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arxiv: 2511.15240 · v2 · pith:P4TV3QWNnew · submitted 2025-11-19 · ✦ hep-th · math-ph· math.AG· math.MP· math.NT

A construction of single-valued elliptic polylogarithms

Konstantin Baune , Johannes Broedel , Yannis Moeckli This is my paper

Pith reviewed 2026-05-21 18:26 UTC · model grok-4.3

classification ✦ hep-th math-phmath.AGmath.MPmath.NT
keywords single-valued elliptic polylogarithmsonce-punctured elliptic curveKnizhnik-Zamolodchikov-Bernard equationelliptic associatorsBrown's constructiontrivial monodromyelliptic Bloch-Wigner dilogarithm
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The pith

Single-valued elliptic polylogarithms arise as functions on the once-punctured elliptic curve when trivial monodromy is imposed on solutions to the Knizhnik-Zamolodchikov-Bernard equation via elliptic associators.

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

The paper gives a general construction for single-valued elliptic polylogarithms defined directly on the once-punctured elliptic curve. It extends Brown's genus-zero method by expressing the trivial-monodromy condition for the Knizhnik-Zamolodchikov-Bernard equation through elliptic associators that employ two representations of a two-letter alphabet. A reader would care because these functions supply a systematic way to obtain single-valued quantities that appear in elliptic integrals and related calculations. The new single-valued condition reduces exactly to Brown's genus-zero condition when the torus degenerates to a sphere, confirming consistency with the known case. Concrete examples, such as the elliptic Bloch-Wigner dilogarithm, illustrate the construction at work.

Core claim

We establish a general construction of single-valued elliptic polylogarithms as functions on the once-punctured elliptic curve. Our formalism is an extension of Brown's construction of genus-zero single-valued polylogarithms to the elliptic curve: the condition of trivial monodromy for solutions to the Knizhnik-Zamolodchikov-Bernard equation is expressed in terms of elliptic associators and involves two representations of a two-letter alphabet. Our elliptic single-valued condition reduces to Brown's genus-zero condition upon degeneration of the torus. We provide several examples for our construction, including the elliptic Bloch-Wigner dilogarithm.

What carries the argument

Elliptic associators that encode the trivial-monodromy condition for the Knizhnik-Zamolodchikov-Bernard equation using two representations of a two-letter alphabet; they generalize Brown's genus-zero associators and enforce single-valuedness on the punctured elliptic curve.

If this is right

  • The elliptic single-valued condition reduces to Brown's genus-zero condition under torus degeneration.
  • The construction supplies explicit single-valued versions of elliptic polylogarithms including the elliptic Bloch-Wigner dilogarithm.
  • The same elliptic associators can be used to define single-valued functions for any two representations of the two-letter alphabet.

Where Pith is reading between the lines

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

  • The same formalism could be tested on higher-weight examples to check whether single-valuedness holds uniformly.
  • If the construction works, it may supply a practical way to extract single-valued parts from elliptic Feynman integrals without separate regularization.
  • The reduction to the sphere suggests a possible degeneration limit that relates elliptic and ordinary polylogarithm identities directly.

Load-bearing premise

The condition of trivial monodromy for solutions to the Knizhnik-Zamolodchikov-Bernard equation can be expressed in terms of elliptic associators involving two representations of a two-letter alphabet.

What would settle it

An explicit loop around a puncture on the elliptic curve for which one of the constructed functions acquires a non-zero monodromy factor would falsify the claim that the construction produces single-valued functions.

Figures

Figures reproduced from arXiv: 2511.15240 by Johannes Broedel, Konstantin Baune, Yannis Moeckli.

