M2 brane in AdS$_4\times S^7/\mathbb Z_k$: 2-loop correction to $\frac{1}{2}$-BPS Wilson loop vs localization in ABJM theory
We study the quantization of an M2 brane wrapping $\mathrm{AdS}_2\times S^1$ inside $\mathrm{AdS}_4\times S^7/\mathbb{Z}_k$, dual to the $\frac{1}{2}$-BPS circular Wilson loop in ABJM theory. The classical action and the one-loop contribution were previously shown to reproduce the gauge-theory localization result. Here we extend the analysis to the two-loop order. In an adapted static and $Îș$-symmetry gauge the world-volume action contains no cubic interaction vertices, so that the two-loop correction is determined entirely by quartic interactions and is manifestly free of logarithmic UV divergences. We find that the bosonic and fermionic contributions combine into a perfect square $ {\mathscr X}^2$ where ${\mathscr X}$ is a linear combination of the bosonic and fermionic 3d Green's functions and their derivatives. The evaluation of $\mathscr X$ requires fixing the $\mathrm{AdS}_2$ boundary quantization of the massless fermionic modes. With the boundary conditions selected by the preserved supersymmetry, $\mathscr X$ vanishes and hence the complete two-loop correction is found to be zero. This result would not be consistent with identifying the M2-brane loop expansion directly with the large-$N$ expansion of the canonical, fixed-$(N,k)$ localization prediction. Instead, it agrees precisely with the recent proposal that the semiclassical M2-brane partition function should be matched to the gauge-theory grand-canonical ensemble, in which the Wilson loop expectation value is perturbatively one-loop exact. The vanishing of the two-loop correction thus provides a non-trivial test of this grand-canonical identification.