D-type Minimal Conformal Matter: Quantum Curves, Elliptic Garnier Systems, and the 5d Descendants

We study the quantization of the 6d Seiberg-Witten curve for D-type minimal conformal matter theories compactified on a two-torus. The quantized 6d curve turns out to be a difference equation established via introducing codimension two and four surface defects. We show that, in the Nekrasov-Shatashvili limit, the 6d partition function with insertions of codimension two and four defects serve as the eigenfunction and eigenvalues of the difference equation, respectively. We further identify the quantum curve of D-type minimal conformal matters with an elliptic Garnier system recently studied in the integrability community. At last, as a concrete consequence of our elliptic quantum curve, we study its RG flows to obtain various quantum curves of 5d ${\rm Sp}(N)+N_f \mathsf{F},N_f\leq 2N+5$ theories.Comment: 36+6 page

Complex crystallographic reflection groups and Seiberg-Witten integrable systems: rank 1 case

We consider generalisations of the elliptic Calogero--Moser systems associated to complex crystallographic groups in accordance to \cite{EFMV11ecm}. In our previous work \cite{Argyres:2021iws}, we proposed these systems as candidates for Seiberg--Witten integrable systems of certain SCFTs. Here we examine that proposal for complex crystallographic groups of rank one. Geometrically, this means considering elliptic curves $T^2$ with $\Z_m$-symmetries, $m=2,3,4,6$, and Poisson deformations of the orbifolds $(T^2\times\mathbb{C})/\Z_m$. The $m=2$ case was studied in \cite{Argyres:2021iws}, while $m=3,4,6$ correspond to Seiberg--Witten integrable systems for the rank 1 Minahan--Nemeshansky SCFTs of type $E_{6,7,8}$. This allows us to describe the corresponding elliptic fibrations and the Seiberg--Witten differential in a compact elegant form. This approach also produces quantum spectral curves for these SCFTs, which are given by Fuchsian ODEs with special properties

Seiberg-Witten geometries for Coulomb branch chiral rings which are not freely generated

Abstract Coulomb branch chiral rings of N = 2 $$\mathcal{N}=2$$ SCFTs are conjectured to be freely generated. While no counter-example is known, no direct evidence for the conjecture is known either. We initiate a systematic study of SCFTs with Coulomb branch chiral rings satisfying non-trivial relations, restricting our analysis to rank 1. The main result of our study is that (rank-1) SCFTs with non-freely generated CB chiral rings when deformed by relevant deformations, always flow to theories with non-freely generated CB rings. This implies that if they exist, they must thus form a distinct subset under RG flows. We also find many interesting characteristic properties that these putative theories satisfy which may behelpful in proving or disproving their existence using other methods

Geometric constraints on the space of N=2 SCFTs. Part III: enhanced Coulomb branches and central charges

Abstract This is the third in a series of three papers on the systematic analysis of rank 1 four dimensional N $$\mathcal{N}$$ = 2 SCFTs. In the first two papers [1, 2] we developed and carried out a strategy for classifying and constructing physical planar rank-1 Coulomb branch geometries of N $$\mathcal{N}$$ = 2 SCFTs. Here we describe general features of the Higgs and mixed branch geometries of the moduli space of these SCFTs, and use this, along with their Coulomb branch geometry, to compute their conformal and flavor central charges. We conclude with a summary of the state of the art for rank-1 N $$\mathcal{N}$$ = 2 SCFTs

Geometric constraints on the space of N N \mathcal{N} = 2 SCFTs. Part I: physical constraints on relevant deformations

Abstract We initiate a systematic study of four dimensional N $$\mathcal{N}$$ = 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 N $$\mathcal{N}$$ = 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 N $$\mathcal{N}$$ = 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 N $$\mathcal{N}$$ = 2 supersymmetry-preserving deformations of unitary N $$\mathcal{N}$$ = 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