Abstract
We consider perturbative heterotic string backgrounds. These are described by an SU(3) structure with torsion, a holomorphic stable vector bundle with a connection A that solves Hermitian-Yang-Mills equation and a three-form H that satisfies an anomaly cancellation condition. Firstly, we are interested in how the string scale affects the infinitesimal moduli problem. This involves studying deformations of background fields that preserve the equations of motion, the action of the symmetry group on them, a convenient choice of gauge-fixing. We determine the Hodge decompositions of the fields deformations perturbatively. Secondly, we consider a perspective in which these backgrounds are fibered over their parameter space. This is the universal geometry of the title. Symmetry transformations are allowed to depend on parameters and the process of defining deformations involves derivatives that are appropriately covariantised. In this formalism all the fields as well as some of the equations have a natural extension to a ‘universal bundle’.