M2-branes and AdS/CFT

TL;DR

The notes analyze the worldvolume theory of coincident M2-branes by developing and validating the ABJM framework: a $U(N)\times U(N)$ Chern-Simons-matter theory with levels $(k,-k)$ that describes $N$ M2-branes at $\mathbb{R}^8/\mathbb{Z}_k$ and is dual to M-theory on $AdS_4\times S^7/\mathbb{Z}_k$. The ABJM construction exhibits ${\cal N}=6$ supersymmetry (enhanced to ${\cal N}=8$ for $k=1,2$ via monopole operators) and captures the correct moduli space $\mathbb{C}^4/\mathbb{Z}_k$, parity, and operator spectrum necessary to match the dual gravity KK states. The gravitational description reveals the $N^{3/2}$ scaling of degrees of freedom in the M2-brane theory and clarifies the role of the orbifold in the dual geometry. Together, these results establish ABJM as a robust holographic description of coincident M2-branes and illuminate the essential function of monopole operators in the symmetry enhancement and spectrum matching.

Abstract

These notes provide a brief introduction to the ABJM theory, the level k U(N) x U(N) superconformal Chern-Simons matter theory which has been conjectured to describe N coincident M2-branes. We discuss its dual formulation in terms of M-theory on AdS_4 x S^7/Z_k and review some of the evidence in favor of the conjecture. We end with a brief discussion of the important role played by the monopole operators.

Figures (2)

• Figure 1: The quiver diagram for the conifold gauge theory.
• Figure 2: The quiver diagram for the $\mathcal{N}=3$ supersymmetric gauge theory which flows to ABJM theory in the IR.