A physical basis for cosmological correlators from cuts

TL;DR

This work shows that FRW cosmological correlators can be understood as twisted integrals whose analytic content is governed by a dual relative twisted cohomology organized by cuts. By pairing the FRW cohomology with a dual basis via the intersection number, the physical subspace—forms whose residues overlap with the FRW integrand—emerges naturally, yielding integrable differential equations that separate kinematic and ε-dependent structures. The authors provide a concrete, diagrammatic method (cut tubings) to enumerate physical cuts and construct the corresponding FRW forms, including a careful treatment of degenerate cuts via linear relations among hyperplane polynomials. The 3-site chain serves as a pedagogical exemplar showing how physical cuts reproduce a 16-element tree-level physical basis, with extensions to larger trees and loops yielding results consistent with the kinematic flow algorithm and prior time-integral approaches. Overall, the paper establishes a geometric, cohomological foundation for predicting the space of physical FRW differential forms and their DEQs, with potential implications for diagrammatic unitarity and coaction in cosmological perturbation theory.

Abstract

Significant progress has been made in our understanding of the analytic structure of FRW wavefunction coefficients, facilitated by the development of efficient algorithms to derive the differential equations they satisfy. Moreover, recent findings indicate that the twisted cohomology of the associated hyperplane arrangement defining FRW integrals overestimates the number of integrals required to define differential equations for the wavefunction coefficient. We demonstrate that the associated dual cohomology is automatically organized in a way that is ideal for understanding and exploiting the cut/residue structure of FRW integrals. Utilizing this understanding, we develop a systematic approach to organize compatible sequential residues, which dictates the physical subspace of FRW integrals for any $n$-site, $\ell$-loop graph. In particular, the physical subspace of tree-level FRW wavefunction coefficients is populated by differential forms associated to cuts/residues that factorize the integrand of the wavefunction coefficient into only flat space amplitudes. After demonstrating the validity of our construction using intersection theory, we develop simple graphical rules for cut tubings that enumerate the space of physical cuts and, consequently, differential forms without any calculation.