Toric AdS4/CFT3 duals and M-theory Crystals

TL;DR

This work extends the brane-tiling intuition to three-dimensional CFTs arising from M2-branes at toric CY$_4$ by developing a crystal model and establishing forward/inverse algorithms that map toric data to a crystal lattice and back. The authors leverage amoeba/alga projections, untwisting, and perfect matchings to connect toric vertices, charges, and BPS meson spectra, and they show that mesons correspond to closed M2-membranes with charges determined by the toric Reeb data. Key contributions include explicit treatments of the inverse/forward maps, the role of perfect matchings in encoding abelian F-terms, and sample computations of meson spectra via a character function $Z(q;X)$ for several CY$_4$ examples. The results provide a concrete combinatorial framework for analyzing AdS$_4$/CFT$_3$ duals in toric backgrounds, enabling spectrum calculations and offering insight into dualities and deformations within the crystal picture.

Abstract

We study the recently proposed crystal model for three dimensional superconformal field theories arising from M2-branes probing toric Calabi-Yau four-fold singularities. We explain the algorithms mapping a toric Calabi-Yau to a crystal and vice versa, and show how the spectrum of BPS meson states fits into the crystal model.

Figures (24)

• Figure 1: The forward/inverse algorithms.
• Figure 2: A toric diagram (solid line) is a convex polyhedron with integer-valued vertices in $\mathbb{R}^3$. Its graph dual (dotted line) gives the fan diagram.
• Figure 3: Toric diagrams, Reeb vectors and R-charges of some simple examples of CY$_4$.
• Figure 4: Amoeba for $\mathbb{C}^3$.
• Figure 5: Alga and the inverse algorithm for $\mathbb{C}^3$.