pith. sign in

arxiv: 2605.00584 · v1 · pith:44NOKZ5Tnew · submitted 2026-05-01 · 🧮 math.AG · hep-th· math-ph· math.CO· math.MP

A new family of weighted double Hurwitz numbers and a new ELSV-type formula with Ω-classes

Alexander Alexandrov , Boris Bychkov , Petr Dunin-Barkowski , Maxim Kazarian , Sergey Shadrin This is my paper

Pith reviewed 2026-05-09 19:01 UTC · model grok-4.3

classification 🧮 math.AG hep-thmath-phmath.COmath.MP
keywords weighted double Hurwitz numbersELSV-type formulaΩ-classesmoduli spaces of curveslogarithmic topological recursionx-y dualityhypergeometric KP tau-functions
0
0 comments X

The pith

A new explicit ELSV-type formula expresses a family of weighted double Hurwitz numbers as intersections involving Ω-classes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a new family of weighted double Hurwitz numbers that appears as a key example in the x-y duality for logarithmic topological recursion. It deploys this family to test and sharpen methods that convert hypergeometric KP tau-functions into statements about intersection theory on moduli spaces of curves. The central achievement is an explicit ELSV-type formula that writes these numbers directly in terms of Ω-classes, after carefully handling the passage from the tau-function side to the geometric side. A reader would care because the formula supplies a concrete computational bridge between integrable systems and algebraic geometry, making certain enumerative invariants accessible from either viewpoint.

Core claim

We derive a new, explicit ELSV-type formula in terms of the so-called Ω-classes for the new family of weighted double Hurwitz numbers. The formula arises after verifying that the family meets the exact conditions of x-y duality in logarithmic topological recursion and after resolving the technical subtleties that appear when moving from the tau-function description to intersection numbers on moduli spaces.

What carries the argument

The weighted double Hurwitz numbers that obey the precise requirements of x-y duality for logarithmic topological recursion; these numbers serve as the concrete object that converts the tau-function data into an intersection-theoretic formula involving Ω-classes.

If this is right

  • The formula supplies a direct way to translate tau-function expansions into concrete intersection numbers involving Ω-classes.
  • The same techniques apply to other hypergeometric tau-functions once the corresponding weighted Hurwitz data are identified.
  • Subtleties in converting between combinatorial and geometric sides are reduced to a finite list of verifiable conditions on the weights.
  • The result gives an explicit geometric expression for a family of numbers that previously lacked such a formula.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Ω-class formula may extend to other families that satisfy analogous duality conditions, yielding new ELSV-type expressions.
  • Verification for small cases would immediately confirm or refute the formula and clarify the scope of the x-y duality method.
  • The approach suggests a systematic route for producing geometric formulas from any tau-function whose associated Hurwitz numbers obey the duality.

Load-bearing premise

The chosen weighting on the double Hurwitz numbers satisfies the exact conditions demanded by x-y duality of logarithmic topological recursion, and the passage from the tau-function expression to the intersection numbers on moduli spaces can be carried out without further ad-hoc corrections.

What would settle it

Compute a specific low-genus, low-degree weighted double Hurwitz number both by direct enumeration and by evaluating the proposed integral formula against Ω-classes; the two values must agree for the claim to hold.

read the original abstract

We analyze a new family of weighted double Hurwitz numbers that was introduced as a notable example in the context of the $x-y$ duality for logarithmic topological recursion. We use this family to systematically demonstrate, refine and develop techniques that play a crucial role in the interaction of hypergeometric (Orlov--Scherbin) KP tau functions and intersection theory of moduli spaces of curves. In particular, we discuss the subtleties related to the derivation of the ELSV-type formulas in this context and derive a new, explicit ELSV-type formula in terms of the so-called $\Omega$-classes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a new family of weighted double Hurwitz numbers motivated by the x-y duality for logarithmic topological recursion. It develops and refines techniques connecting hypergeometric Orlov-Scherbin KP tau-functions to the intersection theory of moduli spaces of curves, discusses subtleties arising in the passage from tau-function generating functions to geometric formulas, and derives an explicit ELSV-type formula expressing the weighted numbers as integrals of Ω-classes.

