Orientifolds of Gepner models without Kähler moduli

One of the main challenges in string theory Calabi-Yau compactifications to four dimensions is the stabilization of the massless complex structure and K\"ahler moduli. In type IIB string theory, complex structure moduli can be stabilized perturbatively by turning on fluxes on the internal space, while there is no perturbative mechanism for K\"ahler moduli stabilization. Since every Calabi-Yau manifold has at least one K\"ahler modulus (the overall volume), there is no hope to stabilize all moduli perturbatively. A way out is given by Landau-Ginzburg/Gepner models string vacua which can have no K\"ahler moduli. To identify the most promising candidates for fully stabilized perturbative string vacua, we provide an exhaustive list of Landau-Ginzburg orientifold models with no K\"ahler moduli, and compute for each model the number of complex structure moduli together with the tadpole charge. From this, we can identify which of these models are genuine candidates for phenomenologically relevant string vacua.