Blobbed topological recursion and KP integrability

TL;DR

This work extends blobbed topological recursion by embedding it into the generalized topological recursion framework and permitting blobs without topological expansions. It proves that non-perturbative and convolution-constructed differentials arise naturally within this extended blobbed TR and establishes KP integrability for blobbed TR when the blobs are KP-integrable, providing a unified, global proof linking several strands of topological recursion and integrable systems. By developing a convolution formalism and a robust graph-based recursion, the paper connects KP tau-function theory, Miwa-variable realizations, and theta-function representations to blobbed TR. The results unify, generalize, and give a new perspective on Borot–Eynard type conjectures about non-perturbative differentials, while clarifying multi-KP integrability and offering practical recursions for computing blobbed TR quantities.

Abstract

We revise the notion of the blobbed topological recursion by extending it to the setting of generalized topological recursion as well as allowing blobs which do not necessarily admit topological expansion. We show that the so-called non-perturbative differentials form a special case of this revisited version of blobbed topological recursion. Furthermore, we prove the KP integrability of the differentials of blobbed topological recursion for the input data that include KP-integrable blobs. This result generalizes, unifies, and gives a new proof of the KP integrability of nonperturbative differentials conjectured by Borot--Eynard and recently proved by the authors.

Theorems & Definitions (56)

• Definition 2.1
• Proposition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• proof : Proof of bilinear relations \ref{['eq:Hirota-KP']}
• Remark 2.5
• Proposition 2.6
• Theorem 2.7
• proof