Pith. sign in

REVIEW 3 major objections 5 minor 6 cited by

Higher-genus multiple zeta values

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Higher-genus multiple zeta values are well-defined A-cycle integrals on Riemann surfaces whose identities go beyond polylogarithm relations.

desk verdict A serious first systematic treatment of higher-genus MZVs, with honest caveats; the depth>1 regularization gap is real but should not block peer review. read the letter →

arxiv 2507.21765 v1 pith:UFVRMRGU submitted 2025-07-29 hep-th math.AGmath.NT

Konstantin Baune , Johannes Broedel , Egor Im , Zhexian Ji , Yannis Moeckli This is my paper
classification hep-thmath.AGmath.NT MSC 11M3211G5514H5530F40
keywords higher-genusmultiplezetavaluespolylogarithmsSchottkyuniformizationiteratedintegralshyperellipticsurfacesellipticdegenerationofRiemannFayidentities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends the theory of multiple zeta values from Riemann surfaces of genus zero and one to compact Riemann surfaces of any genus h>1. The proposed objects, higher-genus multiple zeta values, are iterated integrals of a distinguished family of meromorphic kernels around non-contractible A-cycles; the paper gives a regularization prescription that uses Schottky uniformization to reduce endpoint divergences to the known genus-one case. The paper argues that these numbers encode the surface's complex structure and, unlike their lower-genus relatives, satisfy additional relations: a proven cycle-exchange identity, a proven alternating identity on hyperelliptic surfaces, and two conjectured families backed by numerical checks. If the construction is right, it provides the natural next rung in the genus ladder, with classical multiple zeta values at genus zero, elliptic multiple zeta values at genus one, and genuinely higher-genus invariants above.

What carries the argument

The load-bearing objects are the paper's higher-genus integration kernels, defined as expansion coefficients of a unique flat connection on the universal cover and fixed by quasi-periodicity, residue, and A-cycle axioms. The machinery that carries the argument is Schottky uniformization: the kernels are expanded as Poincaré sums over the Schottky group, with coefficients fixed by a recursion, so regularization of endpoint divergences can be traced back to the known genus-one regularization of Abel's map. A second mechanism is the linear identity (4.6), asserting that differences of diagonal kernels are independent of the auxiliary basepoint; together with shuffle relations it is used to define higher-depth regularized values. Residue-theoretic A-cycle integration on the Schottky cover then turns identities into combinatorial statements about splittings of Schottky words.

What would settle it

A concrete check would be to compute a divergent depth-two hgMZV such as ζ_{A_i}((ii),(jj)) by two independent methods: the epsilon-regulated A-cycle integral and the shuffle-plus-(4.6) reduction to depth-one regularized values, and compare the results. If the two prescriptions ever disagree, the higher-depth values are not well-defined and the identities in Section 8 would need to be re-derived.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that iterated integrals of the higher-genus integration kernels around the A-cycles of a Riemann surface of genus h≥1 define well-defined regularized numbers, called higher-genus multiple zeta values, and that these numbers obey identities not implied by the functional equations of the underlying polylogarithms. The paper proves the cycle-exchange identity of Theorem 13, which equates an integral along one A-cycle with an integral along another plus Bernoulli-number corrections, and the alternating identity of Theorem 14 for alternating labels on hyperelliptic surfaces. It also states as conjectures a depth-two weight-exchange reflection identity and a rule lifting elliptic MZV relations to higher genus, both supported by numerical checks. At depth one the new values reduce to elliptic multiple zeta values and hence to classical zeta values of even argument, with regularized odd cases equal to 0 or 1/2.

Load-bearing premise

The load-bearing premise is that the depth-one regularization prescription can be extended to higher depth via shuffle relations and the linear identity (4.6) without conflict; the paper explicitly states that this compatibility has not been proven.

