Mirror symmetry and line operators

TL;DR

The paper develops a comprehensive framework for half‑BPS line operators in 3d $\mathcal{N}=4$ gauge theories, distinguishing Wilson (B‑type) and vortex (A‑type) lines and analyzing their junction algebras via $Q_A$/$Q_B$ cohomology. It introduces a constructive scheme based on algebraic data $(\mathcal{L}_0,\mathcal{G}_0)$ for vortex lines and raviolo moduli spaces to compute local operators at junctions, including monopole operators, thereby extending the Coulomb‑branch chiral ring to line junctions. The authors verify the approach in abelian theories with complete junction classifications and test it in a nonabelian SQCD context (via a known 3d mirror map from Assel–Gomis), using equivariant intersection cohomology to handle singular spaces. A rich category‑theoretic and geometric picture emerges, with A/B twists, Omega deformation quantization, and a consistent mirror map linking line operators and their junction algebras across dual theories. The work provides a practical, testable framework for computing and understanding the algebraic structure of line operators in 3d $\mathcal{N}=4$ theories, with potential connections to geometric representation theory and Coulomb/Higgs‑branch dynamics.

Abstract

We study half-BPS line operators in 3d N=4 gauge theories, focusing in particular on the algebras of local operators at their junctions. It is known that there are two basic types of such line operators, distinguished by the SUSY subalgebras that they preserve; the two types can roughly be called "Wilson lines" and "vortex lines", and are exchanged under 3d mirror symmetry. We describe a large class of vortex lines that can be characterized by basic algebraic data, and propose a mathematical scheme to compute the algebras of local operators at their junctions --- including monopole operators --- in terms of this data. The computation generalizes mathematical and physical definitions/analyses of the bulk Coulomb-branch chiral ring. We fully classify the junctions of half-BPS Wilson lines and of half-BPS vortex lines in abelian gauge theories with sufficient matter. We also test our computational scheme in a non-abelian quiver gauge theory, using a 3d-mirror-map of line operators from work of Assel and Gomis.

Figures (5)

• Figure 1: Local operators at junctions of lines have products induced by collision.
• Figure 2: The setup used to study local operators bound to vortex lines. We place the 3d theory in a solid cylinder $D\times {\mathbb R}_{t}$, with a vacuum boundary condition wrapping the outside, and a line operator along the axis. The space of local operators ${\mathcal{O}}$ at a point $p$ may be computed as the cohomology of the moduli space of solutions to BPS equations on a "Gaussian pillbox" (or "raviolo") surrounding $p$.
• Figure 3: Endpoints and endomorphisms of Wilson lines
• Figure 4: Introducing an Omega background for rotations about $\ell$.
• Figure 5: Left: the SUSY Hilbert space ${\mathcal{H}}_D({\mathcal{B}},{\mathcal{L}})$ on a disc punctured by ${\mathcal{L}}$, with boundary condition ${\mathcal{B}}$. Right: using this setup to define a representation of the category of line operators on the vector spaces ${\mathcal{H}}_D({\mathcal{B}},{\mathcal{L}})$, for various ${\mathcal{L}}$.