Evidence for an algebra of G2 instantons

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.