Brane boxes and branes on singularities

Brane Box Models of intersecting NS and D5 branes are mapped to D3 branes at ³/Γ singularities and viceversa, in a setup which gives rise to N = 1 supersymmetric gauge theories in four dimensions. The Brane Box Models are constructed on a two-torus. The map is interpreted as T-duality along the two directions of the torus. Some Brane Box Models contain NS fivebranes winding around (p,q) cycles in the torus, and our method provides the geometric T-dual to such objects. An amusing aspect of the mapping is that T-dual configurations are calculated using D = 4 N = 1 field theory data. The mapping to the singularity picture allows the geometrical interpretation of all the marginal couplings in finite field theories. This identification is further confirmed using the AdS/CFT correspondence for orbifold theories. The AdS massless fields coupling to the marginal operators in the boundary appear as stringy twisted sectors of S⁵/Γ. The mapping for theories which are non-finite requires the introduction of fractional D3 branes in the singularity picture.