Hyperfunctions in $A$-model Localization

TL;DR

This work develops a stationary-phase localization framework for topologically $A$-twisted $\mathcal{N}=(2,2)$ GLSMs on $S^{2}$ and derives a novel distributional integral formula for abelian observables. By analyzing the one-loop fluctuations with monopole harmonics, the authors identify a purely imaginary bosonic mode that renders the integral oscillatory, and they show the correct contribution is a distribution times a determinant ratio. They validate the framework on the $A$-twisted $\mathbb{CP}^{N-1}$ model, reproducing the standard selection rule and connecting the distributional description to the Jeffrey–Kirwan contour prescription via hyperfunction theory. This work thus reconciles real-contour distributional localization with complex contour JK localization, offering a robust groundwork for extensions to non-abelian groups and deformations, and highlighting a new role for hyperfunctions in exact QFT computations.

Abstract

We apply localization techniques to topologically $A$-twisted $\mathcal{N}=(2,2)$ supersymmetric theories of vector and chiral multiplets on $S^{2}$ and derive a novel exact formula for abelian observables, described by a distribution integrated along the real line. The distributional integral formula is verified by evaluating the correlator of the $A$-twisted $\mathbb{CP}^{N-1}$ gauged linear sigma model and confirming the standard selection rule. Finally, we use hyperfunctions to demonstrate the equivalence between the distributional and complex contour integral descriptions of the $\mathbb{CP}^{N-1}$ correlator, and find agreement with the Jeffrey-Kirwan residue prescription.