| Summary: | We have investigated matrix factorisations of polynomials corresponding to vari-ous Landau-Ginzburg models with N = 2 supersymmetry. These are non-conformal Lagrangian models with specific super-potentials and are thought to flow to a renor-malisation group fixed point, which correspond to conformal field theories. Matrix factorisations can be used to construct BRST type operators which have a basis of states which correspond to the chiral primaries of the CFTs confirming the corre-spondence. We look at how these matrix factorisations can be created from exact sequences and put this into practice using the homological algebra package, Singu-lar, to create exact sequences/free resolutions from a restricted list of ideals thereby producing a matrix factorisation factory whose only input is the potential. We man-aged to construct all ADE indecomposable matrix factorisations from simple ideals built from generators in the quotient ring. As a side result, this procedure required the development of a simple algorithm to identify isomorphic matrix factorisations. We also make some statements about invertibility of matrix elements and factors in order to discuss and where other Lagrangian, conformal theories, such as Liouville might fit in this correspondence. The main body of work concentrates on the nature of orbifold equivalence. This is an aspect of topological field theories with defects. We analyse the nature of the quantum dimension formula making some interesting discoveries which we use to refine a procedure to find such orbifold equivalences. This procedure was eventually successful, in theory only limited by computer power, and we review the current updated cataloge of orbifold equivalences and discuss the some implications of our findings and observations on such equivalences.
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