Figure 1
Figure 1. Figure 1: Graphical representation of the A- and B-cycle associators defined by Enriquez [25]. The extra phase for the A-associator reflects our convention chosen in this article: the elliptic associators always interpolate between the same tangential basepoints. The dotted line is not a monodromy factor. 3.4 Elliptic associators In § 2.2, the genus-zero monodromy conditions were formulated in terms of the Drinfeld … view at source ↗
Figure 2
Figure 2. Figure 2: Graphical representation of the cycle relation in eq. (4.1). For better readability, we shifted the puncture away from 0. which is the free group on two generators10 A and B. To see this, notice that A 7→ A, B 7→ B and C 7→ ABA−1B −1 defines a group isomorphism. In other words, the non-trivial relation in eq. (4.1) allows to eliminate the generator C, thereby reflecting that its corresponding path is homot… view at source ↗
Figure 3
Figure 3. Figure 3: A- and B-cycle single-valued conditions (4.6) sketched on the fundamental parallelogram (left) and on the torus (right). sets of generators {a, b} and {a ′ , b′} of the same algebra f2(Y). As a small remark, notice that the discussion around eq. (3.7) showed that any solution to the KZB equation (3.4) is related to the solution Γ(z|τ ) through multiplication by a function S(τ ), which is constant in z and … view at source ↗
Figure 4
Figure 4. Figure 4: (a) complex plot of Γ(1; ˜ z|τ ), and (b) 3D plot of Γ˜sv(1; z|τ ) = Υ(ab; z|τ ), both for τ = 3 4 + 5 4˚ı. In (a) we can see the branch cuts and thus the multi-valued nature of Γ(1; ˜ z|τ ), while in (b) all branch cuts have been canceled and we are left with the single-valued real function Γ˜sv(1; z|τ ), which is doubly￾periodic. A-cycle nor the B-cycle. For the monodromies around the cycles, we find exp… view at source ↗
Figure 5
Figure 5. Figure 5: Plots showing (a) the multi-valued function Υ(b 2a; z|τ ) and (b) its single-valued analogue Υ(b 2a; z|τ ) given in eq. (5.6), both for τ = 0.003+1.01˚ı. Numerical difficulties around z = 0 have been cut out of the plots. The expansion yields the expression Υ(b 2 a; z|τ ) = Υ(b 2 a; z|τ ) − Υ(ab2; z|τ ) − Υ(b 2 ; z|τ )  Υ(a; z|τ ) + Im(τ ) π Υ(b; z|τ )  − Υ(b; z|τ )  Υ(ab; z|τ ) + Im(τ ) π Υ(b 2; z|τ ) … view at source ↗
read the original abstract

We establish a general construction of single-valued elliptic polylogarithms as functions on the once-punctured elliptic curve. Our formalism is an extension of Brown's construction of genus-zero single-valued polylogarithms to the elliptic curve: the condition of trivial monodromy for solutions to the Knizhnik-Zamolodchikov-Bernard equation is expressed in terms of elliptic associators and involves two representations of a two-letter alphabet. Our elliptic single-valued condition reduces to Brown's genus-zero condition upon degeneration of the torus. We provide several examples for our construction, including the elliptic Bloch-Wigner dilogarithm.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper claims to establish a general construction of single-valued elliptic polylogarithms as functions on the once-punctured elliptic curve. This extends Brown's genus-zero construction by expressing the trivial monodromy condition for solutions to the Knizhnik-Zamolodchikov-Bernard equation in terms of elliptic associators involving two representations of a two-letter alphabet. The elliptic single-valued condition reduces to Brown's genus-zero condition upon torus degeneration, with explicit examples provided including the elliptic Bloch-Wigner dilogarithm.

Significance. If the result holds, the construction supplies a canonical method for producing single-valued elliptic polylogarithms, extending the theory of multiple polylogarithms and associators to the elliptic curve setting. This is relevant for elliptic motives, periods, and applications in string theory amplitudes. The explicit reduction to the genus-zero case and the inclusion of concrete examples such as the elliptic Bloch-Wigner dilogarithm provide verifiable consistency checks and strengthen the overall framework.

minor comments (2)
  1. [§1] The abstract and introduction refer to 'two representations of a two-letter alphabet' without an immediate explicit definition or comparison to the genus-zero alphabet; adding this in §1 or §2 would improve readability for readers familiar with Brown's construction.
  2. [Example section] In the discussion of the elliptic Bloch-Wigner dilogarithm example, a side-by-side comparison of the resulting single-valued function with its genus-zero counterpart would clarify the degeneration property.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. We are pleased that the referee recognizes the construction as a canonical extension of Brown's genus-zero single-valued polylogarithms to the once-punctured elliptic curve, including the expression via elliptic associators, the reduction to the torus degeneration limit, and the explicit example of the elliptic Bloch-Wigner dilogarithm. These features align precisely with the claims in our abstract and main text.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper constructs single-valued elliptic polylogarithms by expressing the trivial-monodromy condition for KZB solutions via elliptic associators on a two-letter alphabet, extending Brown's genus-zero case with explicit reduction under torus degeneration. Derivations proceed directly from KZB connection properties and associator definitions; examples including the elliptic Bloch-Wigner dilogarithm are internally consistent. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. Dependence on prior external work (Brown, KZB) is standard and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on standard properties of the KZB equation and elliptic associators drawn from prior literature; no free parameters or new invented entities are indicated in the abstract.

axioms (1)
  • domain assumption The Knizhnik-Zamolodchikov-Bernard equation admits solutions whose monodromy can be controlled by elliptic associators
    Invoked when expressing the trivial monodromy condition for single-valuedness.

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Reference graph

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