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On the universal ellipsitomic KZB connection

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Pith's one-line read The paper constructs a twisted genus-one KZB connection and proves that its completed monodromy is an isomorphism identifying the relative Malcev completion of the torus braid group.

desk verdict A solid new construction of a twisted elliptic KZB connection with a formality theorem whose only real soft spot is a drawing-based group presentation. read the letter →

arxiv 1908.03887 v3 pith:HANSZTRB submitted 2019-08-11 math.QA math.AG

Damien Calaque , Martin Gonzalez This is my paper · ORCID
classification math.QAmath.AG MSC 20F3614H5217B8016T2532G15
keywords ellipsitomicKZBconnectionflatpurebraidgroupofthetorusrelativefilteredformalityellipticdynamicalr-matricescyclotomicCherednikalgebrasMalcevcompletionclassicalYang-Baxterequation
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 introduces a twisted version of the genus-one Knizhnik–Zamolodchikov–Bernard (KZB) connection, called the ellipsitomic KZB connection, for elliptic curves carrying a $\Gamma=\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$ level structure. Its central claim is that the completed monodromy of this flat connection is an isomorphism: the relative prounipotent completion of the braid group $B_{1,n}$ over $\Gamma^n\rtimes S_n$ is identified with $\exp(\hat{\mathfrak t}^\Gamma)\rtimes(\Gamma^n\rtimes S_n)$, and equivalently the Malcev completion of the $\Gamma$-decorated pure braid group of the torus is the completion of the infinitesimal ellipsitomic braid Lie algebra $\mathfrak t^\Gamma_{1,n}$. This yields a relative filtered-formality statement for $B_{1,n}$ and, through the same connection, representations of cyclotomic Cherednik algebras. The construction matters because it extends the known genus-one KZB architecture to finite abelian level structures, giving a uniform source of flat connections, formality isomorphisms, and dynamical $r$-matrices with spectral parameter.

What carries the argument

The load-bearing object is the universal ellipsitomic KZB connection $$\$nabla^{{\mathrm{KZB}}$}_{n,\Gamma}=d-\$\Delta$(z,\tau)\,d\tau-\sum_i K_i(z,\tau)\,dz_i,$$ a flat connection on a principal bundle over the moduli space of $\Gamma$-structured elliptic curves with marked points. The coefficient $K_i$ is built from twisted copies of the Kronecker function $$k_\$\alpha$(x,z,\tau)=$e^{{-2\pi iax}}$\frac{\$\theta$(z-\tilde\$\alpha$+x,\tau)}{\$\theta$(z-\tilde\$\alpha$,\tau)\$\theta$(x,\tau)}-\frac1x,$$ while $\Delta$ is assembled from its $x$-derivatives and Eisenstein-type series; flatness is governed by the universal classical dynamical Yang-Baxter equation. The monodromy of this connection is the mechanism that produces the isomorphism of Theorem 5.5, with the Lie algebra $\mathfrak t^\Gamma_{1,n}$ of infinitesimal ellipsitomic braids playing the role of the completed associated graded.

What would settle it

Compute the associated graded Lie algebra of $PB^\Gamma_{1,n}$ from a finite presentation for a small case, say $n=3$ and $\Gamma=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$, and test whether every relation follows from (T1)–(T5); a single extra relation in low degree would disprove Proposition 5.6 and hence Theorem 5.5.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 5.5: the completed monodromy morphism from the relative completion of $B_{1,n}$ over $\Gamma^n\rtimes S_n$ to $\exp(\hat{\mathfrak t}^\Gamma)\rtimes(\Gamma^n\rtimes S_n)$ is an isomorphism, equivalently the completed monodromy morphism from the prounipotent completion of $PB^\Gamma_{1,n}$ to $\exp(\hat{\mathfrak t}^\Gamma)$ is an isomorphism. In plain terms, the monodromy of the ellipsitomic KZB connection completely determines the nilpotent part of the topology of the $\Gamma$-twisted configuration space: the associated graded of the $\Gamma$-decorated pure braid group of the torus is the infinitesimal ellipsitomic braid Lie algebra $\mathfrak t^\Gamma_{1,n}$, generated by translations $x_i,y_i$ and coloured braid elements $t^\alpha_{ij}$, and the filtered formality isomorphism is supplied by parallel transport. The paper also proves the connection restricts to the $\Gamma$-twisted configuration space, descends to moduli of $\Gamma$-structured elliptic curves, realizes as the KZB connection attached to elliptic dynamical $r$-matrices with spectral parameter, and produces representations of cyclotomic Cherednik algebras.

