Conformal Field Theory Between Supersymmetry and Indecomposable Structures

This thesis considers conformal field theory in its supersymmetric extension as well as in its relaxation to logarithmic conformal field theory. Compactification of superstring theory on four-dimensional complex manifolds obeying the Calabi-Yau conditions yields the moduli space of N = (4, 4) superconformal field theories with central charge c = 6 which consists of two continuously connected subspaces. This thesis is concerned with the subspace of K3 compactifications which is not well known yet. In particular, we inspect the intersection point of the Z2 and Z4 orbifold subvarieties within the K3 moduli space, explicitly identify the two corresponding points on the subvarieties geometrically, and give an explicit isomorphism of the three conformal field theory models located at that point, a specific Z2 and a Z4 orbifold model as well as the Gepner model (2)4. We also prove the orthogonality of the two subvarieties at the intersection point. This is the starting point for the programme to investigate generic points in K3 moduli space. We use the coordinate identification at the intersection point in order to relate the coordinates of both subvarieties and to explicitly calculate a geometric geodesic between the two subvarieties as well as its generator. A generic point in K3 moduli space can be reached by such a geodesic originating at a known model. We also present advances on the conformal field theoretic side of deformations along such a geodesic using conformal deformation theory. Since a consistent regularisation of the appearing deformation integrals has not been achieved yet, the completion of this programme is still an open problem. Moreover, we regard a relaxation of conformal field theory to logarithmic conformal field theory. The latter allows the indecomposable action of the L0 Virasoro mode within a representation of the conformal symmetry. In particular, we study general augmented cp,q minimal models which generalise the well-known (augmented) cp,1 model series. We calculate logarithmic nullvectors in both types of models. But most importantly, we investigate the low lying Virasoro representation content and fusion algebra of two general augmented cp,q models, the augmented c2,3 = 0 model as well as the augmented Yang-Lee model at c2,5 = −22/5. These exhibit a much richer structure as the cp,1 models with indecomposable representations up to rank 3. We elaborate several of these new rank 3 representations in great detail and uncover astonishing features. Furthermore, we argue that irreducible representations corresponding to the Kac table domain of the proper minimal models cannot be included into the theory. In particular, the true vacuum representation is rather given by a rank 1 indecomposable but not irreducible subrepresentation of a rank 2 representation. We generalise these generic examples to give the representation content and the fusion algebra of general augmented cp,q models as a conjecture. Finally, we open a new connection between logarithmic conformal field theory and quantum spin chains by relating some representations of the augmented c2,3 = 0 model to the representation content of a c = 0 model which describes an XXZ quantum spin chain.