Geometry of Logarithmic Topological Recursion: Dilaton Equations, Free Energies and Variational Formulas

Abstract: One of the most important applications of topological recursion concerns spectral curves for which the functions $(x,y)$ defining the spectral curve are allowed to have logarithmic singularities. This occurs for instance for Seiberg-Witten curves and mirror curves computing Gromov--Witten invariants of toric Calabi--Yau threefolds. A recently introduced extension of topological recursion, the so-called logarithmic topological recursion, exhibits the correct behavior under certain limits of those spectral curves. In this article, we derive the dilaton equations in the setting of logarithmic topological recursion, as well as variational formulas, and provide a definition of the free energies in situations where standard topological recursion was known to fail. We present examples in which the new definition of the free energies \textit{directly} (without any computation) reproduces the full perturbative part of the Nekrasov--Shatashvili partition function of 4d $\mathcal{N}=2$ pure supersymmetric gauge theory, as well as the all-genus free energies of mirror curves of strip geometries, including in particular the topological vertex and the resolved conifold.

• The paper establishes dilaton equations for logarithmic topological recursion, including essential residue terms at LogTR-vital singularities where standard recursion fails.
• It defines basepoint-independent free energies for all genera h≥1 and derives compatible variations with respect to classical times and logarithmic singularity positions.
• The framework reproduces perturbative Nekrasov–Shatashvili free energies and all-genus strip-geometry results, while leaving x–y duality and higher-order ramification as open problems.

The paper develops the geometric foundations of logarithmic topological recursion (LogTR), an extension of the Eynard–Orantin topological recursion (TR) designed for spectral curves on which the differential $dy$ carries residues at points where $dx$ is regular. The authors, Hock, Marchal and Orantin (2604.25622), derive the dilaton equations for LogTR, use them to define the free energies $F_h=\omega_{h,0}$, and prove variational formulas both with respect to the classical times of $ydx$ and with respect to the so-called LogTR-vital singularities. The resulting free-energy definition reproduces, without further computation, the perturbative Nekrasov–Shatashvili partition function of 4d $\mathcal{N}=2$ pure supersymmetric gauge theory and the all-genus free energies of mirror curves of strip geometries, including the topological vertex and the resolved conifold.

Background: from TR to LogTR

Standard TR [EO07] takes as input a compact Riemann surface $\Sigma$ with a Torelli marking and two meromorphic one-forms $dx,dy$ without residues, and recursively produces symmetric meromorphic $n$-differentials $\omega_{h,n}$. Its applications range from intersection numbers on $\overline{\mathcal{M}}_{g,n}$ and Hurwitz theory to the computation of Gromov–Witten invariants of toric Calabi–Yau threefolds via the BKMP correspondence, where the spectral curve is a mirror curve living in $dx$0. In the latter setting $dx$1 and $dx$2 have simple poles with non-vanishing residues; standard TR still yields correct invariants provided a generic framing parameter is chosen, but fails at special framings and does not commute with singular limits of mirror curves [Bouchard:2011ya].

LogTR [Alexandrov:2023tgl] resolves this by adding, in the recursion for $dx$3 only, a residue contribution at the LogTR-vital singularities: points $dx$4 that are simple poles of $dx$5 with residue $dx$6 at which $dx$7 is regular. The extra term is

$dx$8

with $dx$9. An equivalent, computationally convenient formulation replaces this residue by derivatives of $F_h=\omega_{h,0}$0 evaluated at $F_h=\omega_{h,0}$1, weighted by $F_h=\omega_{h,0}$2. When no LogTR-vital singularity exists, LogTR coincides with standard TR; this is precisely the generic-framing situation. A second, independent motivation comes from the $F_h=\omega_{h,0}$3–$F_h=\omega_{h,0}$4 duality of TR, which was observed to fail exactly in the regimes where standard TR gives incorrect enumerative invariants; enforcing the duality led to the LogTR definition.

The paper works under admissibility assumptions: $F_h=\omega_{h,0}$5 compact, ramification points simple zeros of $F_h=\omega_{h,0}$6, $F_h=\omega_{h,0}$7 regular at ramification points, and zero loci of $F_h=\omega_{h,0}$8 and $F_h=\omega_{h,0}$9 disjoint. The authors note explicitly that generalizations allowing higher-order ramification or relaxing these conditions are left open.

