M5-brane in three-form flux and multiple M2-branes

TL;DR

This work derives a six-dimensional M5-brane world-volume theory from the Bagger-Lambert-Gustavsson framework by promoting fields to depend on a three-dimensional internal space ${\cal N}$ endowed with a Nambu-Poisson bracket. The resulting gauge symmetry is the volume-preserving diffeomorphisms of ${\cal N}$, and the theory naturally contains a self-dual tensor and a tensor multiplet consistent with ${\cal N}=(2,0)$ supersymmetry; a Seiberg-Witten map relates it to the trivial-background theory. A key result is the double dimensional reduction to a noncommutative $U(1)$ gauge theory on a D4-brane in a background $B$-field, with explicit identifications between BLG/M5 parameters and M-theory/IIA data, including the large $C$-field regime where the M5 description is most reliable. The paper also interprets the M5-brane theory as a dynamical theory of the NP structure and discusses implications for the global topology of the internal space, quantization conditions, and extensions to long or multiple M5-branes, as well as connections to vortex-string dynamics on the M5 world-volume.

Abstract

We investigate the Bagger-Lambert-Gustavsson model associated with the Nambu-Poisson algebra as a theory describing a single M5-brane. We argue that the model is a gauge theory associated with the volume-preserving diffeomorphism in the three-dimenisonal internal space. We derive gauge transformations, actions, supersymmetry transformations, and equations of motions in terms of six-dimensional fields. The equations of motions are written in gauge-covariant form, and the equations for tensor fields have manifest self-dual structure. We demonstrate that the double dimensional reduction of the model reproduces the non-commutative U(1) gauge theory on a D4-brane with a small non-commutativity parameter. We establish relations between parameters in the BLG model and those in M-theory. This shows that the model describes an M5-brane in a large C-field background.