Gravity duals of half-BPS Wilson loops

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 $AdS_2 \times S^2 \times S^4 \times \Sigma$, with isometry group $SO(2,1)\times SO(3) \times SO(5)$ in terms of two locally harmonic functions on a Riemann surface $\Sigma$ 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 $\Sigma$, 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 $\Sigma$ 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.

• The paper constructs analytical gravitational duals for half-BPS Wilson loops in Type IIB supergravity using specific symmetry-preserving solutions.
• Solutions are built on manifolds leveraging harmonic functions over hyperelliptic Riemann surfaces, resulting in varying fields and fluxes parameterized by genus.
• These constructed duals exhibit symmetries expected for Wilson loops, offering insights into strong coupling SYM and are numerically verified for genus 1.

Gravity Duals of Half-BPS Wilson Loops in Type IIB Supergravity

The paper by Eric D'Hoker, John Estes, and Michael Gutperle presents an analytical construction of gravitational duals to half-BPS Wilson loops within the framework of Type IIB supergravity. These solutions maintain 16 supersymmetries in $\mathcal{N}=4$ Super Yang-Mills (SYM) with an $SU(N)$ gauge group. The dual geometries emerge from the AdS/CFT correspondence, underpinned by the symmetries $SO(2,1)\times SO(3)\times SO(5)$, characteristic of these half-BPS Wilson loop configurations.

Solution Construction on Hyperelliptic Riemann Surfaces

The authors solve the supergravity equations on $AdS_2 \times S^2 \times S^4 \times \Sigma$ manifolds, where $\Sigma$ is a hyperelliptic Riemann surface with boundary. The method leverages two locally harmonic functions over $\Sigma$, resulting in solutions that exhibit varying dilaton and axion fields alongside non-zero 3-form fluxes. These solutions are parameterized explicitly by a genus $g$ surface $\Sigma$, where all branch points are real, leading to novel geometries dependent upon the genus and the harmonic functions.

Key Features of the Solutions

For genus $g=0$, the solutions revert to the familiar $AdS_5 \times S^5$ geometry with three free parameters. With higher genus $g\ge1$, the general solution presents $2g+5$ free parameters and includes complexities such as $g$ homology 3-spheres and additional $g$ 5-spheres, capable of carrying RR charges corresponding to 3-form and 5-form fluxes. Remarkably, these configurations exhibit symmetries expected in duals to Wilson loops in gauge theory.

Regularity and Topology Conditions

The paper pays careful attention to imposing regularity and topological conditions to ensure non-singular solutions. The solutions must support the topology of an asymptotic $AdS_5 \times S^5$ region with defined boundary conditions at $\Sigma$. The authors elaborate on how these conditions are mathematically realized, ensuring that the solutions do not develop singularities except at one designated region, corresponding to the asymptotic behavior.

Numerical Verification for Genus 1

While analytical expressions form much of the groundwork, the paper employs numerical analysis within the genus $g=1$ parameter space, affirming the regularity of the solutions. By examining specific cases, they verify the solutions remain robust under various parameter choices, cementing confidence in the broader applicability of their constructed geometries.

Implications and Future Speculations

The complexity of these dual configurations implies further insights into the strong coupling regime within SYM theories, enriching our understanding of Wilson loops' holographic behavior. The solutions underscore the utility of geometrical treatment in supergravity as a bridge to decipher the intricacies of quantum gauge theories, hinting at potential extensions into interfaces and other symmetry-breaking scenarios.

Looking ahead, these constructions invite further investigation including their application in broader scenarios such as non-BPS operators or immersed within more generalized supergravity models. The implications span from theoretical explorations in string theory to practical interpretations in quantum field models, where the refinements in supergravity may mirror insights into the behaviors of gauge field interactions under extreme conditions.

In summary, the paper presents a detailed and profound exploration into half-BPS Wilson loop gravity duals using harmonic functions over hyperelliptic surfaces, with implications that resonate through contemporary theoretical physics discourse, laying groundwork for further exploration of supersymmetric solution dynamics and applications.