Abstract
The Coulomb phase of a quantum field theory, when present, illuminates the analysis of its line operators and one-form symmetries. For 4d \( \mathcal{N} \) = 2 field theories the low energy physics of this phase is encoded in the special Kähler geometry of the moduli space of Coulomb vacua. We clarify how the information on the allowed line operator charges and one-form symmetries is encoded in the special Kähler structure. We point out the important difference between the lattice of charged states and the homology lattice of the abelian variety fibered over the moduli space, which, when principally polarized, is naturally identified with a choice of the lattice of mutually local line operators. This observation illuminates how the distinct S-duality orbits of global forms of \( \mathcal{N} \) = 4 theories are encoded geometrically.
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O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
J.P. Ang, K. Roumpedakis and S. Seifnashri, Line Operators of Gauge Theories on Non-Spin Manifolds, JHEP 04 (2020) 087 [arXiv:1911.00589] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
Y. Tachikawa, On the 6d origin of discrete additional data of 4d gauge theories, JHEP 05 (2014) 020 [arXiv:1309.0697] [INSPIRE].
Y. Tachikawa, N = 2 supersymmetric dynamics for pedestrians (2013), https://doi.org/10.1007/978-3-319-08822-8 [arXiv:1312.2684] [INSPIRE].
P.A.M. Dirac, Quantised singularities in the electromagnetic field,, Proc. Roy. Soc. Lond. A 133 (1931) 60.
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. 430 (1994) 485] [hep-th/9407087] [INSPIRE].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
D.S. Freed, Special Kähler manifolds, Commun. Math. Phys. 203 (1999) 31 [hep-th/9712042] [INSPIRE].
M. Del Zotto and I. García Etxebarria, Global Structures from the Infrared, arXiv:2204.06495 [INSPIRE].
C. Closset, S. Schäfer-Nameki and Y.-N. Wang, Coulomb and Higgs Branches from Canonical Singularities: Part 0, JHEP 02 (2021) 003 [arXiv:2007.15600] [INSPIRE].
M. Del Zotto, I. García Etxebarria and S.S. Hosseini, Higher form symmetries of Argyres-Douglas theories, JHEP 10 (2020) 056 [arXiv:2007.15603] [INSPIRE].
C. Closset, S. Giacomelli, S. Schäfer-Nameki and Y.-N. Wang, 5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities, JHEP 05 (2021) 274 [arXiv:2012.12827] [INSPIRE].
L. Bhardwaj, M. Hubner and S. Schäfer-Nameki, 1-form Symmetries of 4d N = 2 Class S Theories, SciPost Phys. 11 (2021) 096 [arXiv:2102.01693] [INSPIRE].
S.S. Hosseini and R. Moscrop, Maruyoshi-Song flows and defect groups of \( {\mathrm{D}}_{\mathrm{p}}^{\mathrm{b}} \)(G) theories, JHEP 10 (2021) 119 [arXiv:2106.03878] [INSPIRE].
M. Buican and H. Jiang, 1-form symmetry, isolated \( \mathcal{N} \) = 2 SCFTs, and Calabi-Yau threefolds, JHEP 12 (2021) 024 [arXiv:2106.09807] [INSPIRE].
C. Closset, S. Schäfer-Nameki and Y.-N. Wang, Coulomb and Higgs branches from canonical singularities. Part I. Hypersurfaces with smooth Calabi-Yau resolutions, JHEP 04 (2022) 061 [arXiv:2111.13564] [INSPIRE].
L. Bhardwaj, S. Giacomelli, M. Hübner and S. Schäfer-Nameki, Relative Defects in Relative Theories: Trapped Higher-Form Symmetries and Irregular Punctures in Class S, arXiv:2201.00018 [INSPIRE].
F. Carta, S. Giacomelli, N. Mekareeya and A. Mininno, Dynamical consequences of 1-form symmetries and the exceptional Argyres-Douglas theories, JHEP 06 (2022) 059 [arXiv:2203.16550] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part I: physical constraints on relevant deformations, JHEP 02 (2018) 001 [arXiv:1505.04814] [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part II: construction of special Kähler geometries and RG flows, JHEP 02 (2018) 002 [arXiv:1601.00011] [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Expanding the landscape of \( \mathcal{N} \) = 2 rank 1 SCFTs, JHEP 05 (2016) 088 [arXiv:1602.02764] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part III: enhanced Coulomb branches and central charges, JHEP 02 (2018) 003 [arXiv:1609.04404] [INSPIRE].
I. García-Etxebarria and D. Regalado, \( \mathcal{N} \) = 3 four dimensional field theories, JHEP 03 (2016) 083 [arXiv:1512.06434] [INSPIRE].
O. Aharony and M. Evtikhiev, On four dimensional N = 3 superconformal theories, JHEP 04 (2016) 040 [arXiv:1512.03524] [INSPIRE].
O. Aharony and Y. Tachikawa, S-folds and 4d N = 3 superconformal field theories, JHEP 06 (2016) 044 [arXiv:1602.08638] [INSPIRE].
M. Caorsi and S. Cecotti, Geometric classification of 4d \( \mathcal{N} \) = 2 SCFTs, JHEP 07 (2018) 138 [arXiv:1801.04542] [INSPIRE].
F. Bonetti, C. Meneghelli and L. Rastelli, VOAs labelled by complex reflection groups and 4d SCFTs, JHEP 05 (2019) 155 [arXiv:1810.03612] [INSPIRE].
