The M2-M5 Brane System and a Generalized Nahm's Equation

TL;DR

The paper addresses describing N M2-branes ending on an M5-brane in M-theory by proposing a generalized Nahm-type equation with $SO(4)$ symmetry implemented via a fuzzy $S^3$ and $N\times N$ matrices. It constructs a funnel-like solution with $X^i(s)$ proportional to a fuzzy $S^3$ generator and a radial profile $f(s)=1/\sqrt{s}$, establishing a large-$N$ scaling where $\lambda^2 N$ is fixed to achieve overlap with the M5 description and matching the self-dual string on the M5-brane. The energy and action are analyzed, revealing a decomposition into membrane and five-brane contributions in the BPS limit, and a linearized fluctuation equation is derived to study stability and dynamics, consistent with membrane and self-dual string limits. The work provides a matrix-based framework for M2–M5 junctions, connects to known D1–D3 results, and highlights open questions about UV completions and the nature of degrees of freedom in multiple M2-branes.

Abstract

We propose an equation that describes M2-branes ending on M5-branes, and which generalizes the description of the D1-D3 system via Nahm's equation. The simplest solution to this equation constructs the transverse geometry in terms of a fuzzy three sphere. We show that the solution passes a number of consistency checks including a consistent reduction to the D1-D3 system, a calculation of the energy of the system, matching to the self-dual string solution in the M5-brane world volume, and a study of simple fluctuations about the ground state configuration. We write down certain terms in the effective action of multiple membranes, which includes a sextic scalar coupling.