A Metric for Heterotic Moduli

Heterotic vacua of string theory are realised, at large radius, by a compact threefold with vanishing first Chern class together with a choice of stable holomorphic vector bundle. These form a wide class of potentially realistic four-dimensional vacua of string theory. Despite all their phenomenological promise, there is little understanding of the metric on the moduli space of these. What is sought is the analogue of special geometry for these vacua. The metric on the moduli space is important in phenomenology as it normalises D-terms and Yukawa couplings. It is also of interest in mathematics, since it generalises the metric, first found by Kobayashi, on the space of gauge field connections, to a more general context. Here we construct this metric, correct to first order in alpha', in two ways: first by postulating a metric that is invariant under background gauge transformations of the gauge field, and also by dimensionally reducing heterotic supergravity. These methods agree and the resulting metric is Kahler, as is required by supersymmetry. Checking that the metric is in fact Kahler is quite intricate and uses the anomaly cancellation equation for the H-field, in an essential way. The Kahler potential nevertheless takes a remarkably simple form: it is Kahler potential for special geometry with the Kahler form replaced by the alpha'-corrected hermitian form.