An N=1 Superfield Action for M2 branes

TL;DR

This work addresses constructing a manifestly maximally supersymmetric 2+1D theory for M2 branes by formulating an ${\cal N}=1$ real superfield action that reproduces the BLG action for a totally antisymmetric 3-algebra with structure constants ${f^{abcd}}$. The main approach uses an octonionic self-dual tensor $C_{IJKL}$ to build a superpotential with ${\rm SO}(7)$ invariance, which enhances to ${\rm SO}(8)$ at the component level for a special choice of couplings, yielding the BL action. A key contribution is showing a super-Higgs mechanism, via a scalar vev, that reduces the M2 theory to the maximally supersymmetric YM theory in 2+1 dimensions for ${SU(2)}$ on D2 branes, thereby connecting M2 and D2 sectors within the same framework. The results suggest a fundamental role for octonions in AdS$_4$/CFT$_3$ and provide a practical ${\cal N}=1$ superspace tool for future explorations of the holographic correspondence.

Abstract

We present an octonionic N=1 superfield action that reproduces in components the action of Bagger and Lambert for M2 branes. By giving an expectation value to one of the scalars we obtain the maximally supersymmetric superfield action for D2 branes.