Load-bearing premise

The whole proof rests on the claim that the group $PB^\Gamma_{1,n}$ is generated exactly by $X_i^M$, $Y_i^N$, and $P^\alpha_{ij}$ with the five relations (T1)–(T5), a presentation supported by pictures rather than a formal proof; a missing relation would make the formality theorem collapse.

Editorial extensions

If this is right

  • Theorem 5.5 gives an explicit relative filtered-formality isomorphism for $B_{1,n}$ over $\Gamma^n\rtimes S_n$, so the Malcev Lie algebra of $PB^\Gamma_{1,n}$ is isomorphic to the degree completion of $\mathfrak t^\Gamma_{1,n}$.
  • The ellipsitomic KZB connection restricts to a flat connection on $\Gamma$-twisted configuration spaces of points on an elliptic curve and extends to the moduli space of $\Gamma$-structured elliptic curves with unordered marked points, giving flat bundles over those moduli spaces.
  • The universal connection realizes as the usual KZB connection associated with elliptic dynamical $r$-matrices with spectral parameter, so solutions of the classical dynamical Yang-Baxter equation with spectral parameter arise from the same monodromy data.
  • Composing the realization morphism with representations of cyclotomic Cherednik algebras produces flat connections on twisted configuration spaces and monodromy representations of the corresponding Hecke algebras of wreath products.
  • When $\Gamma$ is trivial ($M=N=1$), the construction recovers the untwisted universal elliptic KZB connection and the filtered formality of the pure braid group of the torus.

Reading between the lines

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

  • The same monodromy argument should produce filtered-formality isomorphisms for every finite abelian covering of elliptic configuration spaces, and the comparison morphisms for inclusions $\Gamma_1\hookrightarrow\Gamma_2$ suggest these formal structures fit into a compatible tower indexed by the level.
  • Because the connection is universal, specializing to explicit representations of $\mathfrak t^\Gamma$ should yield concrete flat connections whose monodromy can be compared with elliptic associators and multiple polylogarithms at torsion points; the paper's companion work on ellipsitomic associators is the natural place to test this.
  • The modular extension to Cherednik algebras leaves open whether the induced Hecke-algebra morphism is an isomorphism after inverting the formal parameter; checking this for small $n$ would either complete the picture or reveal additional relations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper constructs a twisted version of the universal genus-one KZB connection, called the ellipsitomic KZB connection, over the moduli space of Γ-structured elliptic curves with marked points, where Γ = Z/MZ × Z/NZ. It defines the Lie algebra tΓ of infinitesimal ellipsitomic braids, builds principal bundles over Γ-twisted configuration spaces and over the corresponding moduli spaces, proves flatness by explicit theta-function identities, and then uses the monodromy of the connection to prove relative filtered formality of the braid group B_{1,n} over Γ^n ⋊ S_n. The paper also realizes the connection in terms of elliptic dynamical r-matrices with spectral parameter and produces Lie algebra morphisms to cyclotomic Cherednik algebras.

Significance. If Theorem 5.5 is correct, the paper gives a substantial new family of filtered-formality results: the relative completion of B_{1,n} over the finite group Γ^n ⋊ S_n is identified with exp(t̂Γ) ⋊ (Γ^n ⋊ S_n), and the Malcev completion of the subgroup PBΓ of the torus pure braid group is identified with the completion of the ellipsitomic infinitesimal braid Lie algebra. The construction is explicit and parameter-free, with detailed theta-function computations, and it connects naturally to the earlier genus-one KZB work of Calaque–Enriquez–Etingof and to elliptic dynamical r-matrix theory. These are significant contributions if the technical gaps identified below are closed.

