Abstract
We discuss anomaly cancellation in U(2) gauge theories in four dimensions. For a U(2) gauge theory defined with a spin structure, the vanishing of the bordism group \( {\Omega}_5^{\mathrm{Spin}} \) (BU(2)) implies that there can be no global anomalies, in contrast to the related case of an SU(2) gauge theory. We show explicitly that the familiar SU(2) global anomaly is replaced by a local anomaly when SU(2) is embedded in U(2). There must be an even number of fermions with isospin 2r + 1/2, for r ∈ ℤ≥0, for this local anomaly to cancel. The case of a U(2) theory defined without a choice of spin structure but rather using a spin-U(2) structure, which is possible when all fermions (bosons) have half-integer (integer) isospin and odd (even) U(1) charge, is more subtle. We find that the recently-discovered ‘new SU(2) global anomaly’ is also equivalent, though only at the level of the partition function, to a perturbative anomaly in the U(2) theory, which is this time a combination of a mixed gauge anomaly with a gauge-gravity anomaly. This perturbative anomaly vanishes if there is an even number of fermions with isospin 4r + 3/2, for r ∈ ℤ≥0, recovering the condition for cancelling the new SU(2) anomaly. Alternatively, this perturbative anomaly can be cancelled by a Wess-Zumino term, leaving a low-energy theory with a global anomaly, which can itself be cancelled by coupling to topological degrees of freedom.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, An SU(2) anomaly, Phys. Lett. B 117 (1982) 324.
E. Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985) 197 [INSPIRE].
X.-z. Dai and D.S. Freed, η invariants and determinant lines, J. Math. Phys. 35 (1994) 5155 [Erratum ibid. 42 (2001) 2343] [hep-th/9405012] [INSPIRE].
M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Phil. Soc. 79 (1976) 71.
J. Davighi, B. Gripaios and N. Lohitsiri, Global anomalies in the standard model(s) and beyond, arXiv:1910.11277 [INSPIRE].
Z. Wan and J. Wang, Beyond standard models and grand unifications: anomalies, topological terms and dynamical constraints via cobordisms, arXiv:1910.14668 [INSPIRE].
D. Tong, Line operators in the standard model, JHEP 07 (2017) 104 [arXiv:1705.01853] [INSPIRE].
J. Wang, X.-G. Wen and E. Witten, A new SU(2) anomaly, J. Math. Phys. 60 (2019) 052301 [arXiv:1810.00844] [INSPIRE].
S.J. Avis and C.J. Isham, Generalized spin structures on four-dimensional space-times, Commun. Math. Phys. 72 (1980) 103 [INSPIRE].
A. Back, P.G.O. Freund and M. Forger, New gravitational instantons and universal spin structures, Phys. Lett. B 77 (1978) 181.
S.W. Hawking and C.N. Pope, Generalized spin structures in quantum gravity, Phys. Lett. B 73 (1978) 42.
J. Preskill, Gauge anomalies in an effective field theory, Annals Phys. 210 (1991) 323 [INSPIRE].
I. García-Etxebarria et al., 8d gauge anomalies and the topological Green-Schwarz mechanism, JHEP 11 (2017) 177 [arXiv:1710.04218] [INSPIRE].
M.B. Green and J.H. Schwarz, Anomaly cancellation in supersymmetric D = 10 gauge theory and superstring theory, Phys. Lett. 149B (1984) 117 [INSPIRE].
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. 37B (1971) 95 [INSPIRE].
L. Álvarez-Gaumé and E. Witten, Gravitational anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
S.P. Novikov, On manifolds with free abelian fundamental group and their application, Izv. Ross. Akad. Nauk. Ser. Mat. 30 (1966) 207.
E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].
K. Yonekura, Dai-Freed theorem and topological phases of matter, JHEP 09 (2016) 022 [arXiv:1607.01873] [INSPIRE].
E. Witten and K. Yonekura, Anomaly inflow and the η-Invariant, arXiv:1909.08775 [INSPIRE].
D.S. Freed and M.J. Hopkins, Reflection positivity and invertible topological phases, arXiv:1604.06527 [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 and J. Wang, Higher anomalies, higher symmetries and cobordisms I: classification of higher-symmetry-protected topological states and their boundary fermionic/bosonic anomalies via a generalized cobordism theory, Ann. Math. Sci. Appl. 4 (2019) 107 [arXiv:1812.11967] [INSPIRE].
A. Beaudry and J.A. Campbell, A guide for computing stable homotopy groups, Top. Quant. Theor. Interact. 718 (2018) 89.
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
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: 2001.07731
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Davighi, J., Lohitsiri, N. Anomaly interplay in U(2) gauge theories. J. High Energ. Phys. 2020, 98 (2020). https://doi.org/10.1007/JHEP05(2020)098
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2020)098