Abstract
We discuss the geometric origin of discrete higher form symmetries of quantum field theories in terms of defect groups from geometric engineering in M-theory. The flux non-commutativity in M-theory gives rise to (mixed) ’t Hooft anomalies for the defect group which constrains the corresponding global structures of the associated quantum fields. We analyze the example of 4d \( \mathcal{N} \) = 1 SYM gauge theory in four dimensions, and we reproduce the well-known classification of global structures from reading between its lines. We extend this analysis to the case of 7d \( \mathcal{N} \) = 1 SYM theory, where we recover it from a mixed ’t Hooft anomaly among the electric 1-form center symmetry and the magnetic 4-form center symmetry in the defect group. The case of five-dimensional SCFTs from M-theory on toric singularities is discussed in detail. In that context we determine the corresponding 1-form and 2-form defect groups and we explain how to determine the corresponding mixed ’t Hooft anomalies from flux non-commutativity. Several predictions for non-conventional 5d SCFTs are obtained. The matching of discrete higher-form symmetries and their anomalies provides an interesting consistency check for 5d dualities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Del Zotto, J.J. Heckman, D.S. Park and T. Rudelius, On the Defect Group of a 6D SCFT, Lett. Math. Phys. 106 (2016) 765 [arXiv:1503.04806] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [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].
C. Córdova, T.T. Dumitrescu and K. Intriligator, Exploring 2-Group Global Symmetries, JHEP 02 (2019) 184 [arXiv:1802.04790] [INSPIRE].
D.S. Freed, G.W. Moore and G. Segal, The Uncertainty of Fluxes, Commun. Math. Phys. 271 (2007) 247 [hep-th/0605198] [INSPIRE].
D.S. Freed, G.W. Moore and G. Segal, Heisenberg Groups and Noncommutative Fluxes, Annals Phys. 322 (2007) 236 [hep-th/0605200] [INSPIRE].
I. García Etxebarria, B. Heidenreich and D. Regalado, IIB flux non-commutativity and the global structure of field theories, JHEP 10 (2019) 169 [arXiv:1908.08027] [INSPIRE].
E. Witten, Geometric Langlands From Six Dimensions, arXiv:0905.2720 [INSPIRE].
J.J. Heckman and L. Tizzano, 6D Fractional Quantum Hall Effect, JHEP 05 (2018) 120 [arXiv:1708.02250] [INSPIRE].
S. Gukov, D. Pei, P. Putrov and C. Vafa, 4-manifolds and topological modular forms, arXiv:1811.07884 [INSPIRE].
J. Eckhard, H. Kim, S. Schäfer-Nameki and B. Willett, Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence, JHEP 01 (2020) 101 [arXiv:1910.14086] [INSPIRE].
A. Dabholkar, P. Putrov and E. Witten, Duality and Mock Modularity, SciPost Phys. 9 (2020) 072 [arXiv:2004.14387] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, JHEP 03 (2019) 163 [arXiv:1612.00809] [INSPIRE].
C.-M. Chang, 5d and 6d SCFTs Have No Weak Coupling Limit, JHEP 09 (2019) 016 [arXiv:1810.04169] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, Discrete θ and the 5d superconformal index, JHEP 01 (2014) 079 [arXiv:1310.2150] [INSPIRE].
H. Hayashi, Y. Tachikawa and K. Yonekura, Mass-deformed TN as a linear quiver, JHEP 02 (2015) 089 [arXiv:1410.6868] [INSPIRE].
O. Bergman and G. Zafrir, Lifting 4d dualities to 5d, JHEP 04 (2015) 141 [arXiv:1410.2806] [INSPIRE].
G. Zafrir, Duality and enhancement of symmetry in 5d gauge theories, JHEP 12 (2014) 116 [arXiv:1408.4040] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP 12 (2015) 163 [arXiv:1507.03860] [INSPIRE].
G. Zafrir, Brane webs and O5-planes, JHEP 03 (2016) 109 [arXiv:1512.08114] [INSPIRE].
H.-C. Kim, Line defects and 5d instanton partition functions, JHEP 03 (2016) 199 [arXiv:1601.06841] [INSPIRE].
M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and 6d(1,0) → \( 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} \), JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, A new 5d description of 6d D-type minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, More on 5d descriptions of 6d SCFTs, JHEP 10 (2016) 126 [arXiv:1512.08239] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Equivalence of several descriptions for 6d SCFT, JHEP 01 (2017) 093 [arXiv:1607.07786] [INSPIRE].
