Universal holographic Wilson loops in 3d SCFTs

TL;DR

The paper tackles the subleading structure of 1/2-BPS Wilson loops in two families of 3d $\mathcal{N}=2$ CS-matter theories using holography. It develops a universal semiclassical framework: a probe M2-brane in AdS$_4\times\mathrm{SE}_7$ for family A and a fundamental string in massive IIA AdS$_4\times X_6$ for family B, with one-loop quantization yielding explicit corrections encoded through geometric data such as $\mathrm{vol}(\mathrm{SE}_7)$ and $\mathrm{vol}(\mathrm{SE}_5)$. For family A, the M2-brane partition function predicts a perturbatively exact Wilson loop in a grand-canonical setup that maps to an Airy function in the canonical ensemble, $\langle W(N)\rangle^{\rm p}=e^{-\Gamma_{\text{M2}}}\frac{\mathrm{Ai}(\mathcal{C}^{-1/3}(N-2c-\mathcal{B}))}{\mathrm{Ai}(\mathcal{C}^{-1/3}(N-\mathcal{B}))}$, with $\mathcal{C}=6\mathrm{vol}(\mathrm{SE}_7)/\pi^6$. For family B, the universal one-loop string partition function yields a subleading correction to $\langle W\rangle$ of the form $\langle W\rangle_{\rm B} \approx \Gamma(2/3)^3\frac{n^{2/3}N^{1/3}}{2^{4/3}3^{2/3}\pi\mathrm{vol}(\mathrm{SE}_5)^{1/3}}\exp\left[\pi^{2}\frac{4^{1/3}N^{1/3}}{3^{1/6}n^{1/3}\mathrm{vol}(\mathrm{SE}_5)^{1/3}}\right]$, universal up to the SE$_5$ geometry. The results are verified across explicit examples (ABJM, ADHM, Q$^{1,1,1}$, V$_{5,2}$, M$^{3,2}$) and are amenable to matrix-model checks, offering a predictive bridge between holography, localization, and large-$N$ dynamics in three dimensions.

Abstract

We study the vacuum expectation value of half-BPS Wilson loop operators in two families of superconformal $\mathcal{N}=2$ Chern-Simons-matter theories. The first family is dual to AdS$_{4}$ solutions in M-theory, while the second one has a dual description in massive type IIA string theory. Utilizing the properties of the underlying geometry, we provide a universal description for the semiclassical quantization of a probe M2-brane and fundamental string in the respective holographic dual geometries. As a result, we find the one-loop partition function of both the M2-brane and the string which leads to a prediction for the large $N$ behaviour of the Wilson loops in the dual SCFTs. For theories with M-theory duals, we conjecture the full perturbative completion as a ratio of Airy functions.

Figures (8)

• Figure 1: A schematic visualization of the embedding of the M-theory circle $S^{1}_{M}$ in eleven-dimensional supergravity and its dimensional reduction to type IIA supergravity. We observe that in the type IIA limit where $k\rightarrow\infty$, $S^{1}_{M}$ is reduced to a fixed point in the internal manifold M$_{6}$ of the type IIA geometry. On the field theory side the Wilson loop resides on the equator of $S^{3}$. In the dual gravity picture this translates to a holographic Wilson loop that lies on the Hopf circle $S^{1}\subset S^{3}$. This three-sphere is the conformal boundary of Euclidean AdS$_{4}$, inside of which the dual fundamental string occupies the minimal surface $\Sigma_{2}\cong\text{AdS}_{2}\subset$ AdS$_{4}$, whose conformal boundary is that of $S^{1}$.
• Figure 2: Quiver diagram of ABJM.
• Figure 3: Quiver diagram of ADHM.
• Figure 4: Quiver diagram of $Q^{1,1,1}/\mathbf{Z}_{N_{f}}$.
• Figure 5: Quiver diagram of $\tilde{Q}^{1,1,1}/\mathbf{Z}_{k}$.