Vortex loop operators and quantum M2-branes

We study M2-branes in AdS_4× S^7/\mathbb{Z}_k A d S 4 × S 7 / ℤ k dual to 1/2 and 1/3 BPS vortex loop operators in ABJM theory and compute their one-loop correction beyond the classical M2-brane action. The correction depends only on the parity of k k and is independent of all continuous parameters in the definition of the vortex loops. The result for odd k k agrees with the answers for the 1/2 BPS Wilson loop in the k=1 k = 1 theory and for even k k with the one in the k = 2 k = 2 theory. Combining with the classical part, we find that the natural expansion parameter seems to be 1/\sqrt{kN} 1 / k N rather than 1/\sqrt{N} 1 / N . This provides a further setting where semiclassical quantisation can be applied to M2-branes and produces new results inaccessible by other methods.