D. Xie and S.-T. Yau, Three dimensional canonical singularity and five dimensional \( \mathcal{N} \) = 1 SCFT, JHEP 06 (2017) 134 [arXiv:1704.00799] [INSPIRE].
M. Del Zotto, J.J. Heckman and D.R. Morrison, 6D SCFTs and Phases of 5D Theories, JHEP 09 (2017) 147 [arXiv:1703.02981] [INSPIRE].
S. Alexandrov, S. Banerjee and P. Longhi, Rigid limit for hypermultiplets and five-dimensional gauge theories, JHEP 01 (2018) 156 [arXiv:1710.10665] [INSPIRE].
G. Ferlito, A. Hanany, N. Mekareeya and G. Zafrir, 3d Coulomb branch and 5d Higgs branch at infinite coupling, JHEP 07 (2018) 061 [arXiv:1712.06604] [INSPIRE].
P. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, Towards Classification of 5d SCFTs: Single Gauge Node, arXiv:1705.05836 [INSPIRE].
P. Jefferson, S. Katz, H.-C. Kim and C. Vafa, On Geometric Classification of 5d SCFTs, JHEP 04 (2018) 103 [arXiv:1801.04036] [INSPIRE].
F. Apruzzi, L. Lin and C. Mayrhofer, Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations, JHEP 05 (2019) 187 [arXiv:1811.12400] [INSPIRE].
L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: Rank one, JHEP 07 (2019) 178 [Addendum ibid. 01 (2020) 153] [arXiv:1809.01650] [INSPIRE].
C. Closset, M. Del Zotto and V. Saxena, Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective, SciPost Phys. 6 (2019) 052 [arXiv:1812.10451] [INSPIRE].
M. Del Zotto and G. Lockhart, Universal Features of BPS Strings in Six-dimensional SCFTs, JHEP 08 (2018) 173 [arXiv:1804.09694] [INSPIRE].
L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: Arbitrary rank, JHEP 10 (2019) 282 [arXiv:1811.10616] [INSPIRE].
L. Bhardwaj, Revisiting the classifications of 6d SCFTs and LSTs, JHEP 03 (2020) 171 [arXiv:1903.10503] [INSPIRE].
L. Bhardwaj, Dualities of 5d gauge theories from S-duality, JHEP 07 (2020) 012 [arXiv:1909.05250] [INSPIRE].
L. Bhardwaj, P. Jefferson, H.-C. Kim, H.-C. Tarazi and C. Vafa, Twisted Circle Compactifications of 6d SCFTs, arXiv:1909.11666 [INSPIRE].
L. Bhardwaj, Do all 5d SCFTs descend from 6d SCFTs?, arXiv:1912.00025 [INSPIRE].
L. Bhardwaj and G. Zafrir, Classification of 5d N = 1 gauge theories, JHEP 12 (2020) 099 [arXiv:2003.04333] [INSPIRE].
L. Bhardwaj, More 5d KK theories, arXiv:2005.01722 [INSPIRE].
F. Apruzzi, S. Schäfer-Nameki and Y.-N. Wang, 5d SCFTs from Decoupling and Gluing, JHEP 08 (2020) 153 [arXiv:1912.04264] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, Fibers add Flavor, Part II: 5d SCFTs, Gauge Theories, and Dualities, JHEP 03 (2020) 052 [arXiv:1909.09128] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, 5d Superconformal Field Theories and Graphs, Phys. Lett. B 800 (2020) 135077 [arXiv:1906.11820] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States, JHEP 11 (2019) 068 [arXiv:1907.05404] [INSPIRE].
C. Closset and M. Del Zotto, On 5d SCFTs and their BPS quivers. Part I: B-branes and brane tilings, arXiv:1912.13502 [INSPIRE].
H.-C. Kim, S.-S. Kim and K. Lee, Higgsing and twisting of 6d DN gauge theories, JHEP 10 (2020) 014 [arXiv:1908.04704] [INSPIRE].
P. Benetti Genolini, M. Honda, H.-C. Kim, D. Tong and C. Vafa, Evidence for a Non-Supersymmetric 5d CFT from Deformations of 5d SU(2) SYM, JHEP 05 (2020) 058 [arXiv:2001.00023] [INSPIRE].
H. Hayashi, P. Jefferson, H.-C. Kim, K. Ohmori and C. Vafa, SCFTs, Holography, and Topological Strings, arXiv:1905.00116 [INSPIRE].
