Evidence for an Algebra of $\boldsymbol{G_2}$ Instantons

TL;DR

This paper provides evidence for an algebraic structure of $G_2$ instantons, linking 7d SYM theories, 4d SCFTs, and string dualities, suggesting a new framework for understanding their partition functions.

Contribution

It proposes a novel algebraic framework for $G_2$ instantons, connecting geometric engineering, string dualities, and topological M-theory to organize instanton partition functions.

Findings

Evidence for an algebra of $G_2$ instantons.

Connection between $G_2$ instanton partition functions and algebraic structures.

Identification of dual geometries related to 4d SCFTs and twisted M-theory.

Abstract

In this short note, we present some evidence towards the existence of an algebra of BPS $G_2$ instantons. These are instantonic configurations that govern the partition functions of 7d SYM theories on local $G_2$ holonomy manifolds $\mathcal X$. To shed light on such structure, we begin investigating the relation with parent 4d $\mathcal N=1$ theories obtained by geometric engineering M-theory on $\mathcal X$. The main point of this paper is to substantiate the following dream: the holomorphic sector of such theories on multi-centered Taub-NUT spaces gives rise to an algebra whose characters organise the $G_2$ instanton partition function. As a first step towards this program, we argue by string duality that a multitude of geometries $\mathcal X$ exist that are dual to well-known 4d SCFTs arising from D3 branes probes of CY cones: all these models are amenable to analysis along the lines suggested by Dijkgraaf, Gukov, Neitzke and Vafa in the context of topological M-theory. Moreover, we discuss an interesting relation to Costello's twisted M-theory, which arises at local patches, and is a key ingredient in identifying the relevant algebras.