This is the continuation of a discussion of  Ezra Miller's paper https://arxiv.org/abs/1709.08155.

This is the first part of a discussion of  Ezra Miller's paper https://arxiv.org/abs/1709.08155.

I am interested in the moduli spaces of heterotic vacua. These are closely related to the moduli spaces of stable holomorphic bundles but in which the base and bundle vary simultaneously, together with additional constraints deriving from string theory. I will first summarise some pre-Brexit results we have derived. These include an explicit Kaehler metric and Kaehler potential for both the moduli space and its first cousin, the matter field space. I will secondly describe new, post-Brexit work in which these results are encased within an elegant geometry, which we call a universal heterotic geometry. Beyond compelling aesthetics, the framework is surprisingly useful giving both a concise derivation of our pre-Brexit results as well as some new results. 

For energy functionals composed of competing short- and long-range interactions, minimizers are often conjectured to form essentially periodic patterns on some intermediate lengthscale. However,  not many detailed structural results or proofs of periodicity are known in dimensions larger than 1. We study a functional composed of  the attractive, local interfacial energy of charges concentrated on a hyperplane and the energy of the electric field generated by these charges in the full space, which can be interpreted as a repulsive, non-local functional of the charges. We follow the approach of Alberti-Choksi-Otto and prove that the energy of minimizers of this functional is uniformly distributed  on cubes intersecting the hyperplane, which are sufficiently large with respect to the intrinsic lengthscale.

This is a joint work with A. Julia and F. Otto.

TBA