Non-linear geometry of multiple zeta values

Francis Brown

arxiv: 2604.22735 · v1 · submitted 2026-04-24 · 🧮 math.NT · hep-th· math-ph· math.AG· math.MP

Non-linear geometry of multiple zeta values

Francis Brown This is my paper

Pith reviewed 2026-05-08 09:51 UTC · model grok-4.3

classification 🧮 math.NT hep-thmath-phmath.AGmath.MP

keywords multiple zeta valuesnon-linear geometrydeterminantal representationstropical geometrymoduli spacesFeynman integralsquadratic forms

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The pith

Multiple zeta values have non-linear integral representations with matrix determinants that trace to tropical geometry and moduli spaces.

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

The paper shows that multiple zeta values, usually defined via linear iterated integrals on a three-punctured sphere, also possess distinct non-linear integral forms in which determinants of matrices appear in the denominators. These forms appear across mathematics and physics and originate in tropical geometry, the moduli spaces of tropical curves, Feynman integrals, the general linear group of integer matrices, and the reduction theory of quadratic forms. The author traces these connections and proposes a geometric framework built on the non-linear representations while listing open questions. A reader would care if this framework unifies the many appearances of multiple zeta values under one geometric source rather than treating the linear and non-linear cases separately.

Core claim

Multiple zeta values admit integral representations whose denominators involve determinants rather than complete factorizations into linear terms; these non-linear representations arise from tropical geometry, the moduli spaces of tropical curves, Feynman integrals, the general linear group over the integers, and the reduction theory of quadratic forms, and therefore motivate a geometric framework organized around determinantal integrands.

What carries the argument

The non-linear geometry consisting of determinantal integrands for multiple zeta values, which supplies representations distinct from the standard linear iterated integrals and links them to tropical and moduli-space constructions.

If this is right

Identities and relations among multiple zeta values can be derived from the geometry of tropical curves and quadratic-form reduction.
The appearance of multiple zeta values in Feynman integrals gains a systematic geometric explanation via determinantal forms.
Open questions posed about the framework may resolve how linear and non-linear representations interact or generate each other.
The reduction theory of quadratic forms supplies arithmetic constraints on the values that appear in the non-linear integrals.

Where Pith is reading between the lines

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

The framework may suggest new ways to evaluate multiple zeta values by reducing them to problems in tropical curve moduli.
Similar non-linear representations could exist for other periods that arise in both number theory and quantum field theory.
Treating the determinantal forms as primary might reorganize known tables of multiple zeta values according to geometric invariants rather than weight and depth alone.

Load-bearing premise

The non-linear determinantal representations share a common geometric source in tropical geometry, moduli spaces, and Feynman integrals that can support a single unified framework.

What would settle it

An explicit computation of a multiple zeta value that possesses no determinantal integral representation traceable to tropical curves or the moduli space of tropical curves would undermine the proposed common origin.

Figures

Figures reproduced from arXiv: 2604.22735 by Francis Brown.

**Figure 1.** Figure 1: In quantum electrodynamics, the theory of light and matter, Feynman graphs are built out of two types of edges and one vertex (left). The graph G1 represents, for example, two electrons (straight lines) exchanging a photon (wiggly line) which contributes to the repulsive force experienced between two electrons. The graph G2 represents the same process, but has two ‘loops’, and thus represents a higher-or… view at source ↗

**Figure 2.** Figure 2: The part of the open locus LM◦,trop 2 corresponding to the sunrise graph is the quotient of the above figure, which depicts a closed triangle minus its three corners, by the action of the symmetric group Σ3. The three corners of the triangle map to a single point in ∂LMtrop 2 . 3.7. *Graph polynomial vanishing locus. The space LMtrop g is closely related to the geometry of Feynman integrals. First of all,… view at source ↗

**Figure 3.** Figure 3: A picture of the geometry associated to the sunrise graph G. The graph hypersurface V (ΨG) in P 2 is depicted in red; the coordinate hyperplanes in blue. The domain of integration σG is the coordinate simplex which meets the graph hypersurface in the three corners. By identifying the projective simplex {(x1 : x2 : x3) : xi ≥ 0} with the region in real space {(ℓ1, ℓ2, ℓ3) : P3 i=1 ℓi = 1} , one sees that th… view at source ↗

read the original abstract

Since their rediscovery in the 1990s, multiple zeta values have become ubiquitous in many areas of mathematics and physics. Their standard integral and sum representations can usually be traced back to a single source, namely the iterated integrals on the Riemann sphere with three punctures. We refer to such representations as the \emph{linear} geometry of multiple zeta values, since the denominators of the corresponding integrands factor completely into linear terms. However, there also exist equally important and entirely distinct integral representations for multiple zeta values arising in mathematics and physics, in which matrix determinants appear in the denominator of the integrand. We call this the \emph{non-linear} geometry of multiple zeta values. These lectures trace the origins of this non-linear geometry and provide an introductory journey through a range of topics including tropical geometry, the moduli spaces of tropical curves, Feynman integrals in quantum field theory, the general linear group of integer matrices, and the reduction theory of quadratic forms. In doing so, we propose a geometric framework for multiple zeta values based on such non-linear, determinantal representations and set out a number of open questions for future research.

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.

