Pith. sign in

REVIEW 3 major objections 5 minor 4 cited by

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

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read The paper derives a double-copy (KLT) formula at arbitrary genus: a single-valued 'closed-string' integral equals a bilinear of 'open-string' integrals, with the inverse homology-intersection matrix as kernel.

desk verdict A serious genus-g double-copy paper with real new content and honest scope; the main gap is a homology basis claim that is plausible but not fully proved. read the letter →

arxiv 2509.01598 v1 pith:5DNR6ZCE submitted 2025-09-01 hep-th math-phmath.AGmath.MP

Andrzej Pokraka , Lecheng Ren , Carlos Rodriguez This is my paper
classification hep-thmath-phmath.AGmath.MP
keywords doublecopytwistedhomologycohomologyKLTrelationsgenus-ghypergeometricintegralsintersectionnumbersAbelianKroneckerformshigher-genuspolylogarithms
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

The paper establishes double-copy (KLT-type) quadratic relations for a family of one-fold hypergeometric integrals on a punctured Riemann surface of genus g - integrals that are the building blocks of g-loop open-string amplitudes in the chiral-splitting formalism. Its central claim is that the single-valued 'closed-string' analogue, J = ∫_{Σ*_g} |T(z1)|² φ_l ∧ φ_k, is exactly a bilinear combination of ordinary 'open-string' integrals I = ∫_γ T φ, with coefficients given by the inverse of the matrix of twisted-homology intersection numbers - so that inverse matrix is the genus-g double-copy kernel. To reach this, the paper constructs explicit bases of twisted homology from regularized integration cycles and computes all their intersection numbers, then verifies the identity numerically at genus two and extends it to quasiperiodic Abelian Kronecker forms, which generate integration kernels for higher-genus polylogarithms. If correct, this is the first double-copy formula valid at every genus and a concrete step toward g-loop KLT relations in string theory.

What carries the argument

Four pieces carry the argument. (1) The twist T(z1) = Π_j [E(z1,zj)]^{s1j} exp(2πi Σ_I s1AI ν_I(z1)), a multivalued function on the punctured surface whose monodromies define the local system L_s and the twisted differential ∇_ω; it plays the role of the Koba-Nielsen factor. (2) Twisted homology H1(Σ*_g, L_s), spanned by regularized cycles γ_{2AJ}, γ_{2BJ}, γ_{2j} built from an explicit cell decomposition, subject to exactly one linear relation (4.16). (3) The homology intersection numbers (4.25)-(4.31), assembled into the matrix H_{(2,2)} whose inverse is the double-copy kernel. (4) Twisted Riemann bilinear relations, which convert the compatibility of integration and intersection pairings

What would settle it

Compute the intersection matrix H_{(2,2)} for a genus-three surface from the cell decomposition (4.13)-(4.15) by direct implementation, and check that its determinant matches (4.35) and that the bilinear identity (5.25) holds numerically for several holomorphic differentials and parameter values. A sign error, a missed intersection, or a second linear relation among the cycles would surface as a mismatch or a singular matrix; probing the excluded locus s1j ∈ Z would show whether the double copy breaks there or is recovered by a different basis.

Watch

Extended reading notes

Core claim

The paper's central result is the double-copy identity (5.24)-(5.25): for real monodromy parameters s1•, the single-valued ('closed-string') integral J^{φ_k,φ_l} = ∫_{Σ*_g} |T(z1)|² φ_l ∧ φ_k equals the bilinear sum Σ_{a,b∈K} H^{a,b}_{(2,2)} I^{φ_k}_{γ_b} I^{φ_l}_{γ_a} over twisted cycles, with H^{a,b}_{(2,2)} the inverse of the homology intersection matrix. For holomorphic differentials this reads ∫ |T|² ω_J ∧ ω_I = Σ_{a,b} H^{a,b} (∫_{γ_a} T ω_I)(∫_{γ_b} T ω_J). The structure extends to Abelian Kronecker forms (6.20)-(6.25), giving double-copy formulas for single-valued higher-genus polylogarithm kernels, and is verified numerically at genus two. The authors' reading: the inverse intersect

Load-bearing premise

The load-bearing premise is that the explicitly listed regularized cycles span the twisted homology with exactly one linear relation, so the intersection matrix is invertible and the double-copy kernel is well defined; this is supported by a non-vanishing determinant (4.35) presented through geometric sketches rather than a complete derivation for general g.

