Phases of M2-brane Theories

TL;DR

This work systematically analyzes 2+1 dimensional supersymmetric Chern–Simons quiver theories arising from M2-branes probing toric Calabi–Yau 4-folds. By employing brane tilings and the forward algorithm, the authors extract toric data, Hilbert series, and generator structures for multiple toric phases of each singularity, establishing toric dualities where different phases share identical mesonic moduli spaces. The study carefully distinguishes Master spaces and the mesonic spaces, detailing baryonic charges, global symmetries, and R-charge assignments across phases, and demonstrates how perfect matchings encode the operator content consistently. The results reinforce the role of toric geometry in organizing 2+1D CS theories and provide explicit checks via Hilbert series and generators, highlighting phase-for-phase equivalences and the origin of baryonic symmetries. Overall, the paper advances understanding of toric duality, phase structure, and moduli spaces for M2-brane theories, with implications for AdS4/CFT3 holography and the combinatorial geometry of toric CY4-folds.

Abstract

We investigate different toric phases of 2+1 dimensional quiver gauge theories arising from M2-branes probing toric Calabi-Yau 4 folds. A brane tiling for each toric phase is presented. We apply the 'forward algorithm' to obtain the toric data of the mesonic moduli space of vacua and exhibit the equivalence between the vacua of different toric phases of a given singularity. The structures of the Master space, the mesonic moduli space, and the baryonic moduli space are examined in detail. We compute the Hilbert series and use them to verify the toric dualities between different phases. The Hilbert series, R-charges, and generators of the mesonic moduli space are matched between toric phases.

Figures (24)

• Figure 1: [Phase I of $\mathbb{C}^4$] (i) Quiver diagram for the $\mathscr{C}$ model. (ii) Tiling for the $\mathscr{C}$ model.
• Figure 2: [Phase I of $\mathbb{C}^4$] The fundamental domain of the tiling for the $\mathscr{C}$ model: Assignments of the integers $n_i$ to the edges are shown in blue and the weights for these edges are shown in green.
• Figure 4: The lattice of generators of the $\mathbb{C}^4$ theory.
• Figure 5: [Phase II of $\mathbb{C}^4$] (i) Quiver diagram for the $\mathscr{D}_1\mathscr{H}_1$ model. (ii) Tiling for the $\mathscr{D}_1\mathscr{H}_1$ model.
• Figure 6: [Phase II of $\mathbb{C}^4$] The fundamental domain of tiling for the $\mathscr{D}_1\mathscr{H}_1$ model : Assignments of the integers $n_i$ to the edges are shown in blue and the weights for these edges are shown in green.