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

TL;DR

This paper develops supersymmetric indices for 3d $ =2$ theories on manifolds with boundaries, linking them to elliptic genera and dualities, and explores their reductions and applications in string theory contexts.

Contribution

It introduces and computes supersymmetric indices on $I imes T^2$ with boundary conditions, connecting 3d indices to 2d elliptic genera and dualities, and extends to open string and Gepner model applications.

Findings

Indices computed via localization match duality predictions.

Correlation functions agree with geometric and Landau-Ginzburg phase calculations.

Reduction to 2d yields known supersymmetric index formulas.

Abstract

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.