Michael Borinsky
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View article: Tree-tubings and the combinatorics of resurgent Dyson–Schwinger equations
Tree-tubings and the combinatorics of resurgent Dyson–Schwinger equations Open
We give a novel combinatorial interpretation for the perturbative series solutions to a class of Dyson–Schwinger equations. We show how binary tubings of rooted trees with labels from an alphabet on the tubes, and where the labels satisfy …
View article: Tropicalized quantum field theory and global tropical sampling
Tropicalized quantum field theory and global tropical sampling Open
We explain how to tropicalize scalar quantum field theory and show that tropicalized massive scalar quantum field theory is exactly solvable. This exact solution manifests as a non-linear recursion equation fulfilled by the expansion coeff…
View article: Bivariate exponential integrals and edge-bicolored graphs
Bivariate exponential integrals and edge-bicolored graphs Open
We show that specific exponential bivariate integrals serve as generating functions of labeled edge-bicolored graphs. Based on this, we prove an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights assi…
View article: Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations
Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations Open
We give a novel combinatorial interpretation to the perturbative series solutions for a class of Dyson-Schwinger equations. We show how binary tubings of rooted trees with labels from an alphabet on the tubes, and where the labels satisfy …
View article: On the Euler characteristic of the commutative graph complex and the top weight cohomology of $\mathcal M_g$
On the Euler characteristic of the commutative graph complex and the top weight cohomology of $\mathcal M_g$ Open
We prove an asymptotic formula for the Euler characteristic of Kontsevich's commutative graph complex. This formula implies that the total amount of commutative graph homology grows super-exponentially with the rank and, via a theorem of C…
View article: Tropical Feynman integration in the physical region
Tropical Feynman integration in the physical region Open
The software feyntrop for direct numerical evaluation of Feynman integrals is presented. We focus on the underlying combinatorics and polytopal geometries facilitating these methods. Especially matroids, generalized permutohedra and normal…
View article: Flow Oriented Perturbation Theory
Flow Oriented Perturbation Theory Open
Flow Oriented Perturbation Theory (FOPT) is a novel approach to Feynman diagrams based on the coordinate (position) space description of Quantum Field Theories (QFT). FOPT offers interesting features regarding the computation of higher-loo…
View article: Tropical Feynman integration in the physical region
Tropical Feynman integration in the physical region Open
The software feyntrop for direct numerical evaluation of Feynman integrals is presented. We focus on the underlying combinatorics and polytopal geometries facilitating these methods. Especially matroids, generalized permutohedra and normal…
View article: Flow Oriented Perturbation Theory
Flow Oriented Perturbation Theory Open
Flow Oriented Perturbation Theory (FOPT) is a novel approach to Feynman diagrams based on the coordinate (position) space description of Quantum Field Theories (QFT). FOPT offers interesting features regarding the computation of higher-loo…
View article: The Euler characteristic of the moduli space of graphs
The Euler characteristic of the moduli space of graphs Open
The moduli space of rank n graphs, the outer automorphism group of the free group of rank n and Kontsevich's Lie graph complex have the same rational cohomology. We show that the associated Euler characteristic grows like −e−1/4(n/e)n/(nlo…
View article: Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group
Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group Open
We compute the weight 2 cohomology of the Feynman transforms of the cyclic (co)operads $\mathsf{BV}$ and $\mathsf{HyCom}$, and the top$-2$ weight cohomology of the Feynman transforms of $D\mathsf{BV}$ and $\mathsf{Grav}$. Using a result of…
View article: The ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs
The ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs Open
We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$ stabilizes fo…
View article: Tropical Monte Carlo quadrature for Feynman integrals
Tropical Monte Carlo quadrature for Feynman integrals Open
We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The me…
View article: The Euler characteristic of the moduli space of graphs
The Euler characteristic of the moduli space of graphs Open
The moduli space of rank $n$ graphs, the outer automorphism group of the free group of rank $n$ and Kontsevich's Lie graph complex have the same rational cohomology. We show that the associated Euler characteristic grows like $-e^{-1/4}\,(…
View article: Tropical Feynman integration in the Minkowski regime
Tropical Feynman integration in the Minkowski regime Open
We present a new computer program, feyntrop, which uses the tropical geometric approach to evaluate Feynman integrals numerically. In order to apply this approach in the physical regime, we introduce a new parametric representation of Feyn…
View article: Taming a resurgent ultra-violet renormalon
Taming a resurgent ultra-violet renormalon Open
Perturbative expansions in quantum field theory diverge for at least two reasons: the number of Feynman diagrams increases dramatically with the loop number and the process of renormalization may make the contribution of some diagrams larg…
View article: Flow-oriented perturbation theory
Flow-oriented perturbation theory Open
We introduce a new diagrammatic approach to perturbative quantum field theory, which we call flow-oriented perturbation theory (FOPT). Within it, Feynman graphs are replaced by strongly connected directed graphs (digraphs). FOPT is a coord…
View article: Taming a resurgent ultra-violet renormalon
Taming a resurgent ultra-violet renormalon Open
Perturbative expansions in quantum field theory diverge for at least two reasons: the number of Feynman diagrams increases dramatically with the loop number and the process of renormalization may make the contribution of some diagrams larg…
View article: Bayesian Integrals on Toric Varieties
Bayesian Integrals on Toric Varieties Open
We explore the positive geometry of statistical models in the setting of toric varieties. Our focus lies on models for discrete data that are parameterized in terms of Cox coordinates. We develop a geometric theory for computations in Baye…
View article: Computing Euler characteristics using quantum field theory
Computing Euler characteristics using quantum field theory Open
This paper explains how to use quantum field theory techniques to find formal power series that encode the virtual Euler characteristics of $\mathrm{Out}(F_n)$ and related graph complexes. Finding such power series was a necessary step in …
View article: Graphical functions in even dimensions
Graphical functions in even dimensions Open
Graphical functions are special position space Feynman integrals, which can be used to calculate Feynman periods and one- or two-scale processes at high loop orders. With graphical functions, renormalization constants have been calculated …