We present a method to compute the full non-linear deformations of matrix factorizations for ADE minimal models. This method is based on the calculation of higher products in the cohomology, called Massey products. The algorithm yields a polynomial ring whose vanishing relations encode the obstructions of the deformations of the D-branes characterized by these matrix factorizations. This coincides with the critical locus of the effective superpotential which can be computed by integrating these relations. Our results for the effective superpotential are in agreement with those obtained from solving the A-infinity relations. We point out a relation to the superpotentials of Kazama-Suzuki models. We will illustrate our findings by various examples, putting emphasis on the E 6 minimal model.

We consider matrix factorizations and homological mirror symmetry on the torus T 2 using a Landau-Ginzburg description. We identify the basic matrix factorizations of the Landau-Ginzburg superpotential and compute the full spectrum, taking into account the explicit dependence on bulk and boundary moduli. We verify homological mirror symmetry by comparing three-point functions in the A-model and the B-model.

A T 6 orbifold compactification is discussed from the somewhat unconventional perspective as the large radius limit of a Landau-Ginzburg model. The features of the model are in principle familiar, but the way they enter here is different from the way they enter when using more commonly used methods. It is hoped that the point of view presented here can supplement the understanding of the features used in string compactifications, notably in terms of naturalness and completeness. More precisely, the analyzed T 6 / 4 × 4 features two different kinds of O-planes, branes in the bulk as well as fractional branes, continuous and discrete Wilson lines as well as an orientifold action which can act in different ways on the Wilson lines. The D-branes are desribed by matrix factorizations. This work is also intended to be a showcase for the potentials of matrix factorizations which are for the first time geared to their full level of sophistication in this paper. Throughout the analyis everything is mapped from the B-model side of the LG-model to the A side by mirror symmetry. The work could be extended straightforwardly yet tediously to perform mirror symmetry on a general intersecting brane configuration and to compute Yukawa couplings. The analysis presented here can also be applied to non-toroidal backgrounds with an intersecting brane configuration on it, so I hope that it will be a helpful basis for later applications of mirror symmetry to models exhibiting real world properties.

This work discusses string compactifications on the torus with optional Z 4 × Z 4 or Z 2 × Z 2 orbifold action from the perspective of matrix factorizations. The method is brought to a level where model building on these backgrounds is possible. Whereas branes discussed in the literature typically wrap factorizable cycles, that is, cycles in H 1 (Ì 2 , ) 3 ⊂ H 3 (Ì 6 , ), branes studied here can be in generic homology classes, can have arbitrary position and Wilson line, have full complex structure respectively Kähler moduli dependence and can be subject to any consistent orientifold action. It is shown how any desired D-brane can be constructed systematically. Three-point correlators can be computed as is demonstrated at hand of an example. Their normalization is not discussed.