Exact results in N=8 Chern-Simons-matter theories and quantum geometry

TL;DR

The paper addresses the non-perturbative structure of partition functions for maximally supersymmetric ABJ(M) theories. It shows that localization to a matrix model and a Fermi gas formulation lead to a drastic simplification: only genus-0 and genus-1 topological-string data contribute to the non-perturbative sector, enabling closed forms for the grand potential $J(\mu,k)$ and exact generating functionals for $Z(N,k)$ in terms of Jacobi theta functions on the spectral curve. It further derives exact quantization conditions from theta-function zeros, expressing the spectrum of the associated Fermi gas, and proves that the gauge-theory partition function extends to an entire function of $N$, with implications for quantum geometry of M-theory and potential de Sitter continuations. These results yield precise tests of the non-perturbative framework and reveal deep connections between matrix-model techiques, topological string theory, and modular structures in the AdS/CFT context. The work thus provides a powerful toolkit for exact, non-perturbative insights into M-theory backgrounds and their quantum geometric structure.

Abstract

We show that, in ABJ(M) theories with N=8 supersymmetry, the non-perturbative sector of the partition function on the three-sphere simplifies drastically. Due to this simplification, we are able to write closed form expressions for the grand potential of these theories, which determines the full large N asymptotics. Moreover, we find explicit formulae for the generating functionals of their partition functions, for all values of the rank N of the gauge group: they involve Jacobi theta functions on the spectral curve associated to the planar limit. Exact quantization conditions for the spectral problem of the Fermi gas are then obtained from the vanishing of the theta function. We also show that the partition function, as a function of N, can be extended in a natural way to an entire function on the full complex plane, and we explore some possible consequences of this fact for the quantum geometry of M-theory and for putative de Sitter extensions.

Figures (6)

• Figure 1: The standard contour in the complex plane of the chemical potential, defining the Airy function ${\rm Ai}$.
• Figure 2: The analyticity structure of the genus $g$ free energies leads to two branch cuts in the plane of the 't Hooft coupling, starting at the singularities $\pm \lambda_c$. The circle of radius $|\lambda_c|$ separates two different regions: there is a "short distance phase" for $|\lambda|< |\lambda_c|$, and a "long distance phase" for $|\lambda|> |\lambda_c|$.
• Figure 3: The analyticity structure of the functions $J(\mu, 2)$ (left) and $J(\mu, 2;1)$ (right). Both functions are analytic for ${\rm Re}(\mu) > 2 \log (2)$ (which is the region to the right of the vertical dashed line), but they have branch cuts when ${\rm Re}(\mu) < 2 \log (2)$, indicated here by horizontal dashed lines.
• Figure 4: The contour of integration ${\cal C}$ in (\ref{['naif']}) might be deformed to lie inside the region of analyticity of the modified grand potential, which is the semi-plane to the right of the vertical dashed line.
• Figure 5: A plot of the interpolating function $-\log Z(N,2)$ as a function of real, positive $N$. The dots are the values obtained by evaluating the gauge theory partition function at the positive integers.