Superconformal field theories from crystal lattices

Abstract

We propose a brane configuration for the (2+1)d, $\mathcal{N}=2$ superconformal theories (CFT$_3$) arising from M2-branes probing toric Calabi-Yau 4-fold cones, using a T-duality transformation of M-theory. We obtain intersections of M5-branes on a three-torus which form a 3d bipartite crystal lattice in a way similar to the 2d dimer models for CFT$_4$. The fundamental fields of the CFT$_3$ are M2-brane discs localized around the intersections, and the super-potential terms are identified with the atoms of the crystal. The model correctly reproduces the complete BPS spectrum of mesons and baryons.

Figures (4)

• Figure 1: A toric diagram (solid line) is a convex polyhedron with integer-valued vertices in $\mathbb{R}^3$. Its graph dual (dashed line) gives the fan diagram.
• Figure 2: A partition of the toric diagram.
• Figure 3: The crystal for $\mathbb{C}^4$ has the zincblende structure.
• Figure 4: The crystal for $C(Q(1,1,1))$ has the NaCl structure.