Schwinger, ltd: Loop-tree duality in the parametric representation

TL;DR

This paper develops a new variant of loop-tree duality for Feynman integrals using the Schwinger parametric representation, involving a domain decomposition into cells associated with spanning trees, and explores topological and homological implications.

Contribution

It introduces a novel loop-tree duality formulation based on domain decomposition into cells linked to spanning trees, providing new insights into graph moduli spaces and potential applications in graph cohomology.

Findings

Derived a loop-tree duality variant via domain decomposition

Proved homotopy equivalence between graph moduli space and its spine

Outlined potential applications to Kontsevich's graph cohomology

Abstract

We derive a variant of the loop-tree duality for Feynman integrals in the Schwinger parametric representation. This is achieved by decomposing the integration domain into a disjoint union of cells, one for each spanning tree of the graph under consideration. Each of these cells is the total space of a fiber bundle with contractible fibers over a cube. Loop-tree duality emerges then as the result of first decomposing the integration domain, then integrating along the fibers of each fiber bundle. As a byproduct we obtain a new proof that the moduli space of graphs is homotopy equivalent to its spine. In addition, we outline a potential application to Kontsevich's graph (co-)homology.