Editorial extensions

If this is right

  • The double copy holds at every genus g for the one-fold integrals: the inverse homology intersection matrix is a purely geometric KLT kernel, computable from the surface and the monodromy parameters alone.
  • At genus one the formula reproduces the known Riemann-Wirtinger double copy, matching the proposed one-loop KLT relation; the genus-g result is its natural continuation.
  • The extension to Abelian Kronecker forms puts the double copy inside reach of higher-genus polylogarithm kernels - concretely, the genus-two kernel f̃_2^1 satisfies a double-copy formula (7.17).
  • Under the reality conditions (5.31)/(6.22), |T|² becomes the exponential of the string Green's function G(z1,zj), so the double copy becomes a statement about g-loop chiral-splitting string integrands rather than purely mathematical integrals.
  • Numerically, the double copy provides the analytic continuation of the complex integral J beyond its naive convergence region - the genus-two example continues -1 < s12 < 1 to s12 > 1.

Reading between the lines

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

  • Because the kernel H^{a,b}_{(2,2)} depends only on the surface and the monodromies - not on which twisted 1-form is integrated - the same matrix should control double copies for every pair of cohomology classes on Σ*_g; extracting it once per surface would give all such quadratic identities at once.
  • A sharp next test would be a genus-three verification: at g = 3 the genuinely new I ≠ J intersection numbers (4.28)-(4.31) mix in ways absent at genus two, so a numerical check there would localize any error in the basis or determinant claim.
  • The paper restricts to Abelian (rank-one) quasiperiodic forms; extending the double copy to the full non-Abelian Kronecker generating function would presumably require higher-rank local systems and would deliver double-copy relations for individual higher-genus polylogarithm kernels rather than only symmetric combinations.
  • One implication the authors leave implicit: in degeneration limits of the surface, the inverse intersection matrix should factor, yielding inductive checks of the genus-g kernel against lower-genus (including tree-level) KLT kernels - a structure worth testing since it connects the all-genus formula to known factorization of string amplitudes.
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 studies one-fold hypergeometric integrals on a genus-g Riemann surface punctured at n-1 points, with a multivalued twist T(z1). It constructs regularized twisted cycles, computes homology intersection numbers among them, and uses twisted Riemann bilinear relations to derive a double-copy identity: the ``complex'' integral J^{φ_k,φ_l} = ∫ |T|² φ_l ∧ φ_k is expressed as a bilinear sum of ordinary integrals I^{φ}_{γ} weighted by the inverse of the homology intersection matrix H^{(2,2)}. The same structure is extended to Abelian Kronecker forms. Numerical checks are provided at genus two for both holomorphic differentials and a genus-two polylog kernel, using Schottky uniformization and Crowdy-Marshall methods.

Significance. If the central identity (5.24)/(6.20) holds for all g, the paper provides a concrete genus-g analogue of a KLT formula in which the inverse twisted-homology intersection matrix plays the role of the KLT kernel. The manuscript also contains useful technical contributions: explicit intersection numbers for a new family of twisted cycles, an analytic inverse matrix for genus two, a numerical implementation of the relevant integrals, and apparently the first publicly available evaluation of a genus-two polylogarithm kernel. The main claim is supported by three independent routes — a specialization of cited twisted Riemann bilinear relations, a Stokes-theorem argument in Appendix D, and numerics — but, as detailed below, two of these routes are incomplete for general g.