Editorial extensions

If this is right

  • Depth-one higher-genus multiple zeta values are not new transcendental functions: they equal elliptic multiple zeta values, hence rational multiples of classical zeta values at even weight.
  • Pinching an A-cycle degenerates a genus-h value to a genus-(h-1) value when the entries avoid the pinched direction, giving a concrete degeneration ladder; separating degeneration splits values into a genus-one and a lower-genus part.
  • The cycle-exchange identity (8.6) mixes integrals along different A-cycles, so the values along distinct cycles cannot be treated as independent algebras.
  • The alternating identity (8.17) holds on hyperelliptic surfaces when the total weight is even, and fails in general, showing that surface symmetry produces genuinely new relation classes.
  • If the weight-exchange and lifting conjectures are correct, depth-two values with labels in a single direction are symmetric and reduce to elliptic values, while mixed labels are antisymmetric.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves open whether the depth-one regularization is compatible with the shuffle and linear identities; if it is not, the higher-depth definitions and the Section 8 identities would need adjustment, so a direct compatibility proof is the first thing an interested reader should look for.
  • The coefficient recursion that organizes the Schottky expansions resembles the word combinatorics of an associator; one could try to extract from it a higher-genus symbol or coproduct, which the paper itself lists as an open problem.
  • The alternating identity has no genus-one counterpart, which suggests that hyperelliptic or involutive cover symmetry is a distinct source of relations; the numerical identities around eq. (8.28) hint that many more such relations exist at higher depth.
  • The lifting of eMZV identities to all-equal-label hgMZVs implies that the elliptic part of a hgMZV dominates certain relations even when the higher-genus geometry is non-trivial; a testable consequence would be that any proven eMZV identity lifts number-for-number to genus two.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper defines higher-genus multiple zeta values (hgMZVs) as iterated integrals of Enriquez' meromorphic kernels along A-cycles of a genus-h Riemann surface, using Schottky uniformization to extend the known genus-one regularization. It gives a depth-one regularization formula, extends it to higher depth via shuffle relations and the linear identity (4.6), proves degeneration statements for separating and non-separating limits, and derives several new relations: Fay-type identities, a cycle-exchange identity (Theorem 13), and an alternating identity on hyperelliptic surfaces (Theorem 14). It also states two numerically supported conjectures (weight exchange and lifting of elliptic MZV identities) and provides numerical checks for the proven identities.

Significance. If the construction is fully sound, this is a substantial contribution: it provides the first systematic framework for multiple zeta values on Riemann surfaces of genus h>1, with explicit and often detailed analytic proofs of relations that go beyond the elliptic case, and it gives a coherent Schottky-based computational language. The paper is honest about the limits of its derivations, and the proven Theorems 13 and 14 are nontrivial. The numerical verification of all stated identities for generic genus-two and genus-three examples is also a strength. However, the central well-definedness claim is conditional on an explicitly unproven compatibility of the regularization prescription, so the significance can only be assessed after that gap is resolved.

