arXiv subjects

Marko Berghoff

Publications and source records attributed to Marko Berghoff.

13 recordsLinked to original sources

arXiv2024

Operad of posets 101: The Wix\'arika posets

We study classes of objects whose combinatorics are closely related to those of posets. The framework of operads and operad algebras allows us to make this relationship precise and provides tools for a deeper understanding of their combinatorial structure. In this note, we present a nontrivial example of a suboperad of the operad of posets, called Wix\'arika posets, together with its associated algebras. This example is sufficiently rich to exhibit key structural features of the theory, while remaining accessible and avoiding unnecessary technicalities.

math.CO↗

arXiv2023

Graph complexes from the geometric viewpoint

These notes loosely follow an introductory course on graph complexes, held at Humboldt-Universit\"at zu Berlin in summer 23. Instead of simply typing up my lecture notes I decided to give here an overview over (parts of) the topic (lecture notes can be found on my homepage). We introduce the associative, commutative and Lie graph complexes, and moduli spaces of metric graphs, then discuss how the commutative and Lie graph complexes can be interpreted as cellular chain complexes associated to certain (pairs of) subspaces of the latter, both for ``even" and ``odd" orientations. We explain why this does not work for the associative complex and how to adjust the space of graphs to deal with this case. Along the way we highlight how algebraic properties on one side translate into geometric statements on the other.

math.AT↗

arXiv2022

Hierarchies in relative Picard-Lefschetz theory

We prove a relative version of the Picard-Lefschetz theorem, describing the variation of relative homology groups $H_d(Y_t \setminus A_t,B_t\setminus A_t)$ in the fibers of a smooth fiber bundle $Y \to T$ of complex manifolds with $A\cup B \subset Y$ transverse. From this we derive the vanishing of certain iterated variations, a system of constraints dubbed "hierarchy". As applications, we rederive the known analytic structure of Aomoto polylogarithms and massive one loop Feynman integrals. Moreover, we introduce the "simple type" to prove hierarchy constraints in degenerate cases where the Picard-Lefschetz formula does not apply, e.g. the massless triangle or the ice cream cone Feynman diagram. We compare our findings with a "classical" hierarchy of iterated variations (from 1960's $S$-matrix theory) and show how our setup not only explains, but also refines the latter. In order to do so, we need to further resolve the geometry of Feynman motives: We boldly blow up what no one has blown up before.

math-ph↗

arXiv2022

Schwinger, ltd: Loop-tree duality in the parametric representation

We derive a variant of the loop-tree duality for Feynman integrals in the Schwinger parametric representation. This is achieved by decomposing the integration domain into a disjoint union of cells, one for each spanning tree of the graph under consideration. Each of these cells is the total space of a fiber bundle with contractible fibers over a cube. Loop-tree duality emerges then as the result of first decomposing the integration domain, then integrating along the fibers of each fiber bundle. As a byproduct we obtain a new proof that the moduli space of graphs is homotopy equivalent to its spine. In addition, we outline a potential application to Kontsevich's graph (co-)homology.

hep-th↗

arXiv2022

Virtual posets, shuffle algebras and associators

We provide a method to construct new associators out of Drinfel'd's KZ associator. We obtain two analytic families of associators whose coefficients we can describe explicitly by a generalization of multiple zeta values. The two families contain two different paths that deform the Drinfel'd KZ associator into the trivial associator 1. We show that both paths are injective, that is, all of the associators parametrized by them are different. Our construction is based on the observation that one can recover multiple polylogarithms as generating functions of order polynomials of certain formally constructed posets.

math.QA↗

arXiv2021

An algebra over the operad of posets and structural binomial identities

We study generating functions of strict and non-strict order polynomials of series-parallel posets, called order series. These order series are closely related to Ehrhart series and h*-polynomials of the associated order polytopes. We explain how they can be understood as algebras over a certain operad of posets. Our main results are based on the fact that the order series of chains form a basis in the space of order series. This allows to reduce the search space of an algorithm that finds for a given power series f, if possible, a poset P such that f is the generating function of the order polynomial of P. In terms of Ehrhart theory of order polytopes, the coordinates with respect to this basis describe the number of (internal) simplices in the canonical triangulation of the order polytope of P. Furthermore, we derive a new proof of the reciprocity theorem of Stanley. As an application, we find new identities for binomial coefficients and for finite partitions that allow for empty sets, and we describe properties of the negative hypergeometric distribution.