major comments (3)
  1. [Section 4.3, Eq. (4.35)] The proof of Claim 4.1/Theorem 4.2 is incomplete. The non-vanishing determinant of H^{(2,n)} is the only demonstrated mechanism excluding additional linear relations among the cycles {γ_{2AJ}, γ_{2BJ}, γ_{2j}}, and hence the only justification that H^{(2,2)} in (4.38) is invertible. However, the entries of H^{(2,n)} are not written down; the text simply asserts that the matrix is diagonal and gives a product formula. Appendix A computes [γ_{2a}|γ_{2b}] for cycles based at z2, not the dual cycles ˇγ_{nb} of figure 5 that enter (4.34). Thus the determinant is unverified for general g,n. Since H^{a,b}_{(2,2)} in (5.24) is the KLT kernel, this gap is load-bearing. Please supply the full computation or an alternative proof of the basis property.
  2. [Appendix D, Eq. (D.11)] The elementary Stokes proof is performed explicitly only for genus two: the text after (D.11) states that the general formula is straightforward to generalize and that agreement with (7.7) was verified for g=2, n=3. The paper's central claim (5.24), however, is for all g. No explicit derivation of (D.11) for g>2 is shown, and the general formula is asserted rather than proved. This is a second incomplete justification of the higher-genus result. If the basis gap in Section 4.3 is fixed, this may be less critical, but as written the higher-genus double copy is not fully proved.
  3. [Section 4.3, around Eq. (4.35)-(4.37)] The determinant formula (4.35) contains factors (e^{2πis_{1j}}-1)^{-1} only for j=3,...,n-1, but the linear relation (4.16) has coefficients that can vanish for special values of s_{1AJ} and s_{1BJ}. The paper does not characterize the excluded locus where the chosen subset of cycles fails to be a basis, nor does it state whether the inverse kernel H^{a,b}_{(2,2)} has removable singularities at those points. For a claim about generic s-values this is acceptable, but it should be made precise.
minor comments (5)
  1. [Eq. (2.2)] The notation is inconsistent: [AI|BJ]_{top} = δ_{IJ} = -[BI, AJ]_{top} mixes comma and bar notation. Please standardize.
  2. [Section 7.1, Figure 8] The numerical comparison lacks a discussion of numerical accuracy or convergence criteria. Please state the estimated error of the direct-integration and double-copy results, and specify the number of sample points used.
  3. [Ancillary files] The text repeatedly refers to ancillary files for numerical checks. Please provide a permanent repository link or DOI to allow reproducibility.
  4. [Section 7.2] In the sentence introducing (7.14), “flowing vectors” appears to be a typo for “following vectors.”
  5. [Propositions 3.1 and 6.1] The examples in Section 7 use (g,n)=(2,3), which violates the stated condition n > max(2,2g-1). The text explains why this is harmless, but this should be remarked explicitly in the statement of the propositions to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the double-copy identity is independently defined, numerically verified, and re-derived by a Stokes-theorem proof; the unproved determinant (4.35) is a rigor gap rather than a circular step.

full rationale

The central formula (5.24)/(5.25) is not true by definition. The left-hand side J^{φ_k,φ_l} is independently defined as the integral ∫Σ_g^* |T(z1)|^2 φ_l∧φ_k (5.16), while the right-hand side uses the matrix H^{a,b}_{(2,2)}, defined as the inverse of the homology intersection matrix H_{(2,2)} computed from (4.25)-(4.31). The equality is checked numerically in section 7 (figure 8) and, more importantly, derived from Stokes theorem in Appendix D, eq. (D.11), which the paper explicitly states requires 'no need for the setup of [28] or twisted (co)homology.' Thus the cited double-copy theorem [28] is not load-bearing, and the self-citations to [17] occur only in comparisons of genus-one intersection numbers (A.1) and in outlook comments. The homology-basis statement (Claim 4.1 / Theorem 4.2) is supported by the asserted non-vanishing determinant (4.35). That determinant is stated, not fully derived in the text; an error there could invalidate the invertibility of H_{(2,2)} and hence the kernel used in (5.24). However, this is a completeness/correctness concern, not circularity: the basis claim and the determinant are not assumed to be the double-copy relation, and the determinant is not fitted to make (5.24) hold. The derivation chain does not reduce to its inputs by construction, so the paper receives no circularity penalty.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central construction rests on standard Riemann-surface theory (prime functions, Schottky uniformization, Abelian differentials), on Watanabe's twisted-cohomology dimension theorem (cited), and on the Cho-Matsumoto twisted Riemann bilinear relations (cited). The genuinely new ingredients, the regularized cycle basis and intersection numbers, are derived geometrically in appendix A. The restriction to rank-1 Abelian local systems (section 8) is an explicit limitation, and the quasiperiodic extension of Watanabe's theorem (Prop 6.1) is argued by analogy with the line-bundle case rather than fully proven. No data-fit parameters are involved; the s-variables are kinematics-like inputs. No new physical entities are postulated; the regularized twisted cycles, local systems, and intersection matrices are mathematical constructions internal to the derivation.

