Families of N=2 field theories

TL;DR

This paper surveys how large families of ${\mathcal N}=2$ field theories can be realized via a Lagrangian description, twisted compactification of 6d ${\mathcal N}=(2,0)$ theories on a Riemann surface $C$ (class ${\mathcal S}$), or Calabi–Yau geometric engineering, highlighting overlaps and the utility of each UV perspective. It develops the Seiberg–Witten framework to encode low-energy Coulomb-branch dynamics through a prepotential ${\cal F}$, period pairs $(a^I,a^D_I)$, and central charges $Z_\gamma$, with dualities governed by symplectic transformations; singular loci and wall-crossing refine the BPS spectrum. The class ${\mathcal S}$ construction connects 4d ${\mathcal N}=2$ theories to the geometry of $C$, yielding a rich structure where exactly marginal couplings correspond to complex structure moduli and where $A_1$ theories are described by a quadratic differential $\phi_2$ and its square root spectral cover. General ADE theories extend this to higher-degree differentials tied to Casimirs, with punctures labeled by nilpotent data and outer automorphisms, unifying many Lagrangian and non-Lagrangian theories through a common geometric lens. Collectively, these insights illuminate S-duality networks, moduli spaces, and UV completions, offering a coherent framework to study protected quantities and low-energy dynamics across a broad landscape of ${\mathcal N}=2$ theories.

Abstract

This is the first article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories", ed. J. Teschner. It describes how large families of field theories with N=2 supersymmetry can be described by means of Lagrangian formulations, or by compactification from the six-dimensional theory with (2,0) supersymmetry on spaces of the form $M^4 \times C$, with C being a Riemann surface. The class of theories that can be obtained in this way is called class $\cal S$. This description allows us to relate key aspects of the four-dimensional physics of class $\cal S$ theories to geometric structures on C.