math.CO↗

arXiv2020

On the homology of independence complexes

The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double complex with trivial homology. Filtering this double complex in the right direction induces a spectral sequence that converges to zero and contains on its first page the homology of the independence complexes of $G$ and various subgraphs of $G$, obtained by removing independent sets and their neighborhoods from $G$. It is shown that this spectral sequence may be used to study the homology of $\mathrm{Ind}(G)$. Furthermore, a careful investigation of the sequence's first page exhibits a relation between the cardinality of maximal independent sets in $G$ and the vanishing of certain homology groups of the independence complexes of some subgraphs of $G$. This relation is shown to hold for all paths and cyclic graphs.

math.AT↗

arXiv2020

Graph complexes and Feynman rules

We investigate Feynman graphs and their Feynman rules from the viewpoint of graph complexes. We focus on graph homology and on the appearance of cubical complexes when either reducing internal edges or when removing them by putting them on the massshell.

hep-th↗

arXiv2020

Complexes of marked graphs in gauge theory

We review the gauge and ghost cyle graph complexes as defined by Kreimer, Sars and van Suijlekom in "Quantization of gauge fields, graph polynomials and graph homology" and compute their cohomology. These complexes are generated by labelings on the edges or cycles of graphs and the differentials act by exchanging these labels. We show that both cases are instances of a more general construction of double complexes associated to graphs. Furthermore, we describe a universal model for these kind of complexes which allows to treat all of them in a unified way.

math-ph↗

arXiv2019

Moduli spaces of colored graphs

We introduce moduli spaces of colored graphs, defined as spaces of non-degenerate metrics on certain families of edge-colored graphs. Apart from fixing the rank and number of legs these families are determined by various conditions on the coloring of their graphs. The motivation for this is to study Feynman integrals in quantum field theory using the combinatorial structure of these moduli spaces. Here a family of graphs is specified by the allowed Feynman diagrams in a particular quantum field theory such as (massive) scalar fields or quantum electrodynamics. The resulting spaces are cell complexes with a rich and interesting combinatorial structure. We treat some examples in detail and discuss their topological properties, connectivity and homology groups.

math.AT↗

arXiv2018

Feynman amplitudes on moduli spaces of graphs

This article introduces moduli spaces of coloured graphs on which Feynman amplitudes can be viewed as 'discrete' volume densities. The basic idea behind this construction is that these moduli spaces decompose into disjoint unions of open cells on which parametric Feynman integrals are defined in a natural way. Renormalisation of an amplitude translates then into the task of assigning to every cell a finite volume such that boundary relations between neighboring cells are respected. It is shown that this can be organized systematically using a type of Borel-Serre compactification of these moduli spaces. The key point is that in each compactified cell the newly added boundary components have a combinatorial description that resembles the forest structure of subdivergences of the corresponding Feynman diagram.

math-ph↗

arXiv2016

Wonderful Compactifications in Quantum Field Theory

This article reviews the use of DeConcini-Procesi wonderful models in renormalization of ultraviolet divergences in position space as introduced by Bergbauer, Brunetti and Kreimer. In contrast to the exposition there we employ a slightly different approach; instead of the subspaces in the arrangement of divergent loci, we use the poset of divergent subgraphs as the main tool to describe the whole renormalization process. This is based on an article by Feichtner, where wonderful models were studied from a purely combinatorial viewpoint. The main motivation for this approach is the fact that both, renormalization and the model construction, are governed by the combinatorics of this poset. Not only simplifies this the exposition considerably, but also allows to study the renormalization operators in more detail. Moreover, we explore the renormalization group in this setting by studying how the renormalized distributions behave under a change of renormalization points.

math-ph↗

arXiv2012

S^1-equivariant Morse cohomology

We construct a deformed Morse complex computing the equivariant cohomology of a manifold M endowed with a smooth S^1-action. The deformation of the coboundary operator is given by counting gradient flow lines of a Morse function f that are allowed to "jump" along orbits of the S^1-action for finite time intervalls.

math.AT↗