Exact ABJM Partition Function from TBA

TL;DR

The paper addresses the finite-$N$, finite-$k$ ($k=1$) ABJM partition function on $S^3$ by casting it into a Fermi gas framework and solving a pair of TBA-like equations, yielding exact $Z(N)$ for $N=1$ to $19$ as polynomials in $\pi^{-1}$. A Fredholm-determinant representation for the grand partition function together with recursive expansions in the coupling enables efficient computation of all $Z_ u$ and demonstrates the structure of the exact results. The authors validate the results against the perturbative Airy regime and extract non-perturbative membrane instanton corrections, finding a leading constant $c_0=2$ and a membrane 1-instanton scaling $rac{Z^{(1 ext{-inst})}}{Z^{(pert)}} \sim (2N) e^{-2\pi\sqrt{2N}}$. This provides precise finite-$N$ data for ABJM at $k=1$ and showcases a method to access non-perturbative M-theory corrections from matrix-model representations, with implications for AdS$_4$/CFT$_3$ and M2-brane dynamics.

Abstract

We report on the exact computation of the S^3 partition function of U(N)_k\times U(N)_{-k} ABJM theory for k=1, N=1,...,19. The result is a polynomial in π^{-1} with rational coefficients. As an application of our results we numerically determine the coefficient of the membrane 1-instanton correction to the partition function.

Figures (2)

• Figure 1: In this figure, the dots represent the sequence $Z(N)/Z^\text{(pert)}(N)-1$.
• Figure 2: In these figures, the dots represent the sequence (\ref{['nptest']}) (left) and its 9-th Richardson-like transform $\widetilde{F}^{(9)}(N)$ (right).