Summary: We compute the prepotentials and the geometry of the moduli spaces for a Calabi-Yau manifold and its mirror. In this way we obtain all the sigma model corrections to the Yukawa couplings and moduli space metric for the original manifold. The moduli space is found to be subject to the action of a modular group which, among other operations, exchanges large and small values of radius though the action on the radius is not as simple as \(R \to 1/R\). It is also shown that the quantum corrections to the coupling decompose into a sum over instanton contributions and moreover that this sum converges. In particular there are no ‘sub-instanton’ corrections. This sum over instantons points to a deep connection between the modular group and the rational curves of the Calabi-Yau manifold. The burden of the present work is that a mirror pair of Calabi-Yau manifolds is an exactly soluble superconformal theory, at least as far as the massless sector is concerned. Mirror pairs are also more general than exactly soluble models that have hitherto been discussed since we here solve the theory for all points of the moduli space.
Contents: 1. Introduction; 2. The mirror of \(\mathbb{P}_4(5)\), 2.1 Rudiments of the homology, 2.2 The modular group; 3. The periods; 4. The prepotential, metric and Yukawa coupling, 4.1 The Yukawa coupling; 5. \(\mathbb{P}_4(5)\), the mirror map and quantum corrections, 5.1 The loop term, 5.2 The mirror map, 5.3 The sum over instantons, 5.4 The number of rational curves of large degree, 5.5 Some further remarks on the modular group; 6. A mechanism for supersymmetry breaking; A. A second look at the homology of \(\mathcal W\), A.1 Cycles corresponding to the periods \(\overline {\omega}_j\). B. Further properties of the periods.
For the entire collection see [Zbl 0816.00010].