$x$-$y$ symplectic transformation in Topological Recursion and applications

Alexander Hock (U Oxford)

Apr 25. 2023, 10:30 — 11:30

Topological Recursion (TR) has a wide range of applications. Thus, different properties of TR have considerable meaning in each of these examples. One important property already considered from the beginning is the behavior under the symplectic transformations, i.e. a transformation leaving the symplectic form $dx \wedge dy$ invariant, where $(x,y)$ builds the spectral curve.

I want to review recent developments of the so-called $x$-$y$ symplectic transformation which interchanges the role of $x$ and $y$. Let $\omega_{g,n}$ be generated by TR with the spectral curve $(x,y)$ and $\omega^{vee}_{g,n}$ by $(y,x)$, then I want to present a ''simple'' functional relation between $\omega_{g,n}$ and $\omega^{vee}_{g,n}$. This relation was first computed by myself for a specific class of spectral curves based on the work of Borot, Charbonnier, Garcia-Failde, Leid and Shadrin. Shortly later, Alexandrov, Bychkov, Dunin-Barkowski, Kazarian and Shadrin actually proved that this simple functional relation holds for any spectral curve with meromorphic $x,y$ and simple distinct ramification. The functional relation gives interesting insight in almost all examples governed by TR. We get for instance new formulas for intersection numbers on $\overline{\mathcal{M}}_{g,n}$. Especially, the relation between ordinary vs fully simple maps, which is the $x$-$y$ symplectic transformation, is interesting since it computes the relation between higher order moments and higher order free cumulants in free probability.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Non-commutative Geometry meets Topological Recursion (Workshop)
Gaëtan Borot (HU Berlin)
Elba Garcia Failde (Sorbonne U, Paris)
Harald Grosse (U of Vienna)
Masoud Khalkhali (Western U, Ontario)
Hannah Markwig (U Tübingen)
Raimar Wulkenhaar (U Münster)