Abstract
In this paper we study the expectation value of deformations of the circular Wilson loop in mathcal{N}=4 super Yang-Mills theory. The leading order deformation, known as the Bremsstrahlung function, can be obtained exactly from supersymmetric localization, so our focus is on deformations at higher orders. We find simple expressions for the expectation values for generic deformations at the quartic order at one-loop at weak coupling and at leading order at strong coupling. We also present a very simple algorithm (not requiring integration) to evaluate the two-loop result. We find that an exact symmetry of the strong coupling sigma-model, known as the spectral-parameter independence, is an approximate symmetry at weak coupling, modifying the expectation value starting only at the sextic order in the deformation. Furthermore, we find very simple patterns for how the spectral parameter can appear in the weak coupling calculation, suggesting all-order structures.
Highlights
In this paper we study the expectation value of deformations of the circular Wilson loop in N = 4 super Yang-Mills theory
We find that an exact symmetry of the strong coupling sigma-model, known as the spectral-parameter independence, is an approximate symmetry at weak coupling, modifying the expectation value starting only at the sextic order in the deformation
We presented explicit expressions for the expectation values at one-loop at weak coupling and at leading order at strong coupling
Summary
We recount the construction of Wilson loops from the Pohlmeyer description of Euclidean AdS3 [4, 17], discuss the spectral parameter deformation, and focus on the case of nearly circular Wilson loops. Once f and α are given, the world-sheet embedding in target space follows from the solution of an auxiliary linear differential equation. Note that both the equation for α and the action depend only on the modulus of f (z), so are invariant under f (z) → eiφf (z). Solution, and the boundary curve X(θ), may depend on φ.4 This construction leads to a one-parameter family of curves with the same strong coupling expectation value. Examining the full string world-sheet near the boundary one can find an equation relating the Schwarzian derivative of the boundary contour X(θ) to the boundary values β2(θ), f (eiθ) as well as eiφ:. Inverse procedure, going from X(θ) to f (z) and φ, requires to reparametrize the curve in terms of the correct angle in the unit disc [17].5 Lastly, turning on φ by a spectral deformation, which modifies f (z) by a constant phase, influences one part of the calculation outlined above
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