Known structural properties

The correlators produced by LogTR are meromorphic $ydx$0-differentials, symmetric for $ydx$1, residue-free, with vanishing $ydx$2-periods, satisfying the linear and quadratic loop equations at the ramification points and a logarithmic projection property (LPP) following from the Riemann bilinear identity. For $ydx$3 the correlators agree with standard TR; the LogTR correction affects $ydx$4 for $ydx$5 and propagates through the induction. The paper observes that LogTR is the unique solution to the loop equations together with the "loop equations" at the LogTR-vital singularities and the LPP, in analogy with the characterization of standard TR by loop equations alone.

Dilaton equations and definition of the free energies

The central technical input is a lemma describing the local structure of $ydx$6 near a LogTR-vital singularity: for $ydx$7 holomorphic at $ydx$8,

$ydx$9

This identity, proved by showing that a certain family of local forms built from $\mathcal{N}=2$0 is regular and hence residue-free, is what replaces the standard dilaton mechanism at the logarithmic poles. From it, the dilaton equation for LogTR follows by induction on $\mathcal{N}=2$1:

$\mathcal{N}=2$2

where $\mathcal{N}=2$3 is a local antiderivative of $\mathcal{N}=2$4 near the ramification points. A naive extension of the standard dilaton equation fails for two reasons the paper identifies: residues of $\mathcal{N}=2$5 at logarithmic poles are not defined, and residues at ramification points alone cannot generate the poles of $\mathcal{N}=2$6 at the LogTR-vital singularities. The second line of the formula is therefore essential, and it involves all lower-genus one-point correlators.

Setting $\mathcal{N}=2$7 requires an additional integration by parts to eliminate the ill-defined term involving $\mathcal{N}=2$8, which is replaced by $\mathcal{N}=2$9 (well-defined because $\Sigma$0 is residueless). This yields the definition of the free energies for $\Sigma$1, and for $\Sigma$2:

$\Sigma$3

where $\Sigma$4 is the Bergman tau-function. The definition is independent of the basepoint. The paper does not define $\Sigma$5: the homogeneity argument used in [EO07] fails in the presence of LogTR-vital singularities, and the authors state that $\Sigma$6 should depend on the geometric origin of the curve (toric versus Seiberg–Witten). Notably, the $\Sigma$7 term in the definition was already necessary in prior work [Banerjee:2025qgx] to obtain correct free energies of strip geometries, which supports its inclusion here.

Examples: Seiberg–Witten and strip geometries

Two genus-zero examples with no ramification points illustrate the definition.

Half Seiberg–Witten curve. For $\Sigma$8 and $\Sigma$9, the free energies are

$dx,dy$0

For equal residues $dx,dy$1 this reproduces the perturbative part of the 4d $dx,dy$2 pure gauge theory partition function [Nekrasov:2003rj], computed previously as the $dx,dy$3–$dx,dy$4 dual of the half Seiberg–Witten curve. For generic residues the coefficients become double Bernoulli numbers, the same structures appearing in quantized Riemann–Hilbert problems [Barbieri:2019yya] and refined topological recursion [Kidwai:2023fxs]. The authors are careful here: they state that allowing generic $dx,dy$5 should perhaps be understood as refining the spectral curve on which (Log)TR is performed, rather than applying refined TR to the original curve, and they flag the relation between these refinements as an open question.

Strip geometry mirror curves. For $dx,dy$6, $dx,dy$7, the free energies involve polylogarithms $dx,dy$8; for $dx,dy$9 the curve is the $n$0–$n$1 dual of the mirror curve of strip geometries and the result recovers the known closed-string free energies, i.e. Gromov–Witten invariants of the corresponding toric Calabi–Yau threefolds [Iqbal:2004ne], including the topological vertex and resolved conifold. The formal resummation in $n$2 requires an ordering $n$3 for convergence of the polylogarithms; the authors describe this interchange of series as common but non-rigorous. The resulting expression splits the quadratic denominator into two factors, as in refined topological string theory, though the McMahon-type term is not the refined topological vertex.

Both examples motivate the conjecture that, for admissible genus-zero curves on $n$4 with $n$5 of at most simple poles, the free energies defined here are invariant under $n$6–$n$7 duality. This remains a conjecture, not a theorem.

