On Large N Solution of ABJM Theory

TL;DR

This paper analyzes the large $N$ limit of the BPS Wilson loop in ABJM theory via Kapustin et al.'s localization, deriving saddle-point equations that reproduce the perturbative expansion of $W(\lambda)$ and revealing a consistent large-$\lambda$ growth $W(\lambda)\sim e^{c\sqrt{\lambda}}$ in line with AdS$_4$/CFT$_3$ expectations. A recursive algorithm for perturbative computations is developed, and simplified models (coth-, cosech-models) are studied to illuminate the scaling behavior and density profiles that govern the Wilson loop at strong coupling. An ABJM-specific scaling ansatz with $\phi_i=\tfrac{1}{\pi}[\Lambda\,x_i+i f(x_i)]$ yields $\Lambda(\lambda)\sim\sqrt{\lambda}$ and a density $\rho(x)$ that leads to the exponential growth, providing evidence for a dual string worldsheet in AdS$_4\times\mathbb{CP}^3$. The work thus connects exact localization-based results to the holographic picture and points to directions for exact finite-$\lambda$ solutions and extensions to other Chern-Simons-matter theories.

Abstract

We investigate the large N limit of the expectation value W(λ) of a BPS Wilson loop in ABJM theory, using an integral expression of the partition function obtained recently by Kapustin et.al. Certain saddle-point equations provide the correct perturbative expansion of W(λ). The large λbehavior of W(λ) is also obtained from the saddle-point equations. The result is compatible with AdS/CFT correspondence.

Figures (4)

• Figure 1: Solution of the saddle-point equations for $k=100$.
• Figure 2: Solution of the saddle-point equations for $k=1$.
• Figure 3: Plot of $f(x)$ with $\Lambda=50,100$. $f(x)$ approaches to $x$ as $\Lambda$ becomes large.
• Figure 4: Plot of $g(x)$ with $\Lambda=50,100$. $g(x)$ approaches $x$ point-wise.