# Airy structures and symplectic geometry of topological recursion

## TL;DR

This paper introduces Airy structures as a fundamental framework for understanding topological recursion, linking it to symplectic geometry, quantization, and deformation theory, thus broadening the conceptual foundation of the field.

## Contribution

The paper presents Airy structures as a more fundamental approach to topological recursion, connecting it to symplectic geometry and Poisson surfaces, and explores their quantization and deformation theory.

## Key findings

- Airy structures provide a new foundational perspective on topological recursion.
- Quantization of Airy structures naturally yields topological recursion formulas.
- Deformation theory of spectral curves relates to Airy structures.

## Abstract

We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. The notion of spectral curve is also considered in a more general framework of Poisson surfaces endowed with foliation. We explain how the deformation theory of spectral curves is related to Airy structures. Few other topics (e.g. the Holomorphic Anomaly Equation) are also discussed from the general point of view of Airy structures.