Goldstone modes of higher-form global symmetries, Phys. Rev. Lett. 126 (2021) 071601 [arXiv:2007.15901] [INSPIRE].
H. Watanabe, Counting rules of Nambu-Goldstone modes, Ann. Rev. Condensed Matter Phys. 11 (2020) 169 [arXiv:1904.00569] [INSPIRE].
A.J. Beekman, L. Rademaker and J. van Wezel, An introduction to spontaneous symmetry breaking, SciPost Phys. Lect. Notes 11 (2019) 1 [arXiv:1909.01820] [INSPIRE].
L.A. Gaumé, D. Orlando and S. Reffert, Selected topics in the large quantum number expansion, arXiv:2008.03308 [INSPIRE].
J.C. Baez and J. Huerta, An invitation to higher gauge theory, Gen. Rel. Grav. 43 (2011) 2335 [arXiv:1003.4485] [INSPIRE].
T. Pantev and E. Sharpe, Notes on gauging noneffective group actions, hep-th/0502027 [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].
A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, arXiv:1309.4721 [INSPIRE].
Y. Tachikawa, On gauging finite subgroups, SciPost Phys. 8 (2020) 015 [arXiv:1712.09542] [INSPIRE].
C. Córdova, T.T. Dumitrescu and K. Intriligator, Exploring 2-group global symmetries, JHEP 02 (2019) 184 [arXiv:1802.04790] [INSPIRE].
F. Benini, C. Córdova and P.-S. Hsin, On 2-group global symmetries and their Anomalies, JHEP 03 (2019) 118 [arXiv:1803.09336] [INSPIRE].
Y. Hidaka, M. Nitta and R. Yokokura, Higher-form symmetries and 3-group in axion electrodynamics, Phys. Lett. B 808 (2020) 135672 [arXiv:2006.12532].
Y. Hidaka, M. Nitta and R. Yokokura, Global 3-group symmetry and ’t Hooft anomalies in axion electrodynamics, JHEP 01 (2021) 173 [arXiv:2009.14368] [INSPIRE].
T.D. Brennan and C. Cordova, Axions, higher-groups, and emergent symmetry, arXiv:2011.09600 [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].
M. Stone and P. Goldbart, Mathematics for physics: a guided tour for graduate students, Cambridge University Press, Cambridge U.K. (2009)
N. Yamamoto and R. Yokokura, Topological mass generation in gapless systems, arXiv:2009.07621 [INSPIRE].
V.M. Vyas, V. Srinivasan and P.K. Panigrahi, Some results on topological currents in field theory, Int. J. Mod. Phys. A 34 (2019) 1950096 [arXiv:1411.3099] [INSPIRE].
G.W. Bluman, G.J. Reid and S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806.
A.F. Cheviakov and G.W. Bluman, Multidimensional partial differential equation systems: Generating new systems via conservation laws, potentials, gauges, subsystems, J. Math. Phys. 51 (2010) 103521.
A.F. Cheviakov and G.W. Bluman, Multidimensional partial differential equation systems: Nonlocal symmetries, nonlocal conservation laws, exact solutions, J. Math. Phys. 51 (2010) 103522.
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
H. Watanabe and H. Murayama, Effective Lagrangian for nonrelativistic systems, Phys. Rev. X 4 (2014) 031057 [arXiv:1402.7066] [INSPIRE].
J.O. Andersen, T. Brauner, C.P. Hofmann and A. Vuorinen, Effective Lagrangians for quantum many-body systems, JHEP 08 (2014) 088 [arXiv:1406.3439] [INSPIRE].
J. Nissinen, Field theory of higher-order topological crystalline response, generalized global symmetries and elasticity tetrads, arXiv:2009.14184 [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, 2-group global symmetries and anomalies in six-dimensional quantum field theories, arXiv:2009.00138 [INSPIRE].
E. D’Hoker and S. Weinberg, General effective actions, Phys. Rev. D 50 (1994) R6050 [hep-ph/9409402] [INSPIRE].
E. D’Hoker, Invariant effective actions, cohomology of homogeneous spaces and anomalies, Nucl. Phys. B 451 (1995) 725 [hep-th/9502162] [INSPIRE].
S.C. Anco and G.W. Bluman, Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations, J. Math. Phys. 38 (1997) 3508.
