Elliptic Genera of Two-Dimensional {\mathcal{N} = 2} Gauge Theories with Rank-One Gauge Groups

We compute the elliptic genera of two-dimensional ${\mathcal{N} = (2, 2)}$ and ${\mathcal{N} = (0, 2)}$ -gauged linear sigma models via supersymmetric localization, for rank-one gauge groups. The elliptic genus is expressed as a sum over residues of a meromorphic function whose argument is the holonomy of the gauge field along both the spatial and the temporal directions of the torus. We illustrate our formulas by a few examples including the quintic Calabi–Yau, ${\mathcal{N} = (2, 2)}$ SU(2) and O(2) gauge theories coupled to N fundamental chiral multiplets, and a geometric ${\mathcal{N} = (0, 2)}$ model.