Conformal Invariants for Curves and Surfaces in Three Dimensional Space Forms

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.

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“…which agree with the standard definitions [44]. Another formula for µ a can be obtained by defining the curvatures…”

Section: Jhep02(2018)027supporting

confidence: 76%

“…where T (s) is called the conformal torsion (see [44] for the more standard definition that agrees with the present one 1 ),μ a = µa µ , and the covariant derivative is defined as…”

Section: Jhep02(2018)027mentioning

confidence: 76%

Minimal area surfaces in AdSn+1 and Wilson loops

Huang

Kruczenski

2018

J. High Energ. Phys.

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“…For instance, the well-known four-vertex theorem ensures that every smooth, simple closed curve in Q 2 has at least four vertexes. For an account on this result in its various forms one can consult [8], [21], [23]. Given a 1-generic curve f and a point p, by (14) we can always choose a Darboux frame with the property that p 2 = 1, p 3 = .…”

Section: The Frenet-serret Equations For Curves In Q Nmentioning

confidence: 99%

On the geometry of curves and conformal geodesics in the Möbius space

2011

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This article deals with the study of some properties of immersed curves in the conformal sphere Q n , viewed as a homogeneous space under the action of the Möbius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein's Erlangen program. The core of this article is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler-Lagrange equations for any n, we prove an interesting codimension reduction, namely that every conformal geodesic in Q n lies, in fact, in a totally umbilical 4-sphere Q 4 . We then extend and complete the work in Musso (Math Nachr 165:107-131, 1994) by solving the Euler-Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere.

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“…For n = 2, our generating invariants depend on derivatives up to order 5 as happens in the case of single plane curves (see Eqs. 1 and 2, and [8] for a more detailed discussion about curves and surfaces), so it seems that the method in Sect. 3.2 is optimal (in a sense).…”

Section: Some Remarksmentioning

confidence: 99%

“…Local conformal invariants of curves and surfaces (as well as arbitrary submanifolds) of 2-, 3-(and, in general, n-) dimensional space forms are well understood (see [7,8] and the bibliographies there). For instance, in the case of plane curves γ without vertices (that is critical points of the curvature) all local conformal invariants are generated by a single 1-form ω = |κ |ds (1) (called the conformal length) and a single scalar quantity…”

Section: Introductionmentioning

confidence: 99%