Notes on Euclidean Wilson loops and Riemann theta functions

Abstract:We study the finite term of the holographic entanglement entropy of finite domains with smooth shapes and for four dimensional gravitational backgrounds. Analytic expressions depending on the unit vectors normal to the minimal area surface are obtained for both stationary and time dependent spacetimes. The special cases of AdS 4 , asymptotically AdS 4 black holes, domain wall geometries and Vaidya-AdS backgrounds have been analysed explicitly. When the bulk spacetime is AdS 4 , the finite term is the Willmore energy of the minimal area surface viewed as a submanifold of the three dimensional flat Euclidean space. For the static spacetimes, some numerical checks involving spatial regions delimited by ellipses and non convex domains have been performed. In the case of AdS 4 , the infinite wedge has been also considered, recovering the known analytic formula for the coefficient of the logarithmic divergence.

Section: Other Domainsmentioning

confidence: 99%

On shape dependence of holographic entanglement entropy in AdS4/CFT3

Fonda

Seminara

Tonni

2015

“…Recent progress in [14] provided a new, infinite parameter family of minimal area surfaces that can be used to further explore the AdS/CFT duality. In that work the minimal area surfaces were constructed analytically in terms of Riemann theta functions associated to hyperelliptic Riemman surfaces closely following previous work by M. Babich and A. Bobenko [15,16].…”

Section: Jhep05(2014)037mentioning

confidence: 99%

“…Since the paper is essentially a continuation of the previous one we do not attempt to make this paper self-contained and should be read in conjunction with [14]. On the other hand we give new expressions for the boundary curve and the area that were derived by simplifying the ones in [14]. After that, we present the method to find minimal area surfaces ending on multiple curves and give a few examples with plots of the corresponding surfaces and boundaries.…”

Section: Jhep05(2014)037mentioning

confidence: 99%

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Wilson loops and Riemann theta functions II

Kruczenski

Ziama

2014

Self Cite

In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS 3 ). If the Wilson loop is given by a boundary curve X(s) we define, using the integrable properties of the system, a family of curves X(λ, s) depending on a complex parameter λ known as the spectral parameter. This family has remarkable properties. As a function of λ, X(λ, s) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, X(λ, s) has an essential singularity at the origin λ = 0. The coefficients of the expansion of X(λ, s) around λ = 0, when appropriately integrated along the curve give the area of the corresponding minimal area surface.Furthermore we show that the same construction allows the computation of certain surfaces with one or more boundaries corresponding to Wilson loop correlators. We extend the area formula for that case and give some concrete examples. As the main example we consider a surface ending on two concentric circles and show how the boundary circles can be deformed by introducing extra cuts in the spectral curve.

“…One needs to find the reparametrization between the conformal angle θ of the string worldsheet and the arbitrary parameter s. Then the Schwarzian derivative of X(θ) provides the boundary conditions for the Pohlmeyer functions α, f , and it defines the potential of a Schrödinger-like equation whose solutions encode the shape of the curve. In this context, the one-parameter family of boundary curves expected from integrability [38] can be simply obtained by solving the Schrödinger-like equation with the λ-deformed potential without finding the corresponding minimal surfaces. In [39], this formalism was applied to study Wilson loops perturbatively away from the circular contour and the area of the dual minimal surfaces was found to high orders.…”

Section: Introductionmentioning

confidence: 99%

Minimal area surfaces in AdSn+1 and Wilson loops

Huang

Kruczenski

2018

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The AdS/CFT correspondence relates the expectation value of Wilson loops in N = 4 SYM to the area of minimal surfaces in AdS 5 . In this paper we consider minimal area surfaces in generic Euclidean AdS n+1 using the Pohlmeyer reduction in a similar way as we did previously in Euclidean AdS 3 . As in that case, the main obstacle is to find the correct parameterization of the curve in terms of a conformal parameter. Once that is done, the boundary conditions for the Pohlmeyer fields are obtained in terms of conformal invariants of the curve. After solving the Pohlmeyer equations, the area can be expressed as a boundary integral involving a generalization of the conformal arc-length, curvature and torsion of the curve. Furthermore, one can introduce the λ-deformation symmetry of the contours by a simple change in the conformal invariants. This determines the λ-deformed contours in terms of the solution of a boundary linear problem. In fact the condition that all λ deformed contours are periodic can be used as an alternative to solving the Pohlmeyer equations and is equivalent to imposing the vanishing of an infinite set of conserved charges derived from integrability.