6d strings and exceptional instantons

TL;DR

The paper develops ADHM-like 1d/2d gauge theories to compute Coulomb-branch instanton partition functions for exceptional groups, notably $G_2$ with ${f 7}$ and $SO(7)$ with ${f 8}$, and extends these constructions to elliptic genera of self-dual instanton strings in 6d SCFTs. The core method embeds the exceptional group into a classical subgroup of equal rank, augments the ADHM data with extra matter to model massive fluctuations, and evaluates partition functions via JK-residues and colored Young-diagram combinatorics, with extensive cross-checks against D-brane models, 5d descriptions, and topological-vertex calculations. The work demonstrates Weyl enhancements in the Coulomb sector, derives closed-form residue expressions for $SO(7)$ with spinor matter, and applies the framework to strings of non-Higgsable 6d SCFTs, supported by anomaly inflow analyses. It also discusses limitations arising from UV continua and extra branches, and outlines future directions including broader observables and potential string-theory embeddings. Collectively, the results offer a novel, testable approach to computing instanton data for exceptional groups and their string dynamics in higher-dimensional QFTs.

Abstract

We propose new ADHM-like methods to compute the Coulomb branch instanton partition functions of 5d and 6d supersymmetric gauge theories, with certain exceptional gauge groups or exceptional matters. We study $G_2$ theories with $n_{\bf 7}\leq 3$ matters in ${\bf 7}$, and $SO(7)$ theories with $n_{\bf 8}\leq 4$ matters in the spinor representation ${\bf 8}$. We also study the elliptic genera of self-dual instanton strings of 6d SCFTs with exceptional gauge groups or matters, including all non-Higgsable atomic SCFTs with rank $2$ or $3$ tensor branches. Some of them are tested with topological vertex calculus. We also explore a D-brane-based method to study instanton particles of 5d $SO(7)$ and $SO(8)$ gauge theories with matters in spinor representations, which further tests our ADHM-like proposals.

Figures (16)

• Figure 1: Brane realizations of (a) $SO(2N)$ and (b) $SO(2N+1)$ gauge theories
• Figure 2: Instantons of (a) $SO(2N)$ and (b) $SO(2N+1)$ theories
• Figure 3: Hypermultiplet in the spinor representation of (a) $SO(2N)$ and (b) $SO(2N+1)$.
• Figure 4: Two hypermultipets in the spinor representations of (a) $SO(2N)$ and (b) $SO(2N+1)$.
• Figure 5: (a) 1d quiver and (b) matters for $SO(2N)$. (bold/dashed lines for hyper/Fermi)