Abstract
Seiberg and Witten have shown that in \( \mathcal{N}=2 \) SQCD with N f = 2N c = 4 the S-duality group \( \mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) \) acts on the flavor charges, which are weights of Spin(8), by triality. There are other \( \mathcal{N}=2 \) SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of E 6, E 7 and E 8 instead of Spin(8). The S-duality group \( \mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) \) acts on these weights: what replaces Spin(8) triality for the E 6 , E 7 , E 8 root lattices?
In this paper we answer the question. The action on the matter charges of (a finite central extension of) \( \mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) \) factorizes trough the action of the exceptional Shephard-Todd groups G 4 and G 8 which should be seen as complex analogs of the usual triality group \( {\mathfrak{S}}_3\simeq \mathrm{Weyl}\left({A}_2\right) \). Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type.
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References
S. Cecotti and C. Vafa, Classification of complete \( \mathcal{N}=2 \) supersymmetric theories in 4 dimensions, Surveys in differential geometry 18 (2013) [arXiv:1103.5832] [INSPIRE].
T. Eguchi and K. Hori, N = 2 superconformal field theories in four-dimensions and A-D-E classification, in Saclay 1996, The mathematical beauty of physics [hep-th/9607125] [INSPIRE].
S. Cecotti, Categorical tinkertoys for N = 2 gauge theories, Int. J. Mod. Phys. A 28 (2013) 1330006 [arXiv:1203.6734].
S. Cecotti, M. Del Zotto and S. Giacomelli, More on the N = 2 superconformal systems of type D p (G), JHEP 04 (2013) 153 [arXiv:1303.3149] [INSPIRE].
M. Del Zotto, C. Vafa and D. Xie, Geometric Engineering, Mirror Symmetry and 6d (1, 0) → 4d, N = 2, arXiv:1504.08348 [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].
F. Diamond and J. Shurman, Graduate Texts in Mathematics. Vol. 228: A first course in modular forms, Springer, Berlin Germany (2005).
F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, (1884). Reprinted with an introduction and commentary by P. Slodowy, Birkhäuser Verlag, Basel Switzerland (1993).
G.C. Shephard and J.A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954) 274.
M. Broué, G. Malle and R. Rouquier, On complex reflection groups and their associated braid groups, CMS Conf. Proc. 16 (1995) 1.
R. Stekolshchik, Notes on Coxeter transformations and the McKay correspondence, Springer Monographs in Mathematics, Springer, Heidelberg Germany (2008).
W. Geigle and H. Lenzing, A class of weighted projective curves arising in the representation theory of finite dimensional algebras, in Lectures Notes in Mathematics. Vol. 1273: Singularities, Representation of Algebras, and Vector Bundles, Springer, Berlin Germany (1987).
H. Lenzing, Hereditary categories, in London Mathematical Society Lecture Note Series. Vol. 332: Handbook of tilting theory, L. Angeleri Hügel, D. Happel and H. Krause eds., Cambridge University Press, Cambridge U.K. (2007).
H. Lenzing, Hereditary categories, ICTP lectures 2006. Available on line: http://webusers.imj-prg.fr/bernhard.keller/ictp2006/lecturenotes/lenzing1.pdf, http://webusers.imj-prg.fr/bernhard.keller/ictp2006/lecturenotes/lenzing2.pdf.
H. Lenzing and H. Meltzer, The automorphism group of the derived category for a weighted projective line, Comm. Algebra 28 (2000) 1685.
H. Meltzer, Memoirs of the American Mathematical Society. Vol. 808: Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines, AMS Press, Providence U.S.A. (2004).
M. Barot, D. Kussin and H. Lenzing, The cluster category of a canonical algebra, arXiv:0801.4540.
D. Happel and C.M. Ringel, Lecture Notes in Mathematics. Vol. 1177: The derived category of a tubular algebra, Springer, Berlin Germany (1986), pg. 156.
D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, math/0503632.
D. Kussin, H. Lenzing and H. Meltzer, Triangle singularities, ADE-chains, and weighted projective lines, arXiv:1203.5505.
S. Cecotti, A. Neitzke and C. Vafa, R-Twisting and 4d/2d Correspondences, arXiv:1006.3435 [INSPIRE].
O. Schiffmann, Noncommutative projective curves and quantum loop algebras, math/0205267.
M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435 [INSPIRE].
S. Cecotti and M. Del Zotto, Y systems, Q systems and 4D \( \mathcal{N}=2 \) supersymmetric QFT, J. Phys. A 47 (2014) 474001 [arXiv:1403.7613] [INSPIRE].
T. Bridgeland, Stability conditions on triangulated categories, Ann. Math. 166 (2007) 317 [math/0212237].
M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi and C. Vafa, \( \mathcal{N}=2 \) quantum field theories and their BPS quivers, Adv. Theor. Math. Phys. 18 (2014) 27 [arXiv:1112.3984] [INSPIRE].
S. Cecotti and M. Del Zotto, Galois covers of N = 2 BPS spectra and quantum monodromy, arXiv:1503.07485 [INSPIRE].
C.M. Ringel, Lecture Notes in Mathematics. Vol. 1099: Tame Algebras and Integral Quadratic Forms, Springer, Berlin Germany (1984).
R. Moody, S. Rao and T. Yokonuma, Toroidal Lie algebras and vertex representations, Geom. Dedicata 35 (1990) 283.
I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. Vol. 1: Techniques of Representation Theory, Cambridge University Press, Cambridge U.K. (2006).
M.F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. 7 (1957) 414.
L. Hille and M. Van den Bergh, Fourier-Mukai transforms, in Handbook of tilting theory, L. Angeleri Hügel, D. Happel and H. Krause eds., Cambridge University Press, Cambridge U.K. (2007), pg. 147.
D. Kussin, Memoirs of the American Mathematical Society. Vol. 201: Noncommutative curves of genus zero: related to finite dimensional algebras, AMS Press, Providence U.S.A. (2009).
X.-W. Chen and H. Krause, Introduction to coherent sheaves on weighted projective lines, arXiv:0911.4473.
J. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg (1999).
B. Keller, Cluster algebras, quiver representations and triangulated categories, in London Mathematical Society Lecture Note Series. Vol. 375: Triangulated categories, Cambridge University Press, Cambridge U.K. (2010), pg. 76.
A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006) 572 [math/0402054].
I. Reiten, Tilting theory and cluster algebras, ICTP lectures 2006. Available on line: http://webusers.imj-prg.fr/bernhard.keller/ictp2006/lecturenotes/reiten.pdf.
M. Buican, S. Giacomelli, T. Nishinaka and C. Papageorgakis, Argyres-Douglas Theories and S-duality, JHEP 02 (2015) 185 [arXiv:1411.6026] [INSPIRE].
M. Buican and T. Nishinaka, On the Superconformal Index of Argyres-Douglas Theories, arXiv:1505.05884 [INSPIRE].
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Cecotti, S., Del Zotto, M. Higher S-dualities and Shephard-Todd groups. J. High Energ. Phys. 2015, 35 (2015). https://doi.org/10.1007/JHEP09(2015)035
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DOI: https://doi.org/10.1007/JHEP09(2015)035