Non-planar BCFW Grassmannian Geometries

Abstract

In this paper, we study non-adjacent BCFW recursion relations and their connection to positive geometry. For an adjacent BCFW shift, the $n$-point N${}^k$MHV tree-level amplitude in $\mathcal{N}=4$ SYM theory is expressed as a sum over planar on-shell diagrams, corresponding to canonical dlog forms on the cells in the positive Grassmannian $G_{+}(k,n)$. Non-adjacent BCFW shifts naturally lead to an expansion of the amplitude in terms of a different set of objects, which do not manifest the cyclic ordering and the hidden Yangian symmetry of the amplitude. We show that these terms can be interpreted as dlog forms on the non-planar Grassmannian geometries, generalizing the cells of the positive Grassmannian $G_{+}(k,n)$ to a larger class of objects which live in $G(k,n)$. We focus mainly on the case of NMHV amplitudes and discuss in detail the Grassmannian geometries. We also propose an alternative way to calculate the associated on-shell functions and dlog forms using an intriguing connection between Grassmannian configurations and the geometry in the kinematical space.

arXiv comment68 pages, 105 figures

References in corpus (93)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• New Relations for Gauge-Theory Amplitudes
• Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes
• New Recursion Relations for Tree Amplitudes of Gluons
• Gluon scattering amplitudes at strong coupling
• Perturbative Quantum Gravity as a Double Copy of Gauge Theory
• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
• The Amplituhedron
• Magic identities for conformal four-point integrals
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Local Integrals for Planar Scattering Amplitudes
• On planar gluon amplitudes/Wilson loops duality
• Multi-Leg One-Loop Gravity Amplitudes from Gauge Theory
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• A Duality For The S Matrix
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• Dual Superconformal Invariance, Momentum Twistors and Grassmannians
• The Ultraviolet Behavior of N=8 Supergravity at Four Loops
• Space-time S-matrix and Flux-tube S-matrix at Finite Coupling
• Motivic Amplitudes and Cluster Coordinates
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• The Grassmannian Origin Of Dual Superconformal Invariance
• Into the Amplituhedron
• A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon
• Notes on the scattering amplitude / Wilson loop duality
• Positive Geometries and Canonical Forms
• Enhanced Ultraviolet Cancellations in N = 5 Supergravity at Four Loop
• Ultraviolet Properties of N = 8 Supergravity at Five Loops
• All tree-level amplitudes in N=4 SYM
• Unification of Residues and Grassmannian Dualities
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• Unwinding the Amplituhedron in Binary
• Cluster adjacency properties of scattering amplitudes
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Positive Amplitudes In The Amplituhedron
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps
• Evidence for a Nonplanar Amplituhedron
• Gravity, Twistors and the MHV Formalism
• The Tree Formula for MHV Graviton Amplitudes
• Ultraviolet Cancellations in Half-Maximal Supergravity as a Consequence of the Double-Copy Structure
• Prescriptive Unitarity
• A Note on Polytopes for Scattering Amplitudes
• OPE for all Helicity Amplitudes
• The Momentum Amplituhedron
• Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM
• Local Spacetime Physics from the Grassmannian
• The Origin of the Six-Gluon Amplitude in Planar $\mathcal{N}=4$ SYM
• Hexagonal Wilson Loops in Planar $\mathcal{N}=4$ SYM Theory at Finite Coupling
• The Five-Loop Four-Point Amplitude of N=4 super-Yang-Mills Theory
• Simple loop integrals and amplitudes in N=4 SYM
• Gravity On-shell Diagrams
• The Amplituhedron from Momentum Twistor Diagrams
• Towards the Amplituhedron Volume
• Tree-Level Amplitudes in N=8 Supergravity
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• Folding Amplitudes into Form Factors: An Antipodal Duality
• Note on graviton MHV amplitudes
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Tropical Grassmannians, cluster algebras and scattering amplitudes
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• On-Shell Diagrams for N = 8 Supergravity Amplitudes
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• Notes on Scattering Amplitudes as Differential Forms
• Dual Conformal Structure Beyond the Planar Limit
• Implications of nonplanar dual conformal symmetry
• All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops
• The m=1 amplituhedron and cyclic hyperplane arrangements
• Symbol Alphabets from Plabic Graphs
• Amplitudes at Infinity
• Symbol Alphabets from Tensor Diagrams
• From 4d Ambitwistor Strings to On Shell Diagrams and Back
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• Manifesting enhanced cancellations in supergravity: integrands versus integrals
• UV cancelations in gravity loop integrands
• From Momentum Amplituhedron Boundaries to Amplitude Singularities and Back
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• The Geometry of On-Shell Diagrams
• Amplituhedron meets Jeffrey-Kirwan Residue
• Sign Flip Triangulations of the Amplituhedron
• Amplituhedra, and Beyond
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Amplituhedron-like geometries
• Gravity loop integrands from the ultraviolet
• Yangian Symmetry for the Tree Amplituhedron
• Tree-level scattering amplitudes from the amplituhedron
• $\mathcal{N}=7$ On-Shell Diagrams and Supergravity Amplitudes in Momentum Twistor Space
• $\Delta$-Algebra and Scattering Amplitudes
• Triangulation-free Trivialization of 2-loop MHV Amplituhedron
• All-loop Mondrian Reduction of 4-particle Amplituhedron at Positive Infinity

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