arXiv subjects

Francis Brown

Publications and source records attributed to Francis Brown.

At least 19 recordsLinked to original sources

arXiv2026

The motivic Lie algebra embeds into the cohomology of the general linear group

We show that the motivic Lie algebra of mixed Tate motives over $\mathbb{Z}$ embeds canonically into the unstable compactly-supported cohomology of locally symmetric spaces for $\mathrm{GL}_g(\mathbb{Z})$, and into the weight-zero compactly-supported cohomology of $\mathcal{A}_g$, the moduli space of principally polarized abelian varieties. Our construction passes through tropical geometry and graph complexes: compactly-supported analogues of the Borel classes pull back via the tropical Torelli map to canonical graph cocycles. The latter were recently identified with cocycles studied previously by Rossi and Willwacher. We combine their results with the theory of single-valued periods to conclude that the cocycles map to generators of the motivic Lie algebra. We also compute the canonical graph cocycles for all graphs with $\leq14$ edges.

math.AG↗

arXiv2026

Non-linear geometry of multiple zeta values

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.

math.NT↗

arXiv2026

Mellin transforms, transfinite diameter and rational approximations of integrals

We establish a higher-dimensional irrationality criterion for periods which are presented as Mellin integrals depending on many parameters. The criterion is stated as an upper bound on the multi-variate transfinite diameter of the image of the domain of integration under the Mellin arguments. Most of the paper is devoted to studying notions of transfinite diameter relative to very general multivariate Vandermonde matrices. As a proof of principle, we illustrate how this approach works with detailed computations in the case of a 5-parameter family of integrals for $\zeta(2)$ on $\mathcal{M}_{0,5}$, the moduli space of curves of genus 0 with 5 marked points. This yields a `higher-dimensional' proof of the irrationality of $\zeta(2)$, based on an upper bound for a certain kind of transfinite diameter associated to $\mathcal{M}_{0,5}$.

math.NT↗

arXiv2026

Multivariable Vandermonde determinants, amalgams of matrices and Specht modules

Using results of Fayers on the structure of Specht modules, we prove two different formulae for the determinant of matrices which are obtained by amalgamating the entries of two smaller matrices. In particular, this gives formulae for multivariable Vandermonde determinants as a sum of completely factorising terms, each of which is a Vandermonde determinant in fewer variables. As an application, we deduce an elementary proof of the multiplicativity of the transfinite diameter for products of compact sets.

math.RT↗

arXiv2025

Single-valued periods of meromorphic modular forms and a motivic interpretation of the Gross-Zagier conjecture

A well-known conjecture of Gross and Zagier states that the values of the higher automorphic Green's function at pairs of points with complex multiplication in the upper half-plane are proportional to the logarithm of an algebraic number. It was recently settled in the case of congruence subgroups of the form $\Gamma_0(N)$ by analytic methods. In this paper we provide a geometric and motivic interpretation of the general conjecture, and show that it is a consequence of a standard conjecture in the theory of motives. In addition, we define a new class of matrix-valued higher Green's functions for both odd and even weight modular forms, and show that they are single-valued periods of a motive constructed from a suitable moduli stack of elliptic curves with marked points. The motive has the structure of a biextension involving symmetric powers of the motives of elliptic curves. This suggests a very general extension of the Gross-Zagier conjecture relating values of matrix-valued higher Green's functions at points which do not necessarily have complex multiplication to special values of $L$-functions. In particular, our motivic interpretation of the Gross-Zagier log-algebraicity conjecture enables us to give a completely geometric proof in level 1 and weight 4 by showing that the motive of the moduli stack $\mathcal{M}_{1,3}$ of elliptic curves with 3 marked points is mixed Tate. In the course of this paper we develop many new foundational results on: the theory of weak harmonic lifts, meromorphic modular forms, biextensions of modular motives, and their corresponding algebraic de Rham cohomology and single-valued periods, which may all be of independent interest.

math.NT↗

arXiv2025

Positive geometries and canonical forms via mixed Hodge theory

''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent progress in particle physics, where the corresponding canonical forms are interpreted as the integrands of scattering amplitudes. We recast these concepts in the language of mixed Hodge theory, and identify ''genus zero pairs'' of complex algebraic varieties as a natural and general framework for the study of positive geometries and their canonical forms. In this framework, we prove some basic properties of canonical forms which have previously been proved or conjectured in the literature. We give many examples and study in detail the case of arrangements of hyperplanes and convex polytopes.

math.AG↗

arXiv2024

Unstable cohomology of $\mathsf{GL}_{2n}(\mathbb{Z})$ and the odd commutative graph complex

