Supersymmetric Boundary Conditions in Three Dimensional N = 2 Theories

TL;DR

The paper addresses the construction and classification of half-BPS boundary conditions in three-dimensional $\ N=2$ theories across Landau-Ginzburg models, pure Maxwell theory, and SQED. It develops a brane-based geometric interpretation, showing that A-type boundaries correspond to Lagrangian submanifolds with $\mathrm{Im}\,W$ constant while B-type boundaries correspond to holomorphic submanifolds with $W$ constant; it then analyzes abelian duality to connect Maxwell boundary conditions to a free chiral multiplet, including the role of the boundary theta term. In SQED, the authors discuss a concrete B-type boundary and propose a mirror-boundary description in the XYZ model, arguing that 3d mirror symmetry preserves the A/B-type distinction rather than exchanging them. The work lays groundwork for further explorations of boundary indices, domain walls, and 3d-3d correspondences, and highlights nontrivial consequences of boundaries for dualities and moduli spaces in 3d $\ N=2$ theories.

Abstract

We study supersymmetric boundary conditions in 3-dimensional N = 2 Landau-Ginzburg models and abelian gauge theories. In the Landau-Ginzburg model the boundary conditions that preserve (1,1) supersymmetry (A-type) and (2,0) supersymmetry (B-type) on the boundary are classified in terms of subspaces of the target space ("brane"). An A-type brane is a Lagrangian submanifold on which the imaginary part of the superpotential is constant, while a B-type brane is a holomorphic submanifold on which the superpotential is constant. We also consider the N = 2 Maxwell theory with boundary and the abelian duality. Finally we make some comments on N = 2 SQED with boundary condition and the mirror symmetry.

Figures (5)

• Figure 1: The theory defined in $x^{2}\ge 0$ half-space. $x^{2}=0$ gives the boundary and we consider the supersymmetric boundary conditions on this.
• Figure 2: Membrane, shown in blue, ending on the A-brane, shown in green. They live in the target space. $v^{I}$ is parallel to the tangent direction and $w^{Ja}$ is in the normal direction of the A-brane.
• Figure 3: Membrane, shown in blue ending on the B-brane, shown in orange. They live in the target space. $v^{I}$ is in the tangent direction and $z^{Ja}$ is in the normal direction of B-brane.
• Figure 4: A-type boundary conditions for $\rho$ and $\sigma$. The angle $\alpha$ in a $\sigma$-$\rho$ plane can be identified with the phase appearing in the action of $\gamma^{2}$ on $\lambda$.
• Figure 7: The moduli space of SQED and the XYZ model. It contains three branches. They are the Higgs branch, the Coulomb branch with $\sigma>0$ and the Coulomb branch $\sigma<0$ in terms of SQED. They are $X=Y=0$, $Z=X=0$ and $Y=Z=0$ in terms of the XYZ model. The example of the brane \ref{['example']} is indicated by the red region. It fills the Coulomb branch. It is the same as $Z=0$ region in the moduli space of the XYZ model.