Geometric constraints on the space of N=2 SCFTs II: Construction of special Kähler geometries and RG flows

TL;DR

The work develops a computational program to enumerate and realize all rank-1 ${ m N}=2$ SCFT Coulomb-branch geometries by constructing explicit Seiberg–Witten curves and SW one-forms for 28 deformation patterns, linking maximal Kodaira deformations to submaximal ones via linear mass constraints. It uses the string-web picture to connect flavor data to mass deformations and extracts flavor algebras from Weyl-invariant mass polynomials, yielding explicit curves and one-forms that encode the low-energy physics. RG-flow consistency tests are applied to prune geometries, with several patterns surviving only when allowing frozen rank-0 SCFTs or specific gauged constructions, while others are ruled out as inconsistent. The paper clarifies relations between maximal and submaximal deformations (including a 2-isogeny between two $I_0^*$ cases) and demonstrates that flavor symmetries emerge from the geometry and monodromies, setting the stage for a full catalog of rank-1 ${ m N}=2$ SCFTs and paving the way for the third paper to extract central charges and other data. Overall, it provides explicit SW data and RG-consistency criteria that map the landscape of rank-1 theories and test their physical viability.

Abstract

This is the second in a series of three papers on systematic analysis of rank 1 Coulomb branch geometries of four dimensional $\mathcal{N}$=2 SCFTs. In the first paper we developed a strategy for classifying physical rank-1 CB geometries of $\mathcal{N}$=2 SCFTs. Here we show how to carry out this strategy computationally to construct the Seiberg-Witten curves and one-forms for all the rank-1 SCFTs. Explicit expressions are given for all cases, with the exception of the $N_f$=4 SU(2) gauge theory and the En SCFTs which were previously constructed. Our classification includes all known rank-1 theories plus a new one with an abelian flavor group, plus nine additional theories whose existence is more speculative. Four of those, reported in our first paper, depend on the assumption of new frozen rank-1 SCFTs. Here we also also show that the assumption of the existence of certain rank-0 $\mathcal{N}$=2 SCFTs leads to five additional consistent rank-1 CB geometries.

Figures (3)

• Figure 1: Singularities and their monodromies on the $u$-plane without ($m_i=0$) and with ($m_i\neq 0$) generic mass deformation. The solid points with coordinates $u_a$ are the singularities shown with a choice of "branch cuts" emanating from them. The $K_a\in SL(2,\mathbb{Z})$ are EM duality monodromies associated to the closed paths looping around these singularities starting from a conventional base point given by the open circle.
• Figure 2: A presentation of the singularities on the Coulomb branch of the maximally deformed $II^*$ singularity, with the singularities and their cuts in red, labelled by their EM charge vectors. In blue is a basis found in DeWolfe:1998eu of eight neutral oriented "string webs" connecting the singularities corresponding to simple roots of $E_8$.
• Figure 3: Change of monodromy basis associated with ${\sigma}_2$ which moves the 2nd cut through the 3rd singularity. The new monodromies are related to the old by $\widetilde{K}_1 = K_1$, $\widetilde{K}_2 = K_2^{-1} K_3 K_2$, and $\widetilde{K}_3 = K_2$.