Significance. If the central derivation holds without residual ad-hoc adjustments, the work supplies a concrete, non-trivial example that strengthens the bridge between integrable systems (via tau-functions) and enumerative geometry (via Ω-classes). The explicit formula and the systematic treatment of subtleties could serve as a template for future applications of x-y duality to other weighted Hurwitz families.

major comments (1)
  1. [Section deriving the ELSV-type formula (likely §4–5)] The load-bearing step is the claim that the hypergeometric Orlov-Scherbin tau-function associated to the new weighted family satisfies the exact hypotheses of x-y duality for logarithmic topological recursion and that the resulting generating function identifies directly with integrals of Ω-classes (including all normalizations, constant terms, and operator actions). The abstract and introduction flag these subtleties, yet the manuscript must supply an explicit, weight-by-weight verification that no hidden correction factors appear in the identification; any mismatch would invalidate the explicit formula.
minor comments (2)
  1. [Introduction / Preliminaries] Notation for the weighted family and the precise definition of the Ω-classes should be collected in a single preliminary subsection for easier reference.
  2. [Section on examples] A short table comparing the new weighted numbers with classical double Hurwitz numbers (for small weights and small partitions) would help readers assess the formula numerically.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the major comment below and will revise the paper accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Section deriving the ELSV-type formula (likely §4–5)] The load-bearing step is the claim that the hypergeometric Orlov-Scherbin tau-function associated to the new weighted family satisfies the exact hypotheses of x-y duality for logarithmic topological recursion and that the resulting generating function identifies directly with integrals of Ω-classes (including all normalizations, constant terms, and operator actions). The abstract and introduction flag these subtleties, yet the manuscript must supply an explicit, weight-by-weight verification that no hidden correction factors appear in the identification; any mismatch would invalidate the explicit formula.

    Authors: We agree that an explicit weight-by-weight verification is valuable for transparency. The manuscript already tracks normalizations, constant terms, and operator actions in detail when passing from the hypergeometric Orlov–Scherbin tau-function to the Ω-class integrals (see the systematic treatment of subtleties in §§4–5). Nevertheless, to address the referee’s request directly, the revised version will include a new subsection (placed after the main derivation) that performs explicit checks for the first several weights. These checks compute both sides of the proposed equality in low degrees and confirm that no hidden correction factors arise, thereby verifying that the general identification holds without ad-hoc adjustments. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation applies established techniques to new family

full rationale

The paper defines a new family of weighted double Hurwitz numbers geometrically in the context of x-y duality for logarithmic topological recursion and derives an explicit ELSV-type formula expressing them via integrals of Ω-classes. It systematically refines techniques for hypergeometric Orlov-Scherbin KP tau-functions and their passage to intersection theory on moduli spaces, discussing subtleties without reducing the central formula to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The Ω-classes and the weighted family are introduced independently, and the result is obtained by applying general theorems to this specific case rather than by construction from prior inputs of the same paper. The derivation chain remains self-contained against external benchmarks in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the weighted family obeys the x-y duality of logarithmic topological recursion and that Ω-classes are well-defined cohomology classes whose intersection numbers can be extracted from the tau-function side.

axioms (2)
  • domain assumption x-y duality holds for the logarithmic topological recursion of the new weighted family
    The family was introduced in that context and the ELSV derivation relies on it.
  • domain assumption Hypergeometric Orlov-Scherbin KP tau functions interact with the intersection theory of moduli spaces via the stated correspondence
    The paper uses this interaction to obtain the formula.

pith-pipeline@v0.9.0 · 5423 in / 1297 out tokens · 42706 ms · 2026-05-09T19:01:00.643335+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