major comments (3)
  1. [§4.2.1, §4.2.3, §6.2] The higher-depth regularization of hgMZVs is the load-bearing step of the paper, and it is explicitly left open. In §4.2.1 the authors state that they do not make any statement on the compatibility of the general approach (4.22)–(4.23) with the extension to higher depth by shuffle (3.3) and the linear identity (4.6); §4.2.3 then defines higher-depth hgMPLs using exactly that extension, and §6.2 transfers this definition to hgMZVs. If the two prescriptions disagree, the higher-depth values entering the headline identities (8.2), (8.6), (8.17) and the weight-exchange conjecture are convention-dependent. The authors' caveat on p.22 is therefore not a harmless disclaimer but a gap in the definition of the objects studied in Section 8. I ask for either a proof that the depth-one regularization is compatible with shuffle and (4.6), or an explicit definition of hgMZVs as the output of a provably confluent algorithm, together with a verification that the identities of Section 8 hold for that definition.
  2. [§6.1, Definition 9] Definition 9 says z0 is an arbitrary point on the cycle Aj, but the paper does not prove that the regularized hgMZV is independent of this choice. For depth-one values, independence follows from the explicit result (6.8), but for higher depth the shuffle/linear-identity algorithm depends on z0 through the pole variables and through the path decomposition around Aj. In general, changing the basepoint of a based loop conjugates the Chen iterated integral by the transport along the segment, so noncommutative coefficients can change. Since the notation ζ_Aj(i1,...,ik) suppresses z0, the definition is incomplete unless basepoint independence is established or z0 is declared to be part of the data and the dependence is analyzed.
  3. [§8.2, Theorem 13] The proof of the cycle-exchange identity relies on the prime-form representation of Enriquez' kernels as reviewed in §4.1.3, and in particular on [38, Thm. 1] and [38, Thm. 2] in eqs. (8.11) and (8.13). These are external preprints that are not proved in the present paper. The use of such results is acceptable in principle, but since they are load-bearing for a central new identity, the author should either include self-contained statements of the needed theorems or indicate precisely which parts of the proof would need modification if the external results change. This is a presentation/robustness issue rather than a detected error.
minor comments (5)
  1. [§1, Table 1] The notation [3]ζ_A1(121,1232|G) is introduced before the formal definition in §6.1; this is acceptable for motivation, but a forward reference with equation numbers would improve readability.
  2. [§6.3, eq. (6.10)] The formula for depth-one hgMZVs uses ζ0 ≡ 1/2, which is not a standard convention for ζ(0); the authors should state explicitly that this is a shorthand for the regularized value, to avoid confusion with the ordinary Riemann zeta value.
  3. [§8.5, Conjecture 16] The statement that 'any known eMZV identity' lifts to hgMZVs is very broad. The examples in Example 17 are helpful, but the conjecture would benefit from a precise algebraic form (e.g., polynomial identities in the ζ_Aj symbols) and from a statement about how the normalization factors in footnote 28 are applied uniformly.
  4. [§7.2, eqs. (7.7)–(7.24)] The separating degeneration is described with O(epsilon) estimates, but the paper does not discuss the convergence of the Poincaré series as epsilon → 0. A brief justification that the resummations used in eqs. (7.23) and (7.27) commute with the degeneration limit would make the argument more rigorous.
  5. [References] The numerical package SchottkyTools is cited as work in progress, and no version or numerical data are provided. For reproducibility of the claimed numerical checks, it would be helpful to include the specific genera, Schottky parameters, and truncation orders used in the tests.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: hgMZVs are defined from Enriquez kernels and Schottky uniformization; the unproven shuffle/linear regularization compatibility is a well-definedness gap, not a circular input.

full rationale

The paper's central objects are defined as iterated integrals of Enriquez' kernels over A-cycles (Definition 9), and the claimed new identities are derived from the defining properties of those kernels, the Schottky representation, Fay-like identities, and hyperelliptic symmetry. No parameter is fitted to data and then renamed as a prediction: the depth-one regularization is computed explicitly from the Schottky expansion and Abel's map, and the higher-depth extension is presented as an algorithm using shuffle relations and the linear identity. The authors explicitly flag the one genuine gap: 'we do not make any statement on the compatibility of the general approach described by eqs. (4.22) and (4.23) on the one hand and its application at depth one, extended to higher depth by means of eq. (3.3) and eq. (4.6), on the other hand' (Section 4.2.1). This is an unproven well-definedness assumption, not a circular reduction: the depth-one formula (4.31) is not defined in terms of the higher-depth values whose uniqueness is in question, and the Section 8 identities are checked numerically against an independent evaluation scheme. Reliance on prior work [39, 40, 45, 55], including by some of the same authors, is support from published, independently derived mathematics rather than a self-citation chain forbidding alternatives; the paper even supplies an alternative proof of the depth-one formula 'using Schottky uniformization' rather than merely citing it. The conjectures (Conjectures 15 and 16) are explicitly labeled as such and backed by numerical tests, so they are not dressed up as derivations. I find no step where an output equation equals its input by construction, and therefore no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The construction rests on the established theory of Enriquez' kernels and the Schottky uniformization, plus one unproven compatibility assumption for higher-depth regularization. No free parameters are fitted to data. The only invented entity is the new class of zeta values itself, which is well-defined and numerically checkable.

assumptions (5)
  • domain assumption There exists a unique meromorphic flat connection K(z,x) on the universal cover of Sigma_h satisfying (4.1a)-(4.1c).
    Relied on for the definition of Enriquez' kernels and all subsequent constructions; taken from ref. [39].
  • domain assumption Enriquez' kernels admit the Schottky Poincare series expansion (4.8) with coefficients (4.11).
    Central tool for regularization and computations; taken from ref. [45].
  • ad hoc to paper The depth-one regularization (4.22)-(4.23) is compatible with the shuffle product and the linear identity (4.6) at higher depth.
    Used to define regularized hgMZVs of depth >1; the authors state this compatibility is not proven in Section 4.2.1.
  • domain assumption The Fay-like identities (4.5) hold for Enriquez' kernels at any genus.
    Used to derive hgMZV relations in Section 8.1; from refs. [55,56].
  • standard math Keen's theorem: hyperelliptic Riemann surfaces admit a Schottky group with fixed points paired by P'_j = -P_j.
    Used in the proof of the alternating identity (Section 8.3).
invented entities (1)
  • Higher-genus multiple zeta values (hgMZVs) independent evidence
    purpose: New objects of study, generalizing eMZVs to genus h>1.
    They are precisely defined in Definition 9 and evaluated numerically via SchottkyTools; the numerical values can in principle be checked by independent implementations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Higher-genus multiple zeta values." pith.science (2026). https://pith.science/paper/UFVRMRGU

