F-DATA: Non-Symmetric Topological Recursion

• F-DATA is a structured quintuple of tensors that defines an F-Airy structure, producing vector-valued ancestor potentials for F-CohFTs.
• It differentiates connected and disconnected contributions by employing distinct tensors, thereby establishing a non-symmetric recursion framework.
• The spectral-curve realization and symmetry actions of F-DATA extend classical topological recursion, providing a bridge to F-Givental orbits and extended 2-spin examples.

Searching arXiv for the target paper and closely related F-CohFT/F-topological recursion work to ground the article. F-DATA, in the sense developed for F-topological recursion, denotes the initial data that determine a non-symmetric recursion producing a vector-valued ancestor potential rather than the scalar potentials familiar from Eynard–Orantin topological recursion. In this framework, F-topological recursion (F-TR) is formulated through F-Airy structures or, equivalently, through an F-spectral curve, and it is designed to interact with F-cohomological field theories (F-CohFTs), F-manifolds, and the non-linear F-Givental symmetries studied by Arsie, Buryak, Lorenzoni, and Rossi. The central result is that the ancestor vector potential of an F-CohFT is governed by F-TR in a way stable under the F-Givental orbit, and that topological F-CohFTs, including the extended 2-spin example, admit an explicit F-TR description (Borot et al., 2024).

1. Formal definition of F-DATA

In the algebraic formulation, F-DATA is the quintuple of tensors

$(A,B,C^{\connsymb},C^{\discsymb},D) \in (\mathrm{Sym}^2V,V)\times (V^{\otimes 2},V)\times (V,V^{\otimes 2})\times (\mathrm{Sym}^2V,V)\times V,$

where $V$ is a graded vector space over $\mathbb{C}$. This quintuple defines an F-Airy structure and determines recursively the amplitudes

$$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$

with base cases $F_{0,3}=A$ and $F_{1,1}=D$ (Borot et al., 2024).

The recursion is

$$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$

$+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$

with unstable terms $F_{0,1}=F_{0,2}=0$. In a basis $(e_i)_{i\in I}$, the coefficients satisfy the component recursion displayed in equation (2.6) of the source (Borot et al., 2024).

The associated “vector potential” is

$V$0

This is the primary output of F-TR.

A defining feature of F-DATA is non-symmetry. The amplitudes have one distinguished output slot, and $V$1 and $V$2 are independent tensors. This separates the connected and disconnected contributions already at the level of initial data. Unlike classical Airy structures, no non-linear compatibility constraints are imposed on $V$3 beyond the stated tensor symmetries. This suggests that F-DATA is structurally simpler than the symmetric Airy-structure setting, while producing a more general, non-symmetric output (Borot et al., 2024).

2. Non-symmetric recursion and its relation to classical topological recursion

F-TR is described explicitly as a non-symmetric variant of topological recursion. The non-symmetric aspect appears in three places. First, the amplitudes are $V$4, not fully symmetric $V$5-forms. Second, the recursion distinguishes the output variable from the input variables. Third, the connected and disconnected terms are weighted by different tensors, $V$6 and $V$7 (Borot et al., 2024).

If $V$8 is endowed with a non-degenerate pairing $V$9, then inputs and output can be identified. In the special case in which $\mathbb{C}$0 and the data define an Airy structure, the amplitudes become fully symmetric and the vector potential becomes a gradient,

$\mathbb{C}$1

for a scalar potential $\mathbb{C}$2. This identifies the usual symmetric topological-recursion regime as a special case inside the F-framework (Borot et al., 2024).

This comparison clarifies a common misconception. F-TR is not merely classical topological recursion with relabeled variables. It modifies the algebraic type of the outputs and duplicates the bidifferential and kernel data in the spectral-curve picture. This suggests that the “F” formalism is not a small perturbation of EO recursion but a systematically enlarged non-symmetric theory.

The paper places this construction in the broader context of the correspondence between semisimple CohFTs and topological recursion established by Dunin-Barkowski, Orantin, Shadrin, and Spitz, and extends that correspondence to the F-world. Because a Teleman-style full reconstruction theorem is absent for F-CohFTs, the extension is phrased in orbit-theoretic terms rather than as a complete classification (Borot et al., 2024).

3. Spectral-curve realization of F-DATA

The geometric formulation packages F-DATA into an F-spectral curve

$\mathbb{C}$3

Here $\mathbb{C}$4 is a smooth, possibly non-compact and disconnected complex curve; $\mathbb{C}$5 are meromorphic functions; $\mathbb{C}$6 has finitely many simple ramification points $\mathbb{C}$7; $\mathbb{C}$8 is holomorphic and non-zero at $\mathbb{C}$9; $$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$0 and $$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$1 are bidifferentials on $$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$2, holomorphic away from the diagonal and having a double pole with leading coefficient $$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$3 on the diagonal; and $$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$4 is a set of scalar ramification weights (Borot et al., 2024).

At each branch point $$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$5 there is a local involution $$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$6 satisfying

$$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$7

Two projectors are defined on meromorphic $$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$8-forms with poles only at $$F_{g,1+n}\in (\mathrm{Sym}^n V,V), \qquad g,n\ge 0,\quad 2g-2+(1+n)>0,$$9,

$F_{0,3}=A$0

where $F_{0,3}=A$1 and $F_{0,3}=A$2. They induce decompositions

$F_{0,3}=A$3

The recursion kernels are

$F_{0,3}=A$4

From them one defines two recursion operators,

$F_{0,3}=A$5

$F_{0,3}=A$6

The correlators satisfy

$F_{0,3}=A$7

$F_{0,3}=A$8

with the starred sum excluding unstable $F_{0,3}=A$9 factors (Borot et al., 2024).

