Related papers: Integral of two-loop modular graph functions
For a given function from a set to itself, we can define a directed graph called the functional graph, where the vertices are the elements of the set, and the edges are all the pairs of inputs and outputs for the function. In this article…
Spectral functions at finite temperature and two-loop order are investigated, for a medium consisting of massless particles. We consider them in the timelike and spacelike domains, allowing the propagating particles to be any valid…
We develop a Fourier analysis for a generalization of the class of periodic functions, often referred to as $(\theta, T)$-periodic functions, and prove several properties and inequalities related to the Fourier transform, including a type…
In this work, we investigate the finite basis topologies of two-loop dimensionally regularized Feynman integrals in the `t Hooft-Veltman scheme in the Standard Model. We present a functionally distinct finite basis of Master Integrals which…
This paper is concerned with the problem of decomposing a higher order Lipschitz function on a closed Jordan curve $\Gamma$ into a sum of two polyanalytic functions in each open domain defined by $\Gamma$. Our basic tools are the Hardy…
The main goal of this paper is to study properties of the iterated integrals of modular forms in the upper halfplane, eventually multiplied by $z^{s-1}$, along geodesics connecting two cusps. This setting generalizes simultaneously the…
We relate one-loop scattering amplitudes of massless open- and closed-string states at the level of their low-energy expansion. The modular graph functions resulting from integration over closed-string punctures are observed to follow from…
We evaluated all two-loop conformal integrals of scalar half-BPS six-point functions in $\mathcal{N} = 4$ SYM restricted to a configuration where all points lie on a line. Moreover, we also computed some of these integrals in the…
It is known that one-loop Feynman integrals possess an algebraic structure encoding some of their analytic properties called the coaction, which can be written in terms of Feynman integrals and their cuts. This diagrammatic coaction, and…
We propose a practical recipe to compute the ${s}$-parametrized maps for systems with $SU(1,1)$ symmetry using a connection between the ${Q}$ and ${P} $ symbols through the action of an operator invariant under the group. The particular…
We present analytic expressions in terms of polylogarithmic functions for all three families of planar two-loop five-point Master Integrals with one off-shell leg. The calculation is based on the Simplified Differential Equations approach.…
We establish the Fourier inversion for the smooth vectors in ${\rm L}^2({\rm GL}_2, \omega)$ over a number field $\mathbf{F}$, using minimal knowledge from automorphic representation theory. We point out a possible way to establish Fourier…
We offer a Maple package {\tt Poincare\_Series} for calculating the Poincar\'e series for the algebras of invariants/covariants of binary forms, for the algebras of joint invariants/covariants of several binary forms, for the kernel of…
A 2-factor of a graph $G$ is a 2-regular spanning subgraph of $G$. We present a survey summarising results on the structure of 2-factors in regular graphs, as achieved by various researchers in recent years.
Our aim in this paper is to compute the Poincar\'{e} series of the derivation module of the projective closure of certain affine monomial curves.
Earlier, for an action of a finite group $G$ on a germ of an analytic variety, an equivariant $G$-Poincar\'e series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck…
In our previous work (https://doi.org/10.1002/mana.202000268, Math. Nachr., 2021), we proposed an upper bound of the logarithmic derivative of Selberg's zeta function for the modular groups in the critical strip. The present paper studies…
In this note, a general formula is proved. It expresses the integral on the line of the product of a function $f$ and a periodic function $g$ in terms of the Fourier transform of $f$ and the Fourier coefficients of $g$. This allows the…
We study SYM gauge theories living on ALE spaces. Using localization formulae we compute the prepotential (and its gravitational corrections) for SU(N) supersymmetric ${\cal N}=2, 2^*$ gauge theories on ALE spaces of the $A_n$ type.…
A systematic study of the scalar one-loop two-, three-, and four-point Feynman integrals is performed. We consider all cases of mass assignment and external invariants and derive closed expressions in arbitrary space-time dimension in terms…