@misc{pith2026250721765,
  author       = {Pith},
  title        = {Pith review of: Higher-genus multiple zeta values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UFVRMRGU}},
  note         = {Machine review of arXiv:2507.21765}
}
read the original abstract

Multiple zeta values arise as special values of polylogarithms defined on Riemann surfaces of various genera. Building on the vast knowledge for classical and elliptic multiple zeta values, we explore a canonical extension of the formalism to Riemann surfaces of higher genera, which yields higher-genus multiple zeta values. We provide a regularization prescription for higher-genus polylogarithms, which we extend to higher-genus multiple zeta values. Our regularization uses the Schottky uniformization to trace back higher-genus endpoint regularization to known regularization at genus one. Additionally, we are commenting on relations among higher-genus multiple zeta values implied by degeneration of the underlying geometry, where we distinguish between the two types of separating and non-separating degeneration. Finally, employing functional relations for higher-genus polylogarithms in the Schottky uniformization, we explore relations among higher-genus multiple zeta values and check them against our numerical testing setup. We identify relations for higher-genus multiple zeta values beyond those implied by polylogarithm identities, thereby matching the situation for genus zero and genus one. While we find several known structures for elliptic multiple zeta values to generalize to relations for higher-genus multiple zeta values, there are further classes of relations arising from the interplay and combinatorics of different cycles.

Figures

Figures reproduced from arXiv: 2507.21765 by the authors.

Figure 1
Figure 1. Schottky uniformization (b) of the genus-two Riemann surface in (a). A- and B-cycles are drawn in red and blue, respectively. The fundamental domain F of the Schottky cover is the region outside all circles. homology class3 of Ai for all i ∈ {1, . . . , h}. Accordingly, we define the circle Ci with negative (i.e. clockwise) orientation to be the path on the covering space representing the cycle Ai on the Riemann sur… view at source ↗
Figure 2
Figure 2. A genus-three Schottky cover with a circle setup such that the fixed points fulfill P ′ j = − Pj , making this surface manifestly hyperelliptic. Applying the transformation θ from the proof of Lemma 2 swaps the fixed points, leading to the picture on the right side. for P ′ j , Pj the fixed points of σj . Proof. The proof is essentially carried out in ref. [47]. In particular, ref. [47, Lemma 2] implies that G admit… view at source ↗
Figure 3
Figure 3. Illustration of the residue conditions in Situation 1 – Situation 3. It shows a Schottky cover corresponding to a surface of genus three. The path intuitively corresponds to the element Υ = σ3σ −2 2 σ −1 3 σ 2 2σ3σ1 ∈ G (applied to z0 ∈ F). The condition in Situation 2 amounts to counting the positive intersections of the path and the circles C2 and σ −1 2 C2. Correspondingly, the condition in Situ￾ation 3 correspon… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Degenerations of a genus-two Riemann surface. (a) Non-separating degeneration, where an A-cycle is pinched, resulting in a genus-one surface with two punctures. (b) Separating degeneration, where the surface splits into two lower-genus components. 7.1 Non-separating de…
Figure 5
Figure 5. Figure 5: Figure (a) shows a graphical representation of degenerating a Schottky cover of genus three by shrinking one pair of circles, which corresponds to the non-separating degeneration in Figure 4a. Figure (b) shows the absolute value of the difference between the value of […
Figure 6
Figure 6. Figure 6: Figure (a) depicts a degeneration of a genus-three surface by the rescaling (7.6), which corresponds to the separating degeneration from Figure 4b. The absolute value of the difference between [3]ζA1 (21, 12), which depends on the scaling parameter ϵ, and [2]ζA1 (21, 1…

Discussion (0). Sign in to comment.

