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

Alexander Hock

Abstract

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.

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

Abstract

The functional relation coming from the 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 . We apply the Laplace transformed formula to the Airy curve and the Lambert curve.
Paper Structure (8 sections, 10 theorems, 52 equations)

This paper contains 8 sections, 10 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Formula for $x-y$ Symplectic Transformation in Topological Recursion
  3. $x-y$ Symplectic Transformation
  4. Laplace transform of the $x-y$ symplectic transformation
  5. Logarithmic $x,y$
  6. Application to Intersection Numbers on $\overline{\mathcal{M}}_{g,n}$
  7. Airy curve
  8. Lambert curve

Key Result

Theorem 2.3

Let $x,y$ be two meromorphic functions on a compact Riemann surface with simple distinct ramification points, which generates via TR eq:TR-intro the multi-differentials $\omega_{g,n}$ and $\omega_{g,n}^\vee$ as above. Let $\Phi_n,\Phi_n^\vee, W_n,W_n^\vee$ be as above defined from $\omega_{g,n}$ and and for $I=\{i,i\}$ (and genus zero spectral curve) the special case (for higher genus spectral cu

Theorems & Definitions (24)

  • Example 2.1
  • Definition 2.2
  • Theorem 2.3: Hock:2022pbwAlexandrov:2022ydc
  • proof
  • Example 2.4
  • Corollary 2.5
  • proof
  • Proposition 2.6
  • proof
  • Example 2.7
  • ...and 14 more