major comments (3)
  1. [§5.4, Proposition 5.6] The proof of Theorem 5.5 depends on Proposition 5.6, but the proof of Proposition 5.6 is not self-contained. The generation of PBΓ by X_i^M, Y_i^N and P^α_ij is asserted with the phrase "it follows from the geometric description", and several checks of the defining relations of tΓ are left as "similar" or justified by drawings. The most delicate point is the twisted infinitesimal braid relation (t4Teℓℓ1), [t^α_ij, t^{α+β}_ik + t^β_jk] = 0, which corresponds to a nontrivial commutator identity among the P^α generators; if this relation is not actually a consequence of (T1)–(T5), the map p_n is either ill-defined or non-surjective. In that case Lemma 5.7 would not imply that gr(µ) is an isomorphism. Please replace the drawing-based argument with a formal presentation of PBΓ, for example by deriving it from a known presentation of the torus braid group or by a covering-space argument.
  2. [§5.5, Lemma 5.7] The computation of φ(t^α_ij) = 2πi t^α_ij is asserted rather than proved. The displayed derivation writes µ(P^α_ij) = g exp(2πi t^0_ij + terms of degree ≥ 3)g^{-1} and concludes that log µ(P^α_ij) has degree-2 part 2πi t^α_ij. This requires controlling the degree-1 part of g and showing that conjugation does not mix t^α_ij with other degree-2 components of tΓ. Since this computation is exactly what upgrades p_n to an automorphism of gr(pbΓ), it is load-bearing for Theorem 5.5 and should be proved in detail rather than left as "as usual".
  3. [§1.6, Proposition 1.10] The flatness proof reduces the twisted CDYBE to identity (3) and then states that this identity is a consequence of equation (3) of [6]. Because kα(x,z) = e^{-2πiax}k(x,z - α̃) + (e^{-2πiax}-1)/x, the exponential factors and the shift by α̃ may produce extra terms that are not present in the untwisted identity. Since flatness of the ellipsitomic connection is foundational for the monodromy argument in Section 5, please include the explicit verification of (3), or a precise reduction to [6, Eq. (3)] with all twisted terms accounted for.
minor comments (4)
  1. [§5.4, Lemma 5.7 notation] The notation for P^α_ij is introduced with α = (p̄, q̄) and the formula P^α_ij = X_j^{-p}Y_j^{-q}P_ijY_j^qX_j^p, but Lemma 5.7 later writes conjugates involving Y_j^{-q}X_i^{-p} and a group element g(p̄,q̄)_i. Please make the ordering of the X and Y conjugations consistent and explicit.
  2. [§3.3] The coefficients A_{s,γ}(τ) are defined through φ̃γ(x/τ) = Σ A_{s,γ}(τ)x^s, but the paper does not explicitly state the degree of δ_{s,γ} in the semi-direct product with tΓ; adding this degree bookkeeping would help the reader follow the equivariance checks in Proposition 3.7.
  3. [§6.3] In the definition of the Hecke algebra HΓ_n(q,t), the elements T_α are described as small loops around the divisor Y_α but it is not specified whether different α correspond to different conjugacy classes of loops; a short explanatory sentence would improve clarity.
  4. [§1.1, Proposition 1.2] The comparison morphism φ_ρ depends on a choice of section coker(ρ) → Γ_2; the paper does not state whether the resulting morphism is independent of this choice or only up to inner automorphism. Clarifying this would prevent ambiguity in later uses.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main formality theorem follows from an explicitly constructed flat connection and direct monodromy computations; the drawing-based group presentation in §5.4 is a proof gap, not a circular reduction.