We study a closed differential form on the symmetric space of positive definite matrices, which is defined using the Pfaffian and is $\mathsf{GL}_{2n}(\mathbb{Z})$ invariant up to a sign. It gives rise to an infinite family of unstable classes in the compactly-supported cohomology of the locally symmetric space for $\mathsf{GL}_{2n}(\mathbb{Z})$ with coefficients in the orientation bundle. Furthermore, by applying the Pfaffian forms to the dual Laplacian of graphs, and integrating them over the space of edge lengths, we construct an infinite family of cocycles for the odd commutative graph complex. By explicit computation, we show that the first such cocycle gives a non-trivial class in $H^{-6}(\mathsf{GC}_3)$.

math.AT↗

arXiv2024

Hopf algebras in the cohomology of $\mathcal{A}_g$, $\mathrm{GL}_n(\mathbb{Z})$, and $\mathrm{SL}_n(\mathbb{Z})$

We describe a bigraded cocommutative Hopf algebra structure on the weight zero compactly supported rational cohomology of the moduli space of principally polarized abelian varieties. By relating the primitives for the coproduct to graph cohomology, we deduce that $\dim H^{2g+k}_c(\mathcal{A}_g)$ grows at least exponentially with $g$ for $k = 0$ and for all but finitely many positive integers $k$. Our proof relies on a new result of independent interest; we use a filtered variant of the Waldhausen construction to show that Quillen's spectral sequence abutting to the cohomology of $BK(\mathbb{Z})$ is a spectral sequence of Hopf algebras. From the same construction, we also deduce that $\dim H^{\binom{n}{2} - n - k}(\mathrm{SL}_n(\mathbb{Z}))$ grows at least exponentially with $n$, for $k = -1$ and for all but finitely many non-negative integers $k$.

math.AG↗

arXiv2024

The wheel classes in the locally finite homology of $\mathrm{GL}_n(\mathbb{Z})$, canonical integrals and zeta values

We compute the canonical integrals associated to wheel graphs, and prove that they are proportional to odd zeta values. From this we deduce that wheel classes define explicit non-zero classes in: the locally finite homology of the general linear group $\GL_n(\ZZ)$ in both odd and even ranks, the homology of the moduli spaces of tropical curves, and the moduli space of tropical abelian varieties. We deduce the existence of a doubly infinite family of auxiliary classes in the even commutative graph complex.

math.NT↗

arXiv2023

Bordifications of the moduli spaces of tropical curves and abelian varieties, and unstable cohomology of $\mathrm{GL}_g(\mathbb{Z})$ and $\mathrm{SL}_g(\mathbb{Z})$

We construct bordifications of the moduli spaces of tropical curves and of tropical abelian varieties, and show that the tropical Torelli map extends to their bordifications. We prove that the classical bi-invariant differential forms studied by Cartan and others extend to these bordifications by studying their behaviour at infinity, and consequently deduce infinitely many new non-zero unstable cohomology classes in the cohomology of the general and special linear groups $\mathrm{GL}_g(\mathbb{Z})$ and $\mathrm{SL}_g(\mathbb{Z})$. In particular, we obtain a new and geometric proof of Borel's theorem on the stable cohomology of these groups. In addition, we completely determine the cohomology of the link of the moduli space of tropical abelian varieties within a certain range, and show that it contains the stable cohomology of the general linear group. In the process, we define new transcendental invariants associated to the minimal vectors of quadratic forms, and also show that a certain part of the cohomology of the general linear group $\mathrm{GL}_g(\mathbb{Z})$ admits the structure of a motive. In an appendix, we give an algebraic construction of the Borel-Serre compactification by embedding it in the real points of an iterated blow-up of a projective space along linear subspaces, which may have independent applications.

math.AG↗

arXiv2022

On cellular rational approximations to $\zeta(5)$

We analyse a certain family of cellular integrals, which are period integrals on the moduli space $\mathcal{M}_{0,8}$ of curves of genus zero with eight marked points, and give rise to simultaneous rational approximations to $\zeta(3)$ and $\zeta(5)$. By exploiting the action of a large symmetry group on these integrals, we construct an infinite $effective$ sequence of rational approximations $p/q$ to $\zeta(5)$ satisfying \[ 0<\bigg|\zeta(5)-\frac pq\bigg|<\frac1{q^{0.86}}. \]

math.NT↗

arXiv2022

Generalised graph Laplacians and canonical Feynman integrals with kinematics

To any graph with external half-edges and internal masses, we associate canonical integrals which depend non-trivially on particle masses and momenta, and are always finite. They are generalised Feynman integrals which satisfy graphical relations obtained from contracting edges in graphs, and a coproduct involving both ultra-violet and infra-red subgraphs. Their integrands are defined by evaluating bi-invariant forms which represent stable classes in the cohomology of the general linear group on a generalised graph Laplacian matrix which depends on the external kinematics of a graph.

