Dirac pairings, one-form symmetries and Seiberg-Witten geometries

TL;DR

The work uncovers how line operator data and 1-form symmetries of 4d ${\mathcal N}=2$ theories are encoded in the Coulomb branch, distinguishing the charge lattice ${\Lambda}$ from the Seiberg-Witten homology lattice ${\Lambda}_X$ of the SW curve. It provides a concrete derivation of non-principal Dirac pairings for ${\mathcal N}=4$ sYM and shows how invariant factors determine the possible line lattices, linking them to global structures and 1-form symmetries. The paper then explains how the Coulomb-branch special Kähler geometry can realize multiple SK models compatible with a given geometry, and how the S-duality group seen in the SK data may differ from the full physical S-duality group, with explicit rank-1 examples in SU(2) and SU(3) illustrating these phenomena. Overall, it highlights a geometric framework for understanding how global structures of 4d theories are reflected in, and constrained by, the Seiberg-Witten geometry and its polarizations.

Abstract

The Coulomb phase of a quantum field theory, when present, illuminates the analysis of its line operators and one-form symmetries. For 4d $\mathcal{N}=2$ field theories the low energy physics of this phase is encoded in the special Kähler geometry of the moduli space of Coulomb vacua. We clarify how the information on the allowed line operator charges and one-form symmetries is encoded in the special Kähler structure. We point out the important difference between the lattice of charged states and the homology lattice of the abelian variety fibered over the moduli space, which, when principally polarized, is naturally identified with a choice of the lattice of mutually local line operators. This observation illuminates how the distinct S-duality orbits of global forms of $\mathcal{N}=4$ theories are encoded geometrically.