Abstract
We consider a family of perturbative heterotic string backgrounds. These are complex threefolds X with c1 = 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 xto tilde{x}left(x,yright)kern0.5em mathrm{and}kern0.5em yto 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 mathbb{A} 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 mathcal{O} (α′), and it was shown how this was related to a simple Kähler potential. With the present formalism, we are able to rederive the results of this previously long and involved calculation, in less than a page.
Highlights
The extension of gauge fields has been studied in relation to Donaldson theory and monopole moduli spaces
Heterotic geometry is the geometry associated with the moduli space of a heterotic vacuum of string theory
Heterotic geometry is the analogue of the special geometry of Type II vacua
Summary
Heterotic geometry is the geometry associated with the moduli space of a heterotic vacuum of string theory. In order to do this in a way which takes into account the parameter dependence of the gauge transformations, one introduces a connection Λ = Λa dya on the moduli space that transforms in a manner parallel to A. As H is gauge invariant its variation with respect to the parameters can be given as a partial derivative In this way we arrive at a relation of the form. We have two metrics that arise naturally: the metric gmn on the manifold X and the metric gab on the moduli space M If we combine these into a ‘minimal’ extended metric ds2 = gmnemen + gabdyadyb , and write this out in terms of the basis forms dxm and dya we find the cross term gma = gmncan. The freedom inherent in choice of cam corresponds precisely to the freedom to make coordinate transformations as in (1.11)
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