math-ph↗

arXiv2021

Invariant Differential Forms on Complexes of Graphs and Feynman Integrals

We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential forms in question generate the stable real cohomology of the general linear group, as shown by Borel. By integrating such invariant forms over the space of metrics on a graph, we define canonical period integrals associated to graphs, which we prove are always finite and take the form of generalised Feynman integrals. Furthermore, canonical integrals can be used to detect the non-vanishing of homology classes in the commutative graph complex. This theory leads to insights about the structure of the cohomology of the commutative graph complex, and new connections between graph complexes, motivic Galois groups and quantum field theory.

math.AG↗

arXiv2020

A double integral of dlog forms which is not polylogarithmic

Feynman integrals are central to all calculations in perturbative Quantum Field Theory. They often give rise to iterated integrals of dlog-forms with algebraic arguments, which in many cases can be evaluated in terms of multiple polylogarithms. This has led to certain folklore beliefs in the community stating that all such integrals evaluate to polylogarithms. Here we discuss a concrete example of a double iterated integral of two dlog-forms that evaluates to a period of a cusp form. The motivic versions of these integrals are shown to be algebraically independent from all multiple polylogarithms evaluated at algebraic arguments. From a mathematical perspective, we study a mixed elliptic Hodge structure arising from a simple geometric configuration in $\mathbb{P}^2$, consisting of a modular plane elliptic curve and a set of lines which meet it at torsion points, which may provide an interesting worked example from the point of view of periods, extensions of motives, and L-functions.

hep-th↗

arXiv2019

Single-valued integration and superstring amplitudes in genus zero

We study open and closed string amplitudes at tree-level in string perturbation theory using the methods of single-valued integration which were developed in the prequel to this paper. Using dihedral coordinates on the moduli spaces of curves of genus zero with marked points, we define a canonical regularisation of both open and closed string perturbation amplitudes at tree level, and deduce that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values (resp. single-valued multiple zeta values). Furthermore, we prove the existence of a motivic Laurent expansion whose image under the period map is the open string expansion, and whose image under the single-valued period map is the closed string expansion. This proves the recent conjecture of Stieberger that closed string amplitudes are the single-valued projections of (motivic lifts of) open string amplitudes. Finally, applying a variant of the single-valued formalism for cohomology with coefficients yields the KLT formula expressing closed string amplitudes as quadratic expressions in open string amplitudes.

math.NT↗

arXiv2019

Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions

The goal of this paper is to raise the possibility that there exists a meaningful theory of `motives' associated to certain hypergeometric integrals, viewed as functions of their parameters. It goes beyond the classical theory of motives, but should be compatible with it. Such a theory would explain a recent and surprising conjecture arising in the context of scattering amplitudes for a motivic Galois group action on Gauss' ${}_2F_1$ hypergeometric function, which we prove in this paper by direct means. More generally, we consider Lauricella hypergeometric functions and show on the one hand how the coefficients in their Taylor expansions can be promoted, via the theory of motivic fundamental groups, to motivic multiple polylogarithms. The latter are periods of ordinary motives and admit an action of the usual motivic Galois group, which we call the `local' action. On the other hand, we define lifts of the full Lauricella functions as matrix coefficients in a Tannakian category of twisted cohomology, which inherit an action of the corresponding Tannaka group. We call this the `global' action. We prove that these two actions, local and global, are compatible with each other, even though they are defined in completely different ways. The main technical tool is to prove that metabelian quotients of generalised Drinfeld associators on the punctured Riemann sphere are hypergeometric functions. We also study single-valued versions of these hypergeometric functions, which may be of independent interest.

math.AG↗

arXiv2019

From the Deligne-Ihara conjecture to Multiple Modular Values

This is the write-up of a talk given in honour of Prof. Ihara's 80th Birthday conference in Kyoto in 2018. After briefly reviewing the work of Ihara on the projective line minus 3 points, I outline the main ideas in the proof of the Deligne-Ihara conjecture and provide an update on recent progress in this area and raise some new questions. The second part of the talk outlines the main features of the corresponding theory in genus one, i.e., on the moduli stack of elliptic curves. It contains many concrete and elementary examples to illustrate the main features of the theory and to make it more widely accessible. In particular, the case of non-abelian cocycles associated to a pair of modular forms of level one is discussed in some detail. It also contains some new conjectures which have not previously appeared in print.

math.AG↗

arXiv2019

A multi-variable version of the completed Riemann zeta function and other $L$-functions

We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure on its residues. The special case of two variables can be written as a partial Mellin transform of a real analytic Eisenstein series, which enables us to relate its values at pairs of positive even points to periods of (simple extensions of symmetric powers of the cohomology of) the CM elliptic curve corresponding to the Gaussian integers. In general, the totally even values of these functions are related to new quantities which we call multiple quadratic sums. More generally, we cautiously define multiple-variable versions of motivic $L$-functions and ask whether there is a relation between their special values and periods of general mixed motives. We show that all periods of mixed Tate motives over the integers, and all periods of motivic fundamental groups (or relative completions) of modular groups, are indeed special values of the multiple motivic $L$-values defined here.

math.NT↗