Prove equality of the ultraviolet and infrared elliptic genera in the two-dimensional N=(2,2) comparison theory

Prove the equality between the ultraviolet and infrared elliptic genera of the two-dimensional N=(2,2) supersymmetric gauge theories discussed as the simpler analogue of the maximally supersymmetric theory.

Background

The paper uses two-dimensional N=(2,2) theories as a simpler setting for organizing elliptic-genus computations through elliptic Springer theory, E-pseudo-Levi subgroups, and Jeffrey–Kirwan residues. In this setting, the infrared theory flows to free twisted chiral multiplets, giving a comparatively simple closed-form expression for the elliptic genus.

Despite this simplification, the paper states that the equality between the ultraviolet gauge-theory computation and the infrared expression has not been established. This unresolved equality is presented as an obstacle relevant to the broader program of proving the corresponding ultraviolet–infrared elliptic-genus identity for the N=(8,8) theory.