Parametrization of the spectral curve

The paper then develops the deformation theory. After subtracting the logarithmic part of $n$8 via the prime form, $n$9, the form $\omega_{h,n}$0 is meromorphic and admits a global decomposition

$\omega_{h,n}$1

in terms of irregular times $\omega_{h,n}$2, monodromies $\omega_{h,n}$3, filling fractions $\omega_{h,n}$4, and the log-times $\omega_{h,n}$5. Each classical time $\omega_{h,n}$6 is represented as an integral of the Bergman kernel against a contour and density, $\omega_{h,n}$7, extending the formalism of [EO07]. The spectral curve is thereby parametrized by the singularity positions, the times, and the log-times.

Variational formulas

All variations are taken at fixed $\omega_{h,n}$8, so ramification points do not move. The main results are:

• Classical times. For variations with respect to irregular times, monodromy differences, and filling fractions, the standard TR variational formula carries over verbatim: $\omega_{h,n}$9 for all $\overline{\mathcal{M}}_{g,n}$0, and similarly $\overline{\mathcal{M}}_{g,n}$1 for $\overline{\mathcal{M}}_{g,n}$2. The proofs proceed by induction using the Rauch variational formula for $\overline{\mathcal{M}}_{g,n}$3 and a lemma on the variation of the recursion kernel; the LogTR correction terms are handled by rewriting $\overline{\mathcal{M}}_{g,n}$4.
• LogTR-vital singularities. Variations of the position $\overline{\mathcal{M}}_{g,n}$5 produce a genuinely new formula:

$\overline{\mathcal{M}}_{g,n}$6

Here $\overline{\mathcal{M}}_{g,n}$7 is holomorphic except at $\overline{\mathcal{M}}_{g,n}$8 but carries monodromies along the homology cycles, so global identities such as the Riemann bilinear identity must be applied with care; the authors state this explicitly as a technical constraint of the derivation. The free energies satisfy a corresponding variational formula, with an equivalent regrouped form

$\overline{\mathcal{M}}_{g,n}$9

A structural feature persists across all cases: acting with the variation on the dilaton equation, the variation of $dx$00 cancels the prefactor $dx$01, leaving only the variation of $dx$02 (and of $dx$03 at $dx$04). The paper emphasizes that this compatibility between the dilaton equations and the variational formulas is a nontrivial consistency check of the proposed free-energy definition, since both are derived purely from the recursive definition of LogTR. The variational formulas for the LogTR-vital positions differ substantially from the classical ones, which the authors interpret as evidence that these parameters do not belong to the same class as the standard TR times; they conjecture that LogTR-vital singularities play a role analogous to ramification points, to which they converge in certain limits.

Limitations and open questions

Several limitations are conceded in the paper itself. The admissibility assumptions exclude higher-order ramification and curves where $dx$05 vanishes at a ramification point; extending LogTR to these regimes is deferred. The free energy $dx$06 is undefined, with the authors arguing it necessarily depends on the geometric origin of the curve. The $dx$07–$dx$08 duality invariance of the new free energies is established only in the two examples and stated as a conjecture for general genus-zero curves. The formal $dx$09-resummation in the strip-geometry example relies on a non-rigorous interchange of series. The variational formulas are derived at fixed $dx$10; variations of ramification-point positions are not treated. Finally, the relationship between the double-Bernoulli refinement arising from generic log-times and refined topological recursion or refined topological string theory is left unclear, as is whether it originates from higher-dimensional theories such as Calabi–Yau fivefolds.

The paper also identifies specific directions: quantum curves for spectral curves in $dx$11, where no general results exist and where the variational formulas should feed into KZ-type equations replacing the differential equations of [Quantization_2021]; knot-theoretic spectral curves such as the $dx$12-polynomial; augmentation varieties, for which a modified recursion with a calibrated annulus kernel was proposed in [Gu:2014yba] and which the authors suggest revisiting via LogTR; and rigorous variational formulas for mirror curves, which do not yet exist.

Conclusion

The paper completes a missing layer of the LogTR framework: dilaton equations, a basepoint-independent definition of the free energies $dx$13 for $dx$14, and variational formulas with respect to both the classical times and the LogTR-vital singularities, all shown to be mutually compatible. The definition passes a strong external test by reproducing the perturbative NS free energy of pure $dx$15 gauge theory and the all-genus free energies of strip-geometry mirror curves directly from residues. The structural parallel between the dilaton equations and the variational formulas—where the variation of the higher correlator cancels the Euler-characteristic prefactor—persists in the logarithmic setting, including for variations of the LogTR-vital positions themselves, and supports the view that these positions are deformation parameters of a new type whose systematic study, in particular in comparison with ramification points, is the question the paper leaves most directly open.