The Hitchhiker's Guide to 4d $\mathcal{N}=2$ Superconformal Field Theories

TL;DR

The work surveys the landscape of four-dimensional ${\mathcal N}=2$ superconformal field theories, focusing on three complementary frameworks: (i) field-theoretic Coulomb-branch dynamics encoded by Seiberg–Witten geometry, (ii) Class ${\cal S}$ constructions from 6d ${\mathcal N}=(2,0)$ theories via M5-branes and Hitchin systems, and (iii) geometric engineering in Type IIB/F-theory through Calabi–Yau singularities and S-folds. It details how protected operator data, anomalies, and exact results arise from these interwoven pictures, and explains how SW curves, prepotentials, and elliptic fibrations capture the low-energy EFTs and IR fixed points, including Argyres–Douglas theories. The notes also discuss the AGT correspondence, irregular punctures, and outer-automorphism twists as key generalizations, highlighting how geometry and dualities illuminate strongly coupled sectors that lack Lagrangian descriptions. Overall, the document provides a structured, geometry-rich toolkit to classify, analyze, and compute observables in a broad class of 4d ${\mathcal N}=2$ SCFTs, with implications for central charges, flavor symmetries, and holographic connections.

Abstract

Superconformal field theory with $\mathcal{N}=2$ supersymmetry in four dimensional spacetime provides a prime playground to study strongly coupled phenomena in quantum field theory. Its rigid structure ensures valuable analytic control over non-perturbative effects, yet the theory is still flexible enough to incorporate a large landscape of quantum systems. Here we aim to offer a guidebook to fundamental features of the 4d $\mathcal{N}=2$ superconformal field theories and basic tools to construct them in string/M-/F-theory. The content is based on a series of lectures at the Quantum Field Theories and Geometry School (https://sites.google.com/view/qftandgeometrysummerschool/home) in July 2020.

Figures (15)

• Figure 1: General philosophy behind the approaches to study 4d ${\mathcal{N}}=2$ SCFTs. In the center are the three closely related and complementary approaches to SCFTs discussed in these lecture notes, based on bootstrap philosophy, EFT method and geometric engineering. For each of these approaches, the basic building blocks are listed in the nearby circles. They further give rise to observables with natural interpretations in each of these approaches. Some representatives of these observables are listed in the outer ring, which describe various aspects of the same SCFT and for that reason obey nontrivial mathematical relations.
• Figure 2: The vacuum moduli space for a generic ${\mathcal{N}}=2$ theory.
• Figure 3: Examples of manifest (top two) or emergent (bottom one) SCFTs.
• Figure 4: The anomaly coefficients $k_{rrr}$ and $k_r$ from three-point-functions of the $\mathop{\mathrm{U}}\nolimits(1)_r$ current $J_r$ and stress-energy tensor $T$. In Lagrangian theories, they are determined by the massless Weyl fermions running in the 1-loop triangles.
• Figure 5: Different choice of special coordinates $(a,a_D)$ on the CB and the duality monodromy around a singularity.

Theorems & Definitions (5)

• definition 1: Coulomb branch multiplets
• definition 2: Higgs branch multiplets
• definition 3: Coulomb Branch $\mathcal{M}_{\text{CB}}$
• definition 4: Higgs Branch $\mathcal{M}_{\text{HB}}$
• definition 5: Mixed Branch $\mathcal{M}_{\text{MB}}$