Three-dimensional N=6 SCFT's and their membrane dynamics

TL;DR

This work develops a semiclassical, geometry‑driven analysis of the ${\cal N}=6$ ABJM theory, treating it as a membrane theory dual to M-theory on ${AdS}_4\times S^7/\mathbb{Z}_k$ and extracting a detailed map between field theory data and M-theory geometry. By dimensionally reducing on $S^2\times \mathbb{R}$ and focusing on spherically symmetric configurations, the authors derive the moduli space, chiral ring structure, and eigenvalue dynamics, showing how flux quantization and the Vandermonde measure yield emergent geometric spaces (notably a sphere with a Hopf fibration). At strong coupling they identify geometric objects dual to chiral-ring states (giant gravitons, D0/D4 configurations) and prove an all-orders compatible dispersion relation for giant magnons, $\Delta=\sqrt{(\ell+\tfrac{1}{2})^2+ h(\lambda) \sin^2(p/2)}$, with $h(\lambda)$ interpolating between weak and strong coupling. The results provide nonperturbative evidence for AdS$_4$/CFT$_3$ and illuminate the role of the M-theory circle in the emergent geometry, offering a framework potentially extensible to other 2+1 SCFTs and integrable spin chains.

Abstract

We analyze several aspects of the recent construction of three-dimensional conformal gauge theories by Aharony et al. in various regimes. We pay special attention to understanding how the M-theory geometry and interpretation can be extracted from the analysis of the field theory. We revisit the calculations of the moduli space of vacua and the complete characterization of chiral ring operators by analyzing the field theory compactified on a 2-sphere. We show that many of the states dual to these operators can be interpreted as D-brane states in the weak coupling limit. Also, various features of the dual AdS geometry can be obtained by performing a strong coupling expansion around moduli space configurations, even though one is not taking the planar expansion. In particular, we show that at strong coupling the corresponding weak coupling D-brane states of the chiral ring localize on particular submanifolds of the dual geometry that match the M-theory interpretation. We also study the massive spectrum of fields in the moduli space. We use this to investigate the dispersion relation of giant magnons from the field theory point of view. Our analysis predicts the exact functional form of the dispersion relation as a function of the world-sheet momentum, independently of integrability assumptions, but not the exact form with respect to the 't Hooft coupling. We also get the dispersion relation of bound states of giant magnons from first principles, providing evidence for the full integrability of the corresponding spin chain model at strong 't Hooft coupling.

Figures (5)

• Figure 1: A configuration of two parallel M2-branes. The transverse space to the membranes is a real cone. One membrane is fixed at the tip of the cone while the other is moved to a distance $R=\ell_P^{3/2}v$ away. The off-diagonal modes stretching between the two membranes are called membrane bits, and wrap a preferred circle that we interpret as the M-theory circle. The tension of these bits scales like $\ell_P^{-3}$, suggesting that they are indeed two-dimensional objects, rather than strings. Looking at the scalar potential of the ABJM theory, one can also show that this tension is proportional to $1/k$, indicating that the radius of the M-theory circle gets shrunk by the orbifold. In the string limit, in which this circle disappears, the membrane bits become string bits connecting two D2-branes.
• Figure 2: The membrane bit stretching between two M2-branes wraps completely the $S^1$ fiber over $\mathbb{C}P^3$, that we interpret as the M-theory circle. The orbifold procedure cuts this circle into $k$ segments and shrinks its size by a factor of $1/k$, thus changing the string coupling constant.
• Figure 3: Young tableaux describing the contraction of gauge theory indices for the giant gravitons in the ${\cal N}=6$ Chern-Simons theory. We consider for simplicity the case of maximal giants, i.e. operators in rank $N$ antisymmetric representation of the gauge group (in the figure $N=6$). Unlike ${\cal N}=4$ SYM, where one had just one upper and one lower index and one possible contraction thereof, we have now four indices and two possible contractions. The columns represent respectively $(A^a,\,\overline A_a,\, \overline B^{\dot a},\, B_{\dot a})$ and the boxes with letters inside are the marked boxes. We have then that (a) corresponds to the contraction mixing $A$ fields with $B$ fields while (b) correspond to the contraction of $A$ fields among themselves and $B$ fields among themselves.
• Figure 4: Intersecting D4-branes corresponding to a giant graviton. These intersect on a $\mathbb{C}P^1$ and wrap two different ${\mathbb C} P^2$ submanifold in ${\mathbb C} P^3$. There are four kind of strings stretching between them: the ${|aa \rangle}$ strings correspond to replacing a $A_1$ field with a composite word with the same index structure, and similarly the ${|bb \rangle}$ string corresponds to replacing one $B_1$. On the other hand, the ${|ab \rangle}$ and ${|ba \rangle}$ strings correspond to replacing both $A_1$ and $B_1$ fields at the same time.
• Figure 5: A plot of the function $\left| \Phi\left(\frac{2\pi q}{J}+\varphi-\varphi'\right)\right|^2$ appearing in the expression for the energy (\ref{['energy']}). For large $J$ (here we have set $J=8$), this tends to a delta function centered around $\varphi-\varphi'=2\pi q/J$, thus equating the relative phase between two eigenvalues on the sphere to the world-sheet momentum $p$ of the giant magnon.