Scattering Amplitudes and Wilson Loops in Twistor Space

Abstract

This article reviews the recent progress in twistor approaches to Wilson loops, amplitudes and their duality for N=4 super Yang-Mills. Wilson loops and amplitudes are derived from first principles using the twistor action for maximally supersymmetric Yang-Mills theory. We start by deriving the MHV rules for gauge theory amplitudes from the twistor action in an axial gauge in twistor space, and show that this gives rise to the original momentum space version given by Cachazo, Svrcek and Witten. We then go on to obtain from these the construction of the momentum twistor space loop integrand using (planar) MHV rules and show how it arises as the expectation value of a holomorphic Wilson loop in twistor space. We explain the connection between the holomorphic Wilson loop and certain light-cone limits of correlation functions. We give a brief review of other ideas in connection with amplitudes in twistor space: twistor-strings, recursion in twistor space, the Grassmannian residue formula for leading singularities and amplitudes as polytopes. This article is an invited review for a special issue of Journal of Physics A devoted to `Scattering Amplitudes in Gauge Theories'.

62 pages, 15 figures. v2: typos corrected, additional discussions and references added

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• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Supersymmetric gauge theory and the Yangian
• Gravity from Rational Curves
• Scattering amplitudes: the most perfect microscopic structures in the universe
• The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops
• D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops
• Twistor theory at fifty: from contour integrals to twistor strings
• Conformal and Einstein gravity from twistor actions
• Pure spinor superfields -- an overview
• An Introduction to On-shell Recursion Relations
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• Folding Amplitudes into Form Factors: An Antipodal Duality
• Ambidextrous light transforms for celestial amplitudes
• From the Holomorphic Wilson Loop to `d log' Loop-Integrands of Super-Yang-Mills Amplitudes
• From dlogs to dilogs; the super Yang-Mills MHV amplitude revisited
• Composite Operators in the Twistor Formulation of $\mathcal{N}=4$ SYM Theory
• Graviton scattering in self-dual radiative space-times
• Antipodal Self-Duality for a Four-Particle Form Factor
• Twistor methods for AdS${}_5$
• Twistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory
• Twistor Constructions for Higher-spin Extensions of (Self-dual) Yang-Mills
• On Twistors and Conformal Field Theories from Six Dimensions
• Exact Superconformal and Yangian Symmetry of Scattering Amplitudes
• Correlation functions, null polygonal Wilson loops, and local operators
• Gravity with a cosmological constant from rational curves
• Higher-spin Yang-Mills, amplitudes and self-duality
• All-order celestial OPE in the MHV sector
• Conformal Higher Spin Theory and Twistor Space Actions
• OPE for Super Loops
• Gluon scattering on self-dual radiative gauge fields
• $L_{\infty }$-Algebras, the BV Formalism, and Classical Fields
• Twistor-strings and gravity tree amplitudes
• Supersymmetric celestial OPEs and soft algebras from the ambitwistor string worldsheet
• One-Loop Soft Theorems via Dual Superconformal Symmetry
• The holographic dual of the Penrose transform
• Wilson Loop Duality and OPE for Super Form Factors of Half-BPS Operators
• Evolution of cusped light-like Wilson loops and geometry of the loop space
• Minitwistors and 3d Yang-Mills-Higgs theory
• The SAGEX Review on Scattering Amplitudes, Chapter 15: The Multi-Regge Limit
• On the Geometry of Null Polygons in Full N=4 Superspace
• Twistors and antipodes in de Sitter space
• Collinear and Soft Limits of Multi-Loop Integrands in N=4 Yang-Mills
• Twistor fishnets
• Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity
• Wilson Loop diagrams and Positroids
• WiLE: a Mathematica package for weak coupling expansion of Wilson loops in ABJ(M) theory
• A Two-Loop Four-Point Form Factor at Function Level
• On Closed Twistor String Theory
• Ambitwistor strings and reggeon amplitudes in N=4 SYM
• Twistors, Harmonics and Holomorphic Chern-Simons
• A Celestial Description of Planar Super-Yang-Mills Theory
• Ambitwistor Yang-Mills Theory Revisited