graphical-coaction-frw-integrals

name	graphical-coaction-frw-integrals
category	quantum-computing
description	Graphical coaction methodology for FRW integrals using twisted (co)homology intersection theory — decomposing cosmological integrals into diagram-decorated building blocks.
tags	["quantum","cosmology","cohomology","integrals","graphical-methods","FRW"]
created	2026-06-12T00:00:00.000Z
source	arxiv:2606.13627

Graphical Coaction for FRW Integrals from Twisted (Co)homology

Summary

Constructs a graphical coaction for Friedmann-Robertson-Walker (FRW) integrals at all loop orders in conformally-coupled scalar theories. Uses intersection theory in twisted (co)homology to decompose integrals into building blocks represented as decorations of the original Feynman diagram.

Key Contributions

Graphical Coaction Construction

• Systematic decomposition of FRW integrals into elementary building blocks
• Each block corresponds to a decorated version of the original Feynman diagram
• Works at all loop orders in conformally-coupled scalar theories

Intersection Theory Framework

• Uses (partial/relative) twisted (co)homology groups
• Intersection pairings provide algebraic structure for integral relations
• Enables systematic computation of discontinuities and derivatives

Applications

• Cosmological correlator calculations
• Quantum field theory in curved spacetime
• Loop integral reduction and simplification
• Analytic continuation of cosmological amplitudes

Mathematical Framework

Twisted Cohomology

• Cohomology with coefficients in a local system defined by the integral's multivalued functions
• Provides basis for integral relations and reduction identities

Graphical Rules

• Each diagram decoration encodes specific integral operations
• Coaction maps integrals to tensor products of simpler integrals
• Preserves physical properties (unitarity, analyticity)

When to Use

• Computing cosmological correlation functions
• Simplifying multi-loop Feynman integrals
• Studying analytic structure of cosmological amplitudes
• Building integral reduction algorithms

Implementation Considerations

• Requires understanding of intersection theory
• Diagrammatic rules can be automated
• Compatible with existing Feynman integral software
• Extends to non-conformal polynomial interactions

Related Concepts

• Twisted de Rham cohomology
• Feynman integral reduction
• Cosmological bootstrap
• Intersection numbers
• Symbol and coaction methods