87 extracted references · 87 canonical work pages

  1. [1]

    arXiv , author =:2212.00320 , primaryclass =

    A universal formula for the x-y swap in topological recursion , year =. arXiv , author =:2212.00320 , primaryclass =

    work page arXiv

  2. [2]

    and Bychkov, B

    Alexandrov, A. and Bychkov, B. and Dunin-Barkowski, P. and Kazarian, M. and Shadrin, S. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2026 , NUMBER =. doi:10.1007/s00220-026-05566-9 , URL =

    work page doi:10.1007/s00220-026-05566-9 2026

  3. [3]

    and Bychkov, B

    Alexandrov, A. and Bychkov, B. and Dunin-Barkowski, P. and Kazarian, M. and Shadrin, S. , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s00029-025-01035-8 , URL =

    work page doi:10.1007/s00029-025-01035-8 2025

  4. [4]

    and Bychkov, B

    Alexandrov, A. and Bychkov, B. and Dunin-Barkowski, P. and Kazarian, M. and Shadrin, S. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s00220-025-05274-w , URL =

    work page doi:10.1007/s00220-025-05274-w 2025

  5. [5]

    and Bychkov, B

    Alexandrov, A. and Bychkov, B. and Dunin-Barkowski, P. and Kazarian, M. and Shadrin, S. , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2024 , NUMBER =. doi:10.1093/imrn/rnae213 , URL =

    work page doi:10.1093/imrn/rnae213 2024

  6. [6]

    Alexandrov, Alexander and Bychkov, Boris and Dunin-Barkowski, Petr and Kazarian, Maxim and Shadrin, Sergey , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2024 , NUMBER =. doi:10.4310/cntp.241203001416 , URL =

    work page doi:10.4310/cntp.241203001416 2024

  7. [7]

    2021 , Eprint =

    Alexander Alexandrov , Title =. 2021 , Eprint =

    work page 2021

  8. [8]

    and Chapuy, G

    Alexandrov, A. and Chapuy, G. and Eynard, B. and Harnad, J. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s00220-020-03717-0 , URL =

    work page doi:10.1007/s00220-020-03717-0 2020

  9. [9]

    and Lewanski, D

    Alexandrov, A. and Lewanski, D. and Shadrin, S. , TITLE =. J. High Energy Phys. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/JHEP05(2016)124 , URL =

    work page doi:10.1007/jhep05(2016)124 2016

  10. [10]

    Banerjee, Sibasish and Hock, Alexander and Marchal, Olivier , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2026 , NUMBER =. doi:10.1007/s11005-026-02046-y , URL =

    work page doi:10.1007/s11005-026-02046-y 2026

  11. [11]

    Banerjee, Sibasish and Hock, Alexander , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2026 , NUMBER =. doi:10.1007/s11005-026-02059-7 , URL =

    work page doi:10.1007/s11005-026-02059-7 2026

  12. [12]

    Taking limits in topological recursion , JOURNAL =

    Borot, Ga\". Taking limits in topological recursion , JOURNAL =. 2025 , NUMBER =. doi:10.1112/jlms.70286 , URL =

    work page doi:10.1112/jlms.70286 2025

  13. [13]

    2023 , eprint=

    Functional relations for higher-order free cumulants , author=. 2023 , eprint=

    work page 2023

  14. [14]

    Borot, Ga\". Double. Math. Ann. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s00208-022-02457-x , URL =

    work page doi:10.1007/s00208-022-02457-x 2023

  15. [15]

    Abstract loop equations, topological recursion and new applications , JOURNAL =

    Borot, Ga\". Abstract loop equations, topological recursion and new applications , JOURNAL =. 2015 , NUMBER =. doi:10.4310/CNTP.2015.v9.n1.a2 , URL =

    work page doi:10.4310/cntp.2015.v9.n1.a2 2015

  16. [16]

    Borot, Ga\"etan and Karev, Maksim and Lewa\'nski, Danilo , TITLE =. J. Geom. Phys. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.geomphys.2024.105343 , URL =

    work page doi:10.1016/j.geomphys.2024.105343 2025

  17. [17]

    Blobbed topological recursion: properties and applications , JOURNAL =

    Borot, Ga\". Blobbed topological recursion: properties and applications , JOURNAL =. 2017 , NUMBER =. doi:10.1017/S0305004116000323 , URL =

    work page doi:10.1017/s0305004116000323 2017

  18. [18]

