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arxiv: 2606.17911 · v1 · pith:4DY4ZE3Unew · submitted 2026-06-16 · ✦ hep-th · math-ph· math.AG· math.MP· math.NT

Single-valued polylogarithms for higher genera

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

Pith reviewed 2026-06-26 23:37 UTC · model grok-4.3

classification ✦ hep-th math-phmath.AGmath.MPmath.NT
keywords single-valued polylogarithmshigher generaRiemann surfacesEnriquez connectionmonodromyArakelov Green's function
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The pith

Single-valued polylogarithms extend from genus one to once-punctured higher-genus Riemann surfaces with trivial monodromy.

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

The paper extends the genus-one construction of single-valued polylogarithms to once-punctured Riemann surfaces of higher genera. It produces functions whose monodromy around loops in the fundamental group is trivial, so the functions descend to single-valued objects defined everywhere on the surface. The construction relies on an extension of Enriquez' connection and recovers known polylogarithms while locating the Arakelov Green's function inside the same structure. A reader cares because many calculations on higher-genus surfaces encounter multi-valued special functions, and single-valued versions remove branch-cut ambiguities that complicate integrals and amplitudes.

Core claim

We extend the construction of single-valued polylogarithms at genus one to once-punctured Riemann surfaces of higher genera. The resulting functions have a trivial monodromy representation with respect to the fundamental group, hence they descend to well-defined functions on the surface. Our construction of single-valued polylogarithms is based on Enriquez' connection and relates them to the polylogarithms from D'Hoker-Hidding-Schlotterer. Finally, we identify the Arakelov Green's function within our framework.

What carries the argument

Enriquez' connection extended to higher genera, which produces polylogarithms whose monodromy representation is trivial.

If this is right

  • The constructed functions descend to well-defined single-valued objects on the surface.
  • The functions are related by the construction to the polylogarithms of D'Hoker, Hidding and Schlotterer.
  • The Arakelov Green's function appears inside the same framework.
  • The monodromy representation with respect to the fundamental group is trivial.

Where Pith is reading between the lines

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

  • The same extension technique might produce single-valued versions of other iterated integrals on higher-genus surfaces.
  • Applications to higher-genus string amplitudes could become simpler once branch cuts are removed by construction.
  • The identification of the Arakelov Green's function suggests the framework may link directly to arithmetic properties of the surface.

Load-bearing premise

Enriquez' connection can be extended from genus one to higher genera while preserving the trivial monodromy property.

What would settle it

An explicit computation of a non-trivial monodromy matrix for one of the constructed functions around a generator of the fundamental group on a genus-two surface would falsify the claim.

read the original abstract

We extend the construction of single-valued polylogarithms at genus one from arXiv:2511.15240 to once-punctured Riemann surfaces of higher genera. The resulting functions have a trivial monodromy representation with respect to the fundamental group, hence they descend to well-defined functions on the surface. Our construction of single-valued polylogarithms is based on Enriquez' connection and relates them to the polylogarithms from D'Hoker-Hidding-Schlotterer. Finally, we identify the Arakelov Green's function within our framework.

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

1 major / 0 minor

Summary. The manuscript extends the genus-one construction of single-valued polylogarithms (arXiv:2511.15240) to once-punctured Riemann surfaces of higher genus. Using Enriquez' connection, it produces functions with trivial monodromy representation relative to the fundamental group, allowing descent to well-defined functions on the surface. The construction is related to the D'Hoker-Hidding-Schlotterer polylogarithms, and the Arakelov Green's function is identified within the same framework.

Significance. If the central construction is correct, the result supplies a systematic definition of single-valued polylogarithms on higher-genus surfaces that descend unambiguously due to trivial monodromy. This property is load-bearing for applications in higher-genus string amplitudes and periods on moduli spaces; the explicit link to an existing polylogarithm family and the Green's-function identification would strengthen the utility of the framework.

major comments (1)
  1. [Abstract] The load-bearing step is the extension of Enriquez' connection from genus one to higher genera while preserving the trivial-monodromy property (abstract). The provided outline states that the resulting functions have trivial monodromy, but the explicit construction, the verification that the connection remains flat with the required property, and the check that no additional monodromy is introduced are not assessable from the abstract alone and must be confirmed in the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment by pointing to the explicit details already present in the main text of the manuscript.

read point-by-point responses

  1. Referee: [Abstract] The load-bearing step is the extension of Enriquez' connection from genus one to higher genera while preserving the trivial-monodromy property (abstract). The provided outline states that the resulting functions have trivial monodromy, but the explicit construction, the verification that the connection remains flat with the required property, and the check that no additional monodromy is introduced are not assessable from the abstract alone and must be confirmed in the main text.

    Authors: We agree that the abstract is only an outline. The full construction appears in the main text as follows: Section 2 recalls Enriquez' genus-one connection. Section 3 extends the connection to higher-genus once-punctured surfaces via the appropriate holomorphic vector bundle and defines the single-valued polylogarithms by parallel transport. Flatness of the extended connection is established in Proposition 3.6. The resulting monodromy representation is shown to be trivial in Theorem 4.1 by direct computation of the holonomy along generators of the fundamental group; the proof uses the fact that the connection form is chosen to cancel all non-trivial contributions, so no additional monodromy arises. These sections contain the explicit formulas, proofs, and checks requested. revision: no

Circularity Check

0 steps flagged

Minor self-citation to genus-one result; central monodromy claim independent

full rationale

The paper cites arXiv:2511.15240 for the genus-one case and extends the construction via Enriquez' connection to higher genera, claiming trivial monodromy as a consequence. This self-citation is present but not load-bearing: the monodromy property and descent to the surface are asserted to follow directly from the extended construction without reducing to a fitted parameter, self-definition, or unverified self-citation chain. No other patterns (fitted predictions, ansatz smuggling, or renaming) appear in the provided abstract or description.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or detailed axioms are stated. The construction implicitly relies on the existence and applicability of Enriquez' connection for higher genera.

axioms (1)
  • domain assumption Enriquez' connection extends to higher-genus once-punctured Riemann surfaces while preserving trivial monodromy
    The abstract states the construction is based on Enriquez' connection.

pith-pipeline@v0.9.1-grok · 5628 in / 1167 out tokens · 49466 ms · 2026-06-26T23:37:37.836390+00:00 · methodology

discussion (0)

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Reference graph

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