Gravity duals of half-BPS Wilson loops

TL;DR

The paper constructs the fully back-reacted gravity duals of half-BPS Wilson loops in ${\cal N}=4$ SYM by solving Type IIB BPS equations on ${AdS_2\times S^2\times S^4\times \Sigma}$, reduced to a pair of real harmonic functions $h_1,h_2$ on a Riemann surface $\Sigma$. It shows that regular solutions are parametrized by a hyperelliptic surface of genus $g$, carrying $g$ RR 3-form charges and $g$ RR 5-form charges, with a single asymptotic ${AdS_5\times S^5}$ region; branch-cut collapse relates genus-$g$ solutions to genus-$g-1$ ones in a manner consistent with a Wilson-loop dual. The genus-1 analysis is worked out in detail, including positivity/regularity checks and explicit elliptic-function expressions, illustrating the non-singular interpolation among solutions. The framework provides a concrete holographic realization of Wilson-loop operators in higher representations, clarifying how topology and flux charges encode the dual gauge-theory data and enabling controlled deformations and collapses of branch cuts. Overall, the work achieves an exact, back-reacted, globally regular family of solutions that capture the rich topological and flux structure expected for Wilson-loop holography and offers a precise tool for exploring gauge/gravity duality beyond local probes.

Abstract

We explicitly construct the fully back-reacted half-BPS solutions in Type IIB supergravity which are dual to Wilson loops with 16 supersymmetries in $\mathcal{N}=4$ super Yang-Mills. In a first part, we use the methods of a companion paper to derive the exact general solution of the half-BPS equations on the space $AdS_2 \times S^2 \times S^4 \times Σ$, with isometry group $SO(2,1)\times SO(3) \times SO(5)$ in terms of two locally harmonic functions on a Riemann surface $Σ$ with boundary. These solutions, generally, have varying dilaton and axion, and non-vanishing 3-form fluxes. In a second part, we impose regularity and topology conditions. These non-singular solutions may be parametrized by a genus $g \geq 0$ hyperelliptic surface $Σ$, all of whose branch points lie on the real line. Each genus $g$ solution has only a single asymptotic $AdS_5 \times S^5$ region, but exhibits $g$ homology 3-spheres, and an extra $g$ homology 5-spheres, carrying respectively RR 3-form and RR 5-form charges. For genus 0, we recover $AdS_5 \times S^5$ with 3 free parameters, while for genus $g \geq 1$, the solution has $2g+5$ free parameters. The genus 1 case is studied in detail. Numerical analysis is used to show that the solutions are regular throughout the $g=1$ parameter space. Collapse of a branch cut on $Σ$ subtending either a homology 3-sphere or a homology 5-sphere is non-singular and yields the genus $g-1$ solution. This behavior is precisely expected of a proper dual to a Wilson loop in gauge theory.

Figures (6)

• Figure 1: The genus 1 solution with homology 3-spheres and 5-spheres.
• Figure 2: The $u$-plane (top) and $z$-plane (bottom) for $AdS_5 \times S^5$.
• Figure 3: The geometry of the genus 1 solution.
• Figure 4: Plots of $(- N_1 W)$ versus ${\rm Re}(z)$ for values of $\omega_3 = {i \over 2}, i, 2 i$ and $w_0 = {1 \over 2}, {1 \over 4}, {1 \over 10}$. The different lines in each plot are for evenly spaced values of ${\rm Im}(z)$ with $0 < {\rm Im}(z) < {\rm Im}(\omega_3)$ (red) and ${\rm Im}(z) = 0, {\rm Im}(\omega_3)$ (blue). On $\partial \Sigma$, $(-N_1 W) = 0$ as expected.
• Figure 5: Plots of the dilaton (left) and the metric factor $f_1^2$ (right) for the elliptic case with $\omega_3 = i$ and $w_0 = {1\over 2}$.