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.
Categories: hep-th