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Degenerations of flat connections on Riemann surfaces

    hep-th 2026-07 accept novelty 7.0 of 10

    Enriquez and DHS kernels on genus-h surfaces close under non-separating degeneration to genus h-1 kernels with two punctures whose generators are Bernoulli series in the original Lie algebra elements.

  2. A construction of single-valued elliptic polylogarithms

    hep-th 2025-11 unverdicted novelty 7.0 of 10

    A construction of single-valued elliptic polylogarithms on the punctured elliptic curve is given that reduces to Brown's genus-zero condition upon torus degeneration.

  3. A double copy from twisted (co)homology at genus g

    hep-th 2025-09 conditional novelty 7.0 of 10

    A double-copy (KLT-type) formula for genus-g hypergeometric integrals is derived from twisted homology intersection numbers and verified numerically at genus two.

  4. Single-valued polylogarithms for higher genera

    hep-th 2026-06 unverdicted novelty 6.0 of 10

    Single-valued polylogarithms are constructed on higher-genus once-punctured Riemann surfaces with trivial monodromy, related to prior work and used to identify the Arakelov Green's function.

  5. Towards Motivic Coactions at Genus One from Zeta Generators

    hep-th 2025-08 unverdicted novelty 6.0 of 10

    Proposes motivic coaction formulae for genus-one iterated integrals over holomorphic Eisenstein series using zeta generators, verifies expected coaction properties, and deduces f-alphabet decompositions of multiple mo...

  6. IterInt: Evaluating iterated integrals via differential equations

    hep-ph 2026-06 unverdicted novelty 5.0 of 10

    IterInt package evaluates iterated integrals by transforming them into solvable differential equation systems with built-in regularization.

Reference graph

Works this paper leans on

69 extracted references · 33 canonical work pages · cited by 6 Pith papers

  1. [1]

    De summis serierum reciprocarum

    L. Euler, “De summis serierum reciprocarum”, Commentarii academiae scientiarum Petropolitanae 7, 123 (1740)

    work page

  2. [2]

    On the sums of series of reciprocals

    L. Euler, “On the sums of series of reciprocals (English translation of ‘De summis serierum reciprocarum (1740)’)”, math/0506415

    work page Pith review arXiv

  3. [3]

    Geometry of Configurations, Polylogarithms, and Motivic Cohomology

    A. Goncharov, “Geometry of Configurations, Polylogarithms, and Motivic Cohomology” , Advances in Mathematics 114, 197 (1995)

    work page 1995

  4. [4]

    Multiple polylogarithms, cyclotomy and modular complexes

    A. B. Goncharov, “Multiple polylogarithms, cyclotomy and modular complexes” , Math. Res. Lett. 5, 497 (1998) , arxiv:1105.2076

    arXiv 1998

  5. [5]

    Harmonic polylogarithms

    E. Remiddi and J. A. M. Vermaseren, “Harmonic polylogarithms”, International Journal of Modern Physics A 15, 725–754 (2000)

    work page 2000

  6. [6]

    Multiple polylogarithms and mixed Tate motives

    A. Goncharov, “Multiple polylogarithms and mixed Tate motives” , math/0103059

    arXiv

  7. [7]

    Numerical evaluation of multiple polylogarithms

    J. Vollinga and S. Weinzierl, “Numerical evaluation of multiple polylogarithms” , Computer Physics Communications 167, 177–194 (2005)

    work page 2005

  8. [8]

    Iterated path integrals

    K. Chen, “Iterated path integrals”, Bull. Amer. Math. Soc. 83, 831 (1977)

    work page 1977

Show all 69 references
  1. [9]

    Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k

    J. Borwein, D. Bradley and D. Broadhurst, “Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k.”, The Electronic Journal of Combinatorics [electronic only] 4, Research paper R5 (1997)

    1997

  2. [10]

    Multiple harmonic series

    M. E. Hoffman, “Multiple harmonic series” , Pacific J. Math. 152, 275 (1992)

    1992

  3. [11]

