Global Structure of Moduli Space for BPS Walls

TL;DR

The paper addresses how BPS domain walls in Higgs branches of eight-supercharge theories organize into a global moduli space, revealing that the dimension can exceed naive index counts near special Lagrangian submanifolds $M$ and that the full moduli space is a union of multiple $T^*M_a$ glued along intersections. Using an explicit Abelian, hypertoric GLSM construction, the authors show the Higgs branch contains several compact special Lagrangian submanifolds whose union forms the total wall moduli space, leading to rich wall dynamics such as repulsion/attraction and transmutation when walls cross. They analyze the BPS flow equations, discuss transversality violations, and illustrate how walls can carry cotangent-direction moduli, depending on the base geometry (e.g., $F_n$ vs. $CP^2$). The results illuminate nonperturbative wall dynamics on worldvolumes, highlight geometric mechanisms behind moduli-space structure, and suggest pathways to study quantum corrections and D-brane realizations in related setups.

Abstract

We study the global structure of the moduli space of BPS walls in the Higgs branch of supersymmetric theories with eight supercharges. We examine the structure in the neighborhood of a special Lagrangian submanifold M, and find that the dimension of the moduli space can be larger than that naively suggested by the index theorem, contrary to previous examples of BPS solitons. We investigate BPS wall solutions in an explicit example of M using Abelian gauge theory. Its Higgs branch turns out to contain several special Lagrangian submanifolds including M. We show that the total moduli space of BPS walls is the union of these submanifolds. We also find interesting dynamics between BPS walls as a byproduct of the analysis. Namely, mutual repulsion and attraction between BPS walls sometimes forbid a movement of a wall and lock it in a certain position; we also find that a pair of walls can transmute to another pair of walls with different tension after they pass through.

Figures (11)

• Figure 1: The definition of $\mathcal{H^\pm_A}$. The arrow indicates in which side $\mu_A$ is positive.
• Figure 2: An example with $N_F=4, N=2$.
• Figure 3: Left: Hyperplane arrangement for $T^*\mathbb{C}P^2$; Right: Flows in $T^*\mathbb{C}P^2$ with masses $m^A=(1,0,-1)$. Dashed (blue) lines are the level sets of the Morse function $m^A\mu_A$.
• Figure 4: Hyperplane arrangement for the hypertoric which contains $T^*F_n$.
• Figure 5: Numerically calculated structure of the flow in $n=1$ case. From left to the right: case I) $m^A=(0,0,1,-1)$ ; case II) $m^A=(1,0,0,-1)$ ; case III) $m^A=(-1,0,1,0)$. For all cases $c_I=(1,1)$. Dashed (blue) lines designate the contours of constant $m^A\mu_A$.