Think globally, compute locally

Abstract

We introduce a new formulation of the so-called topological recursion, that is defined globally on a compact Riemann surface. We prove that it is equivalent to the generalized recursion for spectral curves with arbitrary ramification. Using this global formulation, we also prove that the correlation functions constructed from the recursion for curves with arbitrary ramification can be obtained as suitable limits of correlation functions for curves with only simple ramification. It then follows that they both satisfy the properties that were originally proved only for curves with simple ramification.

37 pages, v2: published version

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• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
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• Weighted Hurwitz numbers and topological recursion: an overview
• Weighted Hurwitz numbers and topological recursion
• Rational differential systems, loop equations, and application to the q-th reductions of KP
• Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
• The Eynard--Orantin recursion for the total ancestor potential
• Double Hurwitz numbers: polynomiality, topological recursion and intersection theory
• Super Quantum Airy Structures
• Higher Airy structures, W algebras and topological recursion
• Supereigenvalue Models and Topological Recursion
• Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Quantum curves from refined topological recursion: the genus 0 case
• KP integrability through the $x-y$ swap relation
• Schwinger-Dyson and loop equations for a product of square Ginibre random matrices
• Generalised ordinary vs fully simple duality for $n$-point functions and a proof of the Borot--Garcia-Failde conjecture
• Nahm sums, quiver A-polynomials and topological recursion
• Topological recursion and uncoupled BPS structures II: Voros symbols and the $\tau$-function
• Degenerate and irregular topological recursion
• Complete solution of the LSZ Model via Topological Recursion
• $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
• Super Topological Recursion and Gaiotto Vectors For Superconformal Blocks
• Shifted Witten classes and topological recursion
• Symplectic duality for topological recursion
• Taking limits in topological recursion
• Symplectic (Non-)Invariance of the Free Energy in Topological Recursion
• $\mathcal{N}=1$ Super Topological Recursion
• Refined topological recursion revisited -- properties and conjectures
• The factorial growth of topological recursion
• Topological recursion on transalgebraic spectral curves and Atlantes Hurwitz numbers
• A New Class of Higher Quantum Airy Structures as Modules of $\mathcal{W}({\mathfrak{gl}}_r)$-Algebras