Vortex loop operators and quantum M2-branes

We study M2-branes in AdS_4× S^7/\mathbb{Z}_kAdS4×S7/ℤk 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 kk and is independent of all continuous parameters in the definition of the vortex loops. The result for odd kk agrees with the answers for the 1/2 BPS Wilson loop in the k=1k=1 theory and for even kk with the one in the k = 2k=2 theory. Combining with the classical part, we find that the natural expansion parameter seems to be 1/\sqrt{kN}1/kN 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.