Mirror Symmetry and the Web of Landau-Ginzburg String Vacua

We present some mathematical aspects of Landau-Ginzburg string vacua in terms of toric geometry. The one-to-one correspondence between toric divisors and some of (−1, 1) states in Landau-Ginzburg model is presented for superpotentials of typical types. The Landau-Ginzburg interpretation of non-toric divisors is also presented. Using this interpretation, we propose a method to solve the so-called ”twisted sector problem” by orbifold construction. Moreover,this construction shows that the moduli spaces of the original Landau-Ginzburg string vacua and their orbifolds are connected. By considering the mirror map of Landau-Ginzburg models, we obtain the relation between Mori vectors and the twist operators of our orbifoldization. This consideration enables us to argue the embedding of the Seiberg-Witten curve in the defining equation of the Calabi-Yau manifolds on which the type II string gets compactified. Related topics concerning the Calabi-Yau fourfolds and the extremal transition are discussed.