ABJM theory as a Fermi gas

TL;DR

This work reframes the S^3 partition function of ABJM and related Chern--Simons--matter theories as the partition function of an ideal Fermi gas with Planck constant $\hbar=2\pi k$, enabling a semiclassical (thermodynamic) analysis that yields the universal $N^{3/2}$ growth and, via the first quantum corrections, the complete $1/N$ expansion captured by an Airy function. The formalism provides a direct, elementary route to non-perturbative membrane and worldsheet instanton effects, with membrane instantons appearing as exponential corrections in the chemical potential and worldsheet instantons tied to quantum-mechanical instantons of the effective Hamiltonian. The approach extends naturally to necklace quivers and theories with fundamental matter, yielding Airy resummations under broad conditions and revealing how the dual M-theory and type IIA descriptions emerge in the semiclassical and quantum-corrected phases. In short, the Fermi gas picture offers a powerful, unified framework to study strong-coupling, non-perturbative aspects of M2-brane holography and its matrix-model realizations.

Abstract

The partition function on the three-sphere of many supersymmetric Chern-Simons-matter theories reduces, by localization, to a matrix model. We develop a new method to study these models in the M-theory limit, but at all orders in the 1/N expansion. The method is based on reformulating the matrix model as the partition function of an ideal Fermi gas with a non-trivial, one-particle quantum Hamiltonian. This new approach leads to a completely elementary derivation of the N^{3/2} behavior for ABJM theory and N=3 quiver Chern-Simons-matter theories. In addition, the full series of 1/N corrections to the original matrix integral can be simply determined by a next-to-leading calculation in the WKB or semiclassical expansion of the quantum gas, and we show that, for several quiver Chern-Simons-matter theories, it is given by an Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas corresponds to a strong coupling expansion in type IIA theory, and it is dual to the genus expansion. This allows us to calculate explicitly non-perturbative effects due to D2-brane instantons in the AdS background.

Figures (6)

• Figure 1: A one-dimensional periodic lattice with $\ell$ sites. The transfer matrix $\hat{\rho}$ can be regarded as the quantum propagator for a single particle in Euclidean, discretized time with Hamiltonian $\hat{H}$.
• Figure 2: A quiver with $r$ nodes forming a necklace.
• Figure 3: The Riemann surface of $H_\mathrm{cl}(p,q)=E$ for ABJM theory for large $E$. The four interior tubes form the limiting polygon of the Fermi surface.
• Figure 4: The black thick line depicts the Fermi surface (\ref{['fermi-curve']}). The green thin line depicts the limiting polygon. The red thick dashes mark the boundaries of the patches $\mathcal{U}_R$.
• Figure 5: Quiver for the two-node theory with fundamental matter.

Theorems & Definitions (1)

• Conjecture 1.1