    Values of zeta functions and their applications

    D. Zagier, “Values of zeta functions and their applications” , in: “First European Congress of Mathematics Paris, July 6–10, 1992: Vol. II: Invited Lectures (Part 2)” , Birkh¨ auser Basel (1994), Basel, 497–512p, https://doi.org/10.1007/978-3-0348-9112-7 23

    1994 doi

  4. [12]

    Multiple zeta values and periods of moduli spaces M0,n

    F. Brown, “Multiple zeta values and periods of moduli spaces M0,n”, Ann. Sci. ´Ec. Norm. Sup´ er. (4) 42, 371 (2009), math/0606419

    2009 arXiv

  5. [13]

    Multiple zeta values: from numbers to motives

    J. Fresan and J. Gil, “Multiple zeta values: from numbers to motives” , Clay Mathematics Proceedings, to appear , http://javier.fresan.perso.math.cnrs.fr/mzv.pdf

  6. [14]

    Groupes fondamentaux motiviques de Tate mixte

    P. Deligne and A. B. Goncharov, “Groupes fondamentaux motiviques de Tate mixte” , Ann. Sci. ´Ecole Norm. Sup. (4) 38, 1 (2005)

    2005

  7. [15]

    Mixed Tate motives over Z

    F. Brown, “Mixed Tate motives over Z”, Annals of Mathematics 175, 949 (2012)

    2012

  8. [16]

    Depth-graded motivic multiple zeta values

    F. Brown, “Depth-graded motivic multiple zeta values” , Compositio Mathematica 157, 529–572 (2021) , arxiv:1301.3053

    2021 arXiv

  9. [17]

    Periods of the motivic fundamental groupoid of P1 \ {0, µN , ∞}

    C. Glanois, “Periods of the motivic fundamental groupoid of P1 \ {0, µN , ∞}”, PhD thesis, Universit´ e Pierre et Marie Curie, 2016

    2016

  10. [18]

    The Multiple Zeta Value Data Mine

    J. Blumlein, D. Broadhurst and J. Vermaseren, “The Multiple Zeta Value Data Mine” , Comput.Phys.Commun. 181, 582 (2010) , arxiv:0907.2557

    2010 arXiv

  11. [19]

    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” , arxiv:2203.09099, in: “Snowmass 2021”

    2021 arXiv

  12. [20]

    Localization techniques in quantum field theories

    V. Pestun, M. Zabzine, F. Benini, T. Dimofte, T. T. Dumitrescu, K. Hosomichi, S. Kim, K. Lee, B. Le Floch, M. Mari˜ no, J. A. Minahan, D. R. Morrison, S. Pasquetti, J. Qiu, L. Rastelli, S. S. Razamat, S. S. Pufu, Y. Tachikawa, B. Willett and K. Zarembo, “Localization technique...

    2017 doi

  13. [21]

    The AdS Virasoro-Shapiro amplitude

    L. F. Alday and T. Hansen, “The AdS Virasoro-Shapiro amplitude” , JHEP 2310, 023 (2023) , arxiv:2306.12786

    2023 arXiv

  14. [22]

    Elliptic associators

    B. Enriquez, “Elliptic associators” , Selecta Math. (N.S.) 20, 491 (2014) . 68

    2014

  15. [23]

    Analogues elliptiques des nombres multiz´ etas

    B. Enriquez, “Analogues elliptiques des nombres multiz´ etas”, Bull. Soc. Math. France 144, 395 (2016) , arxiv:1301.3042

    2016 arXiv

  16. [24]

    Elliptic polylogarithms: An analytic theory

    A. Levin, “Elliptic polylogarithms: An analytic theory” , Compositio Mathematica 106, 267 (1997)

    1997

  17. [25]

    Multiple elliptic polylogarithms

    F. Brown and A. Levin, “Multiple elliptic polylogarithms” , arxiv:1110.6917v2

    arXiv

  18. [26]

    Elliptic multiple zeta values and one-loop superstring amplitudes

    J. Broedel, C. R. Mafra, N. Matthes and O. Schlotterer, “Elliptic multiple zeta values and one-loop superstring amplitudes”, JHEP 1507, 112 (2015) , arxiv:1412.5535

    2015 arXiv

  19. [27]

    Relations between derivations arising from modular forms

    A. Pollack, “Relations between derivations arising from modular forms” , Undergraduate thesis, Duke University

  20. [28]