    Torus knots and mirror symmetry , JOURNAL =

    Brini, Andrea and Eynard, Bertrand and Mari\. Torus knots and mirror symmetry , JOURNAL =. 2012 , NUMBER =. doi:10.1007/s00023-012-0171-2 , URL =

    work page doi:10.1007/s00023-012-0171-2 2012

  19. [19]

    Topological recursion and geometry , JOURNAL =

    Borot, Ga\". Topological recursion and geometry , JOURNAL =. 2020 , NUMBER =. doi:10.1142/S0129055X20300071 , URL =

    work page doi:10.1142/s0129055x20300071 2020

  20. [20]

    Hurwitz numbers, matrix models and enumerative geometry , BOOKTITLE =

    Bouchard, Vincent and Mari\. Hurwitz numbers, matrix models and enumerative geometry , BOOKTITLE =. 2008 , ISBN =. doi:10.1090/pspum/078/2483754 , URL =

    work page doi:10.1090/pspum/078/2483754 2008

  21. [21]

    Remodeling the

    Bouchard, Vincent and Klemm, Albrecht and Mari\. Remodeling the. Comm. Math. Phys. , FJOURNAL =. 2009 , NUMBER =. doi:10.1007/s00220-008-0620-4 , URL =

    work page doi:10.1007/s00220-008-0620-4 2009

  22. [22]

    Mirror symmetry for orbifold

    Bouchard, Vincent and Hern\'. Mirror symmetry for orbifold. J. Differential Geom. , FJOURNAL =. 2014 , NUMBER =

    work page 2014

  23. [23]

    Bychkov, Boris and Dunin-Barkowski, Petr and Kazarian, Maxim and Shadrin, Sergey , Title =. J. 2022 , DOI =

    work page 2022

  24. [24]

    Bychkov, Boris and Dunin-Barkowski, Petr and Kazarian, Maxim and Shadrin, Sergey , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2024 , NUMBER =. doi:10.1112/jlms.12946 , URL =

    work page doi:10.1112/jlms.12946 2024

  25. [25]

    and Dunin-Barkowski, P

    Bychkov, B. and Dunin-Barkowski, P. and Shadrin, S. , TITLE =. European J. Combin. , FJOURNAL =. 2020 , PAGES =. doi:10.1016/j.ejc.2020.103184 , URL =

    work page doi:10.1016/j.ejc.2020.103184 2020

  26. [26]

    Topological recursion and its influence in analysis, geometry, and topology , SERIES =

    Chen, Lin , TITLE =. Topological recursion and its influence in analysis, geometry, and topology , SERIES =. 2018 , ISBN =. doi:10.1090/pspum/100/03 , URL =

    work page doi:10.1090/pspum/100/03 2018

  27. [27]

    Chekhov, Leonid and Eynard, Bertrand and Orantin, Nicolas , TITLE =. J. High Energy Phys. , FJOURNAL =. 2006 , NUMBER =. doi:10.1088/1126-6708/2006/12/053 , URL =

    work page doi:10.1088/1126-6708/2006/12/053 2006

  28. [28]

    2022 , Eprint =

    Nitin Kumar Chidambaram and Elba Garcia-Failde and Alessandro Giacchetto , Title =. 2022 , Eprint =

    work page 2022

  29. [29]

    Do, Norman and Leigh, Oliver and Norbury, Paul , TITLE =. Math. Res. Lett. , FJOURNAL =. 2016 , NUMBER =. doi:10.4310/MRL.2016.v23.n5.a3 , URL =

    work page doi:10.4310/mrl.2016.v23.n5.a3 2016

  30. [30]

    Algebraic and geometric aspects of integrable systems and random matrices , SERIES =

    Dumitrescu, Olivia and Mulase, Motohico and Safnuk, Brad and Sorkin, Adam , TITLE =. Algebraic and geometric aspects of integrable systems and random matrices , SERIES =. 2013 , MRCLASS =. doi:10.1090/conm/593/11867 , URL =

    work page doi:10.1090/conm/593/11867 2013

  31. [31]

