(0,4) brane box models

TL;DR

The paper constructs 2d $\mathcal{N}=(0,4)$ quiver gauge theories from D3-brane box configurations bounded by various 5-branes and shows these boxes are T-dual to D1-D5-D5' systems at orbifold singularities. It provides a detailed dictionary between brane box data, orbifold quivers, and D1-D5-D5'-KK$'$ systems, including explicit $E$- and $J$-term structures for the multiplets and their interactions. A central result is the necessity of tetravalent Fermi multiplets at NS-NS' junctions to cancel Abelian gauge anomalies, along with a brane-based construction of $2d$ boundary degrees of freedom that cancel boundary gauge anomalies in 3d $\mathcal{N}=4$ theories. The work also explores D1-branes on orbifolds, the inclusion of flavor branes, and dualities (mirrors) within the brane box framework, offering a robust geometric realization of 2d $(0,4)$ theories and their boundary couplings with potential applications in string theory holography and defect CFTs.

Abstract

Two-dimensional $\mathcal{N}=(0,4)$ supersymmetric quiver gauge theories are realized as D3-brane box configurations (two dimensional intervals) which are bounded by NS5-branes and intersect with D5-branes. The periodic brane configuration is mapped to D1-D5-D5$'$ brane system at orbifold singularity via T-duality. The matter content and interactions are encoded by the $\mathcal{N}=(0,4)$ quiver diagrams which are determined by the brane configurations. The Abelian gauge anomaly cancellation indicates the presence of Fermi multiplets at the NS-NS$'$ junction. We also discuss the brane construction of $\mathcal{N}=(0,4)$ supersymmetric boundary conditions in 3d $\mathcal{N}=4$ gauge theories involving two-dimensional boundary degrees of freedom that cancel gauge anomaly.

Figures (23)

• Figure 1: Brane construction of 3d $\mathcal{N}=4$$U(N_{c})$ gauge theory with $N_{f}$ hypermultiplets and that of 3d $\mathcal{N}=4$$\prod_{i=1}^{n} U(N_{i})$ linear quiver gauge theory with bi-fundamental hypermultiplets. The numbers with red color indicate the linking numbers.
• Figure 2: $\mathcal{N}=(0,2)$ quiver for basic building block of $\mathcal{N}=(0,4)$ quiver gauge theory. A circular node represents the $\mathcal{N}=(0,2)$ gauge multiplet. A pair of blue arrows of $\mathcal{N}=(0,2)$ chiral multiplets $(R,L)$ form $\mathcal{N}=(0,4)$ hypermultiplet. A pair of green arrows of the chiral multiplets $(U,D)$ form $\mathcal{N}=(0,4)$ twisted hypermultiplet. Red diagonal lines $\Delta$ and $\nabla$ are the $\mathcal{N}=(0,2)$ Fermi multiplets. A red loop of adjoint $\mathcal{N}=(0,2)$ Fermi multiplet $\Lambda$ is involved in $\mathcal{N}=(0,4)$ vector multiplets.
• Figure 3: (i) Blue line of $\mathcal{N}=(0,4)$ hyper and green line of $\mathcal{N}=(0,4)$ twisted hypermultiplets. (ii) The corresponding pairs of $\mathcal{N}=(0,2)$ chiral multiplets $(R,L)$ and $(U,D)$.
• Figure 4: (i) A V-shaped configuration of Fermi multiplet in the $\mathcal{N}=(0,4)$ quiver diagram. (ii) The corresponding pair of $\mathcal{N}=(0,2)$ Fermi multiplets $(\Delta, \nabla)$ in the $\mathcal{N}=(0,2)$ quiver diagram.
• Figure 5: Triangular loops of the $E$- and $J$-terms in quiver diagram. The positive and negative signs in the triangles represent the clockwise and anti-clockwise orientations which determine the contributions to the $E$- and $J$-terms.