M2-branes on Orbifolds of the Cone over Q^{1,1,1}

TL;DR

The paper constructs and tests AdS_4/CFT_3 duals for M2-branes at cones over Sasaki–Einstein spaces Q^{1,1,1}, its Z_k orbifolds, and related geometries. It develops N=2 Chern–Simons quiver theories with levels (k,k,-k,-k) whose moduli spaces reproduce the Calabi–Yau cones and demonstrates that orbifolding by Z_k yields AdS_4 × Q^{1,1,1}/Z_k with matching chiral spectra to KK harmonics, aided by a sextic superpotential that constrains operators. The work extends to orbifolds leading to Q^{2,2,2} and M^{3,2}, providing moduli-space and chiral-operator tests in multiple phases and showing consistency with Seiberg-like dualities and toric geometry. It also analyzes geometric resolutions of the cones and their dual gauge theories, connecting Higgsing in the field theory to geometric blow-ups of the Calabi–Yau cones, thereby reinforcing the proposed dualities and offering a framework for building and testing similar AdS_4/CFT_3 pairs.

Abstract

We study the N=2 supersymmetric Chern-Simons quiver gauge theory recently introduced in arXiv:0809.3237 to describe M2-branes on a cone over the well-known Sasaki-Einstein manifold Q^{1,1,1}. For Chern-Simons levels (k,k,-k,-k) we argue that this theory is dual to AdS_4 x Q^{1,1,1}/Z_k. We derive the Z_k orbifold action and show that it preserves geometrical symmetry U(1)_R x SU(2) x U(1), in agreement with the symmetry of the gauge theory. We analyze the simplest gauge invariant chiral operators, and show that they match Kaluza-Klein harmonics on AdS_4 x Q^{1,1,1}/Z_k. This provides a test of the gauge theory, and in particular of its sextic superpotential which plays an important role in restricting the spectrum of chiral operators. We proceed to study other quiver gauge theories corresponding to more complicated orbifolds of Q^{1,1,1}. In particular, we propose two U(N)^4 Chern-Simons gauge theories whose quiver diagrams are the same as in the 4d theories describing D3-branes on a complex cone over F_0, a Z_2 orbifold of the conifold (in 4d the two quivers are related by the Seiberg duality). The manifest symmetry of these gauge theories is U(1)_R x SU(2) x SU(2). We argue that these gauge theories at levels (k,k,-k,-k) are dual to AdS_4 x Q^{2,2,2}/Z_k. We exhibit calculations of the moduli space and of the chiral operator spectrum which provide support for this conjecture. We also briefly discuss a similar correspondence for AdS_4 x M^{3,2}/Z_k. Finally, we discuss resolutions of the cones and their dual gauge theories.

Figures (9)

• Figure 1: (a) toric diagram and (b) proposed quiver diagram for ${\cal C}(Q^{1,1,1})$.
• Figure 2: Quiver diagram for a $\mathbb{Z}_2$ orbifold projection of the $Q^{1,1,1}$ quiver.
• Figure 3: Toric diagram for ${\cal C}(Q^{2,2,2}$).
• Figure 4: Quiver diagrams for M2-branes over ${\cal C}(Q^{2,2,2})$. The quivers are the same as the two Seiberg dual phases for D3-branes over ${\cal C}(F_0)$.
• Figure 5: Quiver diagram for M2-branes over $\mathcal{C}(M^{3,2})$.