Mirror Symmetry, $N=1$ Superpotentials and Tensionless Strings on Calabi-Yau Four-Folds

We study aspects of Calabi--Yau four-folds as compactification manifolds of F-theory, using mirror symmetry. Correlation functions of the topological field theory are determined directly in terms of a natural ring structure of divisors and the period integrals, and subsequently used to extract invariants of moduli spaces of rational curves subject to certain conditions. We then turn to the discussion of physical properties of the space-time theories, for a number of examples which are dual to $E_8\times E_8$ heterotic $N=1$ theories. Non-critical strings of various kinds, with low tension for special values of the moduli, lead to interesting physical effects. We investigate the generation of a non-perturbative superpotential in the four-dimensional theory and relate it to singularities associated with tensionless strings. In other cases non-perturbative effects generate an everywhere non-zero quantum tension leading to a combination of a conventional field theory with light strings hiding at a low energy scale.