Abstract
We revisit the classification of rank-1 4d \( \mathcal{N}=2 \) QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-Néron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs (ε, F∞) where E is a relatively minimal, rational elliptic surface with section, and F∞ a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on (ε, F∞) equivalent to the “safely irrelevant conjecture”. The Mordell-Weil group of E (with the Néron-Tate pairing) contains a canonical root system arising from (−1)-curves in special position in the Néron-Severi group. This canonical system is identified with the roots of the flavor group F: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.
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References
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].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
R.Y. Donagi, Seiberg-Witten integrable systems, Surv. Diff. Geom. 4 (1998) 83 [alg-geom/9705010] [INSPIRE].
P.C. Argyres, M. Crescimanno, A.D. Shapere and J.R. Wittig, Classification of N = 2 superconformal field theories with two-dimensional Coulomb branches, hep-th/0504070 [INSPIRE].
P.C. Argyres and J.R. Wittig, Classification of N = 2 superconformal field theories with two-dimensional Coulomb branches. II, hep-th/0510226 [INSPIRE].
P.C. Argyres, C. Long and M. Martone, The singularity structure of scale-invariant rank-2 Coulomb branches, JHEP 05 (2018) 086 [arXiv:1801.01122] [INSPIRE].
M. Caorsi and S. Cecotti, Geometric classification of 4d N = 2 SCFTs, JHEP 07 (2018) 138 [arXiv:1801.04542] [INSPIRE].
P.C. Argyres and M. Martone, Scaling dimensions of Coulomb branch operators of 4d N = 2 superconformal field theories, arXiv:1801.06554 [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of 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 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 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 N = 2 SCFTs. Part III: enhanced Coulomb branches and central charges, JHEP 02 (2018) 003 [arXiv:1609.04404] [INSPIRE].
P.C. Argyres and M. Martone, 4d N = 2 theories with disconnected gauge groups, JHEP 03 (2017) 145 [arXiv:1611.08602] [INSPIRE].
S. Lang, Number theory III. Diophantine geometry, Encyc. Math. Sci. 60, Springer, Berlin Heidelberg, Germany, (1991).
K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Pauli 40 (1991) 83.
M. Schütt and T. Shioda, Mordell-Weil lattices, draft available at http://www2.iag.uni-hannover.de/~schuett/BookMWL17.pdf, book to be published, (2017).
T. Karayayla, The classification of automorphism groups of rational elliptic surfaces with section, Adv. Math. 230 (2012) 1.
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, 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 compact analytic surfaces, II, Ann. Math. 77 (1963) 563.
K. Kodaira, On compact analytic surfaces, III, Ann. Math. 78 (1963) 1.
J.S. Milne, Elliptic curves, Kea, (2006).
J. Silverman, Advanced topics in the arithmetic of elliptic curves, Grad. Text Math. 151, Springer-Verlag, New York, U.S.A., (1994).
R. Miranda, The basic theory of elliptic surfaces, ETS Editrice, Pisa, Italy, (1989).
W.P. Barth, K. Hulek, C.A.M. Peters and A. Ven, Compact complex surfaces, second edition, Springer, Berlin Heidelberg, Germany, (2004).
F.R. Cossec and I.V. Dolgachev, Enriques surfaces. I, Progr. Math. 76, Birkhäuser, Boston, U.S.A., (1989).
I. Dolgachev, Classical algebraic geometry. A modern view, Cambridge University Press, Cambridge, U.K., (2012).
U. Persson, Configurations of Kodaira fibers on rational elliptic surfaces, Math. Z. 205 (1990) 1.
R. Miranda, Persson’s list of singular fibers for a rational elliptic surface, Math. Z. 205 (1990) 191.
E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Trans. Am. Math. Soc. 6 (1957) 111 [INSPIRE].
J. Wolfard, ABC for polynomials, dessins d’enfants, and uniformization — a survey, in Proceedings der ELAZ-Konferenz 2004, W. Schwarz and J. Steuding eds., Steiner Verlag, Stuttgart, Germany, (2006), pg. 313.
S.K. Lando and A.K. Zvonkin, Graphs on surfaces and their applications, Encyc. Math. Sci. 141, Springer, Berlin Heidelberg, Germany, (2004).
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].
R. Miranda and U. Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986) 537.
P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, U.S.A., (1978).
P. Deligne, Théorie de Hodge, II (in French), Publ. Math. I.H. É.S. 40 (1971) 5.
C. Voisin and L. Schneps, Hodge theory and complex algebraic geometry I, Camb. Stud. Adv. Math. 76, Cambridge University Press, Cambridge, U.K., (2002).
J.-P. Serre, A course in arithmetic, Grad. Texts Math. 7, Springer, New York, U.S.A., (1973).
J.W.S. Cassels, Lectures on elliptic curves, Lond. Math. Soc. Stud. Texts 24, Cambridge University Press, Cambridge, U.K., (1991).
J.H. Silvermann, The arithmetic of elliptic curves, 2nd edition, Grad. Texts Math. 105, Springer, New York, U.S.A., (2009).
S. Lang and A. Néron, Rational points of Abelian varieties over function fields, Amer. J. Math. 81 (1959) 95.
S. Lang, Abelian varieties, Interscience, (1959).
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Caorsi, M., Cecotti, S. Special arithmetic of flavor. J. High Energ. Phys. 2018, 57 (2018). https://doi.org/10.1007/JHEP08(2018)057
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DOI: https://doi.org/10.1007/JHEP08(2018)057