This spectral-curve description exhibits the geometric meaning of F-DATA. Classical EO recursion uses a single bidifferential and a symmetric output. F-TR instead uses two bidifferentials, two kernels, a ramification-weight vector $F_{1,1}=D$0, and a distinguished output slot. The loop equations hold in a corresponding non-symmetric form: for each branch point $F_{1,1}=D$1, the combinations

$F_{1,1}=D$2

and similarly in any input variable are holomorphic near $F_{1,1}=D$3 (Borot et al., 2024).

4. F-CohFTs, topological F-theories, and the F-Givental orbit

An F-CohFT consists of maps

$F_{1,1}=D$4

satisfying $F_{1,1}=D$5-equivariance and compatibility with separating gluing, together with a flat unit $F_{1,1}=D$6. The associated amplitudes are defined on loop spaces $F_{1,1}=D$7 and $F_{1,1}=D$8 using up/down morphisms

$F_{1,1}=D$9

which are two-sided inverses under residue pairing (Borot et al., 2024).

The amplitudes attached to an F-CohFT are

$$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$0

Their vector potential is again

$$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$1

A crucial theorem states that topological F-CohFTs, namely F-CohFTs of cohomological degree $$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$2, are governed by F-TR. For an F-topological field theory $$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$3 with amplitudes

$$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$4

the F-TR initial data are

$$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$5

These define an F-Airy structure computing the theory’s amplitudes up to graph factors (Borot et al., 2024).

The orbit structure is controlled by the F-Givental action. Change of basis $$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$6, Givental $$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$7, and translations $$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$8 act on F-CohFTs and on the corresponding F-TR data. The identification theorem shows that if an F-CohFT is computed by F-TR, then any transform

$$F_{g,1+n}(v_1\otimes \cdots \otimes v_n) = \sum_{m=1}^n B\Big( v_m\otimes F_{g,1+(n-1)}(v_1\otimes \cdots \otimes \widehat{v_m}\otimes \cdots \otimes v_n) \Big)$$9

is also computed by F-TR, with explicitly transformed initial data (Borot et al., 2024).

The paper states this as an if-and-only-if along the F-Givental orbit: 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. Since a full Teleman-style reconstruction theorem is not available, this orbit-stability is the operative replacement. A plausible implication is that F-DATA functions as the recursion-theoretic invariant that can be transported along non-linear symmetries even when a full classification theorem is missing.

5. Symmetries of F-DATA

Three universal actions on F-Airy structures are described: change of bases, Bogoliubov transformations, and translations (Borot et al., 2024).

For a pair $+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$0, the transformed amplitudes are

$+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$1

and the tensors $+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$2 transform accordingly.

Bogoliubov transformations are parametrized by $+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$3. The transformed amplitudes are sums over rooted trees,

$+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$4

and the initial data change by

$+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$5

$+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$6

The corresponding vector potential satisfies the fixed-point equation

$+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$7

Translations are given by insertion of a parameter $+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$8,

$+\frac12 \operatorname{Tr}\Big( C^{\connsymb}\circ F_{g-1,1+(n+1)}(v_1\otimes \cdots \otimes v_n\otimes -) \Big) +\frac12 C^{\discsymb} \left( \sum_{\substack{h+h'=g\J\sqcup J'=[n]}} F_{h,1+|J|}(v_J)\otimes F_{h',1+|J'|}(v_{J'}) \right),$9

and act on the vector potential by

$F_{0,1}=F_{0,2}=0$0

Beyond these universal actions, the paper exhibits large linear symmetry groups of F-CohFTs: the “tick” and “fork” actions. The tick group

$F_{0,1}=F_{0,2}=0$1

acts by inserting auxiliary genus-zero classes. The fork action is built from

$F_{0,1}=F_{0,2}=0$2

These preserve the F-CohFT structure but do not commute with change of basis, translation, or the $F_{0,1}=F_{0,2}=0$3-action. The paper gives explicit non-commutativity formulas, such as

$F_{0,1}=F_{0,2}=0$4

This is one of the principal structural novelties of the framework (Borot et al., 2024).

6. Extended 2-spin example and significance

The main worked example is the extended 2-spin F-CohFT. Its phase space is

$F_{0,1}=F_{0,2}=0$5

with idempotent product

$F_{0,1}=F_{0,2}=0$6

The theory is obtained from a topological F-CohFT $F_{0,1}=F_{0,2}=0$7 by an explicit F-Givental transform

$F_{0,1}=F_{0,2}=0$8

where

$F_{0,1}=F_{0,2}=0$9

$(e_i)_{i\in I}$0

$(e_i)_{i\in I}$1

The corresponding F-TR tensors are computed explicitly. For instance,

$(e_i)_{i\in I}$2

and

$(e_i)_{i\in I}$3

with full formulas for $(e_i)_{i\in I}$4, $(e_i)_{i\in I}$5, and $(e_i)_{i\in I}$6 given in the source (Borot et al., 2024).

The associated local F-spectral curve is

$(e_i)_{i\in I}$7

$(e_i)_{i\in I}$8

$(e_i)_{i\in I}$9

with ramification weights

$V$00

This example shows concretely how the extra non-symmetric data are encoded geometrically: $V$01 remains standard, whereas $V$02 captures the $V$03-matrix correction (Borot et al., 2024).

The broader significance of F-DATA lies in this explicit bridge between F-CohFTs and recursion. The paper demonstrates that topological F-CohFTs are governed by F-TR, that their F-Givental orbits preserve that property, and that spectral-curve realizations can be built in semisimple settings. It also shows that the non-symmetric formalism supports symmetry operations absent from the classical CohFT/TR picture. This suggests that F-DATA is best understood not merely as initial recursion input, but as the organizing datum for a genuinely enlarged recursion-symmetry correspondence in the F-world (Borot et al., 2024).