Instanton contributions to the ABJM free energy from quantum M2 branes

TL;DR

This paper computes the instanton contribution to the ABJM free energy using quantum M2 branes, confirming the exact prefactor through a 1-loop M2 brane partition function calculation, aligning with previous localization results.

Contribution

It provides the first quantum M2 brane computation of the instanton prefactor in ABJM theory, matching the exact localization result and extending the understanding of non-perturbative effects in M-theory.

Findings

Exact prefactor matches the 1-loop M2 brane partition function.

Quantum M2 brane computation reproduces the known instanton contribution.

Results confirm the consistency of M2 brane instanton calculations with localization.

Abstract

We present a quantum M2 brane computation of the instanton prefactor in the leading non-perturbative contribution to the ABJM 3-sphere free energy at large $N$ and fixed level $k$. Using supersymmetric localization, such instanton contribution was found earlier to take the form $F^{\rm inst}(N,k) = - ({\sin^{2} \frac{2\pi}{k}} )^{-1} \, \exp (-2\pi \sqrt\frac{2N}{k}) + \cdots$ . The exponent comes from the action of an M2 brane instanton wrapped on $S^3/{\mathbb Z}_k$, which represents the M-theory uplift of the $\mathbb{C}P^1$ instanton in type IIA string theory on AdS$_4 \times \mathbb{C}P^3$. The IIA string computation of the leading large $k$ term in the instanton prefactor was recently performed in arXiv:2304.12340. Here we find that the exact value of the prefactor $({\sin^{2} \frac{2\pi}{k}})^{-1} $ is reproduced by the 1-loop term in the M2 brane partition function expanded near the $S^3/\mathbb{Z}_k$ instanton configuration. As in the Wilson loop example in arXiv:2303.15207, the quantum M2 brane computation is well defined and produces a finite result in exact agreement with localization.