The fuzzy S^2 structure of M2-M5 systems in ABJM membrane theories

TL;DR

This work shows that fluctuations around ABJM M2–M5 ground states organize as functions on S^2 rather than S^3 in the large-N, large-k limit, revealing a novel fuzzy S^2 realized with bifundamental degrees of freedom. Starting from the M2-brane action, the authors derive a U(1) Yang–Mills theory on R^{2,1}×S^2, consistent with a D4-brane interpretation in Type IIA and a Hopf-fibration picture that discretely realises the S^2 base of S^3. They analyze the SO(4)-covariant fuzzy S^3 construction and show it is not a solution for ABJM with N>2, while detailing the SU(2) harmonic decomposition of bifundamental fluctuations and the resulting S^2 spectrum. The results connect M2–M5 funnel dynamics to a D4-brane worldvolume through Hopf reduction and clarify how ABJM encodes the Hopf-fibration structure, leaving open how to recover the full classical S^3 geometry from a perturbative membrane theory at large N. Overall, the paper provides a consistent ABJM-based route to M2–M5 physics via fuzzy S^2 and D4 reduction, and outlines future directions toward a fuller M5 description or finite-k corrections.

Abstract

We analyse the fluctuations of the ground-state/funnel solutions proposed to describe M2-M5 systems in the level-k mass-deformed/pure Chern-Simons-matter ABJM theory of multiple membranes. We show that in the large N limit the fluctuations approach the space of functions on the 2-sphere rather than the naively expected 3-sphere. This is a novel realisation of the fuzzy 2-sphere in the context of Matrix Theories, which uses bifundamental instead of adjoint scalars. Starting from the multiple M2-brane action, a U(1) Yang-Mills theory on R^{2,1} x S^2 is recovered at large N, which is consistent with a single D4-brane interpretation in Type IIA string theory. This is as expected at large k, where the semiclassical analysis is valid. Several aspects of the fluctuation analysis, the ground-state/funnel solutions and the mass-deformed/pure ABJM equations can be understood in terms of a discrete noncommutative realisation of the Hopf fibration. We discuss the implications for the possibility of finding an M2-brane worldvolume derivation of the classical S^3 geometry of the M2-M5 system. Using a rewriting of the equations of the SO(4)-covariant fuzzy 3-sphere construction, we also directly compare this fuzzy 3-sphere against the ABJM ground-state/funnel solutions and show them to be different.