The equivariant A-twist and gauged linear sigma models on the two-sphere

TL;DR

This work computes exact correlation functions for 2d N=(2,2) GLSMs on the Ω-deformed sphere S^2_Ω by localizing to the Coulomb branch and evaluating a sum over flux sectors via Jeffrey-Kirwan residues. The key result is a holomorphic Coulomb-branch formula expressing ⟨O^(N)(σ_N) O^(S)(σ_S)⟩ as a sum over k with a residue density built from the one-loop determinants, including an ε_Ω-dependent shift of σ at the poles; in the ε_Ω→0 limit this reduces to the A-twist and yields standard quantum cohomology data, including non-abelian generalizations. The paper develops both the derivation and the machinery (JK residues, GLSM chambers, and 1-loop determinants) and confirms the framework through extensive abelian and non-abelian examples, such as the abelian Higgs model, CP^{N_f-1}, the quintic, and Grassmannian complete intersections. It also connects to Higgs-branch localization as an alternative viewpoint and derives recursion relations governing ε_Ω-deformed correlators, providing a robust toolkit for exact topological data in GLSMs with or without mass deformations. The results offer a unifying exact framework for quantum cohomology and mirror-like data across phases, with potential implications for enumerative geometry and CY string phenomenology.

Abstract

We study two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on the $Ω$-deformed sphere, $S^2_Ω$, which is a one-parameter deformation of the $A$-twisted sphere. We provide an exact formula for the $S^2_Ω$ supersymmetric correlation functions using supersymmetric localization. The contribution of each instanton sector is given in terms of a Jeffrey-Kirwan residue on the Coulomb branch. In the limit of vanishing $Ω$-deformation, the localization formula greatly simplifies the computation of $A$-twisted correlation functions, and leads to new results for non-abelian theories. We discuss a number of examples and comment on the $ε_Ω$-deformation of the quantum cohomology relations. Finally, we present a complementary Higgs branch localization scheme in the special case of abelian gauge groups.

Figures (11)

• Figure 1: Two different choices of contours $\Gamma_-$ and $\Gamma_+$ for the ${\hat{D}}$-integral depicted on the ${\hat{D}}$-plane. The poles of the integrand coming from positively charged fields are marked by $\otimes$, while the those coming from negatively charged fields are marked $\times$. The pole at the origin ${\hat{D}} = 0$ is marked by a $\star$.
• Figure 2: Schematic depiction of behavior of poles of $\mathcal{Z}_k (\hat{\sigma},\bar{\hat{\sigma}}, {\hat{D}})$ in the ${\hat{D}}$-plane with respect to variation of $\hat{\sigma}$.
• Figure 3: Deformation of the contour $\Gamma_-$ in the ${\hat{D}}$-plane as a positive pole swoops below the real axis. When negative poles are far enough away (which can always be made the case by taking small ${ \boldsymbol\epsilon_\Omega}$ due to the assumption of projectivity), the contour $\Gamma_-$ can be taken to be parallel along the real axis by taking the imaginary part of $\Gamma_-$ to be $-i \delta$ for $\delta > K |{ \boldsymbol\epsilon_\Omega}|^2$ for an order-one constant $K$.
• Figure 4: Two homologically equivalent ways of deforming of the contour $\Gamma_-$ in the ${\hat{D}}$-plane as a negative pole circles around the origin. One can either split a small portion of the contour and move it around with the pole, or pass it through the origin to obtain $\Gamma_+ + C_0$, where $C_0$ is a tight contour around the origin. When all positive poles are far enough away, the contour $\Gamma_+$ can be made parallel to the real axis, with imaginary part $i \delta$ with $\delta > K |{ \boldsymbol\epsilon_\Omega}|^2$ for a constant $K$ of order-one.
• Figure 5: The behavior of poles of $\mathcal{Z}(\hat{\sigma},\bar{\hat{\sigma}},{\hat{D}})$ in the ${\hat{D}}$-plane as $\hat{\sigma}$ varies inside ${\Delta_{\epsilon,k}}$. The left panel depicts a positive pole colliding with the origin as $\hat{\sigma}$ is taken to a singular point within ${\Delta_{\epsilon,k}}^+$. The $\Gamma_-$-contour integral is smooth and bounded. The right panel depicts a negative pole colliding with the origin as $\hat{\sigma}$ is taken to a singular locus within ${\Delta_{\epsilon,k}}^-$. The $\Gamma_-$-contour integral is singular---the singular element can be isolated as the contour integral around $C_0$.