We study, by means of mirror symmetry, the quantum geometry of the Kähler-class parameters of a number of Calabi-Yau manifolds that have b 11 = 2. Our main interest lies in the structure of the moduli space and in the loci corresponding to singular models. This structure is considerably richer when there are two parameters than in the various one-parameter models that have been studied hitherto. We describe the intrinsic structure of the point in the (compactification of the) moduli space that corresponds to the large complex structure or classical limit. The instanton expansions are of interest owing to the fact that some of the instantons belong to families with continuous parameters. We compute the Yukawa couplings and their expansions in terms of instantons of genus zero. By making use of recent results of Bershadsky et al. we compute also the instanton numbers for instantons of genus one. For particular values of the parameters the models become birational to certain models with one parameter. The compactification divisor of the moduli space thus contains copies of the moduli spaces of one parameter models. Our discussion proceeds via the particular models IP 4

We study ζ-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The ζ-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 ζ-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 ζ-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the ζ-functions are rational functions and the degrees of the numerators and denominators are exchanged between the ζ-functions for the manifold and its mirror. It is clear nevertheless that the ζ-function, as classically defined, makes an essential distinction between Kähler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a 'quantum modification' of the ζ-function that restores the symmetry between the Kähler and complex structure parameters. We note that the ζ-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion.

The complete structure of the moduli space of Calabi-Yau manifolds and the associated Landau-Ginzburg theories, and hence also of the corresponding lowenergy effective theory that results from (2,2) superstring compactification, may be determined in terms of certain holomorphic functions called periods. These periods are shown to be readily calculable for a great many such models. We illustrate this by computing the periods explicitly for a number of classes of Calabi-Yau manifolds. We also point out that it is possible to read off from the periods certain important information relating to the mirror manifolds.

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 α , 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 Kähler, as is required by supersymmetry. Checking the metric is Kähler is intricate and the anomaly cancellation equation for the H field plays an essential role. The Kähler potential nevertheless takes a remarkably simple form: it is the Kähler potential of special geometry with the Kähler form replaced by the α -corrected hermitian form.arXiv:1605.05256v4 [hep-th]

Abstract:We describe the first order moduli space of heterotic string theory compactifications which preserve N = 1 supersymmetry in four dimensions, that is, the infinitesimal parameter space of the Strominger system. We establish that if we promote a connection on T X to a field, the moduli space corresponds to deformations of a holomorphic structure D on a bundle Q. The bundle Q is constructed as an extension by the cotangent bundle T * X of the bundle E = End(V )⊕End(T X)⊕T X with an extension class H which precisely enforces the anomaly cancelation condition. The deformations corresponding to the bundle E are simultaneous deformations of the holomorphic structures on the poly-stable bundles V and T X together with those of the complex structure of X. We discuss the fact that the "moduli" corresponding to End(T X) cannot be physical, but are however needed in our mathematical structure to be able to enforce the anomaly cancelation condition. In the appendix we comment on the choice of connection on T X which has caused some confusion in the community before. It has been shown by Ivanov and others that this connection should also satisfy the instanton equations, and we give another proof of this fact.