---
title: 'F-DATA: Non-Symmetric Topological Recursion'
url: https://www.emergentmind.com/topics/f-data
type: topic
---

# F-DATA: Non-Symmetric Topological Recursion

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 [2406.06304].

## 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 \(F_{1,1}=D\) [2406.06304].

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 [2406.06304].

The associated “vector potential” is
\[
\Phi(x)=\sum_{g,n\ge 0}\frac{\hbar^{g-1}}{n!}F_{g,1+n}(x^{\otimes n})
=
\sum_{g,n\ge 0}\frac{\hbar^{g-1}}{n!}
F^{i_0}_{g;i_1,\dots,i_n}x^{i_1}\cdots x^{i_n}e_{i_0}.
\]
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 \(C^{\connsymb}\) and \(C^{\discsymb}\) 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 \((A,B,C,D)\) 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 [2406.06304].

## 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 \(F_{g,1+n}\), not fully symmetric \(n\)-forms. Second, the recursion distinguishes the output variable from the input variables. Third, the connected and disconnected terms are weighted by different tensors, \(C^{\connsymb}\) and \(C^{\discsymb}\) [2406.06304].

If \(V\) is endowed with a non-degenerate pairing \(\eta\), then inputs and output can be identified. In the special case in which \(C^{\connsymb}=C^{\discsymb}\) and the data define an Airy structure, the amplitudes become fully symmetric and the vector potential becomes a gradient,
\[
\Phi(x)=\nabla F(x),
\]
for a scalar potential \(F\). This identifies the usual symmetric topological-recursion regime as a special case inside the F-framework [2406.06304].

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 [2406.06304].

## 3. Spectral-curve realization of F-DATA

The geometric formulation packages F-DATA into an F-spectral curve
\[
(\Sigma,x,y,\omega_{0,2},\bar{\omega}_{0,2},w).
\]
Here \(\Sigma\) is a smooth, possibly non-compact and disconnected complex curve; \(x,y\) are meromorphic functions; \(x\) has finitely many simple ramification points \(\mathfrak a\); \(dy\) is holomorphic and non-zero at \(\mathfrak a\); \(\omega_{0,2}\) and \(\bar{\omega}_{0,2}\) are bidifferentials on \(\Sigma^2\), holomorphic away from the diagonal and having a double pole with leading coefficient \(1\) on the diagonal; and \(w=(w^\alpha)_{\alpha\in\mathfrak a}\) is a set of scalar ramification weights [2406.06304].

At each branch point \(\alpha\in\mathfrak a\) there is a local involution \(\sigma^\alpha\) satisfying
\[
x\circ \sigma^\alpha=x,\qquad \sigma^\alpha\neq \mathrm{Id}.
\]
Two projectors are defined on meromorphic \(1\)-forms with poles only at \(\mathfrak a\),
\[
\mathscr P^\star[\chi](z_0)
=
\sum_{\alpha\in\mathfrak a}
\operatorname{Res}_{z=\alpha}
\left(\int_\alpha^z \omega_{0,2}^\star(z_0|\cdot)\right)\chi(z),
\qquad \star\in\{\connsymb,\discsymb\},
\]
where \(\omega_{0,2}^{\connsymb}=\omega_{0,2}\) and \(\omega_{0,2}^{\discsymb}=\bar{\omega}_{0,2}\). They induce decompositions
\[
\mathscr M=d\mathscr O\oplus \mathscr M_-^\star.
\]

The recursion kernels are
\[
K^{\star,\alpha}(z_0|z)
=
\frac{\frac12\int_{\sigma^\alpha(z)}^z \omega_{0,2}^\star(z_0|\cdot)}
{\omega_{0,1}(z)-\omega_{0,1}(\sigma^\alpha(z))},
\qquad \omega_{0,1}=y\,dx.
\]
From them one defines two recursion operators,
\[
\mathcal K^{\connsymb}[\chi](z_0)
=
\sum_{\alpha\in\mathfrak a}
w^\alpha
\operatorname{Res}_{z=\alpha}
K^{\connsymb,\alpha}(z_0|z)\,
(\mathscr P^{\connsymb})^{\otimes 2}[\chi](z,\sigma^\alpha(z)),
\]
\[
\mathcal K^{\discsymb}[\chi](z_0)
=
\sum_{\alpha\in\mathfrak a}
\operatorname{Res}_{z=\alpha}
K^{\discsymb,\alpha}(z_0|z)\,
(\mathscr P^{\discsymb})^{\otimes 2}[\chi](z,\sigma^\alpha(z)).
\]

The correlators satisfy
\[
\omega_{g,1+n}(z_0|z_1,\dots,z_n)
=
\mathcal K^{\connsymb}
\big[
\omega_{g-1,1+(n+1)}(\cdot|\cdot,z_1,\dots,z_n)
\big](z_0)
\]
\[
+
\mathcal K^{\discsymb}
\left[
\sum_{\substack{h+h'=g\\J\sqcup J'=[n]}}^{*}
\omega_{h,1+|J|}(\cdot|z_J)\otimes
\omega_{h',1+|J'|}(\cdot|z_{J'})
\right](z_0),
\]
with the starred sum excluding unstable \((0,1)\) factors [2406.06304].