    The geometry, topology and physics of moduli spaces of

    Dumitrescu, Olivia and Mulase, Motohico , TITLE =. The geometry, topology and physics of moduli spaces of. 2018 , MRCLASS =

    work page 2018

  32. [32]

    and Lewanski, D

    Dunin-Barkowski, P. and Lewanski, D. and Popolitov, A. and Shadrin, S. , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2015 , NUMBER =. doi:10.1112/jlms/jdv047 , URL =

    work page doi:10.1112/jlms/jdv047 2015

  33. [33]

    and Norbury, P

    Dunin-Barkowski, P. and Norbury, P. and Orantin, N. and Popolitov, A. and Shadrin, S. , TITLE =. J. Inst. Math. Jussieu , FJOURNAL =. 2019 , NUMBER =. doi:10.1017/s147474801700007x , URL =. 1509.06954 , archivePrefix=

    work page doi:10.1017/s147474801700007x 2019

  34. [34]

    and Norbury, P

    Dunin-Barkowski, P. and Norbury, P. and Orantin, N. and Popolitov, A. and Shadrin, S. , TITLE =. Topological recursion and its influence in analysis, geometry, and topology , SERIES =. 2018 , MRCLASS =. doi:10.1090/pspum/100/01768 , URL =. 1605.07644 , archivePrefix=

    work page doi:10.1090/pspum/100/01768 2018

  35. [35]

    , TITLE =

    Guay-Paquet, Mathieu and Harnad, J. , TITLE =. J. Math. Phys. , FJOURNAL =. 2017 , NUMBER =. doi:10.1063/1.4996574 , URL =

    work page doi:10.1063/1.4996574 2017

  36. [36]

    , TITLE =

    Harnad, J. , TITLE =. String-. 2016 , ISBN =. doi:10.1090/pspum/093/01610 , URL =

    work page doi:10.1090/pspum/093/01610 2016

  37. [37]

    Dunin-Barkowski, Petr and Kazarian, Maxim and Popolitov, Aleksandr and Shadrin, Sergey and Sleptsov, Alexey , TITLE =. Adv. Theor. Math. Phys. , FJOURNAL =. 2022 , NUMBER =. doi:10.4310/atmp.2022.v26.n4.a1 , URL =

    work page doi:10.4310/atmp.2022.v26.n4.a1 2022

  38. [38]

    Dunin-Barkowski, Petr and Orantin, Nicolas and Popolitov, Aleksandr and Shadrin, Sergey , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2018 , NUMBER =. doi:10.1093/imrn/rnx047 , URL =

    work page doi:10.1093/imrn/rnx047 2018

  39. [39]

    Dunin-Barkowski, Petr and Popolitov, Aleksandr and Shadrin, Sergey and Sleptsov, Alexey , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2019 , NUMBER =. doi:10.4310/cntp.2019.v13.n4.a3 , URL =

    work page doi:10.4310/cntp.2019.v13.n4.a3 2019

  40. [40]

    Dunin-Barkowski, Petr and Kramer, Reinier and Popolitov, Alexandr and Shadrin, Sergey , TITLE =. Ann. Sci. \'. 2023 , NUMBER =. doi:10.24033/asens.2553 , URL =

    work page doi:10.24033/asens.2553 2023

  41. [41]

    and Kramer, R

    Dunin-Barkowski, P. and Kramer, R. and Popolitov, A. and Shadrin, S. , TITLE =. J. Geom. Phys. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.geomphys.2018.11.010 , URL =

    work page doi:10.1016/j.geomphys.2018.11.010 2019

  42. [42]

    and Kazarian, M

    Dunin-Barkowski, P. and Kazarian, M. and Orantin, N. and Shadrin, S. and Spitz, L. , TITLE =. Adv. Math. , FJOURNAL =. 2015 , PAGES =. doi:10.1016/j.aim.2015.03.016 , URL =

    work page doi:10.1016/j.aim.2015.03.016 2015

  43. [43]

    and Orantin, N

    Dunin-Barkowski, P. and Orantin, N. and Shadrin, S. and Spitz, L. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00220-014-1887-2 , URL =

    work page doi:10.1007/s00220-014-1887-2 2014

  44. [44]

