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We present a quantum M2 brane computation of the instanton prefactor in theleading non-perturbative contribution to the ABJM 3-sphere free energy at large$N$ and fixed level $k$. Using supersymmetric localization, such instantoncontribution 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 theinstanton prefactor was recently performed in arXiv:2304.12340. Here we findthat the exact value of the prefactor $({\sin^{2} \frac{2\pi}{k}})^{-1} $ isreproduced by the 1-loop term in the M2 brane partition function expanded nearthe $S^3/\mathbb{Z}_k$ instanton configuration. As in the Wilson loop examplein arXiv:2303.15207, the quantum M2 brane computation is well defined andproduces a finite result in exact agreement with localization.

Instanton contributions to the ABJM free energy from quantum M2 branes