Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups

TL;DR

The paper develops a localization-based framework to compute the elliptic genus $Z_{T^2}(\tau,z,\xi)$ for 2D $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,2)$ gauge theories with rank-one gauge groups on the torus. It expresses the genus as a sum over residues of a meromorphic function of the gauge holonomy $u$, carefully treating gauge-zero modes via an auxiliary field $D$ and contour deformations to isolate boundary contributions. The authors derive explicit rank-one formulas, demonstrate consistency with Landau–Ginzburg orbifolds and complete-intersection geometries, and illustrate the method through a suite of examples including the quintic Calabi–Yau, $\mathrm{SU}(2)$ and $\mathrm{O}(2)$ theories with fundamentals, and Distler–Kachru $(0,2)$ models. The results connect physical localization with mathematical elliptic genera of subvarieties in Kähler quotients and support extension to higher-rank gauge groups. Overall, the work provides a practical, residue-based approach to compute elliptic genera across a broad class of 2D gauge theories.

Abstract

We compute the elliptic genera of two-dimensional N=(2,2) and 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, N=(2,2) SU(2) and O(2) gauge theories coupled to N fundamental chiral multiplets, and a geometric N=(0,2) model.

Figures (1)

• Figure 1: Left: poles for $u_* \in \partial\Delta_\varepsilon^{(+)}$ and contour $\Gamma'_-$. Right: poles for $u_* \in \partial\Delta_\varepsilon^{(-)}$ and contour $\Gamma_-$, which is equivalent to the sum of $\Gamma'_+$ and $C_0$.