    Proceedings of the

    Eynard, Bertrand , TITLE =. Proceedings of the. 2014 , MRCLASS =

    work page 2014

  45. [45]

    Eynard, Bertrand , TITLE =. J. High Energy Phys. , FJOURNAL =. 2004 , NUMBER =. doi:10.1088/1126-6708/2004/11/031 , URL =

    work page doi:10.1088/1126-6708/2004/11/031 2004

  46. [46]

    , TITLE =

    Eynard, B. , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2014 , NUMBER =. doi:10.4310/CNTP.2014.v8.n3.a4 , URL =

    work page doi:10.4310/cntp.2014.v8.n3.a4 2014

  47. [47]

    Eynard, Bertrand and Mulase, Motohico and Safnuk, Bradley , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 2011 , NUMBER =. doi:10.2977/PRIMS/47 , URL =

    work page doi:10.2977/prims/47 2011

  48. [48]

    Eynard, Bertrand and Orantin, Nicolas , TITLE =. J. Phys. A , FJOURNAL =. 2009 , NUMBER =. doi:10.1088/1751-8113/42/29/293001 , URL =

    work page doi:10.1088/1751-8113/42/29/293001 2009

  49. [49]

    and Orantin, N

    Eynard, B. and Orantin, N. , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2007 , NUMBER =. doi:10.4310/CNTP.2007.v1.n2.a4 , URL =

    work page doi:10.4310/cntp.2007.v1.n2.a4 2007

  50. [50]

    and Orantin, N

    Eynard, B. and Orantin, N. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00220-015-2361-5 , URL =

    work page doi:10.1007/s00220-015-2361-5 2015

  51. [51]

    Fang, Bohan and Liu, Chiu-Chu Melissa and Zong, Zhengyu , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2020 , NUMBER =. doi:10.1090/jams/934 , URL =

    work page doi:10.1090/jams/934 2020

  52. [52]

    arXiv , author =:2201.05357 , primaryclass =

    On the x - y Symmetry of Correlators in Topological Recursion via Loop Insertion Operator , year =. arXiv , author =:2201.05357 , primaryclass =

    work page arXiv

  53. [53]

    arXiv , author =:2211.08917 , primaryclass =

    A simple formula for the x - y symplectic transformation in topological recursion , year =. arXiv , author =:2211.08917 , primaryclass =

    work page arXiv

  54. [54]

    arXiv , author =:2304.03032 , primaryclass =

    Laplace transform of the x-y symplectic transformation formula in Topological Recursion , year =. arXiv , author =:2304.03032 , primaryclass =

    work page arXiv

  55. [55]

    SciPost Phys

    Hock, Alexander , TITLE =. SciPost Phys. , FJOURNAL =. 2024 , NUMBER =

    work page 2024

  56. [56]

    Hock, Alexander , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s00220-025-05373-8 , URL =

    work page doi:10.1007/s00220-025-05373-8 2025

  57. [57]

    2025 , eprint=

    Quantum Curves in the Context of Symplectic Duality , author=. 2025 , eprint=

    work page 2025

  58. [58]

    2022 , eprint=

    KP hierarchy for Hurwitz-type cohomological field theories , author=. 2022 , eprint=

    work page 2022

  59. [59]

    International Mathematics Research Notices , pages =

    Kazarian, Maxim and Norbury, Paul , title = ". International Mathematics Research Notices , pages =. 2023 , month =. doi:10.1093/imrn/rnad061 , url =

    work page doi:10.1093/imrn/rnad061 2023

  60. [60]

    Lewanski, Danilo and Popolitov, Alexandr and Shadrin, Sergey and Zvonkine, Dimitri , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s11005-016-0928-5 , URL =

    work page doi:10.1007/s11005-016-0928-5 2017

  61. [61]

    Norbury, Paul , TITLE =. Geom. Topol. , FJOURNAL =. 2023 , NUMBER =. doi:10.2140/gt.2023.27.2695 , URL =

    work page doi:10.2140/gt.2023.27.2695 2023

  62. [62]

