Generating AdS string solutions

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.

Section: Jhep11(2014)065mentioning

confidence: 81%

Section: Jhep11(2014)065mentioning

confidence: 99%

Wilson loops and minimal area surfaces in hyperbolic space

Kruczenski

2014

“…2 Pohlmeyer reduction of string actions in AdS 3 and dS 3 Pohlmeyer reduction relates string actions in symmetric spaces like AdS and dS to integrable systems such as the sinh-Gordon or cosh-Gordon equation and multi-component generalizations thereof. In this section, following closely the literature [24,33,37], we review the Pohlmeyer reduction of string actions in AdS 3 and dS 3 spaces that result in the sinh-Gordon or cosh-Gordon equation, depending on circumstances, which are specified below.…”

Section: Jhep07(2016)070mentioning

confidence: 99%

“…These transformations can be used inductively to construct families of kink solutions starting from a given solution, such as the vacuum solution. Special emphasis has been placed on the construction of the NLSM counterparts of kink solutions in AdS backgrounds [33][34][35][36][37].…”

Section: Introductionmentioning

confidence: 99%

On elliptic string solutions in AdS3 and dS3

Bakas

Pastras

2016

Classical string actions in AdS 3 and dS 3 can be connected to the sinh-Gordon and cosh-Gordon equations through Pohlmeyer reduction. We show that the problem of constructing a classical string solution with a given static or translationally invariant Pohlmeyer counterpart is equivalent to solving four pairs of effective Schrödinger problems. Each pair consists of a flat potential and an n = 1 Lamé potential whose eigenvalues are connected, and, additionally, the four solutions satisfy a set of constraints. An approach for solving this system is developed by employing an interesting connection between the specific class of classical string solutions and the band structure of the Lamé potential. This method is used for the construction of several families of classical string solutions, one of which turns out to be the spiky strings in AdS 3 . New solutions include circular rotating strings in AdS 3 with singular time evolution of their radius and angular velocity as well as classical string solutions in dS 3 .

“…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]. It also follows related work where Wilson loops were studied or theta functions were used in similar problems, for example in [17][18][19][20][21][22][23][24][25][26][27][28][29][34][35][36][37][38]. 1 Much of that work was motivated by the relation to scattering amplitudes [28,29] which we do not pursue here.…”

Section: Jhep05(2014)037mentioning

confidence: 99%

Wilson loops and Riemann theta functions II

Kruczenski

Ziama

2014

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.