Logarithmic Topological Recursion (Log-TR)

• Logarithmic Topological Recursion (Log-TR) is an extension of standard topological recursion that incorporates logarithmic singularities of y to address spectral curves with simple pole behaviors.
• It modifies the conventional residue formulas by adding key logarithmic corrections in the 1-point sector, ensuring universal x-y swap symmetry and compatibility with duality transformations.
• Log-TR underpins diverse applications—from weighted Hurwitz theory and gauge theory to mirror symmetry and topological strings—bridging enumerative geometry with quantum curve constructions.

Logarithmic topological recursion, usually abbreviated Log-TR or LogTR, is an extension of Eynard–Orantin topological recursion to spectral curves for which the globally defined objects are the meromorphic $1$-forms $dx$ and $dy$, while the functions $x$ and $y$ themselves may acquire logarithmic branches because $dx$ and/or $dy$ have simple poles with nonzero residues. In this setting, ordinary topological recursion is not stable under the universal $x$-$y$ swap, and the remedy is a modified recursion in which the only direct correction occurs in the $1$-point sector, through prescribed logarithmic principal parts at distinguished poles of $dx$0 (Alexandrov et al., 2023). Later work supplied the geometric infrastructure that ordinary topological recursion had in the meromorphic case—dilaton equations, variational formulas, and a definition of free energies—and showed that the logarithmic formalism is natural in weighted Hurwitz theory, toric Calabi–Yau mirror symmetry, Seiberg–Witten geometry, and quantum-curve constructions (Hock et al., 28 Apr 2026).

1. From ordinary topological recursion to logarithmic spectral data

Ordinary topological recursion starts from a spectral curve $dx$1, with $dx$2 a compact Riemann surface, $dx$3 and $dx$4 meromorphic, $dx$5 having simple ramification points, and $dx$6 the Bergman kernel. In one common convention, the initial data are

$dx$7

and the higher $dx$8 are determined by residues at the ramification points of $dx$9 (Alexandrov et al., 2024).

The logarithmic extension did not arise in isolation. A precursor is the global reformulation of topological recursion on a compact Riemann surface, which proved equivalence with the generalized recursion for arbitrary ramification and showed that higher-ramification correlation functions arise as limits of simple-ramification ones (Bouchard et al., 2012). In that framework, the kernel can already be modified for curves in $dy$0 by replacing $dy$1 and $dy$2 through

$dy$3

with kernel

$dy$4

That construction is meromorphic only on the complement of cuts connecting logarithmic singularities, but it already isolates the mechanism that later becomes central in Log-TR: recursion is still local near ramification, while the global geometry must accommodate logarithmic behavior (Bouchard et al., 2012).

This historical passage is important because Log-TR is not merely a change of variables from $dy$5 to $dy$6. The later theory treats logarithmic singularities as part of the intrinsic geometry of the spectral curve, rather than as an artifact of parametrization. This is the setting in which the ordinary topological-recursion formulas cease to be compatible with the universal $dy$7-$dy$8 swap, and in which the logarithmic correction becomes unavoidable (Alexandrov et al., 2023).

2. Core definition: LogTR-vital singularities and the modified recursion

The modern geometric formulation considers a compact Torelli-marked Riemann surface $dy$9, the Bergman kernel $x$0, and meromorphic $x$1-forms $x$2 and $x$3 such that the ramification points are simple zeros of $x$4, $x$5 is regular at the ramification points, and the zero loci of $x$6 and $x$7 are disjoint (Hock et al., 28 Apr 2026). The poles of $x$8 with nonzero residue are the source of logarithmic branches of $x$9.

A point $y$0 is called a LogTR-vital singularity of $y$1 if $y$2 is a simple pole of $y$3 and $y$4 is regular at $y$5 (Alexandrov et al., 2023). These are exactly the points at which Log-TR differs from ordinary topological recursion. If there are no such points, Log-TR reduces to the usual theory. The same reduction occurs when a logarithmic singularity of $y$6 is also a pole of $y$7; in that case the new logarithmic term vanishes (Hock, 25 Feb 2025).

The defining operator is

$y$8

equivalently $y$9 in another notation (Alexandrov et al., 2023). The logarithmic contribution is prescribed through principal parts of the form

$dx$0

where $dx$1 is the residue of $dx$2 at the LogTR-vital point $dx$3 (Alexandrov et al., 2024).

