KP integrability through the $$x-y$$ swap relation
Alexander S. Alexandrov, Boris Sergeevich Bychkov, Petr Dunin‐Barkowski, Maxim Eduardovich Kazarian, Sergey Viktorovich Shadrin · Selecta Mathematica · 2025
Abstract We discuss a universal relation that we call the $$x-y$$ x - y 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 $$x-y$$ x - y 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.