The moduli of the universal geometry of heterotic moduli

We study the moduli of the universal geometry of $d=4$ $N=1$ heterotic vacua. Universal geometry refers to a family of heterotic vacua fibered over the moduli space. The universal geometry mimics aspects of the original heterotic vacua, in particular holomorphic data such as F-terms, as well as the Green-Schwarz Bianchi identity. Here we study first order deformations of the universal geometry and find this provides a shortcut to computing second order deformations of the original problem. The equations governing the moduli of the universal geometry are remarkably similar to the equations of the underlying heterotic theory and we find a fascinating double extension structure that mirrors the original heterotic problem. As an application we find first order universal deformations determine second order deformations of the original heterotic theory. This gives a shortcut to determining results that are otherwise algebraically unwieldy. The role of the D-terms is closely related to the existence of flat connections on the moduli space. Finally, we re-derive some of these results by direct differentiation - this direct approach requires significantly more calculation.