    Universal KZB equations: the elliptic case

    D. Calaque, B. Enriquez and P. Etingof, “Universal KZB equations: the elliptic case” , in: “Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I” , Birkh¨ auser Boston, Inc., Boston, MA (2009), 165–266p

    2009

  21. [29]

    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. A49, 155203 (2016) , arxiv:1507.02254

    2016 arXiv

  22. [30]

    Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism

    J. Broedel, C. Duhr, F. Dulat and L. Tancredi, “Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism” , JHEP 1805, 093 (2018) , arxiv:1712.07089

    2018 arXiv

  23. [31]

    Functions Beyond Multiple Polylogarithms for Precision Collider Physics

    J. L. Bourjaily et al., “Functions Beyond Multiple Polylogarithms for Precision Collider Physics” , arxiv:2203.07088, in: “Snowmass 2021”

    2021 arXiv

  24. [32]

    Elliptic multiple zeta values datamine

    J. Broedel, N. Matthes and O. Schlotterer, “Elliptic multiple zeta values datamine” , https://tools.aei.mpg.de/emzv/datamine.html

  25. [33]

    Elliptic multiple zeta values

    N. Matthes, “Elliptic multiple zeta values” , PhD thesis, Universit¨ at Hamburg, 2016

    2016

  26. [34]

    Classical and Elliptic Polylogarithms and Special Values of L-Series

    D. Zagier and H. Gangl, “Classical and Elliptic Polylogarithms and Special Values of L-Series” , The Arithmetic and Geometry of Algebraic Cycles, Nato Science Series C 548, 61 (2000)

    2000

  27. [35]

    Functional relations for elliptic polylogarithms

    J. Broedel and A. Kaderli, “Functional relations for elliptic polylogarithms” , Journal of Physics A: Mathematical and Theoretical 53, 245201 (2020) , http://dx.doi.org/10.1088/1751-8121/ab81d7

    2020 doi

  28. [36]

    Elliptic Feynman integrals and pure functions

    J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, “Elliptic Feynman integrals and pure functions”, JHEP 1901, 023 (2019) , arxiv:1809.10698

    2019 arXiv

  29. [37]

    Exploring transcendentality in superstring amplitudes

    E. D’Hoker and M. B. Green, “Exploring transcendentality in superstring amplitudes” , JHEP 1907, 149 (2019) , arxiv:1906.01652

    2019 arXiv

  30. [38]

    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” , arxiv:2502.14769

    arXiv

  31. [39]

    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, 204 (2014) , arxiv:1112.0864

    2014 arXiv

  32. [40]

    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

    arXiv

  33. [41]

    SchottkyTools

    E. Im, “SchottkyTools”, Software package for numerical evaluation of higher-genus polylogarithms. Work in progress

  34. [42]

    Ueber eine specielle Function, welche bei einer bestimmten linearen Transformation ihres Arguments unver¨ andert bleibt

    F. Schottky, “Ueber eine specielle Function, welche bei einer bestimmten linearen Transformation ihres Arguments unver¨ andert bleibt.”, Journal f¨ ur die reine und angewandte Mathematik 1887, 227 (1887)

  35. [43]

    Computational approach to Riemann surfaces

    A. I. Bobenko and C. Klein, “Computational approach to Riemann surfaces” , Springer Berlin, Heidelberg (2011)

    2011

  36. [44]

    Schottky space and Teichm¨ uller disks

    F. Herrlich, “Schottky space and Teichm¨ uller disks”, Handbook of group actions 1, 289 (2015)

    2015

  37. [45]

    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. A 57, 445202 (2024) , arxiv:2406.10051. 69

    2024 arXiv

  38. [46]

    Chapter XII: A particular form of fundamental surface

    H. F. Baker, “Chapter XII: A particular form of fundamental surface” , in: “Abelian functions: Abel’s theorem and the allied theory” , Cornell University Library (1900), 343–374p

    1900

  39. [47]

    On hyperelliptic Schottky groups

    L. Keen, “On hyperelliptic Schottky groups” , Annales Fennici Mathematici 5, 165–174 (1980)

    1980

  40. [48]

    Iterated integrals in quantum field theory

    F. Brown, “Iterated integrals in quantum field theory” , in: “Proceedings, Geometric and Topological Methods for Quantum Field Theory : 6th Summer School: Villa de Leyva, Colombia, July 6-23, 2009”, 188-240p

