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.Comment: 57 pages; v2 blackboard bold font error fixed; v3 minor improvements, typos fixed, references added; v4 version for publication in CM

Mirror Symmetry for Calabi-Yau Hypersurfaces in Weighted P_4 and Extensions of Landau Ginzburg Theory

Recently two groups have listed all sets of weights (k_1,...,k_5) such that the weighted projective space P_4^{(k_1,...,k_5)} admits a transverse Calabi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b_{11},b_{21}) whose mirrors do not occur in the list. By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror. The `missing mirrors' are constructed as hypersurfaces in toric varieties. We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P_4's, ie, hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau--Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories.Comment: 29 pages, plain TeX. Two figures submitted separately as a uuencoded file. A plot at the end of the paper requires an extended memory version of TeX. Instructions for suppressing the plot included at head of source fil

Calabi-Yau Manifolds Over Finite Fields, I

We study Calabi-Yau manifolds defined over finite fields. These manifolds have parameters, which now also take values in the field and we compute the number of rational points of the manifold as a function of the parameters. The intriguing result is that it is possible to give explicit expressions for the number of rational points in terms of the periods of the holomorphic three-form. We show also, for a one parameter family of quintic threefolds, that the number of rational points of the manifold is closely related to as the number of rational points of the mirror manifold. Our interest is primarily with Calabi-Yau threefolds however we consider also the interesting case of elliptic curves and even the case of a quadric in CP_1 which is a zero dimensional Calabi-Yau manifold. This zero dimensional manifold has trivial dependence on the parameter over C but a not trivial arithmetic structure.Comment: 75 pages, 6 eps figure

Calabi-Yau Manifolds Over Finite Fields, II

We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to `see' these curves in the geometry of the quintic. Having these zeta-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the zeta-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are rational functions and the degrees of the numerators and denominators are exchanged between the zeta-functions for the manifold and its mirror. It is clear nevertheless that the zeta-function, as classically defined, makes an essential distinction between Kahler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a `quantum modification' of the zeta-function that restores the symmetry between the Kahler and complex structure parameters. We note that the zeta-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion.Comment: Plain TeX, 50 pages, 4 eps figure

The Heterotic Superpotential and Moduli

We study the four-dimensional effective theory arising from ten-dimensional heterotic supergravity compactified on manifolds with torsion. In particular, given the heterotic superpotential appropriately corrected at $\mathcal{O}(\alpha')$ to account for the Green-Schwarz anomaly cancellation mechanism, we investigate properties of four-dimensional Minkowski vacua of this theory. Considering the restrictions arising from F-terms and D-terms we identify the infinitesimal massless moduli space of the theory. We show that it agrees with the results that have recently been obtained from a ten-dimensional perspective where supersymmetric Minkowski solutions including the Bianchi identity correspond to an integrable holomorphic structure, with infinitesimal moduli calculated by its first cohomology. As has recently been noted, interplay of complex structure and bundle deformations through holomorphic and anomaly constraints can lead to fewer moduli than may have been expected. We derive a relation between the number of complex structure and bundle moduli removed from the low energy theory in this way, and give conditions for there to be no complex structure moduli or bundle moduli remaining in the low energy theory. The link between Yukawa couplings and obstruction theory is also briefly discussed.Comment: 35 pages, minor correction

The Universal Geometry of Heterotic Vacua

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.Comment: 50 pages, 3 figures, version accepted by JHEP, improved abstract and typos correcte