Pohlmeyer reduction and Darboux transformations in Euclidean worldsheet AdS 3

Παπαθανασίου, Γεώργιος

doi:10.1007/jhep08(2012)105

Cited by 6 publications

(6 citation statements)

References 82 publications

(165 reference statements)

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“…Recently, in the case of closed, Euclidean, plane Wilson loops (with constant scalar) an infinite parameter family of analytical solutions was found using Riemann theta functions [28,29] following results from the mathematical literature [30,31] and from previous results for closed strings [32][33][34][35][36][37][38][39][40][41]. This integrability construction for the Wilson loop was further discussed in [42] and also in [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. More recently, certain integrability properties of the near circular Wilson loop were explained in [58].…”

Section: Jhep11(2014)065mentioning

confidence: 93%

Wilson loops and minimal area surfaces in hyperbolic space

Kruczenski

2014

J. High Energ. Phys.

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The AdS/CFT correspondence relates Wilson loops in N = 4 SYM theory to minimal area surfaces in AdS space. If the loop is a plane curve the minimal surface lives in hyperbolic space H 3 (or equivalently Euclidean AdS 3 space). We argue that finding the area of such extremal surface can be easily done if we solve the following problem: given two real periodic functions V 0,1 (s), V 0,1 (s + 2π) = V 0,1 (s) a third periodic function V 2 (s) is to be found such that all solutions to the equation −∂ 2] for any value of λ. This problem is equivalent to the statement that the monodromy matrix is trivial. It can be restated as that of finding a one complex parameter family of curves X(λ, s) where X(λ = 1, s) is the given shape of the Wilson loop and such that the Schwarzian derivative {X(λ, s), s} is meromorphic in λ with only two simple poles. We present a formula for the area in terms of the functions V 0,1,2 and discuss solutions to these equivalent problems in terms of theta functions. Finally, we also consider the near circular Wilson loop clarifying its integrability properties and rederiving its area using the methods described in this paper.

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Section: Jhep11(2014)065mentioning

confidence: 93%

Wilson loops and minimal area surfaces in hyperbolic space

Kruczenski

2014

J. High Energ. Phys.

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“…A closely related approach can also be found in [39] where integrability properties are applied to the computation of Wilson loops as minimal area surfaces. More recent developments can be found in [40][41][42][43][44][45][46][47][48][49][50].…”

Section: Jhep05(2014)037mentioning

confidence: 99%

Wilson loops and Riemann theta functions II

Kruczenski

Ziama

2014

J. High Energ. Phys.

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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.

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“…The Pohlmeyer reduction for classical strings moving in AdS space formulated in terms of an explicit coordinate parametrization has been extensively studied and applied before [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]59]. As we will repeatedly use the resulting set of equations of motion we have outlined their derivation in appendix A, highlighting the features that will be important to us here.…”

Section: Ads Pohlmeyer Reduction In Coordinate Parametrizationmentioning

confidence: 99%

“…It was used to find string solutions with time-like world-sheets corresponding to solitons of the reduced theory, e.g., of the sinh-Gordon model for AdS 3 [46][47][48]. More recently, it has found fruitful applications in the construction of Euclidean minimal surfaces for open strings ending at the AdS boundary [46,[49][50][51][52][53][54][55][56][57][58][59], which are dual to Wilson loops in planar N = 4 super Yang-Mills at strong coupling, and in the study of semiclassical threeand four-point correlation functions for closed-string states [60][61][62][63].…”

Section: Introductionmentioning

confidence: 99%

Pohlmeyer reduction for superstrings in AdS space

Hoare¹,

Tseytlin²

2012

J. Phys. A: Math. Theor.

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The Pohlmeyer reduced equations for strings moving only in the AdS subspace of AdS5 × S 5 have been used recently in the study of classical Euclidean minimal surfaces for Wilson loops and some semiclassical three-point correlation functions. We find an action that leads to these reduced superstring equations. For example, for a bosonic string in AdSn such an action contains a Liouville scalar part plus a K/K gauged WZW model for the group K = SO(n − 2) coupled to another term depending on two additional fields transforming as vectors under K. Solving for the latter fields gives a non-abelian Toda model coupled to the Liouville theory. For n = 5 we generalize this bosonic action to include the S 5 contribution and fermionic terms. The corresponding reduced model for the AdS2 × S 2 truncation of the full AdS5 × S 5 superstring turns out to be equivalent to N = 2 super Liouville theory. Our construction is based on taking a limit of the previously found reduced theory actions for bosonic strings in AdSn × S 1 and superstrings in AdS5 × S 5 . This new action may be useful as a starting point for possible quantum generalizations or deformations of the classical Pohlmeyer-reduced theory. We give examples of simple extrema of this reduced superstring action which represent strings moving in the AdS5 part of the space. Expanding near these backgrounds we compute the corresponding fluctuation spectra and show that they match the spectra found in the original superstring theory.

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Pohlmeyer reduction and Darboux transformations in Euclidean worldsheet AdS 3

Cited by 6 publications

References 82 publications

Wilson loops and minimal area surfaces in hyperbolic space

Wilson loops and minimal area surfaces in hyperbolic space

Wilson loops and Riemann theta functions II

Pohlmeyer reduction for superstrings in AdS space

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