Connected chord diagrams and bridgeless maps

TL;DR

This paper establishes a bijection between rooted connected chord diagrams and rooted bridgeless maps, simplifying complex quantum field theory calculations and providing new combinatorial proofs and interpretations, with applications to lambda calculus.

Contribution

It introduces a bijection between chord diagrams and bridgeless maps, extending to indecomposable diagrams and maps, and applies this to quantum field theory and lambda calculus.

Findings

A bijection between rooted connected chord diagrams and bridgeless maps.

A combinatorial proof of a previously recurrence-derived formula.

A new interpretation of Arquès and Béraud's generating function equation.

Abstract

We present a surprisingly new connection between two well-studied combinatorial classes: rooted connected chord diagrams on one hand, and rooted bridgeless combinatorial maps on the other hand. We describe a bijection between these two classes, which naturally extends to indecomposable diagrams and general rooted maps. As an application, this bijection provides a simplifying framework for some technical quantum field theory work realized by some of the authors. Most notably, an important but technical parameter naturally translates to vertices at the level of maps. We also give a combinatorial proof to a formula which previously resulted from a technical recurrence, and with similar ideas we prove a conjecture of Hihn. Independently, we revisit an equation due to Arqu\`es and B\'eraud for the generating function counting rooted maps with respect to edges and vertices, giving a new bijective interpretation of this equation directly on indecomposable chord diagrams, which moreover can be specialized to connected diagrams and refined to incorporate the number of crossings. Finally, we explain how these results have a simple application to the combinatorics of lambda calculus, verifying the conjecture that a certain natural family of lambda terms is equinumerous with bridgeless maps.