M2 to D2

TL;DR

The paper addresses how to relate the non-Abelian M2-brane theory, formulated as a 2+1D 3-algebra field theory with a Chern-Simons gauge sector, to the better-understood D2-brane Yang-Mills theory. The authors show that giving a scalar VEV in the 3-algebra triggers a generalized Higgs mechanism that promotes the topological CS gauge field to a dynamical Yang-Mills field, yielding maximally supersymmetric SU(2) (and SU($N$) in general) YM on D2-branes, plus a decoupled Abelian sector. This result is demonstrated explicitly for the ${\cal A}_4$ 3-algebra and extended to general 3-algebras by analyzing the fundamental identity and the induced Lie-algebra substructure, offering a constructive path to SU($N$) YM from 3-algebras. The work strengthens the connection between 3-algebra formulations of M2-branes and conventional YM theories, supports a compactification/M-theory interpretation, and raises questions about classifying allowable 3-algebras and the role of FI constraints in higher-dimensional generalizations.

Abstract

We examine the recently proposed "3-algebra" field theory for multiple M2-branes and show that when a scalar field valued in the 3-algebra develops a vacuum expectation value, the resulting Higgs mechanism has the novel effect of promoting topological (Chern-Simons) to dynamical (Yang-Mills) gauge fields. This leads to a precise derivation of the maximally supersymmetric Yang-Mills theory on multiple D2-branes and thereby provides a relationship between 3-algebras and Yang-Mills theories. We discuss the physical interpretation of this result.