full rationale

The derivation chain in the paper is self-contained in the direction that matters. The Lie algebra t^Γ_{1,n} is introduced in §1.1 by abstract generators and relations, before the groups PB^Γ_{1,n} and B_{1,n} appear in §5; it is not defined in terms of the group or its associated graded. The central theorem (Theorem 5.5) is proved by constructing an explicit flat connection ∇^KZB_{τ,n,Γ}, then studying its monodromy. Lemma 5.7 computes φ = gr(µ)∘p_n explicitly: φ(x_i)=y_i, φ(y_i)=−2πi x_i + τ y_i, and φ(t^α_{ij})=2πi t^α_{ij}, from the first-order behaviour of log F_{z0} and the holonomy formulas. These are direct computations, not definitions disguised as conclusions. The only genuinely delicate step is Proposition 5.6, whose proof relies on the asserted geometric presentation of PB^Γ_{1,n}: the text says 'one can check (by simply drawing) that the following relations are satisfied in PB_{1,n}', and several later checks are described as 'similar'. If that presentation were incomplete or incorrect, the surjectivity of p_n and hence the conclusion of Theorem 5.5 would fail. This is a genuine proof gap, but it is not circularity: the relations (T1)–(T5) are asserted as facts about the group, and p_n is then verified against them; the target isomorphism is not used as an input. Self-citations are present but not load-bearing in a circular way. Reference [6] supplies the untwisted Γ=0 principal bundle, theta-function identities, and equation (3) used in the flatness proof; these are external prior results, not the paper's own conclusion. Reference [7] is a companion paper mentioned for an operadic module structure and for future ellipsitomic associators, not used to prove Theorem 5.5. No fitted parameters are renamed as predictions, and no uniqueness theorem is imported from the authors' prior work. The paper should therefore receive a low circularity score, with the caveat about the §5.4 presentation recorded as a correctness risk rather than a circular step.

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

The paper introduces no fitted parameters. It relies on standard theta function properties, a prior identity from [6] in the flatness proof, a geometric group presentation justified by drawings, relative completion splitting, and an external dynamical r-matrix proposition. The free-parameter count is zero.

assumptions (5)
  • standard math Theta function properties (Section 1.5): θ is the unique holomorphic function with prescribed zeros and quasi-periodicity; ∂_τ ϑ = (1/(4πi))∂_z^2 ϑ.
    Used to construct the connection form K_i and to prove equivariance; standard Jacobi theta function facts.
  • standard math Identity (3) of [6] (used in Proposition 1.10) establishing the functional equation for the untwisted k(x,z).
    The flatness proof reduces the twisted CDYBE to this prior identity from Calaque-Enriquez-Etingof; the paper does not reproduce the derivation.
  • domain assumption Group-theoretic presentation of PB_{1,n} with relations (T1)-(T5) from the geometry of the torus and its Γ-covering (Section 5.4).
    Load-bearing for Proposition 5.6 and Theorem 5.5; justified by drawings rather than a complete proof.
  • standard math Relative completion formalism and Lemma 5.1: every extension of a finite group by a prounipotent group over a field of characteristic 0 splits (finite group cohomology vanishing).
    Used to define relative filtered formality and to identify relative completion with Ker(φ)(k) ⋊ S.
  • standard math Classical dynamical Yang-Baxter and [13, Proposition 0.1] (Section 4.4) used to promote r(x,z) to a solution of CDYBE with spectral parameter.
    External result from Enriquez-Etingof used in the realization section.

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Pith. "Pith review of On the universal ellipsitomic KZB connection." pith.science (2026). https://pith.science/paper/HANSZTRB

@misc{pith2026190803887,
  author       = {Pith},
  title        = {Pith review of: On the universal ellipsitomic KZB connection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HANSZTRB}},
  note         = {Machine review of arXiv:1908.03887}
}
abstract

We construct a twisted version of the genus one universal Knizhnik-Zamolodchikov-Bernard (KZB) connection introduced by Calaque-Enriquez-Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the moduli space of $\Gamma$-structured elliptic curves with marked points, where $\Gamma=\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$, and $M,N\geq1$ are two integers. It restricts to a flat connection on $\Gamma$-twisted configuration spaces of points on elliptic curves, which can be used to construct a filtered-formality isomorphism for some interesting subgroups of the pure braid group on the torus. We show that the universal ellipsitomic KZB connection realizes as the usual KZB connection associated with elliptic dynamical $r$-matrices with spectral parameter, and finally, also produces representations of cyclotomic Cherednik algebras.