free parameters (2)
  • Schottky moduli in numerical example (δ1, δ2, q1, q2) = 0.5, 0.5i, 0.15, 0.1
    Hand-chosen test inputs for the genus-2 numerical check (section 7.1), not fitted and not part of the derivation. Varying them does not affect the central claim.
  • s1AI values in the numerical example = set by reality condition (5.31)
    Determined by the constraint that s1BI be real, not fitted to make the double copy hold. They are outputs of a consistency condition, not free fit parameters.
assumptions (7)
  • domain assumption Rank-1 (Abelian) local system: monodromy representation ρ: π1(Σ*_g) → C*, scalar twist T(z1).
    The whole construction uses a scalar twist with multiplicative monodromies (3.10)-(3.14); the Abelian-Kronecker restriction in section 6 and the remarks in section 8 make this restriction explicit.
  • domain assumption Watanabe's theorem: dim H^1(Σ*_g, ∇ω) = 2g+n-3 with the basis (3.23), extended to L_α-quasiperiodic forms (Proposition 6.1).
    Invoked at (3.23), (4.32), and (6.9); the homology dimension and the pairing picture rest on it. The extension to line bundles of trivial Chern class is argued, not fully proven in the paper.
  • standard math Standard prime-function properties: monodromies (2.10), ratios (2.12), Abelian differentials of first, second, and third kind.
    Used to define T(z1) (3.3) and to read off the B-cycle monodromy s1BI (3.6). Treated as background from [37].
  • standard math Twisted Riemann bilinear relations of Cho-Matsumoto [30, theorem 2].
    Proposition 5.1 and hence the double-copy Proposition 5.2 rest on it; appendix D provides an independent elementary derivation of the double copy.
  • standard math Identity [γ_a|γ̄_b] = [γ_a|ˇγ_b] relating complex-conjugated and dualized homology intersection numbers (Mimachi-Yoshida [42,43]).
    Used in (5.15) and (6.19) to reuse the computed intersection matrix for the complex-conjugated cycles; cited, not proven in this paper.
  • domain assumption Boundary conditions: T vanishes on contour boundaries (3.7) and momentum conservation Σ s1j = 0 (3.1).
    Needed for Stokes' theorem (no boundary terms) and for consistency of monodromies (3.15); standard in this literature.
  • domain assumption Convergence of the integrals J and I for the chosen s-ranges and existence of the analytic continuation via the double copy.
    Figure 8 explicitly uses the double copy as an analytic continuation beyond the direct-integration convergence range; the paper states convergence conditions but does not prove the continuation claim in detail.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A double copy from twisted (co)homology at genus g." pith.science (2026). https://pith.science/paper/5DNR6ZCE

@misc{pith2026250901598,
  author       = {Pith},
  title        = {Pith review of: A double copy from twisted (co)homology at genus g},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5DNR6ZCE}},
  note         = {Machine review of arXiv:2509.01598}
}
read the original abstract

We study a family of generalized hypergeometric integrals defined on punctured Riemann surfaces of genus g. These integrals are closely related to g-loop string amplitudes in chiral splitting, where one leaves the loop-momenta, moduli and all but one puncture un-integrated. We study the twisted homology groups associated to these integrals, and determine their intersection numbers. We make use of these homology intersection numbers to write a double-copy formula for the "complex" version of these integrals -- their closed-string analogues. To verify our findings, we develop numerical tools for the evaluation of the integrals in this work. This includes the recently introduced Enriquez kernels -- integration kernels for higher-genus polylogarithms.

Discussion (0). Sign in to comment.

Forward citations

Cited by 4 Pith papers

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

  1. Twisted de Rham theory for string double copy in AdS

    hep-th 2025-12 conditional novelty 8.0 of 10

    Noncommutative twisted de Rham theory derives the intersection number of open-string contours whose inverse is the double-copy kernel for four-point AdS string generating functions.

  2. 4d CFT Correlators from Ambitwistors

    hep-th 2026-06 conditional novelty 7.0 of 10

    Conserved spinning two- and three-point correlators in 4d CFTs are obtained from holomorphic contour integrals on ambitwistor space, with spin, parity, and even a conformally coupled scalar treated uniformly.

  3. 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.

  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.

