Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories

TL;DR

This work addresses constructing gravity duals for 3d Chern-Simons–matter theories by identifying the Calabi–Yau 4-fold cone $C(Y_7)$ with a branch of the moduli space, realized through toric, torus-based dimer models. It introduces a concrete forward algorithm that extracts the toric data of $C(Y_7)$ from the quiver, superpotential, and CS levels by analyzing perfect matchings on brane tilings, yielding the convex hull of vectors $v^{\alpha}$ as the CY$_4$ toric diagram. The authors validate the method with examples, reproducing known toric data for cases like $Y^{p,k}(\mathbb{CP}^2)$ and identifying limitations for $Y^{p,k}(\mathbb{CP}^1\times\mathbb{CP}^1)$, where the conjectured toric data fails to match and Seiberg duality does not commute with the construction. Overall, the paper provides a practical, systematic bridge between 3d CS–matter theories and M-theory backgrounds, enabling the generation of gravity duals and guiding future investigations into dualities and extensions such as orientifolds and marginal deformations.

Abstract

We propose a new method to find gravity duals to a large class of three-dimensional Chern-Simons-matter theories, using techniques from dimer models. The gravity dual is given by M-theory on AdS_4\times Y_7, where Y_7 is an arbitrary seven-dimensional toric Sasaki-Einstein manifold. The cone of Y_7 is a toric Calabi-Yau 4-fold, which coincides with a branch of the vacuum moduli space of Chern-Simons-matter theories.

Figures (9)

• Figure 1: The bipartite graph (a), quiver diagram (b,d) and toric diagram (c) corresponding to the Suspended Pinched Point.
• Figure 2: The perfect matchings of bipartite graphs and their height functions. The height functions gives the coordinates of the toric diagram.
• Figure 3: The choices of paths $p_1,\dots, p_4$ (left) and Chern-Simons levels for SPP (right). On the right figure the flow of the Chern-Simons charge is represented as a green arrow.
• Figure 4: The bipartite graph (a), thee toric diagram (c) and the quiver diagram (b,d) for $C(Y_5)=K_{\mathbb{CP}^2}$.
• Figure 5: The choice of paths $p_1,\dots, p_4$, flow of CS cherge (above) and perfect matchings with corresponding coordinate of the lattice point of convex polytope (below) in the case of $C(Y_5)=K_{\mathbb{CP}^2}$.