Summing Up All Genus Free Energy of ABJM Matrix Model

TL;DR

This work uses localization to formulate the ABJM theory as a two-matrix model and applies holomorphic anomaly techniques to sum all-genus (nonworldsheet-instanton) corrections at strong coupling and finite $N$. The authors show that the all-genus free energy reduces to an Airy function of a renormalized 't Hooft coupling, $\lambda_{\rm ren}=\lambda-\tfrac{1}{24}-\tfrac{\lambda^2}{3N^2}$, after a two-step resummation: first the weight-zero sector and then the remaining sectors via a coupling renormalization, yielding $F^{\rm ABJM}(\lambda,N)=\log\left[2\pi C_1 \mathrm{Ai}\left((-g_s^2 y^3)^{-2/3}\right)\right]+\mathcal{O}\left(e^{-2\pi\sqrt{\lambda-1/24}}\right)$ with $y= x/\sqrt{1+(g_s x)^2/6}$. This Airy-form encapsulates the nonplanar corrections up to the worldsheet instantons and reproduces the AdS radius renormalization in the planar limit, while revealing a residual $-3\lambda^2/(8N^2)$ discrepancy that motivates considering gauge-group refinements (e.g., SU$(N)\times$SU$(N)$) or further SUGRA corrections. The results highlight the efficiency of localization combined with BCOV-type recursion in obtaining nonperturbative, finite-$N$ information and suggest connections to matrix-models of Kontsevich type through the universal Airy structure.

Abstract

The localization technique allows us to compute the free energy of the U(N)_k x U(N)_{-k} Chern-Simons-matter theory dual to type IIA strings on AdS_4 x CP^3 from weak to strong 't Hooft coupling λ= N / k at finite N, as demonstrated by Drukker, Marino, and Putrov. In this note we study further the free energy at large 't Hooft coupling with the aim of testing AdS/CFT at the quantum gravity level and, in particular, sum up all the 1/N corrections, apart from the worldsheet instanton contributions. The all genus partition function takes a remarkably simple form -- the Airy function, Ai (k^{4/3} λ_r), with the renormalized 't Hooft coupling λ_r.