    Norbury, Paul and Scott, Nick , TITLE =. Geom. Topol. , FJOURNAL =. 2014 , NUMBER =. doi:10.2140/gt.2014.18.1865 , URL =

    work page doi:10.2140/gt.2014.18.1865 2014

  63. [63]

    and Spitz, L

    Shadrin, S. and Spitz, L. and Zvonkine, D. , TITLE =. Math. Ann. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00208-014-1082-y , URL =

    work page doi:10.1007/s00208-014-1082-y 2015

  64. [64]

    2009 , Eprint =

    Jian Zhou , Title =. 2009 , Eprint =

    work page 2009

  65. [65]

    Zhou, Jian , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s11005-013-0632-7 , URL =

    work page doi:10.1007/s11005-013-0632-7 2013

  66. [66]

    Special cases of the orbifold version of

    Borot, Ga\". Special cases of the orbifold version of. Michigan Math. J. , FJOURNAL =. 2021 , NUMBER =. doi:10.1307/mmj/1592877614 , URL =

    work page doi:10.1307/mmj/1592877614 2021

  67. [67]

    and Lewanski, D

    Kramer, R. and Lewanski, D. and Popolitov, A. and Shadrin, S. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/tran/7793 , URL =

    work page doi:10.1090/tran/7793 2019

  68. [68]

    and Pandharipande, R

    Johnson, P. and Pandharipande, R. and Tseng, H.-H. , TITLE =. Michigan Math. J. , FJOURNAL =. 2011 , NUMBER =. doi:10.1307/mmj/1301586310 , URL =

    work page doi:10.1307/mmj/1301586310 2011

  69. [69]

    Chiodo, Alessandro , TITLE =. Compos. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1112/S0010437X08003709 , URL =

    work page doi:10.1112/s0010437x08003709 2008

  70. [70]

    and Pandharipande, R

    Janda, F. and Pandharipande, R. and Pixton, A. and Zvonkine, D. , TITLE =. Publ. Math. Inst. Hautes \'. 2017 , PAGES =. doi:10.1007/s10240-017-0088-x , URL =

    work page doi:10.1007/s10240-017-0088-x 2017

  71. [71]

    An intersection-theoretic proof of the

    Giacchetto, Alessandro and Lewa\'. An intersection-theoretic proof of the. Algebr. Geom. , FJOURNAL =. 2023 , NUMBER =

    work page 2023

  72. [72]

    2024 , eprint=

    Stable tree expressions with Omega-classes and Double Ramification cycles , author=. 2024 , eprint=

    work page 2024

  73. [73]

    Ekedahl, Torsten and Lando, Sergei and Shapiro, Michael and Vainshtein, Alek , TITLE =. Invent. Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/s002220100164 , URL =

    work page doi:10.1007/s002220100164 2001

  74. [74]

    , TITLE =

    Givental, Alexander B. , TITLE =. Internat. Math. Res. Notices , FJOURNAL =. 2001 , NUMBER =. doi:10.1155/S1073792801000605 , URL =

    work page doi:10.1155/s1073792801000605 2001

  75. [75]

    Janda, Felix , TITLE =. Algebr. Geom. , FJOURNAL =. 2017 , NUMBER =. doi:10.14231/AG-2017-018 , URL =

    work page doi:10.14231/ag-2017-018 2017

  76. [76]

    2025 , eprint=

    A new spin on polynomial relations among kappa classes , author=. 2025 , eprint=

    work page 2025

  77. [77]

    2025 , eprint=

    Cohomological representations of quantum tau functions , author=. 2025 , eprint=

    work page 2025

  78. [78]

    2026 , addendum=

    A new integrable system associated to a cohomological field theory with a unit (tentative title) , author=. 2026 , addendum=

    work page 2026

  79. [79]

    Beitr\"age Algebra Geom

    Bini, Gilberto , TITLE =. Beitr\"age Algebra Geom. , FJOURNAL =. 2003 , NUMBER =

    work page 2003

  80. [80]

    A new spin on

    Giacchetto, Alessandro and Kramer, Reinier and Lewa\'. A new spin on. Selecta Math. (N.S.) , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s00029-025-01077-y , URL =

    work page doi:10.1007/s00029-025-01077-y 2025

Showing first 80 references.