Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homological Classification of 4d $\mathcal{N}=2$ QFT. Part I: Rank-1 revisited

Matteo Caorsi, Sergio Cecotti

TL;DR

The authors develop a homological framework to classify 4d ${\cal N}=2$ QFTs, focusing on rank-1 theories and reproducing the Argyres–Plesser rank-1 list via 2-CY categories and their base-change/gauging structures. They connect the RT classification to geometric data from rational elliptic surfaces (Mordell–Weil groups) and to the 4d/2d correspondence, with a detailed treatment of discrete gaugings, UV/IR categorical correspondences, and the role of cluster-tilting objects. The work identifies 16 rank-1 2-acyclic theories, 15 additional SCFTs via base-change (and 5 false-gaugings), and analyzes explicit gaugings in the SU(2) with $N_f=4$ context, including AD models and exceptional flavor symmetries. The results establish a precise dictionary between 2-CY categories, their Kodaira type, Grothendieck groups, and the UV flavor/gauge data, offering a robust, scalable RT-based route to rank-1 and setting the stage for higher-rank (Part II) classifications. The paper thus demonstrates that holo-morphic, representation-theoretic, and geometric viewpoints converge to a coherent, computationally tractable classification of rank-1 ${\cal N}=2$ QFTs with implications for mirror symmetry and Fano geometry.

Abstract

Argyres and co-workers started a program to classify all 4d $\mathcal{N}=2$ QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 $\mathcal{N}=2$ QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces. The classification of 4d $\mathcal{N}=2$ 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 $\mathcal{N}=2$ SCFT.

Homological Classification of 4d $\mathcal{N}=2$ QFT. Part I: Rank-1 revisited

TL;DR

The authors develop a homological framework to classify 4d QFTs, focusing on rank-1 theories and reproducing the Argyres–Plesser rank-1 list via 2-CY categories and their base-change/gauging structures. They connect the RT classification to geometric data from rational elliptic surfaces (Mordell–Weil groups) and to the 4d/2d correspondence, with a detailed treatment of discrete gaugings, UV/IR categorical correspondences, and the role of cluster-tilting objects. The work identifies 16 rank-1 2-acyclic theories, 15 additional SCFTs via base-change (and 5 false-gaugings), and analyzes explicit gaugings in the SU(2) with context, including AD models and exceptional flavor symmetries. The results establish a precise dictionary between 2-CY categories, their Kodaira type, Grothendieck groups, and the UV flavor/gauge data, offering a robust, scalable RT-based route to rank-1 and setting the stage for higher-rank (Part II) classifications. The paper thus demonstrates that holo-morphic, representation-theoretic, and geometric viewpoints converge to a coherent, computationally tractable classification of rank-1 QFTs with implications for mirror symmetry and Fano geometry.

Abstract

Argyres and co-workers started a program to classify all 4d QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces. The classification of 4d 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 SCFT.

Paper Structure

This paper contains 100 sections, 28 theorems, 310 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. BPS categories: a cartoon
  3. Triangular case: 4d/2d correspondence.
  4. Properties of the quantum monodromies.
  5. Flavor symmetry.
  6. Kodaira type of a 2-CY category
  7. Kodaira type of an algebra of finite global dimension.
  8. Kodaira type of the category $\mathscr{C}_G$.
  9. Appetizer: derived category of index-1 Fano surfaces (a.k.a. Minahan-Nemeshanski SCFT)
  10. Algebraic-Geometric viewpoint
  11. Physicist's viewpoint
  12. Review of unpublished (in rank 1)
  13. The 16 rank-1 2-acyclic QFT
  14. Subtleties for $A\neq\{1^f\}$.
  15. Comparison with rational elliptic surfaces
  16. ...and 85 more sections

Key Result

Theorem 1

All finite-dimensional Jacobian algebras have the form $\mathrm{End}_\mathscr{C}(T)$ with $\mathscr{C}$ a 2-CY category and $T\in\mathscr{C}$ a cluster-tilting object.

Figures (1)

  • Figure 1: The quiver of the endo-algebras ${\cal A}^{(k)}$ derived equivalent to the category of coherent sheaves on a rational elliptic surface with section.

Theorems & Definitions (54)

  • Remark
  • Theorem : Amiot, Keller
  • Remark 1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2
  • Definition 3.1.1
  • proof
  • proof
  • ...and 44 more