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Francis Brown

Publications and source records attributed to Francis Brown.

50 records · Page 3Linked to original sources
arXiv2013

Proof of the zig-zag conjecture

A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the amplitudes of the zig-zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms. The zig-zag graphs therefore provide the only infinite family of primitive graphs in $ϕ^4_4$ theory (in fact, in any renormalisable quantum field theory in four dimensions) whose amplitudes are now known.

math.NT↗
arXiv2013

Framings for graph hypersurfaces

We present a method for computing the framing on the cohomology of graph hypersurfaces defined by the Feynman differential form. This answers a question of Bloch, Esnault and Kreimer in the affirmative for an infinite class of graphs for which the framings are Tate motives. Applying this method to the modular graphs of Brown and Schnetz, we find that the Feynman differential form is not of Tate type in general. This finally disproves a folklore conjecture stating that the periods of Feynman integrals of primitive graphs in phi^4 theory factorise through a category of mixed Tate motives.

math.AG↗
arXiv2013

Dedekind Zeta motives for totally real fields

Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $π^{n[k:\Q]}ζ^*_k(1-n)$, where $ζ^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculate the groups $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$. In particular, the special value of the Dedekind zeta function is a determinant of volumes of geodesic hyperbolic simplices defined over $k$.

math.AG↗
arXiv2012

Decomposing Feynman rules

We exhibit in the simple example of the Dunce's cap Feynman graph the structure of parametric renormalization on the level of integrands, and exhibit also the decomposition $Φ^R=Φ_{\mathrm{fin}}^{-1}(Θ_0)\star Φ^R_{\mathrm{\makebox{1-s}}}(S/S_0)\starΦ_{\mathrm{fin}}(Θ)$ into angles and scales of renormalized Feynman rules.

hep-th↗
arXiv2012

Properties of c_2 invariants of Feynman graphs

The c_2 invariant of a Feynman graph is an arithmetic invariant which detects many properties of the corresponding Feynman integral. In this paper, we define the c_2 invariant in momentum space and prove that it equals the c_2 invariant in parametric space for overall log-divergent graphs. Then we show that the c_2 invariant of a graph vanishes whenever it contains subdivergences. Finally, we investigate how the c_2 invariant relates to identities such as the four-term relation in knot theory.

math.AG↗
arXiv2011

Angles, scales and parametric renormalization

We decompose renormalized Feynman rules according to the scale and angle dependence of amplitudes. We use parametric representations such that the resulting amplitudes can be studied in algebraic geometry.

hep-th↗
arXiv2011

A K3 in phi4

Inspired by Feynman integral computations in quantum field theory, Kontsevich conjectured in 1997 that the number of points of graph hypersurfaces over a finite field $\F_q$ is a (quasi-) polynomial in $q$. Stembridge verified this for all graphs with $\leq12$ edges, but in 2003 Belkale and Brosnan showed that the counting functions are of general type for large graphs. In this paper we give a sufficient combinatorial criterion for a graph to have polynomial point-counts, and construct some explicit counter-examples to Kontsevich's conjecture which are in $ϕ^4$ theory. Their counting functions are given modulo $pq^2$ ($q=p^n$) by a modular form arising from a certain singular K3 surface.

math.AG↗
arXiv2011

On the decomposition of motivic multiple zeta values

We review motivic aspects of multiple zeta values, and as an application, we give an exact-numerical algorithm to decompose any (motivic) multiple zeta value of given weight into a chosen basis up to that weight.

math.NT↗
arXiv2011

Mixed Tate motives over $\Z$

We prove that the category of mixed Tate motives over $\Z$ is spanned by the motivic fundamental group of $\Pro^1$ minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a $\Q$-linear combination of $ζ(n_1,..., n_r)$ where $n_i\in \{2,3\}$.

math.AG↗
arXiv2009

Spanning forest polynomials and the transcendental weight of Feynman graphs

We give combinatorial criteria for predicting the transcendental weight of Feynman integrals of certain graphs in $ϕ^4$ theory. By studying spanning forest polynomials, we obtain operations on graphs which are weight-preserving, and a list of subgraphs which induce a drop in the transcendental weight.

math-ph↗
arXiv2009

Inversion of series and the cohomology of the moduli spaces $\mathcal{M}^δ_{0,n}$

For $n\geq 3$, let $\mathcal{M}_{0,n}$ denote the moduli space of genus 0 curves with $n$ marked points, and $\overline{\mathcal{M}}_{0,n}$ its smooth compactification. A theorem due to Ginzburg, Kapranov and Getzler states that the inverse of the exponential generating series for the Poincaré polynomial of $H^{\bullet}(\mathcal{M}_{0,n})$ is given by the corresponding series for $H^{\bullet}(\overline{\mathcal{M}}_{0,n})$. In this paper, we prove that the inverse of the ordinary generating series for the Poincaré polynomial of $H^{\bullet}(\mathcal{M}_{0,n})$ is given by the corresponding series for $H^{\bullet}(\mathcal{M}^δ_{0,n})$, where $\mathcal{M}_{0,n}\subset \mathcal{M}^δ_{0,n} \subset \overline{\mathcal{M}}_{0,n}$ is a certain smooth affine scheme.

math.AG↗
arXiv2009

The algebra of cell-zeta values

In this paper, we introduce cell-forms on $\mathcal{M}_{0,n}$, which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space $\mathcal{M}_{0,n}(\mathbb{R})$. We show that the cell-forms generate the top-dimensional cohomology group of $\mathcal{M}_{0,n}$, so that there is a natural duality between cells and cell-forms. In the heart of the paper, we determine an explicit basis for the subspace of differential forms which converge along a given cell $X$. The elements of this basis are called insertion forms, their integrals over $X$ are real numbers, called cell-zeta values, which generate a $\mathbb{Q}$-algebra called the cell-zeta algebra. By a result of F. Brown, the cell-zeta algebra is equal to the algebra of multizeta values. The cell-zeta values satisfy a family of simple quadratic relations coming from the geometry of moduli spaces, which leads to a natural definition of a formal version of the cell-zeta algebra, conjecturally isomorphic to the formal multizeta algebra defined by the much-studied double shuffle relations.

math.NT↗
arXiv2008

The massless higher-loop two-point function

We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph $G$ to evaluate to multiple zeta values. The criterion depends only on the topology of $G$, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with $6^\mathrm{th}$ roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph $K_{3,4}$ at one edge.

math.AG↗