Instanton contributions to the ABJM free energy from quantum M2 branes

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 $F^{\mathrm{inst}}(N,k)=-({\sin }^2\frac{2\pi }{k}{)}^{-1}\exp (-2\pi \sqrt{\frac{2N}{k}})+\cdots$ . 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.

27 pages. v2: comments added