Webs of Walls

TL;DR

The paper develops a comprehensive framework for webs of domain walls as 1/4 BPS states in $d=4$, ${ m \\cal N}=2$ $U(N_{ m C})$ gauge theories with $N_{ m F}$ fundamentals. Using Bogomol'nyi techniques, it derives the 1/4 BPS equations and a master equation for the gauge-invariant data, showing the full moduli space is the complex Grassmann manifold $G_{N_{ m F},N_{ m C}}$ and that genuine 1/4 BPS webs arise after excluding 1/2 BPS and vacuum sectors; in the strong coupling limit the master equation becomes algebraic, yielding explicit solutions. In the Abelian case, wall webs are mapped to grid diagrams dual to the toric diagram ${\bf C}P^{N_{ m F}-1}$, with the wall and junction charges encoded geometrically and with multi-junctions and loops visible for larger $N_{ m F}$. The work also develops a slicing approach to generate 1/4 BPS junctions from 1/2 BPS walls in more general theories and outlines extensions to non-Abelian junctions, effective world-volume theories, and connections to string-theoretic web constructions and the topological vertex.

Abstract

Webs of domain walls are constructed as 1/4 BPS states in d=4, N=2 supersymmetric U(Nc) gauge theories with Nf hypermultiplets in the fundamental representation. Web of walls can contain any numbers of external legs and loops like (p,q) string/5-brane webs. We find the moduli space M of a 1/4 BPS equation for wall webs to be the complex Grassmann manifold. When moduli spaces of 1/2 BPS states (parallel walls) and the vacua are removed from M, the non-compact moduli space of genuine 1/4 BPS wall webs is obtained. All the solutions are obtained explicitly and exactly in the strong gauge coupling limit. In the case of Abelian gauge theory, we work out the correspondence between configurations of wall web and the moduli space CP^{Nf-1}.

Figures (19)

• Figure 1: Honeycomb web of domain walls. This web divides 37 vacua and has 18 external legs and 19 internal faces. The moduli space corresponds to ${\bf C}P^{36}$ whose dimension is 72.
• Figure 2: 1/2 BPS domain wall with a complex masses. The left one shows the configuration in the complex $\Sigma$ plane and the right one shows the configuration in the real space.
• Figure 3: The minimal model for 3-pronged wall junction. The left one is the grid diagram in the complex $\Sigma$ plane and the right one is the web diagram in the configuration space.
• Figure 4: The toric diagram for ${\bf C}P^2$ which corresponds to the total moduli space of the single 3-pronged junction. Points in the face correspond to the 1/4 BPS junctions, points in the edges to the 1/2 BPS single walls and vertices to the vacua.
• Figure 5: Three limits I,II and III of the 1/2 BPS 3-pronged junction. Each limit leads to the different 1/2 BPS walls from the 1/4 BPS junction.