Wilson loops at large $N$ and the quantum M2 brane

TL;DR

This paper demonstrates an exact match between the Wilson loop expectation value in ABJM theory at large N and the quantum M2 brane calculation in M-theory, including both classical and one-loop quantum contributions.

Contribution

It provides the first exact matching of the Wilson loop's numerical prefactor with the dual M2 brane computation, including quantum corrections.

Findings

Classical M2 brane action reproduces the leading exponential factor.

One-loop quantum corrections match the subleading k-dependent prefactor.

First case of exact numerical prefactor matching in holographic Wilson loop calculations.

Abstract

The Wilson loop operator in the $U(N)_k \times U(N)_{-k}$ ABJM theory at large $N$ and fixed level $k$ has a dual description in terms of a wrapped M2 brane in the M-theory background AdS$_4 \times S^7/\mathbb Z_k$. We consider the localization result for the $1\over 2$-BPS circular Wilson loop expectation value $W$ in this regime, and compare it to the prediction of the M2 brane theory. The leading large $N$ exponential factor is matched as expected by the classical action of the M2 brane solution with AdS$_2\times S^1$ geometry. We show that the subleading $k$-dependent prefactor in $W$ is also exactly reproduced by the one-loop term in the partition function of the wrapped M2 brane (with all Kaluza-Klein modes included). This appears to be the first case of an exact matching of the overall numerical prefactor in the Wilson loop expectation value against the dual holographic result. It provides an example of a consistent quantum M2 brane computation, suggesting various generalizations.