A. Bourget, J.F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, Magnetic Quivers from Brane Webs with O5 Planes, JHEP 07 (2020) 204 [arXiv:2004.04082] [INSPIRE].
A. Bourget, S. Cabrera, J.F. Grimminger, A. Hanany and Z. Zhong, Brane Webs and Magnetic Quivers for SQCD, JHEP 03 (2020) 176 [arXiv:1909.00667] [INSPIRE].
S. Cabrera, A. Hanany and M. Sperling, Magnetic quivers, Higgs branches, and 6d N = (1, 0) theories, JHEP 06 (2019) 071 [Erratum ibid. 07 (2019) 137] [arXiv:1904.12293] [INSPIRE].
S. Cabrera, A. Hanany and F. Yagi, Tropical Geometry and Five Dimensional Higgs Branches at Infinite Coupling, JHEP 01 (2019) 068 [arXiv:1810.01379] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Complete prepotential for 5d \( \mathcal{N} \) = 1 superconformal field theories, JHEP 02 (2020) 074 [arXiv:1912.10301] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Rank-3 antisymmetric matter on 5-brane webs, JHEP 05 (2019) 133 [arXiv:1902.04754] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Dualities and 5-brane webs for 5d rank 2 SCFTs, JHEP 12 (2018) 016 [arXiv:1806.10569] [INSPIRE].
H. Hayashi and K. Ohmori, 5d/6d DE instantons from trivalent gluing of web diagrams, JHEP 06 (2017) 078 [arXiv:1702.07263] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6d SCFTs, 5d Dualities and Tao Web Diagrams, JHEP 05 (2019) 203 [arXiv:1509.03300] [INSPIRE].
V. Saxena, Rank-two 5d SCFTs from M-theory at isolated toric singularities: a systematic study, JHEP 04 (2020) 198 [arXiv:1911.09574] [INSPIRE].
I. Garozzo, N. Mekareeya, M. Sacchi and G. Zafrir, Symmetry enhancement and duality walls in 5d gauge theories, JHEP 06 (2020) 159 [arXiv:2003.07373] [INSPIRE].
Y. Tachikawa, On the 6d origin of discrete additional data of 4d gauge theories, JHEP 05 (2014) 020 [arXiv:1309.0697] [INSPIRE].
D.R. Morrison, S. Schäfer-Nameki and B. Willett, Higher-Form Symmetries in 5d, JHEP 09 (2020) 024 [arXiv:2005.12296] [INSPIRE].
M. Dierigl, P.-K. Oehlmann and F. Ruehle, Non-Simply-Connected Symmetries in 6D SCFTs, JHEP 10 (2020) 173 [arXiv:2005.12929] [INSPIRE].
J. de Boer, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan, D.R. Morrison et al., Triples, fluxes, and strings, Adv. Theor. Math. Phys. 4 (2002) 995 [hep-th/0103170] [INSPIRE].
J.A. Minahan and M. Zabzine, Gauge theories with 16 supersymmetries on spheres, JHEP 03 (2015) 155 [arXiv:1502.07154] [INSPIRE].
K. Polydorou, A. Rocén and M. Zabzine, 7D supersymmetric Yang-Mills on curved manifolds, JHEP 12 (2017) 152 [arXiv:1710.09653] [INSPIRE].
N. Iakovidis, J. Qiu, A. Rocén and M. Zabzine, 7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds, JHEP 06 (2020) 026 [arXiv:2003.12461] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
O. Aharony and E. Witten, Anti-de Sitter space and the center of the gauge group, JHEP 11 (1998) 018 [hep-th/9807205] [INSPIRE].
E. Witten, AdS/CFT correspondence and topological field theory, JHEP 12 (1998) 012 [hep-th/9812012] [INSPIRE].
E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].
H. Sati, Framed M-branes, corners, and topological invariants, J. Math. Phys. 59 (2018) 062304 [arXiv:1310.1060] [INSPIRE].
K. Hikami, Decomposition of Witten-Reshetikhin-Turaev invariant: Linking pairing and modular forms, AMS/IP Stud. Adv. Math. 50 (2011) 131 [INSPIRE].
D. Mumford, Tata Lectures on Theta, 1, Modern Birkäuser Classics, Springer, Dordrecht (2007).
I. García-Etxebarria and M. Montero, Dai-Freed anomalies in particle physics, JHEP 08 (2019) 003 [arXiv:1808.00009] [INSPIRE].
