Topological Recursion, explicit ABCD tensors and implementation

TL;DR

This paper provides explicit ABCD tensor expressions for classical spectral curves within Topological Recursion and details an end-to-end algorithmic framework to compute TR invariants. By reformulating TR in terms of A,B,C,D tensors and a basis of 1-forms, it enables concrete, implementable recursion across diverse models, including Airy/KdV, GUE, and elliptic curves. It also extends the theory to higher ramification points and r-spin systems, and introduces a bespoke Python-based implementation to manage finite and infinite tensor data, with a public repository for community contributions. Collectively, the work offers both concrete computational tools and a roadmap for applying TR in enumerative geometry, random matrices, and conformal field theory contexts.

Abstract

We provide explicit expressions of ABCD tensors for the most classical classes of spectral curves. And we discuss algorithmic implementation of Topological Recursion.