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Cut-and-join operators in cohomological field theory and topological recursion

Alexander Alexandrov

Abstract

We construct a cubic cut-and-join operator description for the partition function of the Chekhov-Eynard-Orantin topological recursion for a local spectral curve with simple ramification points. In particular, this class contains partition functions of all semi-simple cohomological field theories. The cut-and-join description leads to an algebraic version of topological recursion. For the same partition functions we also derive N families of the Virasoro constraints and prove that these constraints, supplemented by a deformed dimension constraint, imply the cut-and-join description.

Cut-and-join operators in cohomological field theory and topological recursion

Abstract

We construct a cubic cut-and-join operator description for the partition function of the Chekhov-Eynard-Orantin topological recursion for a local spectral curve with simple ramification points. In particular, this class contains partition functions of all semi-simple cohomological field theories. The cut-and-join description leads to an algebraic version of topological recursion. For the same partition functions we also derive N families of the Virasoro constraints and prove that these constraints, supplemented by a deformed dimension constraint, imply the cut-and-join description.
Paper Structure (29 sections, 32 theorems, 218 equations)

This paper contains 29 sections, 32 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Givental decomposition formula
  3. Main statement
  4. Open problems
  5. Conventions
  6. Cut-and-join operators for KW and BGW tau-functions
  7. Intersection numbers and KdV tau-function
  8. Norbury's classes and KdV tau-function
  9. Virasoro constraints and cut-and-join operators
  10. From translations to Virasoro group elements
  11. Cohomological field theories
  12. Cohomological field theory
  13. Semi-simple CohFTs and normalized canonical basis
  14. Givental--Teleman group
  15. Givental--Teleman group action on the partition functions
  16. ...and 14 more sections

Key Result

Theorem 1

A generalized ancestor potential satisfies

Theorems & Definitions (63)

  • Theorem 1
  • Corollary 1.1
  • Theorem 2
  • Theorem 3: Kon92
  • Theorem 4: NorbPr
  • Theorem 5: KSSKS2
  • Definition 1
  • Corollary 2.1
  • Lemma 2.2: H3
  • Definition 2
  • ...and 53 more