We study three dimensional N=2 supersymmetric theories on I×M2 with 2d N=(0,2) boundary conditions at the boundaries ∂(I×M2)=M2⊔M2, where M2=C or T2. We introduce supersymmetric indices of three dimensional N=2 theories on I×T2 that couple to elliptic genera of 2d N=(0,2) theories at the two boundaries. We evaluate the I×T2 indices in terms of supersymmetric localization and study dualities on the I×M2. We consider the dimensional reduction of I×T2 to I×S1 and obtain the localization formula of 2d N=(2,2) supersymmetric indices on I×S1. We illustrate computations of open string Witten indices based on gauged linear sigma models. Correlation functions of Wilson loops on I×S1 agree with Euler pairings in the geometric phase and also agree with cylinder amplitudes for B-type boundary states of Gepner models in the Landau-Ginzburg phase.