    2009

  41. [49]

    Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series

    J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, “Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series” , JHEP 1808, 014 (2018) , arxiv:1803.10256

    2018 arXiv

  42. [50]

    Elliptic hyperlogarithms

    B. Enriquez and F. Zerbini, “Elliptic hyperlogarithms”, Canadian Journal of Mathematics online, 1–36 (2025) , arxiv:2307.01833

    2025 arXiv

  43. [51]

    Amplitude recursions with an extra marked point

    J. Broedel and A. Kaderli, “Amplitude recursions with an extra marked point” , Commun. Num. Theor. Phys. 16, 75 (2022) , arxiv:1912.09927

    2022 arXiv

  44. [52]

    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. 19, 355 (2025) , arxiv:2306.08644

    2025 arXiv

  45. [53]

    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” , arxiv:2501.07640

  46. [54]

    Towards multiple elliptic polylogarithms

    A. Levin and G. Racinet, “Towards multiple elliptic polylogarithms” , math/0703237

    arXiv

  47. [55]

    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, 093 (2025) , arxiv:2409.08208

    2025 arXiv

  48. [56]

    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

    arXiv

  49. [57]

    Theta functions on Riemann surfaces

    J. Fay, “Theta functions on Riemann surfaces” , Springer (1973)

    1973

  50. [58]

    Analogues of hyperlogarithm functions on affine complex curves

    B. Enriquez and F. Zerbini, “Analogues of hyperlogarithm functions on affine complex curves” , arxiv:2212.03119

    arXiv

  51. [59]

    Surface Drinfeld Torsors I : Higher Genus Associators

    M. Gonzalez, “Surface Drinfeld Torsors I : Higher Genus Associators” , arxiv:2004.07303, https://arxiv.org/abs/2004.07303

    2004 arXiv

  52. [60]

    Galois symmetries of fundamental groupoids and noncommutative geometry

    A. B. Goncharov, “Galois symmetries of fundamental groupoids and noncommutative geometry” , Duke Math. J. 128, 209 (2005) , math/0208144

    2005 arXiv

  53. [61]

    On the decomposition of motivic multiple zeta values

    F. C. S. Brown, “On the decomposition of motivic multiple zeta values” , arxiv:1102.1310, in: “Galois-Teichm¨ uller theory and arithmetic geometry”, Math. Soc. Japan, Tokyo (2012), 31–58p

    2012 arXiv

  54. [62]

    Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes

    C. Duhr, “Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes” , JHEP 1208, 043 (2012) , arxiv:1203.0454

    2012 arXiv

  55. [63]

    Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops

    D. J. Broadhurst and D. Kreimer, “Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops” , Phys. Lett. B 393, 403 (1997) , hep-th/9609128

    1997 arXiv

  56. [64]

    Genus drop in hyperelliptic Feynman integrals

    R. Marzucca, A. J. McLeod, B. Page, S. P¨ ogel and S. Weinzierl, “Genus drop in hyperelliptic Feynman integrals”, Phys. Rev. D 109, L031901 (2024) , arxiv:2307.11497

    2024 arXiv

  57. [65]

    Canonical differential equations beyond genus one

    C. Duhr, F. Porkert and S. F. Stawinski, “Canonical differential equations beyond genus one” , JHEP 2502, 014 (2025) , arxiv:2412.02300

    2025

  58. [66]

    Lectures on two loop superstrings

    E. D’Hoker and D. H. Phong, “Lectures on two loop superstrings” , Conf. Proc. C 0208124, 85 (2002) , hep-th/0211111

    2002 arXiv

  59. [67]

    Cyclic Products of Higher-Genus Szeg¨ o Kernels, Modular Tensors, and Polylogarithms

    E. D’Hoker, M. Hidding and O. Schlotterer, “Cyclic Products of Higher-Genus Szeg¨ o Kernels, Modular Tensors, and Polylogarithms” , Phys. Rev. Lett. 133, 021602 (2024) , arxiv:2308.05044

    2024 arXiv

  60. [68]

    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”, arxiv:2505.07947

  61. [69]

    Tata Lectures on Theta I, II

    D. Mumford, “Tata Lectures on Theta I, II” , Birkh¨ auser Boston, MA (1983, 1984). 70

    1983

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.