There are two complementary formulations. The first is loop-theoretic: LogTR is blobbed topological recursion together with a modified projection property. For $dx$4, the correlators have poles only at the critical points of $dx$5; for $dx$6 and $dx$7, one additionally allows the prescribed logarithmic principal parts at the vital poles of $dx$8 (Alexandrov et al., 2024). The second is a direct recursive residue formula. In the geometric formulation, for $dx$9,

$dy$0

where the first line is the usual Eynard–Orantin residue term and the second line is the logarithmic correction (Hock et al., 28 Apr 2026).

A decisive structural feature is that the new term appears only in the $dy$1 sector. For $dy$2, the $dy$3 are the same as in standard topological recursion; the new singular behavior enters through $dy$4 and then propagates recursively (Hock et al., 28 Apr 2026). Higher-order poles of $dy$5 do not contribute to the logarithmic projection in the same way, and simple poles of $dy$6 that are also poles of $dy$7 are not LogTR-vital (Alexandrov et al., 2023).

3. Universal $dy$8-$dy$9 swap and symplectic duality

The main theorem establishing Log-TR as the correct extension is that it satisfies the universal $x$0-$x$1 swap relation. If $x$2 and one applies LogTR to both $x$3 and the swapped pair, then the exact same combinatorial universal swap relation known in the meromorphic theory still holds, now for the logarithmic correlators (Alexandrov et al., 2023). The formula is expressed through a graph expansion in which the logarithmic corrections are absorbed into the $x$4-point input. This result provides a broad generalization and proof of a conjecture of Hock (Alexandrov et al., 2023).

The $x$5-$x$6 swap theorem is more than a formal symmetry statement. It determines which singularities must be inserted into the recursion for the swapped system to have the correct principal parts. In particular, it is precisely the LogTR-vital points that supply the missing terms. Without those terms, the swapped correlators do not have the correct local behavior (Alexandrov et al., 2023).

A further development is symplectic duality. For a transformation

$x$7

the corresponding symplectic-dual system is not a primitive operation external to the theory; it can be factored into $x$8-$x$9 duality, a correction in the $y$0-point sector, and $y$1-$y$2 duality again. In dual variables, the correction takes the form

$y$3

From this factorization one obtains invertibility, a group property, compatibility with topological recursion, and preservation of KP integrability (Alexandrov et al., 2024).

This duality viewpoint has strong practical consequences. In large classes of genus-zero examples, it yields uniform proofs of TR or LogTR for weighted double Hurwitz numbers by reducing them to systems with much simpler dual data (Alexandrov et al., 2024). It also clarifies why some naive $y$4-deformations fail: the logarithmic correction must be inserted with the specific $y$5-operator dictated by the duality formalism (Alexandrov et al., 1 May 2026).

4. Dilaton equations, variational formulas, and free energies

One of the central missing pieces for the logarithmic theory was the analogue of the dilaton equation. In Log-TR, for $y$6, $y$7, $y$8, and $y$9 defined locally near the ramification points, the dilaton equation becomes

$1$0

The second line is the genuinely logarithmic contribution; it has no analogue in ordinary topological recursion (Hock et al., 28 Apr 2026).

Variational formulas also survive in a controlled form. For deformations with respect to the ordinary times $1$1 and filling fractions $1$2, while keeping the log-times and the positions of the LogTR-vital singularities fixed, one has

$1$3

exactly as in the usual theory (Hock et al., 28 Apr 2026). The same structure extends to the free energies.

The definition of free energies is subtler. In ordinary topological recursion one formally sets $1$4 in the dilaton equation, but in the logarithmic case this naive step fails because $1$5 is logarithmically singular at the vital points. The logarithmic theory therefore defines, for $1$6,

$1$7

This regularized definition is independent of the basepoint $1$8 and is compatible with the variational formulas (Hock et al., 28 Apr 2026).

The free energies are not generally invariant under swapping $1$9 and $dx$00. For $dx$01, the difference $dx$02 admits a residue formula supported at logarithmic and non-logarithmic singularities of $dx$03 and $dx$04, including cases where the singularities are logarithmic (Hock, 25 Feb 2025). When the dual side is trivial—for example when $dx$05 is unramified—this formula collapses to an explicit closed expression for $dx$06 itself (Hock, 25 Feb 2025).

