Abstract
We study the strong coupling behaviour of $1/4$-BPS circular Wilson loops (a family of "latitudes") in ${\cal N}=4$ Super Yang-Mills theory, computing the one-loop corrections to the relevant classical string solutions in AdS$_5\times$S$^5$. Supersymmetric localization provides an exact result that, in the large 't Hooft coupling limit, should be reproduced by the sigma-model approach. To avoid ambiguities due to the absolute normalization of the string partition function, we compare the $ratio$ between the generic latitude and the maximal 1/2-BPS circle: Any measure-related ambiguity should simply cancel in this way. We use Gel'fand-Yaglom method to calculate the relevant functional determinants, that present some complications with respect to the standard circular case. After a careful numerical evaluation of our final expression we still find disagreement with the localization answer: The difference is encoded into a precise "remainder function". We comment on the possible origin and resolution of this discordance.
Highlights
In [7]1 using the Gel’fand-Yaglom method, reconsidered in [9] with a different choice of boundary conditions and reproduced in [10]2 with the heat-kernel technique
In this paper we calculated the ratio between the AdS5 × S5 superstring one-loop partition functions of two supersymmetric Wilson loops with the same topology
We address the question whether such procedure — which should eliminate possible ambiguities related to the measure of the partition function, under the assumption that the latter only depends on worldsheet topology — leads to a long-sought agreement with the exact result known via localization at this order, formula (4.19)
Summary
The classical string surface describing the strong coupling regime of the 1/4-BPS latitude was first found in [17] and discussed in details in [15, 18]. Parametrizes a string worldsheet, ending on a unit circle at the boundary of AdS5 and on a latitude sitting at polar angle θ0 on a two-sphere inside the compact space.. The ansatz (2.2) does not propagate along the time direction and defines an Euclidean surface embedded in a Lorentzian target space It satisfies the equation of motions (supplemented by the Virasoro constraints in the Polyakov formulation) when we set sinh ρ(σ) =. The 1/2-BPS circular case falls under this class of Wilson loops when the latitude in S2 shrinks to a point for θ0 = 0, which implies θ(σ) = 0 and σ0 = +∞ from (2.3)–(2.4) In this case the string propagates only in AdS3. Κg stands for the geodesic curvature of the boundary at σ = 0 and ds is the invariant line element With this subtraction, we have the value of the regularized classical area. The (upper-sign) solution dominates the string path integral and is responsible for the leading exponential behaviour in (1.2) and so, in the following, we will restrict to the upper signs in (2.3)
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