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Symmetries of F-cohomological field theories and F-topological recursion

Gaëtan Borot, Alessandro Giacchetto, Giacomo Umer

TL;DR

This work develops F-topological recursion (F-TR), a non-symmetric analogue of topological recursion built from F-Airy structures and a vector potential, and connects it to F-cohomological field theories (F-CohFTs). It identifies a rich symmetry group extending the F-Givental action, including a novel tick action, and shows that F-TR governs ancestor vector potentials for F-CohFTs in the F-Givental orbit, up to translation and basis changes. The paper also proposes a spectral-curve formulation for F-TR, deriving F-Airy data from F-spectral curves and establishing a correspondence with semisimple F-CohFTs of the LRTOmega0 type. The extended 2-spin example illustrates the orbit dictionary and the interplay of linear and nonlinear symmetries. Overall, the results extend the topological-recursion/CohFT correspondence to the F-world and open questions about global spectral curves and reconstruction theorems for F-CohFTs.

Abstract

We define F-topological recursion (F-TR) as a non-symmetric version of topological recursion, which associates a vector potential to some initial data. We describe the symmetries of the initial data for F-TR and show that, at the level of the vector potential, they include the F-Givental (non-linear) symmetries studied by Arsie, Buryak, Lorenzoni, and Rossi within the framework of F-manifolds. Additionally, we propose a spectral curve formulation of F-topological recursion. This allows us to extend the correspondence between semisimple cohomological field theories (CohFTs) and topological recursion, as established by Dunin-Barkowski, Orantin, Shadrin, and Spitz, to the F-world. In the absence of a full reconstruction theorem à la Teleman for F-CohFTs, this demonstrates that F-TR holds for the ancestor vector potential of a given F-CohFT if and only if it holds for some F-CohFT in its F-Givental orbit. We turn this into a useful statement by showing that the correlation functions of F-topological field theories (F-CohFTs of cohomological degree 0) are governed by F-TR. We apply these results to the extended 2-spin F-CohFT. Furthermore, we exhibit a large set of linear symmetries of F-CohFTs, which do not commute with the F-Givental action.

Symmetries of F-cohomological field theories and F-topological recursion

TL;DR

This work develops F-topological recursion (F-TR), a non-symmetric analogue of topological recursion built from F-Airy structures and a vector potential, and connects it to F-cohomological field theories (F-CohFTs). It identifies a rich symmetry group extending the F-Givental action, including a novel tick action, and shows that F-TR governs ancestor vector potentials for F-CohFTs in the F-Givental orbit, up to translation and basis changes. The paper also proposes a spectral-curve formulation for F-TR, deriving F-Airy data from F-spectral curves and establishing a correspondence with semisimple F-CohFTs of the LRTOmega0 type. The extended 2-spin example illustrates the orbit dictionary and the interplay of linear and nonlinear symmetries. Overall, the results extend the topological-recursion/CohFT correspondence to the F-world and open questions about global spectral curves and reconstruction theorems for F-CohFTs.

Abstract

We define F-topological recursion (F-TR) as a non-symmetric version of topological recursion, which associates a vector potential to some initial data. We describe the symmetries of the initial data for F-TR and show that, at the level of the vector potential, they include the F-Givental (non-linear) symmetries studied by Arsie, Buryak, Lorenzoni, and Rossi within the framework of F-manifolds. Additionally, we propose a spectral curve formulation of F-topological recursion. This allows us to extend the correspondence between semisimple cohomological field theories (CohFTs) and topological recursion, as established by Dunin-Barkowski, Orantin, Shadrin, and Spitz, to the F-world. In the absence of a full reconstruction theorem à la Teleman for F-CohFTs, this demonstrates that F-TR holds for the ancestor vector potential of a given F-CohFT if and only if it holds for some F-CohFT in its F-Givental orbit. We turn this into a useful statement by showing that the correlation functions of F-topological field theories (F-CohFTs of cohomological degree 0) are governed by F-TR. We apply these results to the extended 2-spin F-CohFT. Furthermore, we exhibit a large set of linear symmetries of F-CohFTs, which do not commute with the F-Givental action.
Paper Structure (33 sections, 15 theorems, 188 equations, 1 figure)

This paper contains 33 sections, 15 theorems, 188 equations, 1 figure.

Table of Contents

  1. Introduction
  2. F-Airy structures
  3. Definition of F-Airy structures in finite dimension
  4. A graphical interpretation
  5. F-topological field theories
  6. Infinite-dimensional settings
  7. Actions on F-Airy structures
  8. Change of bases
  9. Bogoliubov transformation
  10. Translation
  11. F-cohomological field theories
  12. F-cohomological field theories
  13. Known symmetries of F-CohFTs
  14. Change of basis.
  15. R-action.
  16. ...and 18 more sections

Key Result

Proposition 2.3

The F-TR amplitudes are given by where the second sum ranges over all colourings $c \colon \mathsf{E}^{\varnothing}(\bm{G}) \to I$ satisfying $c(\ell_k) = i_k$.

Figures (1)

  • Figure 1: Example of a stable tree in $\mathbb{T}_{3,1+4}$ and corresponding notation. The genus decoration is depicted inside the vertices.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 36 more