Related papers: Algebraic geometry informs perturbative quantum fi…

Feynman diagrams constitute one of the essential ingredients for making precision predictions for collider experiments. Yet, while the simplest Feynman diagrams can be evaluated in terms of multiple polylogarithms -- whose properties as…

Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…

Complex geometry and supergeometry are closely entertwined in superstring perturbation theory, since perturbative superstring amplitudes are formulated in terms of supergeometry, and yet should reduce to integrals of holomorphic forms on…

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps in a class of commutative Von Neumann regular rings including Boolean rings and p-rings, that we have called CFG-rings. In those rings, the…

We propose a strategy to study the analytic structure of Feynman parameter integrals where singularities of the integrand consist of rational irreducible components. At the core of this strategy is the identification of a selected stratum…

We give a general procedure to construct algebro-geometric Feynman rules, that is, characters of the Connes-Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the…

The calculation of higher-order corrections in Quantum Field Theories is a challenging task. In particular, dealing with multiloop and multileg Feynman amplitudes leads to severe bottlenecks and a very fast scaling of the computational…

String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of…

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…

A Lie algebra is said to be quadratic if it admits a symmetric invariant and non-degenerated bilinear form. Semisimple algebras with the Killing form are examples of these algebras, while orthogonal subspaces provide abelian quadatric…

We employ the recently discovered Hopf algebra structure underlying perturbative Quantum Field Theory to derive iterated integral representations for Feynman diagrams. We give two applications: to massless Yukawa theory and quantum…

It was observed that hyperlogarithms provide a tool to carry out Feynman integrals. So far, this method has been applied successfully to finite single-scale processes. However, it can be employed in more general situations. We give examples…

The phenomenology for the deep spatial geometry of loop quantum gravity is discussed. In the context of a simple model of an atom of space, it is shown how purely combinatorial structures can affect observations. The angle operator is used…

After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry and mathematical physics. This allows us to define…

Certain integrable models are described by pairs (X,Y) of ADET Dynkin diagrams. At high energy these models are expected to have a conformally invariant limit. The S-matrix of the model determines algebraic equations, whose solutions are…

We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A…

To a simple graph we associate a so-called graph series, which can be viewed as the Hilbert--Poincar\'e series of a certain infinite jet scheme. We study new $q$-representations and examine modular properties of several examples including…

Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of observables and of a representation of A on a measure space over the space of generalized connections. This representation is…

In recent years, the intersection of algebra, geometry, and combinatorics with particle physics and cosmology has led to significant advances. Central to this progress is the twofold formulation of the study of particle interactions and…

One-loop amplitudes are to a large extent determined by their unitarity cuts in four dimensions. We show that the remaining rational terms can be obtained from the ultraviolet behaviour of the amplitude, and determine universal form factors…