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 \(w\), and a distinguished output slot. The loop equations hold in a corresponding non-symmetric form: for each branch point \(\alpha\), the combinations
\[
\omega_{g,1+n}(z_0|z_1,\dots,z_n)+
\omega_{g,1+n}(\sigma^\alpha(z_0)|z_1,\dots,z_n)
\]
and similarly in any input variable are holomorphic near \(\alpha\) [2406.06304].

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

An F-CohFT consists of maps
\[
\Omega_{g,1+n}:V_0^{\otimes n}\to H^{\mathrm{even}}(\overline{\mathcal M}_{g,1+n})\otimes V_0
\]
satisfying \(S_n\)-equivariance and compatibility with separating gluing, together with a flat unit \(e\in V_0\). The associated amplitudes are defined on loop spaces \(V_+=V_0[u]\) and \(V_-=V_0[u^{-1}]\,du/u\) using up/down morphisms
\[
\mathscr U:V_-\to V_+,\qquad \mathscr D:V_+\to V_-,
\]
which are two-sided inverses under residue pairing [2406.06304].

The amplitudes attached to an F-CohFT are
\[
F_{g,1+n}(f_1\otimes \cdots \otimes f_n)(u_0)
=
\int_{\overline{\mathcal M}_{g,1+n}}
\mathscr U(u_0,\psi_0)\,
\big[
\Omega_{g,1+n}(f_1(\psi_1)\otimes \cdots \otimes f_n(\psi_n))
\big].
\]
Their vector potential is again
\[
\Phi(x)=\sum_{g,n\ge 0,\ 2g-2+(1+n)>0}\frac{\hbar^{g-1}}{n!}F_{g,1+n}(x^{\otimes n}).
\]

A crucial theorem states that topological F-CohFTs, namely F-CohFTs of cohomological degree \(0\), are governed by F-TR. For an F-topological field theory \((V_0,\cdot,w)\) with amplitudes
\[
\mathscr F_{g,1+n}(v_1\otimes \cdots \otimes v_n)
=
v_1\cdots v_n\cdot w^g,
\]
the F-TR initial data are
\[
A=B=C^{\discsymb}: v_1\otimes v_2\mapsto v_1\cdot v_2,
\qquad
C^{\connsymb}: v\mapsto v\otimes w,
\qquad
D=\frac12 w.
\]
These define an F-Airy structure computing the theory’s amplitudes up to graph factors [2406.06304].

The orbit structure is controlled by the F-Givental action. Change of basis \(L\in GL(V_0)\), Givental \(R(u)\), and translations \(T(u)\in u^2V_0\llbracket u\rrbracket\) 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
\[
\widehat L\,\widehat R\,\widehat T\,\Omega
\]
is also computed by F-TR, with explicitly transformed initial data [2406.06304].

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 [2406.06304].

For a pair \((\lambda_{\mathrm s},\lambda_{\mathrm t})\in GL(V)\times GL(V)\), the transformed amplitudes are
\[
{}^\lambda F_{g,1+n}
=
\lambda_{\mathrm t}\circ F_{g,1+n}\circ
(\lambda_{\mathrm s}^{-1})^{\otimes n},
\]
and the tensors \((A,B,C^{\connsymb},C^{\discsymb},D)\) transform accordingly.

Bogoliubov transformations are parametrized by \(\beta\in \mathrm{End}(V)\). The transformed amplitudes are sums over rooted trees,
\[
{}^\beta F_{g,1+n}
=
\sum_{\mathbf T\in \mathbb T_{g,1+n}}
\left(\bigotimes_{v\in V(\mathbf T)}F_{g(v),1+n(v)}\right)\circ_{\mathbf T}
\left(\bigotimes_{e\in E(\mathbf T)}\beta\right),
\]
and the initial data change by
\[
{}^\beta A=A,\qquad
{}^\beta B=B+A\circ (\mathrm{Id}_V\otimes \beta),
\qquad
{}^\beta C^{\connsymb}=C^{\connsymb},
\]
\[
{}^\beta C^{\discsymb}
=
C^{\discsymb}
+
B\circ (\beta\otimes \mathrm{Id}_V)
+
B\circ (\beta\otimes \mathrm{Id}_V)\circ \sigma_{1,2}
+
A\circ \beta^{\otimes 2},
\qquad
{}^\beta D=D.
\]
The corresponding vector potential satisfies the fixed-point equation
\[
{}^\beta \Phi(x)=\Phi\big(x+(\beta\circ {}^\beta\Phi)(x)\big).
\]

