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Abstract

This thesis addresses several problems related to low-energy supersymmetry and quantum gravity. The first of those is the gauge coupling unication problem within the Minimal Supersymmetric Standard Model. We derive a so called holomorphic master formula which we use in order to analyze the two-loop problem using one-loop renormalization group equations. In terms of model-building our solution relies on the inclusion of extra vector-like matter at intermediate energies which couples to the two Higgs doublets through Yukawa interactions. We find that the extra Yukawa couplings can ameliorate the two-loop discrepancy, an effect which is encoded in the wave-function renormalization factors of the Higgs superfields. Next we address and solve the so called mu-problem by combining two well-known mechanisms for generating a mu-term in the MSSM superpotential - we add a gauge singlet superfield which acquires a vacuum expectation value in its scalar component, on the one hand, and extra vector-like messenger superfields which couple directly to the Higgs sector, on the other. A key ingredient of our construction is the non-trivial hierarchy between the soft masses in the Higgs sector which puts our model into the class of models known as lopsided gauge mediation. This so called inverted hierarchy modifies the RG running of the soft singlet mass ensuring that the effective mu-term can attain sufficiently large values, i.e. mu > 100 GeV, as required by phenomenology. Finally we discuss an application of the so called functional renormalization group to quantum gravity. We construct a diffeomorphism invariant and gauge-fixing independent flow equation which we use in order to analyze the phase diagram of quantum gravity. We solve this equation in several non-trivial approximations, starting with the usual background field approximation. We confirm the presence of a non-trivial UV fixed point and also analyze the infrared sector of the theory which exhibits an infrared fixed point related to classical gravity.