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

Abstract

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 $\kappa$-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 ${\mathcal{X}}^2$ where $\mathcal{X}$ is a linear combination of the bosonic and fermionic 3d Green's functions and their derivatives. The evaluation of $\mathcal{X}$ requires fixing the ${\mathrm{AdS}}_2$ boundary quantization of the massless fermionic modes. With the boundary conditions selected by the preserved supersymmetry, $\mathcal{X}$ 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.

32 pages. v2: minor corrections