[en] This thesis addresses two different topics within the field of string theory. In the first part it is shown how Hodge-theoretic methods in conjunction with open string mirror symmetry can be used to compute non-perturbative effective superpotential couplings for type II/F-theory compactifications with D-branes and fluxes on compact Calabi-Yau manifolds. This is achieved by studying the at structure of operators which derives from the open/closed Β-model geometry. We analyze the variation of mixed Hodge structure of the relative cohomology induced by a family of divisors, which is wrapped by a D7-brane. This leads to a Picard-Fuchs system of differential operators, which can be used to compute the moduli dependence of the superpotential couplings as well as the mirror maps at various points in the open/closed deformation space. These techniques are used to obtain predictions for genuine A-model Ooguri-Vafa invariants of special Lagrangian submanifolds in compact Calabi-Yau geometries and real enumerative invariants of on-shell domain wall tensions. By an open/closed duality the system of differential equations can also be obtained from a gauged linear σ-model, which describes a non-compact Calabi-Yau four-fold compactification without branes. This is used in the examples of multi-parameter models to study the various phases of the combined open/closed deformation space. It is furthermore shown how the brane geometry can be related to a F-theory compactification on a compact Calabi-Yau four-fold, where the Hodge-theoretic techniques can be used to compute the G-flux induced Gukov-Vafa-Witten potential. The dual F-theory picture also allows to conjecture the form of the Kaehler potential on the full open/closed deformation space. In the second part we analyze the background dependence of theories which derive from multiple wrapped M5-branes. Using the Kontsevich-Soibelman wall-crossing formula and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes which are wrapped around a rigid divisor inside a compact Calabi-Yau manifold. The non-holomorphicity of the modified elliptic genus in this situation is traced back to the contribution of non-trivial BPS bound states of M5-branes. As a byproduct a result from pure mathematics obtained by Goettsche is re-derived in physical terms, which concerns the structure of the moduli spaces of stable sheaves on certain complex surfaces. The anomaly equation fulfilled by the modified elliptic genus is shown to be consistent with the holomorphic anomaly equations observed in the context of N=4 topological Vafa-Witten theory on P² and theories of E-strings obtained from wrapping M5-branes on del Pezzo surfaces. In selected examples it is argued how the holomorphic anomaly equation supplemented with appropriate boundary conditions can be used to explicitly compute the BPS degeneracies for certain charges. This work is based on the original publications. (orig.)

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