Field-theoretic origin of non-perturbative fugacity constraints in 2d S^2 reductions

Develop a purely field-theoretic derivation and explanation of the non-perturbative constraints on global-symmetry fugacities that arise in the two-dimensional N=(0,2) elliptic genus after twisted compactification on S^2 of four-dimensional N=1 theories, clarifying in particular why axial symmetries are forbidden when no fields have R=1 and how these constraints relate to KK-monopole–induced constraints in 4d/3d reductions.

Background

In the discussion of reducing 4d N=1 dualities to 2d N=(0,2) via twisted compactification on S2, the authors emphasize that consistent reductions require assigning integer, non‑negative R‑charges. They highlight that, in 2d, axial symmetries may appear unless certain non‑perturbative constraints are imposed on fugacities, and these constraints depend sensitively on whether fields with R=1 are present.

The authors note that these constraints resemble those arising from KK‑monopole superpotentials in 4d/3d reductions, but state that they lack a purely field‑theoretic explanation for their generation in the 2d setting. Establishing such an explanation would clarify the mechanism enforcing the absence of axial symmetries and strengthen the theoretical foundations of the 2d dualities derived from 4d models.

References

This translates into a constraint on the fugacities in the elliptic genus, similarly to the constraints imposed by the KK monopoles in the 4d/3d reduction. Such constraints are enforced at non-perturbative level and we do not have a pure field theoretical explanation on their generation (we refer the reader to for discussions in this direction).

— Six Easy Pieces: interplays among dualities in 4d, 3d and 2d  (2512.10495 - Amariti et al., 11 Dec 2025) in Section 2d dualities (early discussion of S^2 reduction and fugacity constraints)