Translations are given by insertion of a parameter \(\tau\in V\),
\[
{}^\tau F_{g,1+n}
=
\sum_{m\ge 0}\frac{1}{m!}
F_{g,1+n+m}(\mathrm{Id}^{\otimes n}\otimes \tau^{\otimes m}),
\]
and act on the vector potential by
\[
{}^\tau\Phi(x)=\Phi(x+\tau)-\hbar^{-1}\big({}^\tau G+{}^\tau H(x)\big).
\]

Beyond these universal actions, the paper exhibits large linear symmetry groups of F-CohFTs: the “tick” and “fork” actions. The tick group
\[
\mathfrak{tick}
=
\prod_{k\ge 2}\big(H^{\mathrm{even}}(\overline{\mathcal M}_{0,k})\otimes (V_0\llbracket u\rrbracket)^{\otimes k}\big)^{S_k}
\]
acts by inserting auxiliary genus-zero classes. The fork action is built from
\[
\mathfrak{fork}
=
\prod_{k\ge 2}\mathfrak{fork}_k,
\qquad
\mathfrak{fork}_k=
\big(V_0,H^{\mathrm{even}}(\overline{\mathcal M}_{0,1+k})\otimes (V_0\llbracket u\rrbracket)^{\otimes k}\big)^{S_k}.
\]
These preserve the F-CohFT structure but do not commute with change of basis, translation, or the \(R\)-action. The paper gives explicit non-commutativity formulas, such as
\[
\widehat T\,\widehat\Psi\,\Omega
=
\widehat\Psi\,\widehat\Sha_{\Psi,T}\,\widehat T\,\Omega.
\]
This is one of the principal structural novelties of the framework [2406.06304].

## 6. Extended 2-spin example and significance

The main worked example is the extended 2-spin F-CohFT. Its phase space is
\[
V_0=\mathbb Ce_1\oplus \mathbb Ce_2
\]
with idempotent product
\[
e_\beta\cdot e_\gamma=\delta^\alpha_{\beta,\gamma}e_\alpha,
\qquad
w=-s^2 e_2,
\qquad
e=e_1+e_2.
\]
The theory is obtained from a topological F-CohFT \(\Omega^{s,0}\) by an explicit F-Givental transform
\[
\overline\Omega^s=\widehat L\,\widehat R\,\widehat T\,\Omega^{s,0},
\]
where
\[
L=
\begin{pmatrix}
1 & 0\\
\frac1s & -\frac1s
\end{pmatrix},
\]
\[
R(u)=\mathrm{Id}_{V_0}-\sum_{m\ge 1}
\begin{pmatrix}
0&0\\
\frac{(2m-1)!!}{s^{2m}}&0
\end{pmatrix}u^m,
\]
\[
T(u)=-\sum_{m\ge 2}\frac{(2m-3)!!}{s^{2m-2}}\,e_2\,u^m.
\]
The corresponding F-TR tensors are computed explicitly. For instance,
\[
A^{(\alpha,i)}_{(\beta,j),(\gamma,k)}
=
\Big[
\delta^\alpha_1\delta^1_{\beta,\gamma}
+
\delta^\alpha_2(
\delta^{(1,2)}_{(\beta,\gamma)}
+
\delta^{(2,1)}_{(\beta,\gamma)}
-
s\delta^2_{\beta,\gamma})
\Big]\delta^i_0\delta^0_{j,k},
\]
and
\[
D^{(\alpha,i)}=\frac{1}{24}\delta^\alpha_2\left(s\delta^i_1-\frac1s\delta^i_0\right),
\]
with full formulas for \(B\), \(C^{\connsymb}\), and \(C^{\discsymb}\) given in the source [2406.06304].

The associated local F-spectral curve is
\[
x(\zeta)=\frac{\zeta^2}{2}\,e,
\qquad
y(\zeta)=-\zeta\,e_1+\frac{\ln(s-\zeta)}{s}\,e_2,
\]
\[
\omega_{0,2}(\zeta_1|\zeta_2)
=
(e_1\otimes e_1+e_2\otimes e_2)\,
\frac{d\zeta_1\,d\zeta_2}{(\zeta_1-\zeta_2)^2},
\]
\[
\bar\omega_{0,2}(\zeta_1|\zeta_2)
=
\omega_{0,2}(\zeta_1|\zeta_2)
+
e_2\otimes e_1
\sum_{k_1,k_2\ge 0}
\frac{(2k_1+2k_2+1)!!}{(2k_1-1)!!(2k_2-1)!!}
\zeta_1^{2k_1}(-1)^{k_2}\zeta_2^{2k_2}
\,d\zeta_1\,d\zeta_2,
\]
with ramification weights
\[
w=(0,-s^2).
\]
This example shows concretely how the extra non-symmetric data are encoded geometrically: \(\omega_{0,2}\) remains standard, whereas \(\bar\omega_{0,2}\) captures the \(R\)-matrix correction [2406.06304].

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* [2406.06304].

Source: https://www.emergentmind.com/topics/f-data