Instanton Effects in ABJM Theory from Fermi Gas Approach

TL;DR

The paper advances the nonperturbative understanding of the ABJM partition function by extending exact Z_k(N) calculations to select Chern–Simons levels using the Fermi gas formalism, enabling precise extraction of worldsheet and D2-instanton corrections. Worldsheet corrections are derived from topological string theory on local F_0 via GV invariants, while a leading analytic D2-instanton term is proposed to cancel divergences and is validated against exact numerics and TBA results. The nonperturbative structure is organized through Airy-function representations for the partition function, and a periodicity-induced oscillatory component is clarified. The work also opens questions about higher D2-instanton effects, possible D2–worldsheet bound states, and the deep link between ABJM grand partition functions and nonperturbative topological-string completions.

Abstract

We study the instanton effects of the ABJM partition function using the Fermi gas formalism. We compute the exact values of the partition function at the Chern-Simons levels k=1,2,3,4,6 up to N=44,20,18,16,14 respectively, and extract non-perturbative corrections from these exact results. Fitting the resulting non-perturbative corrections by their expected forms from the Fermi gas, we determine unknown parameters in them. After separating the oscillating behavior of the grand potential, which originates in the periodicity of the grand partition function, and the worldsheet instanton contribution, which is computed from the topological string theory, we succeed in proposing an analytical expression for the leading D2-instanton correction. Just as the perturbative result, the instanton corrections to the partition function are expressed in terms of the Airy function.

Figures (13)

• Figure 1: (a) The perturbative, (b) 1-instanton, (c) 2-instanton and (d) 3-instanton corrections of the grand potential $J_k(\mu)$ at $k=1$. The dots represent the grand potential obtained by the exact partition function while the solid lines represent the fitted functions.
• Figure 2: (a) The perturbative, (b) 1-instanton, (c) 2-instanton and (d) 3-instanton corrections of the grand potential $J_k(\mu)$ at $k=2$. The dots represent the grand potential obtained by the exact partition function while the solid lines represent the fitted functions.
• Figure 3: (a) The perturbative, (b) 1-instanton, (c) 2-instanton and (d) 3-instanton corrections of the grand potential $J_k(\mu)$ at $k=3$. The dots represent the grand potential obtained by the exact partition function while the solid lines represent the fitted functions.
• Figure 4: (a) The perturbative, (b) 1-instanton, (c) 2-instanton and (d) 3-instanton corrections of the grand potential $J_k(\mu)$ at $k=4$. The dots represent the grand potential obtained by the exact partition function while the solid lines represent the fitted functions.
• Figure 5: (a) The perturbative, (b) 1-instanton, (c) 2-instanton and (d) 3-instanton corrections of the grand potential $J_k(\mu)$ at $k=6$. The dots represent the grand potential obtained by the exact partition function while the solid lines represent the fitted functions.