A Metric for Heterotic Moduli

TL;DR

This paper constructs a first-order alpha'-corrected Kahler metric on the moduli space of heterotic string vacua, generalizing special geometry and relevant for both phenomenology and mathematics.

Contribution

It provides the first explicit construction of the Kahler metric on heterotic moduli space, incorporating alpha' corrections via two consistent methods.

Findings

The metric is Kahler and invariant under gauge transformations.

The Kahler potential simplifies to a form analogous to special geometry with alpha' corrections.

The construction uses anomaly cancellation and supergravity reduction techniques.

Abstract

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.