Branes between Geometry and Gauge Theory

The subject of this work is part of the theory of strings and in particular concerned with geometrical aspects of the realization of gauge theory within string theory. In particular we study the gauge theory of D3‐branes at generalised singularities like orbifolds and the orbifolded conifold singularities. The introduction of the so called diamonds leads to a consistent picture. The action of mirror symmetry for this class of models is discussed. In a second part the field theory of the M5‐brane will be investigated on the basis of the superembedding approach. The main result here will be a rather explicit identification of the self dual three form field with the parameters of the complex structure of the embedding space. The heuristic material collected from the previous chapters states that the information contained in supersymmetric 3‐cycles can be partially extracted from supersymmetric 2‐cycles. Thus we study the case of brane boxes, which can be obtained by sewing together usual N = 2 setups of D4 and NS5‐branes. These systems satisfy the condition of uniform bending and the lift to a supersymmetric 3‐cycle can be performed explicitly. In the last part a new superconformal N = 1 gauge theories will be constructed which arise as certain deformations of N = 4 SYM and have a dual description due to the AdS/CFT correspondence.