# Elliptic Genera of 2d (0,2) Gauge Theories from Brane Brick Models

## TL;DR

This paper computes the elliptic genus of 2d (0,2) gauge theories from brane brick models and shows it matches the geometric elliptic genus, revealing insights into quantum effects and anomaly cancellation.

## Contribution

It introduces a new localization formula for elliptic genus computation and demonstrates its consistency with geometric predictions in brane brick models.

## Key findings

- Matching of gauge theory and geometric elliptic genus.
- Proposal of an anomaly cancelling term in the elliptic genus formula.
- Example of triality relating different brane brick models with identical elliptic genus.

## Abstract

We compute the elliptic genus of abelian 2d (0,2) gauge theories corresponding to brane brick models. These theories are worldvolume theories on a single D1-brane probing a toric Calabi-Yau 4-fold singularity. We identify a match with the elliptic genus of the non-linear sigma model on the same Calabi-Yau background, which is computed using a new localization formula. The matching implies that the quantum effects do not drastically alter the correspondence between the geometry and the 2d (0,2) gauge theory. In theories whose matter sector suffers from abelian gauge anomaly, we propose an ansatz for an anomaly cancelling term in the integral formula for the elliptic genus. We provide an example in which two brane brick models related to each other by Gadde-Gukov-Putrov triality give the same elliptic genus.