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A review of the T_N theory and its cousins

Yuji Tachikawa

TL;DR

This article provides a cohesive survey of the 4d N=2 T_N theory and its cousins, deriving their properties from a uniform 6d (2,0) origin and class S construction. It consolidates central charges, superconformal indices (notably in the Schur limit and its refinements), moduli spaces, and chiral ring relations, emphasizing the role of puncture closures (partial and complete) and Argyres–Seiberg dualities as organizing principles. Key contributions include explicit formulas for central charges under general g and puncture data, a Schur-index expression linked to 2d q-deformed Yang–Mills, and dimension counts of Higgs/Coulomb branches via Slodowy slices and Hitchin systems. The findings deepen understanding of T_N as a building block for class S and its dualities, with implications for non-Lagrangian dynamics, holography, and exact protected data in these theories.

Abstract

The T_N theory is a four-dimensional N=2 superconformal field theory that has played a central role in the analysis of supersymmetric dualities in the last few years. The aim of this review is to collect known properties of the T_N theory and its cousins in one place as a quick reference.

A review of the T_N theory and its cousins

TL;DR

This article provides a cohesive survey of the 4d N=2 T_N theory and its cousins, deriving their properties from a uniform 6d (2,0) origin and class S construction. It consolidates central charges, superconformal indices (notably in the Schur limit and its refinements), moduli spaces, and chiral ring relations, emphasizing the role of puncture closures (partial and complete) and Argyres–Seiberg dualities as organizing principles. Key contributions include explicit formulas for central charges under general g and puncture data, a Schur-index expression linked to 2d q-deformed Yang–Mills, and dimension counts of Higgs/Coulomb branches via Slodowy slices and Hitchin systems. The findings deepen understanding of T_N as a building block for class S and its dualities, with implications for non-Lagrangian dynamics, holography, and exact protected data in these theories.

Abstract

The T_N theory is a four-dimensional N=2 superconformal field theory that has played a central role in the analysis of supersymmetric dualities in the last few years. The aim of this review is to collect known properties of the T_N theory and its cousins in one place as a quick reference.

Paper Structure

This paper contains 24 sections, 132 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The $T_N$ theory and its cousins
  3. Generalities on 4d $\mathcal{N}{=}2$ theories
  4. The $T_N$ theory
  5. Partial closure of punctures
  6. Structure of the NG bosons under the partial closures
  7. Complete closure
  8. Argyres-Seiberg duality
  9. Central charges
  10. Generalities on the central charges and anomalies
  11. The central charge of the $T_N$ theory
  12. Effect of the complete closure
  13. Effect of the partial closure to $[N-1,1]$
  14. General formula
  15. Superconformal index
  16. ...and 9 more sections

Figures (5)

  • Figure 2.1: 6d theory $S_{\mathrm{SU}(N)}$ on a genus-2 surface and its particular limit.
  • Figure 2.2: Two ways of decomposing a four-punctured sphere.
  • Figure 2.3: Bringing out two full punctures out of a surface.
  • Figure 2.4: We partially close the puncture $B$ to the type $[N-1,1]$. We then partially close the puncture $C$ to the type $[N-1,1]$. The sphere containing two punctures of type $[N-1,1]$ is tricky to analyze.
  • Figure 5.1: The $T_N$ theory with three segments of 5d super Yang-Mills attached.