Wilson loop in AdS$_3 \times S^3 \times T^4$ from quantum M2 brane

Type IIB string theory on AdS$_3 \times S^3\times T^4$ with RR flux as the near-horizon limit of the D1-D5 solution is expected to be dual to a (4,4) supersymmetric 2d CFT parametrized by the integers $Q_1,Q_5$ and other moduli. It is related by T-duality to type IIA string theory in the near-horizon limit of the D2-D4 solution which admits an uplift to the 11d AdS$_3 \times S^3\times T^5$ background which is the near-horizon limit of the M2-M5 solution. We point out that this relation allows one to use the quantum M2-brane description to probe ``non-planar'' corrections in the dual 2d CFT, in close analogy with the ABJM theory case (described by M-theory on AdS$_4 \times S^7/\mathbb{Z}_k$). We consider an analog of a supersymmetric Wilson loop (line defect) expectation value represented by type IIA string partition function expanded around AdS$_2\subset$AdS$_3$ minimal surface. Its M-theory analog is the M2 brane partition function expanded near AdS$_2\times S^1$. We compute the 1-loop contribution $Z_1$ to the M2 brane partition function and find that in contrast to the ABJM case in arXiv:2303.15207 (where $Z_1= (2\sin{\frac{2\pi}{ k}})^{-1} = \frac{k}{ 4 \pi} +\frac{\pi}{ 6k} +...$ contains an infinite series of higher genera string corrections, $k^{-1} \sim \frac{g_s}{ \sqrt {\rm T}}$), here it is given solely by the leading string-theory contribution $Z_1= \frac{\kappa}{ \sqrt{2\pi}}$ where $\kappa \sim \sqrt{Q_5}$ plays a role analogous to $k$. We also discuss a generalization to the mixed flux case which is straightforward from the 11d perspective.