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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$. Weintroduce supersymmetric indices of three dimensional $\mathcal{N}=2$ theorieson $I \times T^2$ 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 termsof supersymmetric localization and study dualities on the $I \times M_2$. Weconsider the dimensional reduction of $I \times T^2$ to $I \times S^1$ andobtain the localization formula of 2d $\mathcal{N}=(2,2)$ supersymmetricindices on $I \times S^1$. We illustrate computations of open string Wittenindices based on gauged linear sigma models. Correlation functions of Wilsonloops on $I \times S^1$ agree with Euler pairings in the geometric phase andalso agree with cylinder amplitudes for B-type boundary states of Gepner modelsin the Landau-Ginzburg phase.

Supersymmetric indices on $I \times T^2$, elliptic genera and dualities with boundaries