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 $\partial (I\times M_2)=M_2\bigsqcup M_2$, 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.

我们研究三维 $\mathcal{ñ}=2$ 关于超对称理论 $\mathrm{一世}\times {\mathrm{中号}}_2$ 与2d $\mathcal{ñ}=（0，2）$ 边界条件 $\partial （\mathrm{一世}\times {\mathrm{中号}}_2）={\mathrm{中号}}_2\bigsqcup {\mathrm{中号}}_2$，在哪里 ${\mathrm{中号}}_2=\mathbb{C}$ 要么 ${Ť}^2$。我们引入三维超对称指数$\mathcal{ñ}=2$ 关于的理论 $\mathrm{一世}\times {Ť}^2$ 那对夫妇的二维椭圆族 $\mathcal{ñ}=（0，2）$两个边界的理论。我们评估$\mathrm{一世}\times {Ť}^2$ 关于超对称定位的指数和对偶的研究对偶 $\mathrm{一世}\times {\mathrm{中号}}_2$。我们考虑减少尺寸$\mathrm{一世}\times {Ť}^2$ 至 $\mathrm{一世}\times {\mathrm{小号}}^{\mathrm{1个}}$ 并获得2d的本地化公式 $\mathcal{ñ}=（2，2）$ 上的超对称指数 $\mathrm{一世}\times {\mathrm{小号}}^{\mathrm{1个}}$。我们说明了基于规范线性西格玛模型的开放字符串维滕指数的计算。Wilson环上的相关函数$\mathrm{一世}\times {\mathrm{小号}}^{\mathrm{1个}}$ 在几何阶段与Euler配对一致，并且在Landau-Ginzburg阶段与Gepner模型的B型边界状态的圆柱振幅一致。