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On gravitational description of Wilson lines

Oleg Lunin

TL;DR

This work constructs a gravitational description for supersymmetric Wilson lines in ${\cal N}=4$ SYM by deriving Type IIB supergravity solutions that are regular, asymptotically $AdS_5\times S^5$, and uniquely specified by a single harmonic function $\Phi$ in two dimensions via the Laplace equation $\nabla^2\Phi=0$. Regularity imposes simple Neumann boundary conditions on the axis $y=0$, which have a natural interpretation as the dissolved flux on D3 and D5 branes, yielding a topologically nontrivial geometry with non-contractible $3$- and $5$-cycles. The full solution is encoded in $\Phi$ through auxiliary potentials and a perturbative scheme around the $AdS_5\times S^5$ background, with explicit results for the AdS5×S5 case and a systematic method for constructing more general geometries. Analytic continuations reveal equivalent descriptions and connections to other bubbling-type backgrounds, while the brane-topology analysis ties the geometric data to D3- and D5-brane configurations, offering a coherent brane-backed holographic picture of Wilson lines. Overall, the paper advances a unified, metric-level description of Wilson-line states in holography, linking harmonic data, regularity, and brane fluxes into a solvable gravitational framework.

Abstract

We study solutions of Type IIB supergravity, which describe the geometries dual to supersymmetric Wilson lines in N=4 super-Yang-Mills. We show that the solutions are uniquely specified by one function which satisfies a Laplace equation in two dimensions. We show that if this function obeys a certain Dirichlet boundary condition, the corresponding geometry is regular, and we find a simple interpretation of this boundary condition in terms of D3 and D5 branes which are dissolved in the geometry. While all our metrics have AdS_5 x S^5 asymptotics, they generically have nontrivial topologies, which can be uniquely specified by a set of non-contractible three- and five-spheres.

On gravitational description of Wilson lines

TL;DR

This work constructs a gravitational description for supersymmetric Wilson lines in SYM by deriving Type IIB supergravity solutions that are regular, asymptotically , and uniquely specified by a single harmonic function in two dimensions via the Laplace equation . Regularity imposes simple Neumann boundary conditions on the axis , which have a natural interpretation as the dissolved flux on D3 and D5 branes, yielding a topologically nontrivial geometry with non-contractible - and -cycles. The full solution is encoded in through auxiliary potentials and a perturbative scheme around the background, with explicit results for the AdS5×S5 case and a systematic method for constructing more general geometries. Analytic continuations reveal equivalent descriptions and connections to other bubbling-type backgrounds, while the brane-topology analysis ties the geometric data to D3- and D5-brane configurations, offering a coherent brane-backed holographic picture of Wilson lines. Overall, the paper advances a unified, metric-level description of Wilson-line states in holography, linking harmonic data, regularity, and brane fluxes into a solvable gravitational framework.

Abstract

We study solutions of Type IIB supergravity, which describe the geometries dual to supersymmetric Wilson lines in N=4 super-Yang-Mills. We show that the solutions are uniquely specified by one function which satisfies a Laplace equation in two dimensions. We show that if this function obeys a certain Dirichlet boundary condition, the corresponding geometry is regular, and we find a simple interpretation of this boundary condition in terms of D3 and D5 branes which are dissolved in the geometry. While all our metrics have AdS_5 x S^5 asymptotics, they generically have nontrivial topologies, which can be uniquely specified by a set of non-contractible three- and five-spheres.

Paper Structure

This paper contains 15 sections, 189 equations, 4 figures.

Table of Contents

  1. Introduction.
  2. Wilson lines and brane probes.
  3. Summary of supergravity solution.
  4. Example: $AdS_5\times S^5$.
  5. Perturbative solution.
  6. Analytic continuations.
  7. Back to the brane probes.
  8. Discussion
  9. Solving gravity equations
  10. Formulation of the problem.
  11. Choosing coordinates and evaluating the metric.
  12. Evaluating the fluxes.
  13. Summarizing supergravity solution.
  14. Constructing spinors on the sphere.
  15. Perturbative solution at large distances.

Figures (4)

  • Figure 1: Two different pictures for fundamental string ending on D3 brane: the naive configuration (a) and the description in terms of spike introduced in CalMald (b). We will argue that the latter picture is responsible for existence of regular supergravity solution.
  • Figure 2: A pictorial representation of the boundary conditions (\ref{['PhiBC']}) on $y=0$ line: the dark region corresponds to shrinking $S^2$ and the light regions correspond to contracting $S^4$. Since we are looking for solutions with $AdS_5\times S^5$ asymptotics, the dark segments are contained in a finite region of the line.
  • Figure 3: A pictorial representation of the boundary conditions (\ref{['PhiBC']}) on $(x,y)$ plane (a) and the lines on which their counterparts (\ref{['AnalContBC3']}) are imposed (b). Figure (c) gives an example of such lines which correspond to $AdS_5\times S^5$.
  • Figure 4: The geometries described in this paper have non--trivial topologies which are characterized by non--contractible 3-- and 5--cycles. To construct a five--cycle, one looks at a contour depicted in figure (a) and fibrates $S^4$ over it. The three--cycles are constructed in a similar ways using contours from figure b and $S^2$.