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

Abstract

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, $k^{-1}\sim \frac{g_s}{\sqrt{\mathrm{T}}}$), 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.

v4: Appendix F on AdS3 x S3 x S3 x S1 x S1 case added