  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.
