Exact results on ABJ theory and the refined topological string

TL;DR

The paper addresses non-perturbative information in ABJ theory by exploiting localization to a matrix model and recasting it as a Fermi gas, enabling exact partition function computations for several $(k,M,N)$. The central finding is that the ABJ grand potential is completely governed by the refined topological string on local $P^1\times P^1$, with both worldsheet and membrane instantons encoded via the Nekrasov–Shatashvili limit and its NS free energy. This framework reproduces the gravity-side expectations (classical SUGRA and one-loop corrections) in the large-$N$ limit and naturally incorporates non-perturbative membrane instantons consistent with AdS/CFT. The results yield a finite, duality-covariant non-perturbative structure, with precise cancellations of divergences between instanton sectors and a clean Airy-function representation for the perturbative part. Overall, the work provides a detailed, topologically enriched picture of non-perturbative ABJ physics and showcases refined topological string theory as a powerful organizing principle for M-theory duals of M2-brane systems.

Abstract

We study the partition function of the ABJ theory, which is the N=6 superconformal Chern-Simons matter theory with gauge group U(N)xU(N+M) and Chern-Simons levels (k,-k). We exactly compute the ABJ partition function on a three sphere for various k, M and N via the Fermi gas approach. By using these exact data, we show that the ABJ partition function is completely determined by the refined topological string on local P^1 x P^1, including membrane instanton effects in the M-theory dual.

Figures (4)

• Figure 1: (Left) Explanation of the integral contours $C_1 ,C_2 ,C_3$ and $C_4$. The real positive parameter $\Lambda$ is taken to infinity. (Right) Explanation of the integral contours $C_5 ,C_6 ,C_7$ and $C_8$.
• Figure 2: (Left) The free energy $\log{\hat{Z}}^{(N,N+M)}(k)$ is plotted to $N^{3/2}$. (Right) The ratio between the exact free energy $\log{\hat{Z}}^{(N,N+M)}(k)$ and the classical SUGRA free energy $F_{\rm SUGRA}$ is plotted against $1/N$. The symbols and dashed lines denote the exact data and fitting functions in the large $N$ region, respectively.
• Figure 3: $\hat{Z}_{\rm np}^{(N,N+M)} (k)$ is plotted to $2\pi \sqrt{2N/k}$ in semi-log scale. (Left) A plot for $(k,M)=(2,1)$. (Right) Plots for $(k,M)=(6,1),(6,2)$ and $(6,3)$. The blue circle, purple square and yellow diamond symbols show the cases for $M=1$, $M=2$ and $M=3$, respectively.
• Figure 4: The quantity $e^{14\pi \sqrt{2N}{k}} |\hat{Z}_k^{(M)} -\hat{Z}_{\rm 7-inst}|/ \hat{Z}_{\rm pert}$ is plotted to $2\pi\sqrt{2N/k}$ both in semi-log scale (Left) and log-log scale (Right) for $(k,M)=(2,1),(3,1),(4,1),(4,2),(6,1),(6,2)$ and $(6,3)$. The blue circle, purple square and yellow diamond symbols show the cases for $M=1$, $M=2$ and $M=3$, respectively.