Higher S-dualities and Shephard-Todd groups

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Argyres and co-workers started a program to classify all 4d N = 2 QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 N = 2 QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces.The classification of 4d N = 2 QFT is also conjectured to be equivalent to the representation theoretic (RT) classification of all 2-Calabi-Yau categories with suitable properties. Since the RT approach smells to be much simpler than the Special-Geometric one, it is worthwhile to check this expectation by reproducing the rank-1 result from the RT side. This is the main purpose of the present paper. Along the route we clarify several issues and learn new details about the rank-1 SCFT. In particular, we relate the rank-1 classification to mirror symmetry for Fano surfaces.In the follow-up paper we apply the RT methods to higher rank 4d N = 2 SCFT.

Section: Review Of Base-change/discrete Gaugingmentioning

confidence: 99%

Homological classification of 4d $$ \mathcal{N} $$ = 2 QFT. Rank-1 revisited

Caorsi

Cecotti

2019

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“…15 These operators correspond to the mathematicians' telescopic (endo)functors in the corresponding derived category [46]. For a review in the present physical context, see [47]. In the categoric language, the BPS states of SU (2) with N f quarks are the stable objects in the derived category D b (coh PN f ) of coherent sheaves on the orbifold of P 1 with N f double points.…”

Section: Jhep11(2017)013mentioning

confidence: 99%

Superconformal index, BPS monodromy and chiral algebras

Cecotti

Song

Vafa

et al. 2017

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We show that specializations of the 4d N = 2 superconformal index labeled by an integer N is given by Tr M N where M is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras A N . This generalizes the recent results for the N = −1 case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on S 2 × T 2 where we turn on 1 2 N units of U(1) r flux on S 2 .

“…Complex reflection groups appeared in the mathematical physics literature previously in e.g. [34,35,38,56].…”

Section: A Complex Reflection Groupsmentioning

confidence: 99%

Reflection groups and 3d $$ \mathcal{N} $$> 6 SCFTs

Tachikawa

Zafrir

2019

We point out that the moduli spaces of all known 3d N =8 and N =6 SCFTs, after suitable gaugings of finite symmetry groups, have the form C 4r /Γ where Γ is a real or complex reflection group depending on whether the theory is N =8 or N =6, respectively.Real reflection groups are either dihedral groups, Weyl groups, or two sporadic cases H 3,4 . Since the BLG theories and the maximally supersymmetric Yang-Mills theories correspond to dihedral and Weyl groups, it is strongly suggested that there are two yet-to-be-discovered 3d N =8 theories for H 3,4 .We also show that all known N =6 theories correspond to complex reflection groups collectively known as G(k, x, N). Along the way, we demonstrate that two ABJM theories (SU(N) k × SU(N) −k )/Z N and (U(N) k × U(N) −k )/Z k are actually equivalent.3 For a nice summary, the readers are referred to a beautiful talk by Córdova at Strings 2018 [23]. 4 We provide the basics of the theory of reflection groups in Appendix. A.