5. Enumerative geometry, gauge theory, and topological strings

One of the major applications of Log-TR is weighted double Hurwitz theory. Through symplectic duality and $dx$07-$dx$08 duality, LogTR yields a uniform proof of recursion for large families of weighted double Hurwitz numbers, encompassing simple Hurwitz, orbifold Hurwitz, monotone and strictly monotone Hurwitz, polynomially and rationally weighted Hurwitz, hypermaps, Ooguri–Vafa, Mariño–Vafa, and related variants (Alexandrov et al., 2024). A particularly instructive example is Family III of Orlov–Scherbin hypergeometric KP tau functions. There one takes

$dx$09

$dx$10

$dx$11

with $dx$12 a nonconstant rational function. The corresponding Orlov–Scherbin differentials satisfy LogTR on $dx$13, and this leads to a new ELSV-type formula involving $dx$14-classes, namely Chiodo classes (Alexandrov et al., 1 May 2026).

In gauge theory and Seiberg–Witten-type geometry, the logarithmic theory becomes a direct computational tool. For spectral curves with logarithmic $dx$15 or $dx$16, the free-energy formula derived from Log-TR proves that the free energies for the Gaiotto curve coincide with the perturbative part of the Nekrasov partition function of $dx$17 pure supersymmetric gauge theory (Hock, 25 Feb 2025). The later geometric theory strengthens this picture by giving examples in which the new definition of the free energies directly reproduces the full perturbative part of the Nekrasov–Shatashvili partition function of $dx$18d $dx$19 pure gauge theory, without auxiliary computation (Hock et al., 28 Apr 2026).

Mirror symmetry for toric Calabi–Yau threefolds provides another natural habitat. For strip geometries the mirror curve is

$dx$20

or, in a standard parametrization,

$dx$21

These curves naturally live in $dx$22, so logarithmic variables are intrinsic. Log-TR reduces to standard topological recursion for generic framing, but for special framings, especially $dx$23, the naive remodeling picture fails and Log-TR supplies the correct replacement (Banerjee et al., 8 Oct 2025).

In this strip-geometry setting, LogTR and universal $dx$24-$dx$25 duality produce both open and closed topological-string data. The open wave function can be defined by integrating the LogTR correlators, the dual wave function can be resummed into products of $dx$26-Pochhammer symbols, and one obtains quantum curves, Heine-type $dx$27-hypergeometric series, and $dx$28-Barnes type integral representations (Banerjee et al., 8 Oct 2025). For the closed sector, the free energies of general strip geometries can be derived directly from LogTR and $dx$29-$dx$30 duality, with explicit polylogarithmic formulas encoding the corresponding GW, DT, and $dx$31d BPS data (Banerjee et al., 21 Aug 2025). The later geometric treatment shows that the logarithmic free-energy definition directly reproduces the all-genus free energies of mirror curves of strip geometries, including the topological vertex and the resolved conifold (Hock et al., 28 Apr 2026).

6. Scope, reductions, and terminological boundaries

Log-TR is best viewed as the minimal extension of Eynard–Orantin topological recursion that allows prescribed logarithmic singularities of $dx$32 and the corresponding extra $dx$33-point principal parts (Alexandrov et al., 2024). It reduces to ordinary topological recursion when there are no LogTR-vital singularities, and also in situations—common for generic framings of toric Calabi–Yau mirror curves—in which the logarithmic poles of $dx$34 coincide with poles of $dx$35 (Hock, 25 Feb 2025). The theory is designed to behave correctly under singular limits of spectral curves, which is one reason it is well adapted to degeneration problems and to duality transformations (Hock et al., 28 Apr 2026).

A common source of confusion is nomenclature. In homotopy theory, “log $dx$36” denotes logarithmic topological restriction homology built from log $dx$37, with operators $dx$38, $dx$39, and $dx$40, and with comparisons to the log de Rham–Witt complex (Andriopoulos, 2024). Closely related work studies residue sequences in logarithmic THH, TR, and TC, proving equivalences between logarithmic constructions and localization cofiber terms (Lundemo, 17 Jun 2025). These are different theories from logarithmic topological recursion in the Eynard–Orantin sense. Another unrelated use of “logarithmic recursion” occurs in formal calculus: the recursion identity for formal iterated logarithms and iterated exponentials explicitly does not develop topological recursion and has no Eynard–Orantin spectral-curve structure (Robinson, 2010).

Within its own domain, however, Log-TR now has a coherent geometric status. It preserves the loop-equation formalism, restores the universal $dx$41-$dx$42 swap in the presence of logarithmic singularities, supports a calculus of variations and free energies, and organizes a wide range of enumerative and physical examples that lie outside the reach of standard topological recursion in its strictly meromorphic form (Alexandrov et al., 2023).