S.C. Anco and D. The, Symmetries, conservation laws, and cohomology of Maxwell’s equations using potentials, Acta Appl. Math. 89 (2005) 1 [math-ph/0501052].
A.F. Cheviakov, Conservation properties and potential systems of vorticity-type equations, J. Math. Phys. 55 (2014) 033508.
N. Seiberg, Field theories with a vector global symmetry, SciPost Phys. 8 (2020) 050 [arXiv:1909.10544] [INSPIRE].
D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, arXiv:1810.05338 [INSPIRE].
H. Leutwyler, Nonrelativistic effective Lagrangians, Phys. Rev. D 49 (1994) 3033 [hep-ph/9311264] [INSPIRE].
H. Leutwyler, On the foundations of chiral perturbation theory, Annals Phys. 235 (1994) 165 [hep-ph/9311274] [INSPIRE].
R.A. Bertlmann, Anomalies in quantum field theory, Clarendon Press, Oxford U.K. (1996).
O. Kaymakcalan, S. Rajeev and J. Schechter, Nonabelian anomaly and vector meson decays, Phys. Rev. D 30 (1984) 594 [INSPIRE].
A. Manohar and G.W. Moore, Anomalous inequivalence of phenomenological theories, Nucl. Phys. B 243 (1984) 55 [INSPIRE].
L. Álvarez-Gaumé and P.H. Ginsparg, The structure of gauge and gravitational anomalies, Annals Phys. 161 (1985) 423 [Erratum ibid. 171 (1986) 233] [INSPIRE].
J.L. Manes, Differential geometric construction of the gauged Wess-Zumino action, Nucl. Phys. B 250 (1985) 369 [INSPIRE].
E. Witten, Global aspects of current algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
J. Davighi and B. Gripaios, Homological classification of topological terms in sigma models on homogeneous spaces, JHEP 09 (2018) 155 [Erratum ibid. 11 (2018) 143] [arXiv:1803.07585] [INSPIRE].
Y. Lee, K. Ohmori and Y. Tachikawa, Revisiting Wess-Zumino-Witten terms, SciPost Phys. 10 (2021) 061 [arXiv:2009.00033] [INSPIRE].
K. Yonekura, General anomaly matching by Goldstone bosons, arXiv:2009.04692 [INSPIRE].
J. Davighi, B. Gripaios and O. Randal-Williams, Differential cohomology and topological actions in physics, arXiv:2011.05768 [INSPIRE].
T. Brauner and H. Kolešová, Gauged Wess-Zumino terms for a general coset space, Nucl. Phys. B 945 (2019) 114676 [arXiv:1809.05310].
A.G. Abanov, Topology, geometry and quantum interference in condensed matter physics, arXiv:1708.07192 [INSPIRE].
T. Frankel, The geometry of physics: an introduction, Cambridge University Press, Cambridge U.K. (2012).
M. Greiter, F. Wilczek and E. Witten, Hydrodynamic relations in superconductivity, Mod. Phys. Lett. B 03 (1989) 903.
D.T. Son, Low-energy quantum effective action for relativistic superfluids, hep-ph/0204199 [INSPIRE].
T. Brauner, G. Filios and H. Kolešová, Chiral soliton lattice in QCD-like theories, JHEP 12 (2019) 029 [arXiv:1905.11409] [INSPIRE].
F. Wilczek, Two applications of axion electrodynamics, Phys. Rev. Lett. 58 (1987) 1799 [INSPIRE].
T. Brauner and N. Yamamoto, Chiral soliton lattice and charged pion condensation in strong magnetic fields, JHEP 04 (2017) 132 [arXiv:1609.05213] [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].
S.E. Hjelmeland and U. Lindström, Duality for the nonspecialist, hep-th/9705122 [INSPIRE].
A. Karch and D. Tong, Particle-Vortex Duality from 3d Bosonization, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].
J. Fröhlich and U.M. Studer, Gauge invariance and current algebra in nonrelativistic many body theory, Rev. Mod. Phys. 65 (1993) 733 [INSPIRE].
N. Iqbal and N. Poovuttikul, 2-group global symmetries, hydrodynamics and holography, arXiv:2010.00320 [INSPIRE].
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: 2012.00051
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
Brauner, T. Field theories with higher-group symmetry from composite currents. J. High Energ. Phys. 2021, 45 (2021). https://doi.org/10.1007/JHEP04(2021)045
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2021)045