Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS4/CFT3

TL;DR

We address the problem of constructing 3d $\mathcal{N}=2$ CS quivers dual to M-theory on CY$_4$ cones $C(Y^{p,q}(\mathbb{CP}^2))$ with torsion flux. Our approach KK-reduces to Type IIA, interprets wrapped D6-branes on exceptional CP$^2$ as fractional branes, and uses exact monopole-operator charges to fix the quantum chiral ring and the moduli space, which contains the CY$_4$ cone and its crepant resolutions. We derive the full family of CS quivers dual to $Y^{p,q}$, including parity-related counterparts and the effects of Freed-Witten anomalies, and show the moduli space matches the geometric CY$_4$ structure via both monopole data and a semiclassical analysis of the FI terms and one-loop corrected D-terms. The results illuminate how torsion $G_4$ flux and D6-branes shape 3d duals, resolve partial resolutions, and provide a concrete bridge between M-theory backgrounds and 3d SCFT data with toric geometry. Overall, this advances explicit AdS$_4$/CFT$_3$ realizations with intricate flux sectors and clarifies the interplay between monopoles, brane charges, and CY$_4$ moduli.

Abstract

We study the quantum moduli space of N=2 Chern-Simons quivers with generic ranks and CS levels, proving along the way exact formulas for the charges of bare monopole operators. We then derive N=2 Chern-Simons quiver theories dual to AdS_4 x Y^{p,q}(CP2) M-theory backgrounds, for the whole family of Sasaki-Einstein seven-manifolds and for any value of the torsion G_4 flux. The derivation of the gauge theories relies on the reduction to type IIA string theory, in which M2-branes become D2-branes while the conical geometry maps to RR flux and D6-branes wrapped on compact four-cycles. M5-branes on torsion cycles map to flux and wrapped D4-branes. The moduli space of the quiver is shown to contain the corresponding CY_4 cone and all its crepant resolutions.

Figures (5)

• Figure 1: Toric diagram of $C(Y^{p,q})$ (drawn for $Y^{3,2}$).
• Figure 2: The $(n_0, n_1)$ plane. The torsion group $\Gamma$ is obtained by quotienting this $\mathbb{Z}^2$ lattice by the sublattice generated by the periodicity vectors $(q, p)$ and $(3q, q)$ shown in red. The shaded area is a choice of fundamental domain, where the opposite sides are identified according to the red vectors (for instance the lower boundary of the area $[0,0]$ is identified with the upper boundary of the area $[1,0]$). The three parallelograms denoted $[-1, 0]$, $[0,0]$ and $[1,0]$ correspond to the three windows needed to cover $\Gamma$ once, as will be explained in Section 5. The dashed blue lines correspond to loci where either $b^-$ or $b^+$ is half integer, as indicated.
• Figure 3: Quiver diagram of the theory for M2-branes on the real cone over $Y^{p,q}({\mathbb{C}\mathbb{P}^2})$, the same quiver for D3-branes at $\mathbb{C}^3/\mathbb{Z}_3$.
• Figure 4: Toric diagram of a toric CY$_4$ singularity leading to ABJM with $p$ flavors, for $p=3$ and $h=1$, and quiver diagram of the flavored ABJM theory.
• Figure 5: Topology of the complex dimension parametrized by bare monopole operators in flavored ABJM for $p=3$ flavors with non-vanishing real masses.