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

Abstract: Type IIB string theory on AdS$_3 \times S^3\times T<sup>4$ 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$_3 \times S^3\times T<sup>5$ 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$_4 \times S<sup>7/\mathbb{Z}_k$). 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$_2\times S<sup>1$. 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 $$Z_1= (2\sin{\frac{2π}{ k}})^{-1} = \frac{k}{ 4 π} +\fracπ{ 6k} +...$$ 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 $Z_1= \fracκ{ \sqrt{2π}}$ where $κ\sim \sqrt{Q_5}$ plays a role analogous to $k$. We also discuss a generalization to the mixed flux case which is straightforward from the 11d perspective.

• The paper demonstrates a quantum M2-brane method that computes the Wilson loop exactly at one-loop order, avoiding non-planar string corrections.
• It employs T-duality and an M-theory uplift to transform the problem into a M2-M5 brane framework, using spectral ζ-function regularization for the partition function.
• Supersymmetric cancellations in the fluctuation spectrum eliminate higher-genus corrections, confirming non-renormalization properties in the AdS3×S3×T4 background.

Wilson Loop in AdS$_3 \times S^3 \times T^4$ from Quantum M2-Brane

Introduction and Context

This work establishes an M-theory framework for analyzing quantum corrections to supersymmetric Wilson loops (WL) in type IIA string theory on $\mathrm{AdS}_3 \times S^3 \times T^4$ with RR flux, via the quantum partition function of an M2 brane in $AdS_3 \times S^3 \times T^5$. The construction leverages the T-duality between the IIB D1-D5 and IIA D2-D4 near-horizon solutions, together with the 11d uplift to M2-M5 systems, extending techniques previously demonstrated in the $AdS_4/CFT_3$ (ABJM) correspondence [Giombi-Tseytlin, (Giombi et al., 2023)]. The strategy is to compute the one-loop determinant for fluctuations about an $AdS_2 \times S^1$ M2-brane configuration, which in the IIA limit descends to the evaluation of WL expectation values in the dual 2d $(4,4)$ superconformal CFT.

The methodology builds on established semiclassical and one-loop analyses for string and M2-brane probes in holographic setups, notably in $AdS_5 \times S^5$ and $AdS_4 \times \mathbb{CP}^3$ supergravities [Giombi-Tseytlin, (2020.08512)], and generalizes the calculation of the string/M2-brane partition function near minimal surfaces.

M2-M5 Supergravity Solution and Its Reductions

The starting point is the intersecting M2-M5 solution in 11d supergravity, parametrized by integer charges $Q_2$, $Q_5$, and geometric moduli encoding the $\mathrm{AdS}_3 \times S^3 \times T^4$0 torus [Tseytlin, 1996bh]. The near-horizon geometry yields $\mathrm{AdS}_3 \times S^3 \times T^4$1, with a natural identification of the circular $\mathrm{AdS}_3 \times S^3 \times T^4$2 as the "11th dimension". Upon dimensional reduction (identifying the 11th direction with one of the $\mathrm{AdS}_3 \times S^3 \times T^4$3 directions), one retrieves the type IIA D2-D4 system; after T-duality, the D1-D5 frame is recovered in type IIB. The effective M2-brane tension and action, relevant for semiclassical expansion, are given in terms of these parameters, with explicit scaling in $\mathrm{AdS}_3 \times S^3 \times T^4$4 and $\mathrm{AdS}_3 \times S^3 \times T^4$5, and the string coupling entering through the geometric moduli.

The M2-brane probe for the Wilson loop maps to an $\mathrm{AdS}_3 \times S^3 \times T^4$6 minimal surface in $\mathrm{AdS}_3 \times S^3 \times T^4$7, with the brane wrapping $\mathrm{AdS}_3 \times S^3 \times T^4$8. In the IIA framework, this corresponds precisely to the string worldsheet describing the WL in $\mathrm{AdS}_3 \times S^3 \times T^4$9.

One-Loop Partition Function: Spectrum and Evaluation

The central technical result is the explicit computation of the quadratic fluctuation spectrum for the M2-brane probe. In static gauge, the spectrum consists of bosonic and fermionic towers labeled by momentum $AdS_3 \times S^3 \times T^5$0 along the $AdS_3 \times S^3 \times T^5$1. Notably, the mass spectra for each $AdS_3 \times S^3 \times T^5$2 assemble into long $AdS_3 \times S^3 \times T^5$3 AdS$AdS_3 \times S^3 \times T^5$4 supermultiplets, such that—theoretically—the total logarithmic UV divergences in the one-loop determinant vanish under $AdS_3 \times S^3 \times T^5$5-function regularization:

Figure 1: $AdS_3 \times S^3 \times T^5$6 plotted versus the parameter $AdS_3 \times S^3 \times T^5$7, demonstrating the regularization of the one-loop correction across parameter space.

