Towards M2-brane Theories for Generic Toric Singularities

TL;DR

This work constructs several $(2+1)d$ ${\cal N}=2$ Chern-Simons theories whose moduli spaces are non-compact toric CY$_4$'s not derivable from any $(3+1)d$ CFT, providing explicit examples such as the cone over $Q^{111}$. Guided by crystal models, the authors verify matter content, superpotentials, and RG flows, including a Klebanov-Witten–type flow connecting $C(dP_3)\times \mathbb{C}$ to $C(Q^{111})$ and additional pairs related by similar flows. They also explore partial resolution in CS theories, showing how FI terms and Higgsing affect CS levels and moduli spaces, and they map a network of inter-model connections, including orbifolds like $\mathbb{C}^3/(\mathbb{Z}_N\times \mathbb{Z}_N)\times \mathbb{C}$. Overall, the results corroborate crystal-model predictions and advance a program to classify and connect pure $(2+1)d$ theories with toric CY$_4$ moduli spaces, moving toward a general construction akin to dimer models in higher dimensions.

Abstract

We construct several examples of (2+1) dimensional N=2 supersymmetric Chern-Simons theories, whose moduli space is given by non-compact toric Calabi-Yau four-folds, which are not derivable from any (3+1) dimensional CFT. One such example is the gauge theory associated with the cone over Q^{111}. For several examples, we explicitly confirm the matter content, superpotential interactions and RG flows suggested by crystal models. Our results provide additional support to the idea that crystal models are relevant for describing the structure of these CFTs.

Figures (16)

• Figure 1: Crystal of $\mathbb{C}^4$ adopted from Lee:2007kv.
• Figure 2: 3d toric diagrams and their projection to the 2d toric diagrams of the parents. (a) is the well known ABJM theory. It projects down to the conifold, from which it borrows the quiver and superpotential. How the 2d toric diagram is 'inflated' into a 3d one depends on the choice of CS levels. The number appearing in the toric diagram denotes the multiplicity of the particular node. The multiplicity is one unless otherwise stated. (b) admits two projections (indicated in green and red) to toric diagrams coming from (3+1)d theories.
• Figure 3: Quiver diagram for $Q^{111}$.
• Figure 4: Toric diagram for the $k=(1,1,-1,-1)$ theory.
• Figure 5: Quiver diagram for $C(dP_3)\times \mathbb{C}$.