Desk Editor's Note private letter to a colleague

Brown's lecture notes collect determinantal integral reps for MZVs from tropical geometry, moduli spaces, and Feynman integrals, frame them as non-linear geometry, and list open questions without proving new results.

read the letter

This is lecture notes that pull together integral representations of multiple zeta values in which determinants appear in the denominators. Brown traces these to tropical geometry, moduli spaces of curves, Feynman integrals in QFT, and reduction of quadratic forms, then sketches a geometric framework around them and poses open questions for later work. The main contribution is the explicit contrast with the usual linear geometry coming from iterated integrals on the thrice-punctured sphere, plus the suggestion that the determinantal forms share a common geometric origin across these domains. The notes do a clear job of showing how the same objects surface in different settings and why that might matter. The limitation is straightforward: there are no new theorems, derivations, or verifications here. Everything rests on representations already in the literature, and the proposed framework functions as an organizing perspective rather than a source of fresh identities or equivalences. The common-source claim is offered as motivation, not as something demonstrated. This is useful for specialists already working on MZVs who want a compact overview of the non-linear side and some questions worth thinking about. It is less useful if you need concrete new results or proofs. A serious editor should send it to referees because the synthesis is coherent and the questions are substantive, even though the piece is exploratory rather than conclusive.

Referee Report

0 major / 3 minor

Summary. The manuscript consists of lecture notes that collect examples of determinantal integral representations for multiple zeta values arising in tropical geometry, moduli spaces of curves, Feynman integrals, the general linear group, and reduction theory of quadratic forms. It contrasts these non-linear representations with the standard linear ones from iterated integrals on the thrice-punctured Riemann sphere, proposes a geometric framework based on the non-linear case, and lists open questions for future research.

Significance. The compilation of examples from disparate areas provides a useful perspective on MZVs beyond the linear setting. The explicit framing of a non-linear geometric viewpoint and the enumeration of open questions are constructive, even if the framework remains at the level of a unifying proposal rather than a set of new theorems. No machine-checked proofs or parameter-free derivations are present, consistent with the expository lecture-notes format.

minor comments (3)

[Introduction] The assertion in the abstract that the non-linear representations are 'equally important' would be strengthened by a brief discussion, with references, of their concrete computational or structural advantages in at least one concrete MZV identity.
[Framework proposal] The proposed geometric framework is described at a conceptual level; a short subsection that isolates its key distinguishing features (e.g., a list of axioms or a diagram relating the cited sources) would help readers assess its novelty relative to existing MZV literature.
[Open questions] Open questions at the end could be cross-referenced to the specific examples or sections in which the relevant phenomena first appear, improving navigability for readers interested in particular topics such as tropical curves or Feynman integrals.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our lecture notes and for recommending minor revision. We are pleased that the compilation of determinantal integral representations across tropical geometry, moduli spaces, Feynman integrals, and related areas is viewed as providing a useful perspective, and that the non-linear geometric framework and open questions are considered constructive.

Circularity Check

0 steps flagged

No significant circularity; framework proposal with open questions

full rationale

The manuscript is structured as lecture notes that collect known determinantal integral representations for MZVs from tropical geometry, moduli spaces, and Feynman integrals, then offers a unifying perspective and lists open questions. The linear vs. non-linear distinction is introduced by explicit construction (complete linear factorization of denominators versus appearance of matrix determinants), with no derivation or prediction that reduces to its own inputs. No load-bearing self-citation chain, fitted parameter renamed as prediction, or uniqueness theorem imported from prior work is present; the central claim is a proposed geometric viewpoint rather than a closed theorem whose validity depends on unverified steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the proposal is conceptual and does not detail mathematical foundations.

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discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 4 canonical work pages

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Statistics of feynman amplitudes inϕ 4-theory.Journal of High Energy Physics, 2023(11):160,

[Bal23] Paul-Hermann Balduf. Statistics of feynman amplitudes inϕ 4-theory.Journal of High Energy Physics, 2023(11):160,

2023

[2] [2]

Brandt, J

[BBC+20] M. Brandt, J. Bruce, M. Chan, M. Melo, G. Moreland, and C. Wolfe. On the top-weight rational cohomology ofA g.Preprint: arXiv:2012.02892,

work page arXiv 2012

[3] [3]

34 [Bro10] Francis C. S. Brown. On the periods of some Feynman integrals.Preprint: arXiv:0910.0114,

work page Pith review arXiv

[4] [4]

Bordifications of the moduli spaces of tropical curves and abelian varieties, and unstable cohomology of GL g(Z) and SL g(Z).Preprint: arXiv:2309.12753,

[Bro23] Francis Brown. Bordifications of the moduli spaces of tropical curves and abelian varieties, and unstable cohomology of GL g(Z) and SL g(Z).Preprint: arXiv:2309.12753,

work page arXiv

[5] [5]

[Del89] P. Deligne. Le groupe fondamental de la droite projective moins trois points. InGalois groups overQ(Berkeley, CA, 1987), volume 16 ofMath. Sci. Res. Inst. Publ., pages 79–297. Springer, New York,

1987

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[RW14] Carlo Antonio Rossi and Thomas Willwacher. P. Etingof’s conjecture about Drinfeld associ- ators.Preprint: arXiv:1404.2047,

work page arXiv 2047