Two Dimensional (0,2) Theories and Resolved $A_n$ singularities

We study (0,2) two-dimensional theories in type IIB configurations with D5 branes wrapping blow-up ${\mathbf{P}}^1$ cycles of deformed resolutions for $A_n$ singularities or in T-dual IIA configurations with suspended D4 branes. We consider deformations of four dimensional $\mathcal{N}=2,{\prod }_{i=1}^{n}U(N_i)$ theories with general superpotentials for the adjoint and bifundamental fields together with fundamental flavours and reduce to two dimensions on a two torus in the presence of magnetic fluxes and FI terms.