Elliptic genera of 2d N=2 gauge theories

TL;DR

This work provides a general, localization-based framework for computing elliptic genera of 2d N=(2,2) and N=(0,2) gauge theories. The central result is a universal formula expressing the elliptic genus as a Jeffrey–Kirwan residue sum of the one-loop determinant on the moduli space of flat connections on T^2, with phase dependence encoded by η. The authors validate the method through numerous concrete GLSM examples (K3, Calabi–Yau complete intersections in Grassmannians, Rø dland, and Gulliksen–Negård models) and establish dualities (e.g., U(k) vs U(N−k), SU(k) vs SU(N−k)) by matching elliptic genera across phases. They also connect their physical computations with the mathematical JK-residue framework, showing exact agreement with established formulas for complete intersections and Grassmannian geometries. Overall, the paper provides a powerful, phase-agnostic tool for exact’s genus calculations with broad implications for geometry, dualities, and string theoretic models.

Abstract

We compute the elliptic genera of general two-dimensional N=(2,2) and N=(0,2) gauge theories. We find that the elliptic genus is given by the sum of Jeffrey-Kirwan residues of a meromorphic form, representing the one-loop determinant of fields, on the moduli space of flat connections on T^2. We give several examples illustrating our formula, with both Abelian and non-Abelian gauge groups, and discuss some dualities for U(k) and SU(k) theories. This paper is a sequel to the authors' previous paper arXiv:1305.0533.

Figures (5)

• Figure 1: Elliptically fibered K3. Left: charge covectors in $\mathfrak{h}^*$, with the three phases indicated. Right: a real slice of the singular hyperplanes in $\mathfrak{M}$.
• Figure 2: The resolved $\mathbb{W}\mathbb{P}^4_{1,1,2,2,2}[8]$ model. Left: charge covectors in $\mathfrak{h}^*$, with the four phases indicated. Right: a real slice of the singular hyperplanes in $\mathfrak{M}$.
• Figure 3: The Rø dland model. Left: charge covectors in $\mathfrak{h}^*$ (Grassmannian phase indicated). Right: a real slice of the singular hyperplanes in $\mathfrak{M}$.
• Figure 4: Charge covectors of the Gulliksen-Negård model in $\mathfrak{h}^*$.
• Figure 5: Charge covectors of $U(k)$ with fundamentals $Q$ and anti-fundamental $\tilde{Q}$, for $k=2$. We included the fields $P_s$ considered in section \ref{['sec: U(k) conformal']}, and our choice of covector $\eta$.