<jats:title>Abstract</jats:title> <jats:p>The interaction between geometry in the adjacent dimensions 2, 3 and 4 is a theme which runs through a great deal of the work by mathematicians on gauge theory over the past few years. In this article we will examine the possibility of developing this theme in higher dimensions. We will find extensions following two intertwining threads. One thread, which we say more about here, replaces real variables by complex variables, and hence operates in complex dimensions 2, 3, 4. The other thread involves, from one point of view, replacing the quaternions by the octonians, and operates in the realm of manifolds with “exceptional holonomy”. The picture we will find pulls together various ideas which have been touched on in the literature but the striking analogies which emerge do not seem to be well-known. Our treatment will be informal throughout this article—our main aim is to advertise the potential for research in these directions. A great deal of technical work is needed to develop these ideas in detail, and a more thorough and wide-ranging account of the Calabi-Yau story will appear in the D.Phil. thesis of the second author. The first author would like to emphasise the debt due to other mathematicians in forming parts of the picture described here; particularly Dominic Joyce, Simon Salamon and Christopher Lewis for lessons on exceptional holonomy. A substantial part of this picture is essentially due to Joyce and Lewis, and again futher details will appear in the doctoral thesis of Lewis.</jats:p>