E. Witten, Topological Tools in Ten-dimensional Physics, Int. J. Mod. Phys. A 1 (1986) 39 [INSPIRE].
A. Hatcher, Algebraic Topology, Algebraic Topology, Cambridge University Press (2002).
E. Sharpe, Undoing decomposition, Int. J. Mod. Phys. A 34 (2020) 1950233 [arXiv:1911.05080] [INSPIRE].
Y. Tanizaki and M. Ünsal, Modified instanton sum in QCD and higher-groups, JHEP 03 (2020) 123 [arXiv:1912.01033] [INSPIRE].
W. Gu, E. Sharpe and H. Zou, Notes on two-dimensional pure supersymmetric gauge theories, arXiv:2005.10845 [INSPIRE].
G. Barthel and K.H. Fieseler, Invariant divisors and homology of compact complex toric varieties, J. Math. Sci. 82 (1996) 3615.
D. Cox, J. Little and H. Schenck, Toric Varieties, Graduate studies in mathematics, American Mathematical Society (2011).
V. Danilov, The geometry of toric varieties, Russ. Math. Surv. 33 (1978) 97.
S. Friedl, Algebraic Topology I — V.5, https://www.uni-regensburg.de/Fakultaeten/nat_ Fak_I/friedl/papers/2020_friedl-algebraic-topology.pdf (2020).
D.A. Cox, J.B. Little and H.K. Schenck, Toric Varieties, Graduate Studies in Mathematics, AMS (2011).
S. Reffert, The Geometer’s Toolkit to String Compactifications, in Conference on String and M-theory Approaches to Particle Physics and Cosmology, (2007) [arXiv:0706.1310] [INSPIRE].
F. Denef, Les Houches Lectures on Constructing String Vacua, Les Houches 87 (2008) 483 [arXiv:0803.1194] [INSPIRE].
K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa et al., Mirror symmetry, vol. 1 of Clay mathematics monographs, AMS, Providence, U.S.A. (2003).
Y.-H. He, R.-K. Seong and S.-T. Yau, Calabi-Yau Volumes and Reflexive Polytopes, Commun. Math. Phys. 361 (2018) 155 [arXiv:1704.03462] [INSPIRE].
W. Stein et al., Sage Mathematics Software (Version 9.1), The Sage Development Team (2020).
I. García-Etxebarria and B. Heidenreich, S-duality in \( \mathcal{N} \) = 1 orientifold SCFTs, Fortsch. Phys. 65 (2017) 1700013 [arXiv:1612.00853] [INSPIRE].
J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Sasaki-Einstein metrics on S2 × S3, Adv. Theor. Math. Phys. 8 (2004) 711 [hep-th/0403002] [INSPIRE].
O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
J. Eckhard, S. Schäfer-Nameki and Y.-N. Wang, Trifectas for TN in 5d, JHEP 07 (2020) 199 [arXiv:2004.15007] [INSPIRE].
L. Foscolo and M. Haskins, New g2-holonomy cones and exotic nearly Kähler structures on S6 and S3 × S3, Annals Math. 185 (2017) 59 [arXiv:1501.07838] [INSPIRE].
B.S. Acharya and S. Gukov, M theory and singularities of exceptional holonomy manifolds, Phys. Rept. 392 (2004) 121 [hep-th/0409191] [INSPIRE].
G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
R. Bryand and S. Salamon, On the construction of some complete metrices with expectional holonomy, Duke Math. J. 58 (1989) 829 [INSPIRE].
G.W. Gibbons, D.N. Page and C.N. Pope, Einstein Metrics on S3, R3 and R4 Bundles, Commun. Math. Phys. 127 (1990) 529 [INSPIRE].
M. Atiyah, J.M. Maldacena and C. Vafa, An M-theory flop as a large N duality, J. Math. Phys. 42 (2001) 3209 [hep-th/0011256] [INSPIRE].
B.S. Acharya, Confining strings from G2 holonomy space-times, hep-th/0101206 [INSPIRE].
B.S. Acharya and C. Vafa, On domain walls of N = 1 supersymmetric Yang-Mills in four-dimensions, hep-th/0103011 [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: 2005.12831
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
Albertini, F., Del Zotto, M., García Etxebarria, I. et al. Higher form symmetries and M-theory. J. High Energ. Phys. 2020, 203 (2020). https://doi.org/10.1007/JHEP12(2020)203
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)203