The Correlahedron

Abstract

We introduce a new geometric object, the correlahedron, which we conjecture to be equivalent to stress-energy correlators in planar N=4 super Yang-Mills. Re-expressing the Grassmann dependence of correlation functions of n chiral stress-energy multiplets with Grassmann degree 4k in terms of 4(n+k)-linear bosonic variables, the resulting expressions have an interpretation as volume forms on a Gr(n+k,4+n+k) Grassmannian, analogous to the expressions for planar amplitudes via the amplituhedron. The resulting volume forms are to be naturally associated with the correlahedron geometry. We construct such expressions in this bosonised space both directly, in general, from Feynman diagrams in twistor space, and then more invariantly from specific known correlator expressions in analytic superspace. We give a geometric interpretation of the action of the consecutive lightlike limit and show that under this the correlahedron reduces to the squared amplituhedron both as a geometric object as well as directly on the corresponding volume forms. We give an explicit easily implementable algorithm via cylindrical decompositions for extracting the squared amplituhedron volume form from the squared amplituhedron geometry with explicit examples and discuss the analogous procedure for the correlators.

38 pages. minor corrections

References in corpus (7)

• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
• Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps
• MHV Diagrams in Momentum Twistor Space
• On BCFW shifts of integrands and integrals
• The Amplituhedron from Momentum Twistor Diagrams
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Cited by in corpus (20)

• Unwinding the Amplituhedron in Binary
• The Momentum Amplituhedron
• The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• Amplituhedron meets Jeffrey-Kirwan Residue
• Amplituhedra, and Beyond
• Amplituhedron-like geometries
• Higher-Point Integrands in N=4 super Yang-Mills Theory
• Internal boundaries of the loop amplituhedron
• Multi-Particle Amplitudes from the Four-Point Correlator in Planar N=4 SYM
• The all-loop conjecture for integrands of reggeon amplitudes in N=4 SYM
• Open associahedra and scattering forms
• The twistor Wilson loop and the amplituhedron
• A note on NMHV form factors from the Graßmannian and the twistor string
• Higher-Point Correlators in N=4 SYM: Generating Functions
• Wilson loops in SYM $N=4$ do not parametrize an orientable space
• Leading singularities and chambers of Correlahedron
• Positive Geometry for Stringy Scalar Amplitudes
• The Geometry of BCFW for ABJM Loop Integrands
• Higher-Point Correlators in N=4 SYM: Ten-Dimensional Null Polygons