The Correlahedron

TL;DR

The paper introduces the correlahedron, a new geometric object conjectured to represent stress-energy correlators in planar N=4 super Yang-Mills, extending the amplituhedron concept to correlation functions.

Contribution

It defines the correlahedron as a geometric volume form in a bosonized Grassmannian space, linking correlation functions to geometric objects similar to the amplituhedron for scattering amplitudes.

Findings

Correlahedron volume forms are constructed from Feynman diagrams and known correlator expressions.

Under lightlike limits, the correlahedron reduces to the squared amplituhedron.

An explicit algorithm for extracting the squared amplituhedron volume form is provided.

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.