Abstract
We initiate a systematic study of four dimensional \( \mathcal{N} \) = 2 superconformal field theories (SCFTs) based on the analysis of their Coulomb branch geometries. Because these SCFTs are not uniquely characterized by their scale-invariant Coulomb branch geometries we also need information on their deformations. We construct all inequivalent such deformations preserving \( \mathcal{N} \) = 2 supersymmetry and additional physical consistency conditions in the rank 1 case. These not only include all the ones previously predicted by S-duality, but also 16 additional deformations satisfying all the known \( \mathcal{N} \) = 2 low energy consistency conditions. All but two of these additonal deformations have recently been identified with new rank 1 SCFTs; these identifications are briefly reviewed.
Some novel ingredients which are important for this study include: a discussion of RG-flows in the presence of a moduli space of vacua; a classification of local \( \mathcal{N} \) = 2 supersymmetry-preserving deformations of unitary \( \mathcal{N} \) = 2 SCFTs; and an analysis of charge normalizations and the Dirac quantization condition on Coulomb branches.
This paper is the first in a series of three. The second paper [1] gives the details of the explicit construction of the Coulomb branch geometries discussed here, while the third [2] discusses the computation of central charges of the associated SCFTs.
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P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N = 2 SCFTs. Part II: construction of special Kähler geometries and RG flows, JHEP 02 (2018) 002 [arXiv:1601.00011] [INSPIRE].
P. Argyres, M. Lotito, Y. Länd M. Martone, Geometric constraints on the space of N = 2 SCFTs. Part III: enhanced Coulomb branches and central charges, JHEP 02 (2018) 003 [arXiv:1609.04404] [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. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
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].
L. Álvarez-Gaumé and S.F. Hassan, Introduction to S duality in N = 2 supersymmetric gauge theories: a pedagogical review of the work of Seiberg and Witten, Fortsch. Phys. 45 (1997) 159 [hep-th/9701069] [INSPIRE].
Y. Tachikawa, N = 2 supersymmetric dynamics for pedestrians, Lect. Notes Phys. 890 (2013) 1 [arXiv:1312.2684].
P.C. Argyres, Y. Lü and M. Martone, Seiberg-Witten geometries for Coulomb branch chiral rings which are not freely generated, JHEP 06 (2017) 144 [arXiv:1704.05110] [INSPIRE].
K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964) 751.
K. Kodaira, On the structure of compact complex analytic surfaces. II, III, Amer. J. Math. 88 (1966) 682.
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
O.J. Ganor, Toroidal compactification of heterotic 6D noncritical strings down to four-dimensions, Nucl. Phys. B 488 (1997) 223 [hep-th/9608109] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E 6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].
J.A. Minahan and D. Nemeschansky, Superconformal fixed points with E n global symmetry, Nucl. Phys. B 489 (1997) 24 [hep-th/9610076] [INSPIRE].
P.C. Argyres and J.R. Wittig, Infinite coupling duals of N = 2 gauge theories and new rank 1 superconformal field theories, JHEP 01 (2008) 074 [arXiv:0712.2028] [INSPIRE].
P.C. Argyres and M. Martone, 4D \( \mathcal{N} \) = 2 theories with disconnected gauge groups, JHEP 03 (2017) 145 [arXiv:1611.08602] [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
I. García-Etxebarria and D. Regalado, \( \mathcal{N} \) = 3 four dimensional field theories, JHEP 03 (2016) 083 [arXiv:1512.06434] [INSPIRE].
O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Z3-twisted D4 theory, arXiv:1601.02077 [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. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
A.D. Shapere and Y. Tachikawa, Central charges of N = 2 superconformal field theories in four dimensions, JHEP 09 (2008) 109 [arXiv:0804.1957] [INSPIRE].
C. Beemet al., Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP 05 (2015) 020 [arXiv:1408.6522] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The \( \mathcal{N} \) = 2 superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].
S. Hellerman and S. Maeda, On the large R-charge expansion in \( \mathcal{N} \) = 2 superconformal field theories, arXiv:1710.07336 [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Deformations of superconformal theories, JHEP 11 (2016) 135 [arXiv:1602.01217] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of superconformal symmetry in diverse dimensions, arXiv:1612.00809 [INSPIRE].
D.S. Freed, Special Kähler manifolds, Commun. Math. Phys. 203 (1999) 31 [hep-th/9712042] [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].
I. Antoniadis, H. Partouche and T.R. Taylor, Spontaneous breaking of N = 2 global supersymmetry, Phys. Lett. B 372 (1996) 83 [hep-th/9512006] [INSPIRE].
D.J. Amit and L. Peliti, On dangerous irrelevant operators, Annals Phys. 140 (1982) 207 [INSPIRE].
S. Gukov, Counting RG flows, JHEP 01 (2016) 020 [arXiv:1503.01474] [INSPIRE].
D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, Exactly marginal deformations and global symmetries, JHEP 06 (2010) 106 [arXiv:1005.3546] [INSPIRE].
F.A. Dolan and H. Osborn, On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41 [hep-th/0209056] [INSPIRE].
D. Gaiotto, N. Seiberg and Y. Tachikawa, Comments on scaling limits of 4d N = 2 theories, JHEP 01 (2011) 078 [arXiv:1011.4568] [INSPIRE].