Reference graph

Works this paper leans on

61 extracted references · 30 canonical work pages · cited by 4 Pith papers

  1. [1]

    Matsubara-Heo, S

    S.-J. Matsubara-Heo, S. Mizera and S. Telen, Four lectures on Euler integrals, SciPost Phys. Lect. Notes 75 (2023) 1, [ 2306.13578]

    arXiv 2023

  2. [2]

    D. A. Kosower, R. Monteiro and D. O’Connell, The SAGEX review on scattering amplitudes Chapter 14: Classical gravity from scattering amplitudes , J. Phys. A 55 (2022) 443015, [ 2203.13025]

    arXiv 2022

  3. [3]

    Buonanno, M

    A. Buonanno, M. Khalil, D. O’Connell, R. Roiban, M. P. Solon and M. Zeng, Snowmass White Paper: Gravitational Waves and Scattering Amplitudes , in Snowmass 2021 , 4, 2022, 2204.05194

    arXiv 2021

  4. [4]

    Aomoto, M

    K. Aomoto, M. Kita, T. Kohno and K. Iohara, Theory of hypergeometric functions . Springer, 2011

    work page 2011

  5. [5]

    Yoshida, Hypergeometric functions, my love: modular interpretations of configuration spaces , vol

    M. Yoshida, Hypergeometric functions, my love: modular interpretations of configuration spaces , vol. 32. Springer Science & Business Media, 2013

    work page 2013

  6. [6]

    Mizera, Inverse of the String Theory KLT Kernel , JHEP 06 (2017) 084, [ 1610.04230]

    S. Mizera, Inverse of the String Theory KLT Kernel , JHEP 06 (2017) 084, [ 1610.04230]

    arXiv 2017

  7. [7]

    Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations , JHEP 08 (2017) 097, [1706.08527]

    S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations , JHEP 08 (2017) 097, [1706.08527]

    arXiv 2017

  8. [8]

    Mizera, Scattering Amplitudes from Intersection Theory , Phys

    S. Mizera, Scattering Amplitudes from Intersection Theory , Phys. Rev. Lett. 120 (2018) 141602, [1711.00469]

    arXiv 2018

Show all 61 references
  1. [9]

    Mastrolia and S

    P. Mastrolia and S. Mizera, Feynman Integrals and Intersection Theory , JHEP 02 (2019) 139, [1810.03818]

    2019 arXiv

  2. [10]

    Caron-Huot and A

    S. Caron-Huot and A. Pokraka, Duals of Feynman integrals. Part I. Differential equations , JHEP 12 (2021) 045, [ 2104.06898]

    2021 arXiv

  3. [11]

    Caron-Huot and A

    S. Caron-Huot and A. Pokraka, Duals of Feynman Integrals. Part II. Generalized unitarity , JHEP 04 (2022) 078, [ 2112.00055]. – 52 –

    2022 arXiv

  4. [12]

    Gasparotto, S

    F. Gasparotto, S. Weinzierl and X. Xu, Real time lattice correlation functions from differential equations, JHEP 06 (2023) 128, [ 2305.05447]

    2023 arXiv

  5. [13]

    Frellesvig and T

    H. Frellesvig and T. Teschke, General relativity from intersection theory , Phys. Rev. D 110 (2024) 044028, [2404.11913]

    2024 arXiv

  6. [14]

    C. Duhr, F. Porkert and S. F. Stawinski, Canonical differential equations beyond genus one , JHEP 02 (2025) 014, [ 2412.02300]

    2025

  7. [15]

    Stieberger, One-Loop Double Copy Relation in String Theory , Phys

    S. Stieberger, One-Loop Double Copy Relation in String Theory , Phys. Rev. Lett. 132 (2024) 191602, [2310.07755]

    2024 arXiv

  8. [16]

    Mazloumi and S

    P. Mazloumi and S. Stieberger, One-loop double copy relation from twisted (co)homology , JHEP 10 (2024) 148, [ 2403.05208]

    2024 arXiv

  9. [17]

    Bhardwaj, A

    R. Bhardwaj, A. Pokraka, L. Ren and C. Rodriguez, A double copy from twisted (co)homology at genus one, JHEP 07 (2024) 040, [ 2312.02148]

    2024 arXiv

  10. [18]

    Z. Bern, J. J. M. Carrasco, W.-M. Chen, H. Johansson, R. Roiban and M. Zeng, Five-loop four-point integrand of N = 8 supergravity as a generalized double copy , Phys. Rev. D 96 (2017) 126012, [1708.06807]

    2017 arXiv

  11. [19]

    V. S. Dotsenko and V. A. Fateev, Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c < 1, Nucl. Phys. B 251 (1985) 691–734

    1985

  12. [20]

    Kawai, D

    H. Kawai, D. C. Lewellen and S. H. H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1–23

    1986

  13. [21]

    Aomoto, On the Complex Selberg Integral , The Quarterly Journal of Mathematics 38 (12, 1987) 385–399

