Hyperfunctions in $A$-model Localization

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 ${\mathbb{C}\mathbb{P}}^{N-1}$ 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 ${\mathbb{C}\mathbb{P}}^{N-1}$ correlator, and find agreement with the Jeffrey-Kirwan residue prescription.