Airy structures and symplectic geometry of topological recursion
Maxim Kontsevich, Yan Soibelman
TL;DR
This work reframes topological recursion as deformation quantization of quadratic Airy structures, shifting the focus from spectral curves to tensor data on a space of meromorphic 1-forms. By introducing four tensors A,B,C,ε and developing both classical and quantum Airy structures (including infinite-dimensional generalizations), the authors connect TR to a systematic algebraic framework with DQ-modules and a canonical wave function. They extend the formalism to generalized spectral curves on Poisson surfaces with foliation, establish local affine/slq-embedding structures for moduli of spectral curves, and formulate a Holomorphic Anomaly Equation within this quantization. The paper also proposes a modified TR that avoids local involutions, identifies a common Airy substructure with conventional TR, and offers speculative directions toward dg-Airy structures and Calabi–Yau quantization, signaling a broad, principled foundation for TR beyond traditional curve-based data.
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.