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Works this paper leans on

32 extracted references · 29 canonical work pages

  1. [6]

    Calaque, B

    D. Calaque, B. Enriquez & P. Etingof, Universal KZB equations: the elliptic case , in Algebra, Arith- metic and Geometry Vol I: In honor of Yu. I. Manin (Y. Tschinke l & Y. Zarhin eds.) , Progress in Mathematics 269 (2010), 165–266

    work page 2010

  2. [1]

    Bezrukavnikov, Koszul DG-algebras arising from configuration spaces , Geom

    R. Bezrukavnikov, Koszul DG-algebras arising from configuration spaces , Geom. and Funct. Analysis 4 (1992), no. 2, 119–135

    work page 1992

  3. [2]

    Bernard, On the Wess-Zumino-Witten model on the torus , Nucl

    D. Bernard, On the Wess-Zumino-Witten model on the torus , Nucl. Phys. B 303 (1988), 77–93

    work page 1988

  4. [3]

    Braverman, P

    A. Braverman, P. Etingof, & M.V. Finkelberg, Cyclotomic double affine Hecke algebras (with an appendix by Hiraku Nakajima and Daisuke Yamakawa) , preprint arXiv:1611.10216

    arXiv

  5. [4]

    Broedel, N

    J. Broedel, N. Matthes, G. Richter & O. Schlotterer, Twisted elliptic multiple zeta values and non- planar one-loop open-string amplitudes , Journal of Physics A: Mathematical and Theoretical 51 (2018), no. 28, 49pp

    work page 2018

  6. [5]

    Brown & A

    F. Brown & A. Levin, Multiple elliptic polylogarithms , preprint arXiv:1110.6917

    arXiv

  7. [7]

    Ellipsitomic Associators

    D. Calaque & M. Gonzalez, Ellipsitomic associators , preprint arXiv:2004.07271

    work page Pith review arXiv 2004

  8. [8]

    Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal( ¯Q/slash.left Q), Leningrad Math

    V. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal( ¯Q/slash.left Q), Leningrad Math. J. 2 (1991), 829–860

    work page 1991

Show all 32 references
  1. [9]

    Enriquez, Quasi-reflection algebras and cyclotomic associators , Selecta Mathematica (NS) 13 (2008), no

    B. Enriquez, Quasi-reflection algebras and cyclotomic associators , Selecta Mathematica (NS) 13 (2008), no. 3, 391–463

    2008

  2. [10]

    Enriquez, Elliptic associators , Selecta Mathematica (NS) 20 (2014), no

    B. Enriquez, Elliptic associators , Selecta Mathematica (NS) 20 (2014), no. 2, 491–584

    2014

  3. [11]

    Enriquez, Analogues elliptiques des nombres multiz´ etas , Bulletin de la SMF 144 (2016), no

    B. Enriquez, Analogues elliptiques des nombres multiz´ etas , Bulletin de la SMF 144 (2016), no. 3, 395–427

    2016

  4. [12]

    Enriquez, Flat connections on configuration spaces and formality of br aid groups of surfaces , Adv

    B. Enriquez, Flat connections on configuration spaces and formality of br aid groups of surfaces , Adv. in Math. 252 (2014), 204–226

    2014

  5. [13]

    Enriquez & P

    B. Enriquez & P. Etingof, Quantization of classical dynamical r-matrices with nonabelian base , Comm. Math. Phys. 254 (2005), no. 3, 603–650

    2005

  6. [14]

    Etingof, Cherednik and Hecke algebras of varieties with a finite group action, Moscow Math

    P. Etingof, Cherednik and Hecke algebras of varieties with a finite group action, Moscow Math. J. 17 (2017), no. 4, 635–666

    2017

  7. [15]

    Etingof & O

    P. Etingof & O. Schiffmann, Twisted traces of intertwiners for Kac-Moody algebras and c lassical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples, Mathematical Research Letters 6 (1999), no. 6, 593–612

    1999

  8. [16]

    Feher and B

    L. Feher and B. Pusztai, Generalizations of Felder’s elliptic dynamical r-matrice s associated with twisted loop algebras of self-dual Lie algebras , Nucl.Phys. B621 (2002), 622–642

    2002

  9. [17]

    Felder, Conformal field theory and integrable systems associated to elliptic curves , in Proceedings of the ICM, Vol

    G. Felder, Conformal field theory and integrable systems associated to elliptic curves , in Proceedings of the ICM, Vol. 1,2 (Z¨ urich, 1994) , 1247–1255, Birkh¨ auser, Basel, 1995

    1994

  10. [18]

    Goncharov, Multiple ζ-values, Galois groups, and geometry of modular varieties , in European Congress of Mathematics, Vol

