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

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 X . To shed light on such structure, we begin investigating the relation with parent 4d N = 1 theories obtained by geometric engineering M-theory on 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 X exist that are dual to well-known 4d SCFTs arising from D3 branes probes of CY cones: all these models are amenable to an 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.