The Moduli of the Universal Geometry of Heterotic Moduli

We study the moduli of the universal geometry of $$d=4$$ d = 4 $$N=1$$ N = 1 heterotic vacua. Universal geometry refers to a family of heterotic vacua fibred 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.