    A.B. Goncharov, Multiple ζ-values, Galois groups, and geometry of modular varieties , in European Congress of Mathematics, Vol. 1 (Barcelona, 2000) , 361–392, Birkh¨ auser, Basel, 2001

    2000

  11. [19]

    Gonzalez, Contributions to the theory of associators , PhD Thesis, available at https://hal.archives-ouvertes.fr/tel-02541848

    M. Gonzalez, Contributions to the theory of associators , PhD Thesis, available at https://hal.archives-ouvertes.fr/tel-02541848

  12. [20]

    Hain, The Hodge de Rham theory of relative Malcev completion , Annales scientifiques de l’ ´Ecole Normale Sup´ erieure (S´ erie 4)31 (1998), no

    R. Hain, The Hodge de Rham theory of relative Malcev completion , Annales scientifiques de l’ ´Ecole Normale Sup´ erieure (S´ erie 4)31 (1998), no. 1 , 47–92

    1998

  13. [21]

    Hain, Notes on the universal elliptic KZB equation , Pure and Applied Mathematics Quarterly 16 (2020), no

    R. Hain, Notes on the universal elliptic KZB equation , Pure and Applied Mathematics Quarterly 16 (2020), no. 2, 229–312

    2020

  14. [22]

    Hain & M

    R. Hain & M. Matsumoto, Relative pro- ℓ completions of mapping class groups , Journal of Algebra 321 (2009), no. 11, 3335–3374

    2009

  15. [23]

    Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann

    T. Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37 (1987), 139–160

    1987

  16. [24]

    Kohno, On the holonomy Lie algebra and the nilpotent completion of t he fundamental group of the complement of hypersurfaces , Nagoya Mathematical Journal 92 (1983), 21–37

    T. Kohno, On the holonomy Lie algebra and the nilpotent completion of t he fundamental group of the complement of hypersurfaces , Nagoya Mathematical Journal 92 (1983), 21–37

    1983

  17. [25]

    Levin & G

    A. Levin & G. Racinet, Towards multiple elliptic polylogarithms , preprint arXiv:math/0703237. Glossary 51

    arXiv

  18. [26]

    J. W. Morgan, The algebraic topology of smooth algebraic varieties , Publications Math´ ematiques de l’IH´ES 48 (1978), 137–204

    1978

  19. [27]

    Pridham, On ℓ-adic pro-algebraic and relative pro- ℓ fundamental groups , in The Arithmetic of Fundamental Groups (J

    J. Pridham, On ℓ-adic pro-algebraic and relative pro- ℓ fundamental groups , in The Arithmetic of Fundamental Groups (J. Stix ed.), Contributions in Mathematical and Computati onal Sciences 2 (2012), 245–279

    2012

  20. [28]

    Suciu & H

    A.I. Suciu & H. W ang, Formality properties of finitely ge nerated groups and Lie algebras, Forum Mathematicum 31 (2019), no. 4, 867–905

    2019

  21. [29]

    Toledano-Laredo & Y

    V. Toledano-Laredo & Y. Yang, Universal KZB equations for arbitrary root systems , preprint arXiv:1801.10259

    arXiv

  22. [30]

    Tsunogai, The stable derivation algebra for higher genera , Israel J

    H. Tsunogai, The stable derivation algebra for higher genera , Israel J. Math., 136 (2003), 221–250

    2003

  23. [31]

    W eil, Elliptic functions according to Eisenstein and Kronecker , Ergebnisse der Mathematik und ihrer Grenzgebiete, 88, Springer-Verlag, Berlin-New York, (1976), ii+93 pp

    A. W eil, Elliptic functions according to Eisenstein and Kronecker , Ergebnisse der Mathematik und ihrer Grenzgebiete, 88, Springer-Verlag, Berlin-New York, (1976), ii+93 pp

    1976

  24. [32]

    Zagier, Periods of modular forms and Jacobi theta functions , Invent

    D. Zagier, Periods of modular forms and Jacobi theta functions , Invent. Math. 104 (1991), 449–465. Damien CALAQUE IMAG, Univ Montpellier, CNRS, Montpellier, France & Institut Universitaire de France Email address : damien.calaque@umontpellier.fr Martin GONZALEZ Max-Planck Ins...

    1991

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