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

TL;DR

This paper explores how the geometry of Coulomb branches in 4d $ abla=2$ theories encodes line operator charges and one-form symmetries, revealing geometric insights into S-duality and global forms.

Contribution

It clarifies the encoding of line operators and one-form symmetries in special K"ahler geometry and distinguishes between charge lattices and homology lattices in this context.

Findings

Identifies the geometric encoding of line operator charges.

Highlights the difference between charge lattice and homology lattice.

Provides insights into S-duality orbits of global forms.

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 K\"ahler 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 K\"ahler 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.