Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion

The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications; for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal power series in $\hbar$. We apply the Laplace transformed formula to the Airy curve and the Lambert curve which provides simple formulas for $\psi$-class intersections numbers and Hodge integrals on $\overline{\mathcal{M}}_{g,n}$.

Keywords

topological recursion, enumerative algebraic geometry, intersection numbers, moduli space of complex curves

2010 Mathematics Subject Classification

Primary 05A15, 14N10. Secondary 14H70, 30F30.

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This work supported by a Walter Benjamin Fellowshop funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 465029630.

Received 15 May 2023

Accepted 31 July 2023

Published 24 January 2024