W. McKay and J. Patera, Tables of dimensions, indices, and branching rules for representations of simple Lie algebras, Lecture notes in pure and applied mathematics, Marcel Dekker, U.S.A. (1981)
J. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge U.K. (1990).
P.C. Argyres, M.R. Plesser and N. Seiberg, The moduli space of vacua of N = 2 SUSY QCD and duality in N = 1 SUSY QCD, Nucl. Phys. B 471 (1996) 159 [hep-th/9603042] [INSPIRE].
Y. Wang, private communication.
P.C. Argyres, K. Maruyoshi and Y. Tachikawa, Quantum Higgs branches of isolated N = 2 superconformal field theories, JHEP 10 (2012) 054 [arXiv:1206.4700] [INSPIRE].
R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel and R. Varnhagen, W algebras with two and three generators, Nucl. Phys. B 361 (1991) 255 [INSPIRE].
J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: a graduate course for physicists, Cambridge University Press, Cambridge U.K. (2003).
J. McOrist, I.V. Melnikov and B. Wecht, Global symmetries and \( \mathcal{N} \) = 2 SUSY, Lett. Math. Phys. 107 (2017) 1545 [arXiv:1312.3506] [INSPIRE].
G. Sierra and P.K. Townsend, The gauge invariant N = 2 supersymmetric σ model with general scalar potential, Nucl. Phys. B 233 (1984) 289 [INSPIRE].
C.M. Hull, A. Karlhede, U. Lindström and M. Roček, Nonlinear σ models and their gauging in and out of superspace, Nucl. Phys. B 266 (1986) 1 [INSPIRE].
N.J. Hitchin, A. Karlhede, U. Lindström and M. Rocek, Hyper-Kähler metrics and supersymmetry, Commun. Math. Phys. 108 (1987) 535.
E. Witten, An SU(2) anomaly, Phys. Lett. 117B (1982) 324 [INSPIRE].
K.A. Intriligator, N. Seiberg and S.H. Shenker, Proposal for a simple model of dynamical SUSY breaking, Phys. Lett. B 342 (1995) 152 [hep-ph/9410203] [INSPIRE].
E. Poppitz and M. Ünsal, Chiral gauge dynamics and dynamical supersymmetry breaking, JHEP 07 (2009) 060 [arXiv:0905.0634] [INSPIRE].
P.C. Argyres and J. Wittig, Mass deformations of four-dimensional, rank 1, N = 2 superconformal field theories, J. Phys. Conf. Ser. 462 (2013) 012001 [arXiv:1007.5026] [INSPIRE].
G. Shephard and J. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954) 274.
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955) 778.
S.R. Coleman and J. Mandula, All possible symmetries of the S matrix, Phys. Rev. 159 (1967) 1251 [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].
A.J. Bordner, E. Corrigan and R. Sasaki, Generalized Calogero-Moser models and universal LAX pair operators, Prog. Theor. Phys. 102 (1999) 499 [hep-th/9905011] [INSPIRE].
J.C. Hurtubise and E. Markman, Calogero-Moser systems and Hitchen systems, Commun. Math. Phys. 223 (2001) 533 [math/9912161] [INSPIRE].
M.R. Gaberdiel and B. Zwiebach, Exceptional groups from open strings, Nucl. Phys. B 518 (1998) 151 [hep-th/9709013] [INSPIRE].
M.R. Gaberdiel, T. Hauer and B. Zwiebach, Open string-string junction transitions, Nucl. Phys. B 525 (1998) 117 [hep-th/9801205] [INSPIRE].
O. DeWolfe and B. Zwiebach, String junctions for arbitrary Lie algebra representations, Nucl. Phys. B 541 (1999) 509 [hep-th/9804210] [INSPIRE].
O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach, Uncovering the symmetries on [p, q] seven-branes: beyond the Kodaira classification, Adv. Theor. Math. Phys. 3 (1999) 1785 [hep-th/9812028] [INSPIRE].
T. Hauer, A. Iqbal and B. Zwiebach, Duality and Weyl symmetry of 7-brane configurations, JHEP 09 (2000) 042 [hep-th/0002127] [INSPIRE].
P. Argyres, D. Kulkarni, C. Long, M. Lotito, Y. Lü and M. Martone, to appear.
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
P.A.M. Dirac, Quantized singularities in the electromagnetic field, Proc. Roy. Soc. Lond. A 133 (1931) 60 [INSPIRE].
J.S. Schwinger, A magnetic model of matter, Science 165 (1969) 757 [INSPIRE].
D. Zwanziger, Quantum field theory of particles with both electric and magnetic charges, Phys. Rev. 176 (1968) 1489 [INSPIRE].
R. Donagi and E. Markman, Cubics, integrable systems, and Calabi-Yau threefolds, alg-geom/9408004.
V.K. Dobrev and V.B. Petkova, All positive energy unitary irreducible representations of extended conformal supersymmetry, Phys. Lett. B 162 (1985) 127 [INSPIRE].
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Argyres, P., Lotito, M., Lü, Y. et al. Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part I: physical constraints on relevant deformations. J. High Energ. Phys. 2018, 1 (2018). https://doi.org/10.1007/JHEP02(2018)001
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DOI: https://doi.org/10.1007/JHEP02(2018)001