P.C. Argyres, A. Bourget and M. Martone, Classification of all \( \mathcal{N} \) ≥ 3 moduli space orbifold geometries at rank 2, SciPost Phys. 9 (2020) 083 [arXiv:1904.10969] [INSPIRE].
G. Shephard and J. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954) 274.
J. Kaidi, M. Martone and G. Zafrir, Exceptional moduli spaces for exceptional \( \mathcal{N} \) = 3 theories, JHEP 08 (2022) 264 [arXiv:2203.04972] [INSPIRE].
I. García-Etxebarria and D. Regalado, Exceptional \( \mathcal{N} \) = 3 theories, JHEP 12 (2017) 042 [arXiv:1611.05769] [INSPIRE].
J.S. Schwinger, Magnetic charge and quantum field theory, Phys. Rev. 144 (1966) 1087 [INSPIRE].
D. Zwanziger, Exactly soluble nonrelativistic model of particles with both electric and magnetic charges, Phys. Rev. 176 (1968) 1480 [INSPIRE].
D. Zwanziger, Quantum field theory of particles with both electric and magnetic charges, Phys. Rev. 176 (1968) 1489 [INSPIRE].
S. Coleman, The magnetic monopole fifty years later, in A. Zichichi ed., The Unity of the Fundamental Interactions, Springer, Boston, MA, USA (1983), pp. 21–117 https://doi.org/10.1007/978-1-4613-3655-6_2.
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
K.G. Wilson, Confinement of Quarks, Phys. Rev. D 10 (1974) 2445 [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
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].
F. Albertini, M. Del Zotto, I. García Etxebarria and S.S. Hosseini, Higher Form Symmetries and M-theory, JHEP 12 (2020) 203 [arXiv:2005.12831] [INSPIRE].
F. Apruzzi, F. Bonetti, I.G. Etxebarria, S.S. Hosseini and S. Schäfer-Nameki, Symmetry TFTs from String Theory, arXiv:2112.02092 [INSPIRE].
F. Apruzzi, Higher Form Symmetries TFT in 6d, arXiv:2203.10063 [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
E. Witten and D.I. Olive, Supersymmetry Algebras That Include Topological Charges, Phys. Lett. B 78 (1978) 97 [INSPIRE].
H. Osborn, Topological Charges for N = 4 Supersymmetric Gauge Theories and Monopoles of Spin 1, Phys. Lett. B 83 (1979) 321 [INSPIRE].
W.N. Franzsen, Automorphisms of Coxeter groups, Ph.D. thesis, University of Sydney, School of Mathematics and Statistics (2001), https://www.maths.usyd.edu.au/u/PG/Theses/franzsen.pdf.
G. ’t Hooft, Magnetic Monopoles in Unified Gauge Theories, Nucl. Phys. B 79 (1974) 276 [INSPIRE].
A.M. Polyakov, Particle Spectrum in Quantum Field Theory, JETP Lett. 20 (1974) 194 [INSPIRE].
R.Y. Donagi, Seiberg-Witten integrable systems, alg-geom/9705010 [INSPIRE].
N. Seiberg, Notes on theories with 16 supercharges, Nucl. Phys. B Proc. Suppl. 67 (1998) 158 [hep-th/9705117] [INSPIRE].
J. Humphreys, Reflection groups and Coxeter groups, Cambridge studies in advanced mathematics, Cambridge University Press (1990).
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955) 778.
P.C. Argyres, A. Bourget and M. Martone, On the moduli spaces of 4d \( \mathcal{N} \) = 3 SCFTs I: triple special Kähler structure, arXiv:1912.04926 [INSPIRE].
P.C. Argyres, A. Kapustin and N. Seiberg, On S-duality for non-simply-laced gauge groups, JHEP 06 (2006) 043 [hep-th/0603048] [INSPIRE].
E. D’Hoker and D.H. Phong, Calogero-Moser systems in SU(N) Seiberg-Witten theory, Nucl. Phys. B 513 (1998) 405 [hep-th/9709053] [INSPIRE].
E. D’Hoker and D.H. Phong, Spectral curves for superYang-Mills with adjoint hypermultiplet for general Lie algebras, Nucl. Phys. B 534 (1998) 697 [hep-th/9804126] [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
M. Martone, Testing our understanding of SCFTs: a catalogue of rank-2 \( \mathcal{N} \) = 2 theories in four dimensions, JHEP 07 (2022) 123 [arXiv:2102.02443] [INSPIRE].
P. Argyres and M. Martone, The rank-2 scale-invariant Coulomb branch classification problem II: curves with additional automorphisms, to appear.
T. Hungerford, Algebra, Graduate Texts in Mathematics, Springer, New York, U.S.A. (2003).
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley (1978).
Wikipedia contributors, Smith normal form, Wikipedia, The Free Encyclopedia (2022).
D. Speyer, How do you construct a symplectic basis on a lattice?, MathOverflow (2009).
J. Cassels, An introduction to the geometry of numbers, Springer (1971).
B. Gruber, Alternative formulae for the number of sublattices, Acta Crystallogr. A 53 (1997) 807.
Y. Zou, Gaussian binomials and the number of sublattices, Acta Crystallogr. A 62 (2006) 409 [math/0610684].
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Argyres, P.C., Martone, M. & Ray, M. Dirac pairings, one-form symmetries and Seiberg-Witten geometries. J. High Energ. Phys. 2022, 20 (2022). https://doi.org/10.1007/JHEP09(2022)020
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DOI: https://doi.org/10.1007/JHEP09(2022)020