    K. Aomoto, On the Complex Selberg Integral , The Quarterly Journal of Mathematics 38 (12, 1987) 385–399

    1987

  14. [22]

    Z. Bern, J. J. M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes , Phys. Rev. D 78 (2008) 085011, [ 0805.3993]

    2008 arXiv

  15. [23]

    Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, The duality between color and kinematics and its applications , J. Phys. A 57 (2024) 333002, [ 1909.01358]

    2024 arXiv

  16. [24]

    Stieberger, A Relation between One-Loop Amplitudes of Closed and Open Strings (One-Loop KLT Relation), 2212.06816

    S. Stieberger, A Relation between One-Loop Amplitudes of Closed and Open Strings (One-Loop KLT Relation), 2212.06816

    arXiv

  17. [25]

    Mano and H

    T. Mano and H. Watanabe, Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral, Proceedings of the American Mathematical Society 140 (2012) 3867–3881

    2012

  18. [26]

    Ghazouani and L

    S. Ghazouani and L. Pirio, Moduli spaces of flat tori and elliptic hypergeometric functions , M´ emoires de la Soci´ et´ e math´ ematique de France(2016) , [ 1605.02356]

    2016 arXiv

  19. [27]

    Y. Goto, Intersection numbers of twisted homology and cohomology groups associated to the Riemann-Wirtinger integral, International Journal of Mathematics 34 (June, 2022) 2350005, [2206.03177]

    2022 arXiv

  20. [28]

    Hanamura and M

    M. Hanamura and M. Yoshida, Hodge structure on twisted cohomologies and twisted Riemann inequalities I, Nagoya mathematical journal 154 (1999) 123–139

    1999

  21. [29]

    Watanabe, Twisted cohomology of a punctured Riemann surface , Kumamoto J

    H. Watanabe, Twisted cohomology of a punctured Riemann surface , Kumamoto J. Math. 29 (2016) 55–63

    2016

  22. [30]

    Cho and K

    K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann ’s period relations I, Nagoya Mathematical Journal 139 (1995) 67–86. – 53 –

    1995

  23. [31]

    Bernard, On the Wess-Zumino-Witten models on Riemann surfaces , Nuclear Physics B 309 (1988) 145–174

    D. Bernard, On the Wess-Zumino-Witten models on Riemann surfaces , Nuclear Physics B 309 (1988) 145–174

    1988

  24. [32]

    Enriquez, Flat connections on configuration spaces and braid groups of surfaces , Advances in Mathematics 252 (2014) 204–226, [ 1112.0864]

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

    2014 arXiv

  25. [33]

    Enriquez and F

    B. Enriquez and F. Zerbini, Construction of Maurer-Cartan elements over configuration spaces of curves, 2110.09341

    arXiv

  26. [34]

    Baune, J

    K. Baune, J. Broedel, E. Im, A. Lisitsyn and F. Zerbini, Schottky–Kronecker forms and hyperelliptic polylogarithms, J. Phys. A 57 (2024) 445202, [ 2406.10051]

    2024 arXiv

  27. [35]

    Lisitsyn, Representations of Kronecker forms at higher genus , Master’s thesis, ETH Z¨ urich, 2024

    A. Lisitsyn, Representations of Kronecker forms at higher genus , Master’s thesis, ETH Z¨ urich, 2024

    2024

  28. [36]

    D’Hoker, M

    E. D’Hoker, M. Hidding and O. Schlotterer, Constructing polylogarithms on higher-genus Riemann surfaces, Commun. Num. Theor. Phys. 19 (2025) 355–413, [ 2306.08644]

    2025 arXiv

  29. [37]

    D. A. Hejhal, Theta functions, kernel functions and Abelian integrals , vol. 129. American Mathematical Soc., 1972

    1972

  30. [38]

    D’Hoker and O

    E. D’Hoker and O. Schlotterer, Meromorphic higher-genus integration kernels via convolution over homology cycles, J. Phys. A 58 (2025) 33LT01, [2502.14769]

    2025 arXiv

  31. [39]

    Twisted Cohomology and Feynman Integrals

    P. de Maat, “Twisted Cohomology and Feynman Integrals.” Master’s Thesis, Utrecht University, 2021

    2021

  32. [40]

    Casali, S

    E. Casali, S. Mizera and P. Tourkine, Monodromy relations from twisted homology , JHEP 12 (2019) 087, [1910.08514]

    2019 arXiv

  33. [41]

