The Universal Geometry of Heterotic Vacua

TL;DR

This paper introduces a universal geometric framework for perturbative heterotic string backgrounds, unifying various results into a tensor formulation and simplifying the derivation of moduli space metrics.

Contribution

It develops a geometric formalism extending gauge fields over parameter spaces, unifying heterotic vacua deformations into a single tensor framework.

Findings

Unified heterotic vacua deformations into a universal geometry.

Rederived moduli space metric with simplified tensor formalism.

Connected gauge extension ideas to Donaldson theory and monopole moduli spaces.

Abstract

We consider a family of perturbative heterotic string backgrounds. These are complex threefolds X with c_1 = 0, each with a gauge field solving the Hermitian Yang-Mill's equations and compatible B and H fields that satisfy the anomaly cancellation conditions. Our perspective is to consider a geometry in which these backgrounds are fibred over a parameter space. If the manifold X has coordinates x, and parameters are denoted by y, then it is natural to consider coordinate transformations x \to \tilde{x}(x,y) and y \to \tilde{y}(y). Similarly, gauge transformations of the gauge field and B field also depend on both x and y. In the process of defining deformations of the background fields that are suitably covariant under these transformations, it turns out to be natural to extend the gauge field A to a gauge field \IA on the extended (x,y)-space. Similarly, the B, H, and other fields are also extended. The total space of the fibration of the heterotic structures is the Universal Geometry of the title. The extension of gauge fields has been studied in relation to Donaldson theory and monopole moduli spaces. String vacua furnish a richer application of these ideas. One advantage of this point of view is that previously disparate results are unified into a simple tensor formulation. In a previous paper, by three of the present authors, the metric on the moduli space of heterotic theories was derived, correct through order \alpha', and it was shown how this was related to a simple Kahler potential. With the present formalism, we are able to rederive the results of this previously long and involved calculation, in less than a page.