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

Abstract

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.

Higher form symmetries of Argyres-Douglas theories

“…4 For instance, the free factor of D (1) in equation (12) corresponds to the continuous zero-form symmetries of these SCFTs. For such zero-form symmetries this group can enhance to become non-Abelian -see [24,25] for conditions about the enhancement of the flavor symmetries in terms of the corresponding categories of BPS states. It is also interesting to remark that in the language of those papers, the Grothendieck group of the cluster category associated to the BPS quiver precisely coincides with the full D (1) we compute in this paper, referred to as the 't Hooft group in [26,27] -see section 2.3 below.…”

VOAs and Rank-Two Instanton SCFTs

“…The numerical constant of proportionality in (2.10) can be taken to define the normalization of ω. 11 More explicitly,…”

“…1 A useful characteristic by which one may organize this menagerie of theories is their rank, i.e., the complex dimension of their Coulomb branch of vacua. A series of incrementally refined papers culminated in a conjectured classification and characterization of all rank-one theories [7][8][9][10] (see also [11]). For higher ranks, a similar feat has not yet been achieved, though for partial progress see [12,13].…”

S-fold magnetic quivers

“…This paper aims to study the Higgs branch of 4d N = 2 superconformal field theories (SCFTs) defined as the world-volume theories on D3 branes probing certain Type IIB backgrounds called S-folds [1][2][3][4][5][6]. At rank 1 these theories are well known; they were constructed through compactifications of 6d N = (1, 0) theories in [7][8][9], their Coulomb JHEP02(2021)054 branches were studied in detail [10][11][12][13][14][15], and their Higgs branches were recently studied through magnetic quivers in [16]. The concept of magnetic quivers proves useful to study Higgs branches of theories, both Lagrangian and non-Lagrangian [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33].…”