Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant

TL;DR

This work develops a sign-problem-free Monte Carlo approach to the ABJM matrix model obtained from localization, enabling simulations at arbitrary $N$ and $k$. It confirms the M-theory scaling $F\sim -\frac{\pi}{3}\sqrt{2k}\,N^{3/2}$ and reveals smooth interpolation between planar and M-theory regimes, highlighting constant-map contributions that reconcile previously observed discrepancies. The authors further connect the constant-map corrections with the Fermi gas results, showing that $A(k)-\tfrac{1}{2}\log 2$ captures the all-genus constant-map effects, thereby linking distinct expansions through analytic continuation. Overall, the method provides nonperturbative insights into ABJM dynamics and paves the way for testing AdS$_4$/CFT$_3$ beyond the planar limit and into quantum string/M-theory regimes.

Abstract

We show that the ABJM theory, which is an N=6 superconformal U(N)*U(N) Chern-Simons gauge theory, can be studied for arbitrary N at arbitrary coupling constant by applying a simple Monte Carlo method to the matrix model that can be derived from the theory by using the localization technique. This opens up the possibility of probing the quantum aspects of M-theory and testing the AdS_4/CFT_3 duality at the quantum level. Here we calculate the free energy, and confirm the N^{3/2} scaling in the M-theory limit predicted from the gravity side. We also find that our results nicely interpolate the analytical formulae proposed previously in the M-theory and type IIA regimes. Furthermore, we show that some results obtained by the Fermi gas approach can be clearly understood from the constant map contribution obtained by the genus expansion. The method can be easily generalized to the calculations of BPS operators and to other theories that reduce to matrix models.

Figures (12)

• Figure 1: The free energy of the ABJM theory for $N=2$ is plotted against the Chern-Simons level $k$. The circles and triangles represent our Monte Carlo result and the exact result (\ref{['exactN2']}), respectively.
• Figure 2: The normalized free energy $F/N^2$ is plotted against $1/N^2$ for various values of $\lambda$ (Left). In the right panel, we zoom up the plot for $\lambda=1$. The data can be nicely fitted to $F(N,\lambda)/N^2=f_0(\lambda)+f_1(\lambda)/N^2 -\frac{1}{6}\log{N}$, which enables us to make a reliable extrapolation to the planar $N\rightarrow \infty$ limit.
• Figure 3: (Left) The free energy in the planar limit $f_0(\lambda)=\lim_{N\to\infty} F(N,\lambda)/N^2$ extracted from fig. \ref{['fig:extrapolation_planar']} is plotted against $1/\sqrt{\lambda}$. Our results seem to interpolate the DMP result at strong coupling and the perturbative result at weak coupling. (Right) The difference between our result and the DMP result, i.e., $\lim_{N\rightarrow\infty}(F-F_{\rm DMP})/N^2$, is plotted against $1/\lambda^2$. The data points can be fitted to a straight line, which implies (\ref{['d_planar']}) and (\ref{['d_planar2']}).
• Figure 4: (Left) The free energy is plotted against $N^{3/2}$ for $k=1,2,4,6,8$. The data points can be fitted to straight lines, which implies $F \sim N^{3/2}$ as $N$ increases. (Right) The normalized free energy $F/N^{3/2}$ is plotted against $1/N$. The data can be nicely fitted to straight lines, which enables us to make extrapolations to the M-theory limit reliably.
• Figure 5: The M-theory limit of the free energy $\lim_{N\rightarrow\infty}F/N^{3/2}$ extracted from fig. \ref{['fig:M_extrapolation']} (Right) is plotted against $\sqrt{k}$. Our data are in good agreement with the result (\ref{['M-N3-2']}) predicted from eleven-dimensional supergravity, which is represented by the solid line.