    Casali, S

    E. Casali, S. Mizera and P. Tourkine, Loop amplitudes monodromy relations and color-kinematics duality, JHEP 03 (2021) 048, [ 2005.05329]

    2021 arXiv

  34. [42]

    Mimachi and M

    K. Mimachi and M. Yoshida, Intersection numbers of twisted cycles and the correlation functions of the conformal field theory II , Commun. Math. Phys. 234 (2003) 339–358, [ math/0208097]

    2003 arXiv

  35. [43]

    Mimachi, K

    K. Mimachi, K. Ohara and M. Yoshida, Intersection numbers for loaded cycles associated with Selberg-type integrals, Tohoku Mathematical Journal, Second Series 56 (2004) 531–551

    2004

  36. [44]

    Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization , Ph.D

    S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization , Ph.D. thesis, Princeton, Inst. Advanced Study, 2020. 1906.02099. 10.1007/978-3-030-53010-5

    2020 arXiv

  37. [45]

    D’Hoker and D

    E. D’Hoker and D. H. Phong, The Geometry of String Perturbation Theory , Rev. Mod. Phys. 60 (1988) 917

    1988

  38. [46]

    D’Hoker, C

    E. D’Hoker, C. R. Mafra, B. Pioline and O. Schlotterer, Two-loop superstring five-point amplitudes. Part I. Construction via chiral splitting and pure spinors , JHEP 08 (2020) 135, [ 2006.05270]

    2020 arXiv

  39. [47]

    C. R. Mafra and O. Schlotterer, Towards the n-point one-loop superstring amplitude. Part II. Worldsheet functions and their duality to kinematics , JHEP 08 (2019) 091, [ 1812.10970]

    2019 arXiv

  40. [48]

    Broedel, C

    J. Broedel, C. R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112, [ 1412.5535]

    2015 arXiv

  41. [49]

    D’Hoker, B

    E. D’Hoker, B. Enriquez, O. Schlotterer and F. Zerbini, Relating flat connections and polylogarithms on higher genus Riemann surfaces , 2501.07640

  42. [50]

    D. G. Crowdy and J. S. Marshall, Computing the Schottky-Klein prime function on the Schottky double of planar domains , Computational Methods and Function Theory 7 (2007) 293–308

    2007

  43. [51]

    Crowdy, E

    D. Crowdy, E. Kropf, C. Green and M. Nasser, The schottky–klein prime function: a theoretical and computational tool for applications , IMA Journal of Applied Mathematics 81 (2016) 589–628. – 54 –

    2016

  44. [52]

    D’Hoker and O

    E. D’Hoker and O. Schlotterer, Worldsheet fermion correlators, modular tensors and higher genus integration kernels, 2505.07947

  45. [53]

    L. F. Alday, M. Nocchi and A. S. Sangar´ e, Stringy KLT Relations on AdS, 2504.19973

  46. [54]

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

  47. [55]

    Baune, J

    K. Baune, J. Broedel, E. Im, Z. Ji and Y. Moeckli, Higher-genus multiple zeta values , 2507.21765

    arXiv

  48. [56]

    Crowdy, Geometric function theory: a modern view of a classical subject , Nonlinearity 21 (2008) T205

    D. Crowdy, Geometric function theory: a modern view of a classical subject , Nonlinearity 21 (2008) T205

    2008

  49. [57]

    Crowdy, Solving problems in multiply connected domains

    D. Crowdy, Solving problems in multiply connected domains . SIAM, 2020

    2020

  50. [58]

    Sepp¨ al¨ a,Myrberg’s numerical uniformization of hyperelliptic curves , Annales Fennici Mathematici 29 (2004) 3–20

    M. Sepp¨ al¨ a,Myrberg’s numerical uniformization of hyperelliptic curves , Annales Fennici Mathematici 29 (2004) 3–20

    2004

  51. [59]

    L. N. Trefethen, Series solution of Laplace problems , The ANZIAM Journal 60 (2018) 1–26

    2018

  52. [60]

    L. N. Trefethen, Polynomial and rational convergence rates for Laplace problems on planar domains , Proceedings of the Royal Society A 480 (2024) 20240178

    2024

  53. [61]

    Baune, J

    K. Baune, J. Broedel, E. Im, A. Lisitsyn and Y. Moeckli, Higher-genus Fay-like identities from meromorphic generating functions , SciPost Phys. 18 (2025) 093. – 55 –

    2025

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.