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Towards Classification of 5d SCFTs: Single Gauge Node

Patrick Jefferson, Hee-Cheol Kim, Cumrun Vafa, Gabi Zafrir

TL;DR

This work tackles the problem of classifying five-dimensional superconformal field theories arising from gauge theories with simple gauge groups. It proposes a refined set of necessary criteria—centered on the convergence of the S^5 partition function, positivity of masses and monopole string tensions, and a positive-definite Coulomb-branch metric—defining a physical Coulomb subregion C_{ m phys} bounded by rational hyperplanes. The authors formalize a Main Conjecture stating that if the metric is positive on {}_{ m phys}, the theory has a nontrivial UV fixed point, and they outline three equivalent conjectures to facilitate practical classification. Using these criteria, they classify standard (large-rank) and exceptional (rank 8) theories, revealing linear bounds on matter content and Chern-Simons levels and identifying numerous known and novel 5d SCFTs, as well as connections to 6d uplifts and geometric transitions. The framework integrates gauge theory, brane constructions, and Calabi–Yau geometry to map the space of consistent 5d fixed points and highlights exotic theories requiring careful treatment of irrational boundaries and flop transitions.

Abstract

We propose a number of apparently equivalent criteria necessary for the consistency of a 5d SCFT in its Coulomb phase and use these criteria to classify 5d SCFTs arising from a gauge theory with simple gauge group. These criteria include the convergence of the 5-sphere partition function; the positivity of particle masses and monopole string tensions; and the positive definiteness of the metric in some region in the Coulomb branch. We find that for large rank classical groups simple classes of SCFTs emerge where the bounds on the matter content and the Chern-Simons level grow linearly with rank. For classical groups of rank less than or equal to 8, our classification leads to additional cases which do not fit in the large rank analysis. We also classify the allowed matter content for all exceptional groups.

Towards Classification of 5d SCFTs: Single Gauge Node

TL;DR

This work tackles the problem of classifying five-dimensional superconformal field theories arising from gauge theories with simple gauge groups. It proposes a refined set of necessary criteria—centered on the convergence of the S^5 partition function, positivity of masses and monopole string tensions, and a positive-definite Coulomb-branch metric—defining a physical Coulomb subregion C_{ m phys} bounded by rational hyperplanes. The authors formalize a Main Conjecture stating that if the metric is positive on {}_{ m phys}, the theory has a nontrivial UV fixed point, and they outline three equivalent conjectures to facilitate practical classification. Using these criteria, they classify standard (large-rank) and exceptional (rank 8) theories, revealing linear bounds on matter content and Chern-Simons levels and identifying numerous known and novel 5d SCFTs, as well as connections to 6d uplifts and geometric transitions. The framework integrates gauge theory, brane constructions, and Calabi–Yau geometry to map the space of consistent 5d fixed points and highlights exotic theories requiring careful treatment of irrational boundaries and flop transitions.

Abstract

We propose a number of apparently equivalent criteria necessary for the consistency of a 5d SCFT in its Coulomb phase and use these criteria to classify 5d SCFTs arising from a gauge theory with simple gauge group. These criteria include the convergence of the 5-sphere partition function; the positivity of particle masses and monopole string tensions; and the positive definiteness of the metric in some region in the Coulomb branch. We find that for large rank classical groups simple classes of SCFTs emerge where the bounds on the matter content and the Chern-Simons level grow linearly with rank. For classical groups of rank less than or equal to 8, our classification leads to additional cases which do not fit in the large rank analysis. We also classify the allowed matter content for all exceptional groups.

Paper Structure

This paper contains 4 sections, 10 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Effective Prepotentials on the Coulomb branch
  3. New Criteria
  4. Main conjecture

Figures (4)

  • Figure 1: Five-brane web for the $SU(2)\times SU(2)$ gauge theory.
  • Figure 2: Appearance of massless degrees of freedom in the $SU(2)\times SU(2)$ gauge theory.
  • Figure 3: Duality of $SU(2)\times SU(2)$ theory and $SU(3)$ theory with two fundamental hypermultiplets, $N_{\textbf{F}}=2$.
  • Figure 4: