Pith. sign in

REVIEW 4 minor 63 references

Flat connections for polylogarithms close under non-separating degeneration of Riemann surfaces, with residues fixed by Bernoulli series.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-11 04:13 UTC pith:WSPFJ2LM

load-bearing objection Clean inductive proof that Enriquez kernels close under A_h pinching, with the connection degenerating to the lower-genus one plus two punctures whose residues are the familiar Bernoulli composites; DHS leading-order match follows the same pattern.

arxiv 2607.05656 v1 pith:WSPFJ2LM submitted 2026-07-06 hep-th math.AGmath.NT

Degenerations of flat connections on Riemann surfaces

Mattia Biancotto , Eric D'Hoker , Axel Kleinschmidt , Michele Santagata , Oliver Schlotterer This is my paper
classification hep-th math.AGmath.NT MSC 14H5532G1511G5517B37 PACS 11.25.Db02.40.Tt
keywords Enriquez connectionDHS kernelsnon-separating degenerationpolylogarithmsRiemann surfacesBernoulli generating functionsmodular tensorsflat connections
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Polylogarithm integration kernels on a compact Riemann surface of any genus are built from flat connections that take values in infinite-dimensional Lie algebras. This paper shows that those kernels remain closed when a non-contractible cycle pinches and the surface degenerates to lower genus with two new punctures. The multivariable Enriquez connection of genus h becomes exactly the Enriquez connection of genus h-1, but with two extra punctures whose residues are composite generators built from the original ones by the same Bernoulli generating functions already known at genus one. The single-valued DHS kernels obey the identical pattern at leading order in the real degeneration parameter that controls modular tensors. The result supplies an explicit analytic bridge between iterated integrals on surfaces of different topology, which is needed both for unitarity checks in string amplitudes and for successive solution of differential equations that mix geometries in Feynman integrals.

Core claim

Under non-separating pinching of the cycle A_h, every multivariable Enriquez kernel of genus h degenerates to a linear combination of Enriquez kernels of genus h-1 (or vanishes), according to the five explicit rules (23a–e). Consequently the full connection reduces to the lower-genus Enriquez connection plus two punctures whose residues are the composite generators t_ia = B_ih a_i^h / (1 – e^{-2\pi i B_ih}) and t_ib = B_ih a_i^h / (1 – e^{2\pi i B_ih}). The same pattern governs the leading t^0 term of the DHS kernels after the dictionary that replaces meromorphic kernels by single-valued Green-function derivatives.

What carries the argument

The recursive A-cycle convolution formulae (9) and (11) that define higher-rank Enriquez kernels from lower-rank ones. They convert the degeneration of the prime form and Abelian differentials into an inductive proof that the kernels close, and they produce the Bernoulli generating functions that fix the composite residues.

Load-bearing premise

The inductive proof uses only the preferred fundamental domain and the special homology basis in which the pinched cycle is A_h; the general B-cycle and modular-orbit cases for Enriquez kernels are left open.

What would settle it

Compute the degeneration of a rank-3 or rank-4 Enriquez kernel for a concrete genus-2 surface by direct residue evaluation of the prime-form formula and check whether it matches the Bernoulli-weighted combination predicted by (23d,e).

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper establishes that the multivariable Enriquez kernels and connection on a compact Riemann surface of genus h close under non-separating degeneration obtained by pinching the cycle A_h. Explicit degeneration formulae (23a–e) express the genus-h kernels in terms of genus-(h-1) Enriquez kernels (or their trace parts) on a surface with two additional punctures; the connection itself degenerates to the Enriquez connection of genus h-1 with composite residues t_ia and t_ib given by the Bernoulli generating series (31), recovering the known genus-1 CEE case. Sequential pinching of all A-cycles yields the KZ connection on the sphere with 2h nodal points. Parallel leading-order (t^0) formulae (56) are derived for the single-valued DHS kernels via the dictionary (55), inheriting modular covariance.

Significance. The results supply the first fully explicit, analytically controlled link between flat connections (and therefore the associated polylogarithms and iterated integrals) on surfaces of different genera. The inductive proofs rest only on the already-published A-cycle recursions (9),(11), residue calculus and classical Bernoulli identities, with no free parameters or circular normalisations. This is directly useful for unitarity checks and low-energy expansions in string perturbation theory, for the differential equations of Feynman integrals that mix geometries, and for the boundary behaviour of modular graph tensors and higher-genus associators. The clean recovery of the Bernoulli composite generators known from genus one, together with the modular covariance of the DHS side, constitutes a genuine advance.

minor comments (4)
  1. The abstract and opening of §I state that the kernels “close” without immediately qualifying that the Enriquez statements are proved only for A_h-pinching (the modular-orbit and B-cycle cases being deferred). A single clarifying sentence would prevent over-reading.
  2. Appendix, after (A12)–(A14): the Bernoulli identity used for even ℓ is standard, yet a one-line reference to its generating-function origin would help non-specialist readers.
  3. Notation for the change of basis (50) that renders DHS kernels single-valued in the punctures is introduced only in §VII; a forward pointer earlier would improve readability.
  4. A few extracted-text artefacts (e.g., “NON-SEP ARA TING”) suggest residual PDF-conversion issues; a final proof-reading pass is advisable.

Circularity Check

0 steps flagged

No significant circularity: kernel degenerations follow by direct induction on the established A-cycle recursions, with Bernoulli residues emerging from residue calculus rather than being assumed.

full rationale

The load-bearing claims are the five degeneration rules (23a–e) for Enriquez kernels and the resulting connection formula (30) with composite generators (31). These are obtained by evaluating the recursive A-cycle convolution formulae (9) and (11) under Fay’s degeneration of the prime form and Abelian differentials (16)–(15). The Appendix supplies a complete induction on rank: base cases of ranks 2 and 3 are computed explicitly from the prime-form integral (7) and residue evaluation around the pinched cycle A_h; the inductive step substitutes the lower-rank degenerations into (9) and recovers the claimed right-hand sides, with the Bernoulli numbers appearing solely as the coefficients of the residue integrals (A10)–(A12). No parameter is fitted to data, no uniqueness theorem is imported to force the form of the residues, and the known h=1 Bernoulli series is recovered as a corollary rather than presupposed. Self-citations supply only the definitions of the kernels and the recursion itself (standard prior results); the degeneration calculation is independent and self-contained within the paper. The DHS leading-order formulae (56) are obtained by the same residue analysis after the dictionary (55) and inherit modular covariance, again without circular reduction. Scope limitations (B-cycle cases deferred) are explicitly flagged and do not affect the internal logic of the stated theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper works entirely inside the existing framework of Enriquez and DHS connections, Fay’s degeneration of Riemann surfaces, and classical properties of the prime form, Arakelov Green function and Bernoulli numbers. No free parameters are fitted; the only background results used are standard or previously published by overlapping authors and are cited with explicit equations.

axioms (4)
  • domain assumption Existence and uniqueness (up to gauge) of the Enriquez connection satisfying the stated monodromy and residue conditions on a compact Riemann surface of genus h.
    Taken as given from Enriquez (2014) and subsequent works; used throughout §§II–VI.
  • domain assumption The recursive A-cycle convolution formulas (9) and (11) that define higher-rank Enriquez kernels from lower-rank ones.
    Cited from D’Hoker–Schlotterer (2025); form the inductive engine of the Appendix.
  • standard math Fay’s local parametrization of a non-separating degeneration (annuli identification z_a z_b = q, period matrix block form).
    Classical; used in §III to set up the degeneration.
  • standard math Classical generating-function identities for Bernoulli numbers, in particular the bilinear sum (A12).
    Used in the even-ℓ inductive step of the Appendix.

pith-pipeline@v1.1.0-grok45 · 26530 in / 2494 out tokens · 23907 ms · 2026-07-11T04:13:57.394623+00:00 · methodology

0 comments
read the original abstract

The integration kernels for polylogarithm functions on a compact Riemann surface of arbitrary genus $h$ are shown to close as the surface undergoes a non-separating degeneration to one of genus $h{-}1$. Explicit formulas are obtained for the non-separating degeneration of the multivariable Enriquez connection for genus $h$ with an arbitrary number of variables to the Enriquez connection for genus $h{-}1$ with two additional punctures whose Lie algebra generators are related to the original ones by the characteristic Bernoulli generating functions known from the degeneration at $h=1$. Analogous degeneration formulas are obtained for the single-valued DHS kernels at the leading order in the real degeneration parameter that is adapted to relating modular tensors at genus $h$ and $h{-}1$.

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

63 extracted references · 19 canonical work pages · 4 internal anchors

  1. [1]

    and especially the second paragraph of section 4.2 in [17]). IV. DEGENERA TION OF ENRIQUEZ KERNELS The behaviour of Enriquez kernels under the above degeneration may be organized according to the rank R=r+1ofbg I1···Ir J(x, y)and its subset of indices equal toh. For this purpose, we use Greek lettersµ, ν,· · ·= 1,· · ·, h−1for indices different fromhand c...

    work page

  2. [2]

    Amplitudes and Algebraic Geometry

    Using the generating functions for the Bernoulli numbers, ±x e±2πix −1 = ∞X ℓ=0 (2πi)ℓ−1 Ber∓ ℓ ℓ! xℓ (29) the degeneration of the connection componentsbKi can be brought into the intuitive form, bKi(x)→K i(x)+X(x i, pa;B i)tia+X(xi, pb;B i)tib (30) where the composite generatorstia,t ib are defined as, tia = Bih 1−e −2πiBih ah i , t ib = Bih 1−e 2πiBih a...

    work page 2026

  3. [3]

    J. A. Lappo-Danilevsky,Mémoires sur la théorie des systèmes des équations différentielles linéaires. Vol. III. Chelsea Publishing Company, New York, 1953

    work page 1953

  4. [4]

    Geometry of Configurations, Polylogarithms, and Motivic Cohomology,

    A. B. Goncharov, “Geometry of Configurations, Polylogarithms, and Motivic Cohomology,”Adv. Math. 4 The components ofb iI ,a I i entering the residuest ia,t ib of (31) also need to be adapted to the change of basis (50) towards term- by-term single-valuedness inp a,p b. 114(1995) 197–318

    work page 1995

  5. [5]

    Elliptic polylogarithms: An analytic theory,

    A. Levin, “Elliptic polylogarithms: An analytic theory,” Compositio Mathematica106(1997) 267–282.https: //api.semanticscholar.org/CorpusID:117413459

    work page 1997

  6. [6]

    Universal KZB equations I: the elliptic case

    D. Calaque, B. Enriquez, and P. Etingof, “Universal KZB equations: the elliptic case,” inAlgebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, vol. 269 ofProgr. Math., pp. 165–266. Birkhäuser Boston, Boston, MA, 2009. 9 arXiv:math/0702670 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  7. [7]

    Towards multiple elliptic polylogarithms,

    A. Levin and G. Racinet, “Towards multiple elliptic polylogarithms,”arXiv:math/0703237 [math.NT]

    Pith/arXiv arXiv

  8. [8]

    Elliptic hyperlogarithms,

    B. Enriquez and F. Zerbini, “Elliptic hyperlogarithms,” Canadian Journal of Mathematics78no. 3, (2025) 733–768,arXiv:2307.01833 [math.AG]

    Pith/arXiv arXiv 2025

  9. [9]

    Multiple Elliptic Polylogarithms,

    F. Brown and A. Levin, “Multiple Elliptic Polylogarithms,”arXiv:1110.6917 [math.NT]

    Pith/arXiv arXiv

  10. [10]

    Weinzierl,Feynman Integrals

    S. Weinzierl,Feynman Integrals. A Comprehensive Treatment for Students and Researchers. UNITEXT for Physics. Springer, 2022.arXiv:2201.03593 [hep-th]

    Pith/arXiv arXiv 2022

  11. [11]

    Functions Beyond Multiple Polylogarithms for Precision Collider Physics,

    J. L. Bourjailyet al., “Functions Beyond Multiple Polylogarithms for Precision Collider Physics,” in Snowmass 2021. 3, 2022.arXiv:2203.07088 [hep-ph]

    Pith/arXiv arXiv 2021

  12. [12]

    Snowmass White Paper: String Perturbation Theory,

    N. Berkovits, E. D’Hoker, M. B. Green, H. Johansson, and O. Schlotterer, “Snowmass White Paper: String Perturbation Theory,” inSnowmass 2021. arXiv:2203.09099 [hep-th]

    Pith/arXiv arXiv 2021

  13. [13]

    D’Hoker and J

    E. D’Hoker and J. Kaidi,Modular Forms and String Theory. Cambridge University Press, 2024

    work page 2024

  14. [14]

    Flat connections on configuration spaces and braid groups of surfaces,

    B. Enriquez, “Flat connections on configuration spaces and braid groups of surfaces,”Advances in Mathematics 252(2014) 204–226,arXiv:1112.0864 [math.GT]

    Pith/arXiv arXiv 2014

  15. [15]

    Constructing polylogarithms on higher-genus Riemann surfaces,

    E. D’Hoker, M. Hidding, and O. Schlotterer, “Constructing polylogarithms on higher-genus Riemann surfaces,”Commun. Num. Theor. Phys.19no. 2, (2025) 355–413,arXiv:2306.08644 [hep-th]

    Pith/arXiv arXiv 2025

  16. [16]

    Single-valued flat connections in several variables on arbitrary Riemann surfaces,

    E. D’Hoker and O. Schlotterer, “Single-valued flat connections in several variables on arbitrary Riemann surfaces,”arXiv:2602.01461 [hep-th]

    arXiv

  17. [17]

    Relating Flat Connections and Polylogarithms on Higher Genus Riemann Surfaces,

    E. D’Hoker, B. Enriquez, O. Schlotterer, and F. Zerbini, “Relating Flat Connections and Polylogarithms on Higher Genus Riemann Surfaces,”Commun. Math. Phys.407no. 3, (2026) 43,arXiv:2501.07640 [hep-th]

    arXiv 2026

  18. [18]

    Towards a global flat connection on moduli spaces of marked Riemann surfaces

    E. D’Hoker, B. Enriquez, O. Schlotterer and F. Zerbini, “Towards a global flat connection on moduli spaces of marked Riemann surfaces.”. In preparation

    work page

  19. [19]

    Higher genus modular graph functions, string invariants, and their exact asymptotics,

    E. D’Hoker, M. B. Green, and B. Pioline, “Higher genus modular graph functions, string invariants, and their exact asymptotics,”Commun. Math. Phys.366no. 3, (2019) 927–979,arXiv:1712.06135 [hep-th]

    Pith/arXiv arXiv 2019

  20. [20]

    Asymptotics of theD 8R4 genus-two string invariant,

    E. D’Hoker, M. B. Green, and B. Pioline, “Asymptotics of theD 8R4 genus-two string invariant,”Commun. Num. Theor. Phys.13(2019) 351–462, arXiv:1806.02691 [hep-th]

    Pith/arXiv arXiv 2019

  21. [21]

    String theory integrands and supergravity divergences,

    B. Pioline, “String theory integrands and supergravity divergences,”JHEP02(2019) 148,arXiv:1810.11343 [hep-th]

    Pith/arXiv arXiv 2019

  22. [22]

    Exploring transcendentality in superstring amplitudes

    E. D’Hoker and M. B. Green, “Exploring transcendentality in superstring amplitudes,”JHEP07 (2019) 149,arXiv:1906.01652 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  23. [23]

    One-loop matrix elements of effective superstring interactions:α’-expanding loop integrands,

    A. Edison, M. Guillen, H. Johansson, O. Schlotterer, and F. Teng, “One-loop matrix elements of effective superstring interactions:α’-expanding loop integrands,” JHEP12(2021) 007,arXiv:2107.08009 [hep-th]

    Pith/arXiv arXiv 2021

  24. [24]

    Unitarity cuts of the worldsheet,

    L. Eberhardt and S. Mizera, “Unitarity cuts of the worldsheet,”SciPost Phys.14no. 2, (2023) 015, arXiv:2208.12233 [hep-th]

    Pith/arXiv arXiv 2023

  25. [25]

    Two-Loop QED Corrections to the Scattering of Four Massive Leptons,

    M. Delto, C. Duhr, L. Tancredi, and Y. J. Zhu, “Two-Loop QED Corrections to the Scattering of Four Massive Leptons,”Phys. Rev. Lett.132no. 23, (2024) 231904,arXiv:2311.06385 [hep-ph]

    Pith/arXiv arXiv 2024

  26. [26]

    Emergence of Calabi–Yau manifolds in high-precision black-hole scattering,

    M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer, and J. Usovitsch, “Emergence of Calabi–Yau manifolds in high-precision black-hole scattering,”Nature641no. 8063, (2025) 603–607,arXiv:2411.11846 [hep-th]

    Pith/arXiv arXiv 2025

  27. [27]

    On the electron self-energy to three loops in QED,

    C. Duhr, F. Gasparotto, C. Nega, L. Tancredi, and S. Weinzierl, “On the electron self-energy to three loops in QED,”JHEP11(2024) 020,arXiv:2408.05154 [hep-th]

    Pith/arXiv arXiv 2024

  28. [28]

    Analytic two-loop amplitudes for di-jet andγ+jet production mediated by a heavy-quark loop,

    F. Coro, C. Nega, L. Tancredi, and F. J. Wagner, “Analytic two-loop amplitudes for di-jet andγ+jet production mediated by a heavy-quark loop,”JHEP01 (2026) 090,arXiv:2509.15315 [hep-ph]

    arXiv 2026

  29. [29]

    Analytic two-loop amplitudes forqq→γγ and gg→γγmediated by a heavy-quark loop,

    M. Becchetti, F. Coro, C. Nega, L. Tancredi, and F. J. Wagner, “Analytic two-loop amplitudes forqq→γγ and gg→γγmediated by a heavy-quark loop,”JHEP 06(2025) 033,arXiv:2502.00118 [hep-ph]

    Pith/arXiv arXiv 2025

  30. [30]

    Construction of Maurer-Cartan elements over configuration spaces of curves,

    B. Enriquez and F. Zerbini, “Construction of Maurer-Cartan elements over configuration spaces of curves,”arXiv:2110.09341 [math.AG]

    Pith/arXiv arXiv

  31. [31]

    Meromorphic higher-genus integration kernels via convolution over homology cycles,

    E. D’Hoker and O. Schlotterer, “Meromorphic higher-genus integration kernels via convolution over homology cycles,”J. Phys. A58no. 33, (2025) 33LT01, arXiv:2502.14769 [hep-th]

    Pith/arXiv arXiv 2025

  32. [32]

    Schottky–Kronecker forms and hyperelliptic polylogarithms,

    K. Baune, J. Broedel, E. Im, A. Lisitsyn, and F. Zerbini, “Schottky–Kronecker forms and hyperelliptic polylogarithms,”J. Phys. A57no. 44, (2024) 445202, arXiv:2406.10051 [hep-th]

    Pith/arXiv arXiv 2024

  33. [33]

    J. D. Fay,Theta Functions on Riemann Surfaces, vol. 352 ofLecture Notes in Mathematics. Springer-Verlag, Berlin-Heidelberg-New York, 1973

    work page 1973

  34. [34]

    Higher-genus multiple zeta values,

    K. Baune, J. Broedel, E. Im, Z. Ji, and Y. Moeckli, “Higher-genus multiple zeta values,”J. Phys. A58 no. 46, (2025) 465401,arXiv:2507.21765 [hep-th]

    Pith/arXiv arXiv 2025

  35. [35]

    Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality,

    E. D’Hoker, C. R. Mafra, B. Pioline, and O. Schlotterer, “Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality,”JHEP02(2021) 139,arXiv:2008.08687 [hep-th]

    Pith/arXiv arXiv 2021

  36. [36]

    Fay identities for polylogarithms on higher-genus Riemann surfaces,

    E. D’Hoker and O. Schlotterer, “Fay identities for polylogarithms on higher-genus Riemann surfaces,” arXiv:2407.11476 [hep-th]

    arXiv

  37. [37]

    Higher-genus Fay-like identities from meromorphic generating functions,

    K. Baune, J. Broedel, E. Im, A. Lisitsyn, and Y. Moeckli, “Higher-genus Fay-like identities from meromorphic generating functions,”SciPost Phys.18 no. 3, (2025) 093,arXiv:2409.08208 [hep-th]

    Pith/arXiv arXiv 2025

  38. [38]

    Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces,

    E. D’Hoker and O. Schlotterer, “Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces,”arXiv:2602.09108 [hep-th]

    arXiv

  39. [39]

    Calculus on arithmetic surfaces,

    G. Faltings, “Calculus on arithmetic surfaces,”Ann. of Math. (2)119no. 2, (1984) 387–424

    work page 1984

  40. [40]

    Bosonization in Arbitrary Genus,

    L. Alvarez-Gaume, G. W. Moore, P. C. Nelson, C. Vafa, and J. b. Bost, “Bosonization in Arbitrary Genus,” Phys. Lett. B178(1986) 41–47

    work page 1986

  41. [41]

    Analogues elliptiques des nombres multizétas,

    B. Enriquez, “Analogues elliptiques des nombres multizétas,”Bull. Soc. Math. France144no. 3, (2016) 395–427,arXiv:1301.3042 [math.NT]

    Pith/arXiv arXiv 2016

  42. [42]

    From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop,

    J. Broedel, O. Schlotterer, and F. Zerbini, “From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop,”JHEP01(2019) 155,arXiv:1803.00527 [hep-th]

    Pith/arXiv arXiv 2019

  43. [43]

    Zerbini,Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes

    F. Zerbini,Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes. PhD thesis, Bonn U., 2017.arXiv:1804.07989 [math-ph]. 10

    Pith/arXiv arXiv 2017

  44. [44]

    The Enriquez connection for higher genus polylogarithms,

    T. Ichikawa, “The Enriquez connection for higher genus polylogarithms,”JHEP02(2026) 140, arXiv:2510.00486 [math.AG]

    work page arXiv 2026

  45. [45]

    Some tensor field on the Teichmüller space,

    N. Kawazumi, “Some tensor field on the Teichmüller space,” 2016.http://www.ms.u-tokyo.ac.jp/ ~kawazumi/OIST1610_v1.pdf. Lecture at MCM2016, OIST

    work page 2016

  46. [46]

    Differential forms and functions on the moduli space of Riemann surfaces,

    N. Kawazumi, “Differential forms and functions on the moduli space of Riemann surfaces,” 2017. http://www.ms.u-tokyo.ac.jp/~kawazumi/ 1701Strasbourg_v1.pdf. in Séminaire Algèbre et topologie, Université de Strasbourg

    work page 2017

  47. [47]

    Identities among higher genus modular graph tensors,

    E. D’Hoker and O. Schlotterer, “Identities among higher genus modular graph tensors,”Commun. Num. Theor. Phys.16no. 1, (2022) 35–74,arXiv:2010.00924 [hep-th]

    Pith/arXiv arXiv 2022

  48. [48]

    Elliptic associators,

    B. Enriquez, “Elliptic associators,”Selecta Math. (N.S.) 20no. 2, (2014) 491–584,arXiv:1003.1012 [math.QA]

    Pith/arXiv arXiv 2014

  49. [49]

    Surface drinfeld torsors i: Higher genus associators,

    M. Gonzalez, “Surface drinfeld torsors i: Higher genus associators,”arXiv:2004.07303 [math.QA]

    Pith/arXiv arXiv 2004

  50. [50]

    Drinfeld associators and Kashiwara-Vergne associators in higher genera,

    T. Taniguchi, “Drinfeld associators and Kashiwara-Vergne associators in higher genera,” arXiv:2511.00473 [math.QA]

    Pith/arXiv arXiv

  51. [51]

    Relations between elliptic multiple zeta values and a special derivation algebra,

    J. Broedel, N. Matthes, and O. Schlotterer, “Relations between elliptic multiple zeta values and a special derivation algebra,”J. Phys.A49no. 15, (2016) 155203,arXiv:1507.02254 [hep-th]

    Pith/arXiv arXiv 2016

  52. [52]

    Elliptic multizetas and the elliptic double shuffle relations,

    P. Lochak, N. Matthes, and L. Schneps, “Elliptic multizetas and the elliptic double shuffle relations,” International Mathematics Research Notices2021 (2020) 695–753,arXiv:1703.09410 [math.NT]

    Pith/arXiv arXiv 2020

  53. [53]

    A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals,

    F. Brown, “A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals,” Forum of Mathematics, Sigma8(2020) 1, arXiv:1708.03354 [math.NT]

    Pith/arXiv arXiv 2020

  54. [54]

    Non-holomorphic modular forms from zeta generators,

    D. Dorigoni, M. Doroudiani, J. Drewitt, M. Hidding, A. Kleinschmidt, O. Schlotterer, L. Schneps, and B. Verbeek, “Non-holomorphic modular forms from zeta generators,”JHEP10(2024) 053,arXiv:2403.14816 [hep-th]

    Pith/arXiv arXiv 2024

  55. [55]

    Canonicalizing Zeta Generators: Genus Zero and Genus One,

    D. Dorigoni, M. Doroudiani, J. Drewitt, M. Hidding, A. Kleinschmidt, O. Schlotterer, L. Schneps, and B. Verbeek, “Canonicalizing Zeta Generators: Genus Zero and Genus One,”Commun. Math. Phys.407 no. 1, (2026) 12,arXiv:2406.05099 [math.QA]

    arXiv 2026

  56. [56]

    Cyclic Products of Higher-Genus Szegö Kernels, Modular Tensors, and Polylogarithms,

    E. D’Hoker, M. Hidding, and O. Schlotterer, “Cyclic Products of Higher-Genus Szegö Kernels, Modular Tensors, and Polylogarithms,”Phys. Rev. Lett.133 no. 2, (2024) 021602,arXiv:2308.05044 [hep-th]

    Pith/arXiv arXiv 2024

  57. [57]

    Worldsheet fermion correlators, modular tensors and higher genus integration kernels,

    E. D’Hoker and O. Schlotterer, “Worldsheet fermion correlators, modular tensors and higher genus integration kernels,”SciPost Phys.19no. 4, (2025) 091, arXiv:2505.07947 [hep-th]

    arXiv 2025

  58. [58]

    Superstring loop amplitudes from the field theory limit

    Y. Geyer, R. Monteiro, and R. Stark-Muchão, “Superstring Loop Amplitudes from the Field Theory Limit,”Phys. Rev. Lett.127no. 21, (2021) 211603, arXiv:2106.03968 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  59. [59]

    Superstring amplitudes from BCJ numerators at one loop

    Y. Geyer, J. Guo, R. Monteiro, and L. Ren, “Superstring amplitudes from BCJ numerators at one loop,”JHEP03(2025) 017,arXiv:2410.19663 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  60. [60]

    Superstring amplitudes from the field-theory limit: an n-point map at one loop,

    R. Monteiro and L. Ren, “Superstring amplitudes from the field-theory limit: an n-point map at one loop,” JHEP01(2026) 057,arXiv:2510.02455 [hep-th]

    work page arXiv 2026

  61. [61]

    Genus drop in hyperelliptic Feynman integrals,

    R. Marzucca, A. J. McLeod, B. Page, S. Pögel, and S. Weinzierl, “Genus drop in hyperelliptic Feynman integrals,”Phys. Rev. D109no. 3, (2024) L031901, arXiv:2307.11497 [hep-th]

    Pith/arXiv arXiv 2024

  62. [62]

    Canonical differential equations beyond genus one,

    C. Duhr, F. Porkert, and S. F. Stawinski, “Canonical differential equations beyond genus one,”JHEP02 (2025) 014,arXiv:2412.02300 [hep-th]. Appendix In this appendix, we shall prove the general formulas (23) for the non-separating degeneration of Enriquez kernels. The proof of each one of the five relations in (23), which we repeat here for convenience, bg...

    arXiv 2025

  63. [63]

    sinceBer k vanishes fork≥3 odd and the degeneration ofbg⃗hℓ−k h(x, y)vanishes fork≥2even by the inductive assumption. Finally, the last line follows from: 1 2 I Bϵ(pa) dt χpat(x) +χ pbt(x) χpapb(t) =− I Bϵ(pa) dt χ(x, t)χ papb(t) =−πiχ papb(x)(A10) 5 Note that there is an implicit summation overβ, but not overα. In other words,χ β papb (x)δα β δ⃗ µ β =χ α...

    work page