Vortex Loop Operators, M2-branes and Holography

TL;DR

The paper defines and analyzes vortex loop operators $V_C$ in the ${\cal N}=6$ Chern-Simons theory (ABJM-like) as disorder loop observables with vortex-like singularities along a curve, and provides a detailed holographic map to M-theory on AdS$_4\times S^7/\mathbb{Z}_k$ where these loops correspond to M2-branes ending on the boundary. It classifies 1/2, 1/3, and 1/6 BPS loops in both Abelian and non-Abelian theories, computes their semiclassical vacuum expectation values and correlators with chiral primary operators and the stress tensor, and matches these to bulk calculations using probe M2-branes and their backreacted “bubbling” geometries. The strong-coupling results yield nontrivial exponential behavior for circular loops, with detailed dependence on the loop data and ’t Hooft coupling, illustrating rich structure and subtleties in AdS$_4$/CFT$_3$ compared to higher-dimensional counterparts. The work lays a foundation for further precision tests of holography in three dimensions through nonlocal operators and bubbling geometries, and suggests avenues for localization-based checks and phase-structure studies in Chern-Simons-matter theories.

Abstract

We construct vortex loop operators in the three-dimensional N = 6 supersymmetric Chern-Simons theory recently constructed by Aharony, Bergman, Jafferis and Maldacena. These disorder loop operators are specified by a vortex-like singularity for the scalar and gauge fields along a one dimensional curve in spacetime. We identify the 1/2, 1/3 and 1/6 BPS loop operators in the Chern-Simons theory with excitations of M-theory corresponding to M2-branes ending along a curve on the boundary of AdS_4 x S^7/Z_k. The vortex loop operators can also be given a purely geometric description in terms of regular "bubbling" solutions of eleven dimensional supergravity which are asymptotically AdS_4 x S^7/Z_k.