Heterotic String Data and Theta Functions

The data required for heterotic string theory with gauge group G, which for anomaly cancellation reasons must either be E8 x E8 or Spin(32)/Z2, consist of the following: a ten-dimensional space-time X and a principal G-bundle P. The fields in the theory consist of: a metric on X, a connection on P and a B-field which is, roughly speaking, a two-form on X. As in a gauge theory, these fields are considered up to gauge equivalence. This means that the connection is considered up to bundle isomorphism, that the metric is considered up to diffeomorphism equivalence, and that the two-form B is considered up to equivalence by adding exact two-forms. The fact that B has a gauge invariance implies that it is not in fact a globally defined two-form but rather must be given as a gerbe-like object. The purpose of this paper is to give a precise definition of the mathematical nature of the B-field following the recent work of Freed and then to work out a description of the moduli space of 8-dimensional Euclidean invariant classical solutions to the string Equations of Motion in the case when space-time is compactified along a two-torus.