The finite piece of the one-loop effective action $AdS_3 \times S^3 \times T^5$8 is, for sufficiently large $AdS_3 \times S^3 \times T^5$9 (or small type IIA coupling), given by

$AdS_4/CFT_3$0

with $AdS_4/CFT_3$1 the effective Kaluza-Klein parameter controlling the $AdS_4/CFT_3$2 radius and RR flux quantum. The partition function and thus the WL expectation value is: $AdS_4/CFT_3$3 where $AdS_4/CFT_3$4 is the on-shell M2-brane action and $AdS_4/CFT_3$5 is the dimensionless M2 tension.

For smaller $AdS_4/CFT_3$6 values (non-perturbative regime), modifications arise due to the presence of exceptional low-lying $AdS_4/CFT_3$7 modes with $AdS_4/CFT_3$8, reflected in a nontrivial structure for $AdS_4/CFT_3$9 (see Figure 1).

Comparison to ABJM and Higher-Genus Corrections

A central result—contrasting with the ABJM ($AdS_2 \times S^1$0) setting—is that the M2-brane one-loop factor is exactly proportional to the leading string theory result, with no infinite $AdS_2 \times S^1$1 expansion (no higher-genus string corrections in $AdS_2 \times S^1$2), as opposed to the $AdS_2 \times S^1$3 prefactor in ABJM [Giombi-Tseytlin, (Giombi et al., 2023)]. The absence of subleading corrections persists for the entire class of WL insertions discussed here, and matches the structure found in certain $AdS_2 \times S^1$4 duals of 3d SCFTs [Gautason-Nix, (Gautason et al., 6 Nov 2025)].

This result suggests that in the specific $AdS_2 \times S^1$5 background with RR flux, the expectation value of the 1/2-BPS Wilson line operator is one-loop exact to all orders in the $AdS_2 \times S^1$6 expansion, at least in the large $AdS_2 \times S^1$7 (planar) regime. This is strongly reminiscent of the behavior of protected observables such as the central charge (from $AdS_2 \times S^1$8 free energy) in large $AdS_2 \times S^1$9 holography.

Generalization: Mixed Flux Backgrounds and Other Observables

The analysis is extended in the appendix to $(4,4)$0 backgrounds with mixed NSNS-RR flux, accessible via modular transformations of the $(4,4)$1 fiber in the 11d perspective. The $(4,4)$2 M2 solution continues to exist with rescaled parameters, and the one-loop structure remains analogous, with the effective parameter $(4,4)$3 replaced by its mixed-flux version depending on $(4,4)$4 (integer fluxes) and geometric moduli. The pure NSNS background is found to be singular in this probe computation.

Potential directions for further development include extension to two-loop corrections (which remain open even in the $(4,4)$5 context [Beccaria-Kurlyand-Tseytlin, (Beccaria et al., 13 Jan 2026)]), application to non-BPS "non-planar" anomalous dimensions, and generalization to surface defect observables in higher-dimensional holographic duals.

Implications

From the AdS/CFT perspective, this result provides a nontrivial check and refinement of the dictionary between string/M-theory observables and protected 2d CFT quantities in the D1-D5 system. The M2-brane quantization methodology, pioneered in ABJM [Giombi-Tseytlin, (Giombi et al., 2023); Beccaria-Giombi-Tseytlin, (Beccaria et al., 2023)], is now shown to extend qualitatively to the AdS$(4,4)$6/CFT$(4,4)$7 duality, but with much simpler non-planar structure for the observable considered.

Practically, this enables precise predictions for the strong-coupling expansion of supersymmetric WLs in candidate $(4,4)$8 2d SCFTs with symmetric orbifold deformations (with recent advances in CFT data extraction from worldsheet amplitudes [Jiang, (Jiang, 26 Jan 2026); Chester-Zhong, (Chester et al., 2024)] potentially providing further testing ground). The absence of higher-genus corrections may further serve to constrain holographic CFT construction and the structure of non-planar corrections in the duality.

Conclusion

This paper demonstrates that the M-theory uplift of type IIA string theory on $(4,4)$9 enables a controlled computation of the quantum corrections to supersymmetric Wilson loops via the semiclassical quantization of the M2-brane. The key technical outcome is that the one-loop contribution is one-loop exact in the large RR flux (planar) regime, with all higher-genus corrections vanishing for the observables studied, in sharp contrast to the $AdS_5 \times S^5$0 (ABJM) case. The methodology is robust, extends to mixed-flux backgrounds, and has significant implications for non-planar holography in AdS$AdS_5 \times S^5$1/CFT$AdS_5 \times S^5$2 contexts.