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Degenerate and irregular topological recursion

Alexander Alexandrov, Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, Sergey Shadrin

Abstract

We use the theory of $x-y$ duality to propose a new definition / construction for the correlation differentials of topological recursion; we call it "generalized topological recursion". This new definition coincides with the original topological recursion of Chekhov-Eynard-Orantin in the regular case and allows, in particular, to get meaningful answers in a variety of irregular and degenerate situations.

Degenerate and irregular topological recursion

Abstract

We use the theory of duality to propose a new definition / construction for the correlation differentials of topological recursion; we call it "generalized topological recursion". This new definition coincides with the original topological recursion of Chekhov-Eynard-Orantin in the regular case and allows, in particular, to get meaningful answers in a variety of irregular and degenerate situations.
Paper Structure (36 sections, 18 theorems, 58 equations)

This paper contains 36 sections, 18 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Various kinds of topological recursion
  3. Towards a new definition through x-y duality
  4. Advantages and properties of the new approach
  5. Organization of the paper
  6. Notation
  7. Acknowledgments
  8. Generalized topological recursion
  9. Original CEO topological recursion
  10. Initial data of generalized topological recursion
  11. Definition of generalized topological recursion
  12. x-y duality
  13. Recollection
  14. Explicit formulas
  15. Formal properties of the x-y duality relation
  16. ...and 21 more sections

Key Result

Proposition 2.14

The differentials of generalized topological recursion $\omega^{(g)}_{n}$ (in the sense of Definition def:TRgeneral, that is, defined by eq:newproj--eq:newTR) are symmetric in all $n$ arguments.

Theorems & Definitions (56)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8: Generalized topological recursion
  • Remark 2.9
  • Remark 2.10
  • ...and 46 more