  We study three dimensional $\mathcal{N}=2$ supersymmetric theories on $I
\times M_2$ with 2d $\mathcal{N}=(0,2)$ boundary conditions at the boundaries
$\partial (I \times M_2)=M_2 \sqcup M_2$, where $M_2=\mathbb{C}$ or $ T^2$. We
introduce supersymmetric indices of three dimensional $\mathcal{N}=2$ theories
on $I \times T^2$ that couple to elliptic genera of 2d $\mathcal{N}=(0,2)$
theories at the two boundaries. We evaluate the $I \times T^2$ indices in terms
of supersymmetric localization and study dualities on the $I \times M_2$. We
consider the dimensional reduction of $I \times T^2$ to $I \times S^1$ and
obtain the localization formula of 2d $\mathcal{N}=(2,2)$ supersymmetric
indices on $I \times S^1$. We illustrate computations of open string Witten
indices based on gauged linear sigma models. Correlation functions of Wilson
loops on $I \times S^1$ 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.
