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Classification of 4d N=2 gauge theories

Lakshya Bhardwaj, Yuji Tachikawa

TL;DR

The paper delivers a complete framework to classify four-dimensional $N{=}2$ UV-complete gauge theories built from semisimple gauge groups and hypermultiplets by encoding theories as associated graphs of nodes (gauge factors) and polygons (hypermultiplets). It shows that UV-complete theories fall into three categories, and that the graph is restricted to either a single loop or a trunk with branches, leading to a finite set of allowed trunks and decorations. The authors explicitly enumerate possible nodes, $1$-gons, $2$-gons, and the rare $3$-gons, reduce non-conformal theories to conformal seeds via mass deformations, and provide a detailed two-part classification covering all semisimple cases. They summarize the status of Seiberg-Witten solutions across the resulting theories, highlighting where curves are known (via Donagi-Witten, class S, or instanton counting) and where open cases remain, and they offer an explicit algorithm and Mathematica implementation to generate the full catalog. The work provides a comprehensive map of traditional $N{=}2$ UV-complete gauge theories, serving as a foundation for future SW analyses and deeper structural insights into duality and class S connections.

Abstract

We classify all possible four-dimensional N=2 supersymmetric UV-complete gauge theories composed of semi-simple gauge groups and hypermultiplets. We also give appropriate references for all theories with known Seiberg-Witten solutions.

Classification of 4d N=2 gauge theories

TL;DR

The paper delivers a complete framework to classify four-dimensional UV-complete gauge theories built from semisimple gauge groups and hypermultiplets by encoding theories as associated graphs of nodes (gauge factors) and polygons (hypermultiplets). It shows that UV-complete theories fall into three categories, and that the graph is restricted to either a single loop or a trunk with branches, leading to a finite set of allowed trunks and decorations. The authors explicitly enumerate possible nodes, -gons, -gons, and the rare -gons, reduce non-conformal theories to conformal seeds via mass deformations, and provide a detailed two-part classification covering all semisimple cases. They summarize the status of Seiberg-Witten solutions across the resulting theories, highlighting where curves are known (via Donagi-Witten, class S, or instanton counting) and where open cases remain, and they offer an explicit algorithm and Mathematica implementation to generate the full catalog. The work provides a comprehensive map of traditional UV-complete gauge theories, serving as a foundation for future SW analyses and deeper structural insights into duality and class S connections.

Abstract

We classify all possible four-dimensional N=2 supersymmetric UV-complete gauge theories composed of semi-simple gauge groups and hypermultiplets. We also give appropriate references for all theories with known Seiberg-Witten solutions.

Paper Structure

This paper contains 75 sections, 45 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Defining data of $\mathcal{N}{=}2$ gauge theories
  3. The classification, part I
  4. Listing available nodes
  5. Listing available $n$-gons
  6. Reduction to the classification of conformal theories
  7. Classification of theories with 3-gons
  8. Status of the Seiberg-Witten solutions
  9. Classification of theories with simple gauge groups
  10. $\mathcal{N}{=}2^*$ theories
  11. Status of the Seiberg-Witten solutions
  12. Theories with other hypermultiplets: exceptional gauge groups
  13. Status of the Seiberg-Witten solutions
  14. Theories with other hypermultiplets: $G=\mathrm{USp}(2n)$
  15. Status of the Seiberg-Witten solutions
  16. ...and 60 more sections

Figures (1)

  • Figure 1: A subjective view of $\mathcal{N}{=}2$ supersymmetric theories