We discuss a universal relation that we call the xโy swap relation, which
plays a prominent role in the theory of topological recursion, Hurwitz theory,
and free probability theory. We describe in a very precise and detailed way the
interaction of the xโy swap relation and KP integrability. As an application,
we prove a recent conjecture that relates some particular instances of
topological recursion to the Mironov-Morozov-Semenoff matrix integrals.Comment: 27 pages; minor correction