"Anomaly" in n=infinity Alday-Maldacena Duality for Wavy Circle

Abstract

If the Alday-Maldacena version of string/gauge duality is formulated as an equivalence between double loop and area integrals a la arXiv: 0708.1625, then this pure geometric relation can be tested for various choices of n-side polygons. The simplest possibility arises at n=infinity, with polygon substituted by an arbitrary continuous curve. If the curve is a circle, the minimal surface problem is exactly solvable. If it infinitesimally deviates from a circle, then the duality relation can be studied by expanding in powers of a small parameter. In the first approximation the Nambu-Goto (NG) equations can be linearized, and the peculiar NG Laplacian plays the central role. Making use of explicit zero-modes of this operator (NG-harmonic functions), we investigate the geometric duality in the lowest orders for small deformations of arbitrary shape lying in the plane of the original circle. We find a surprisingly strong dependence of the minimal area on regularization procedure affecting "the boundary terms" in minimal area. If these terms are totally omitted, the remaining piece is regularization independent, but still differs by simple numerical factors like 4 from the double-loop integral which represents the BDS formula so that we stop short from the first non-trivial confirmation of the Alday-Maldacena duality. This confirms the earlier-found discrepancy for two parallel lines at n=infinity, but demonstrates that it actually affects only a finite number (out of infinitely many) of parameters in the functional dependence on the shape of the boundary, and the duality is only slightly violated, which allows one to call this violation an anomaly.

25 pages, no figures; overall coefficients restored, an Appendix added

References in corpus (42)

• Wilson loops in large N field theories
• Macroscopic strings as heavy quarks: Large-N gauge theory and anti-de Sitter supergravity
• Modeling the fifth dimension with scalars and gravity
• Gluon scattering amplitudes at strong coupling
• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• On the Holographic Renormalization Group
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• Novel Local CFT and Exact Results on Perturbations of N=4 Super Yang Mills from AdS Dynamics
• MHV Amplitudes in N=4 Super Yang-Mills and Wilson Loops
• Gravitational Stability and Renormalization-Group Flow
• Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes
• On planar gluon amplitudes/Wilson loops duality
• Comments on gluon scattering amplitudes via AdS/CFT
• Hexagon Wilson loop = six-gluon MHV amplitude
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• Spacetime and the Holographic Renormalization Group
• A remark on the AdS/CFT correspondence and the renormalization group flow
• RG-Flow, Gravity and the Cosmological Constant
• The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude
• Geometric Holography, the Renormalization Group and the c-Theorem
• Simplifying superstring action on AdS_5 x S^5
• Strong-coupling expansion of cusp anomaly and gluon amplitudes from quantum open strings in AdS_5 x S^5
• Generating AdS String Solutions
• Dimensionally regulated one-loop box scalar integrals with massless internal lines
• Wavy Wilson Line and AdS/CFT
• Regge behavior of gluon scattering amplitudes in N=4 SYM theory
• New Dual Conformally Invariant Off-Shell Integrals
• Infrared Limit of Gluon Amplitudes at Strong Coupling
• Implications of multi-Regge limits for the Bern-Dixon-Smirnov conjecture
• Four-Loop Collinear Anomalous Dimension in N = 4 Yang-Mills Theory
• Planar quark scattering at strong coupling and universality
• Scattering amplitudes in strongly coupled N=4 SYM from semiclassical strings in AdS
• On collinear factorization of Wilson loops and MHV amplitudes in N=4 SYM
• Some properties of the Alday-Maldacena minimum
• Some Implications of Perturbative Approach to AdS/CFT Correspondence
• Conformal SO(2,4) Transformations of the One-Cusp Wilson Loop Surface
• On Renormalization Group in Abstract QFT
• Boundary Ring: a way to construct approximate NG solutions with polygon boundary conditions: I. Z_n-symmetric configurations
• Gluon scattering in $\mathcal{N}=4$ Super Yang-Mills at finite temperature
• Boundary Ring or a Way to Construct Approximate NG Solutions with Polygon Boundary Conditions. II. Polygons which admit an inscribed circle
• Comment on the Alday-Maldacena solution in calculating scattering amplitude via AdS/CFT
• On coincidence of Alday-Maldacena-regularized $\sigma$-model and Nambu-Goto areas of minimal surfaces

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• A Numerical Study of Gluon Scattering Amplitudes in N=4 Super Yang-Mills Theory at Strong Coupling
• Discretized Minimal Surface and the BDS Conjecture in N=4 Super Yang-Mills Theory at Strong Coupling
• Amplitudes in the N=4 SYM from Quantum Geometry of the Momentum Space
• Deviation from Alday-Maldacena Duality For Wavy Circle