Related papers: Meromorphic modular forms and the three-loop equal…
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…
Regarding the resolution of singularities for the differential equations of Painlev\'e type, there are important differences between the second-order Painlev\'e equations and those of higher order. Unlike the second-order case, in higher…
We give sufficient conditions for three- or four-dimensional truncated Poincare-Dulac normal forms of resonance degree two to be meromorphically nonintegrable when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues…
In this paper we study generalizations of quadratic form Poincar\'e series, which naturally occur as outputs of theta lifts. Integrating against them yields evaluations of higher Green's functions. For this we require a new regularized…
We introduce a novel, systematic method for the complete symbolic reduction of multi-loop Feynman integrals, leveraging the power of generating functions. The differential equations governing these generating functions naturally yield…
In this manuscript, we elaborate on a procedure to derive $\epsilon$-factorised differential equations for multi-scale, multi-loop classes of Feynman integrals that evaluate to special functions beyond multiple polylogarithms. We…
This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define…
We perform a comprehensive study of a certain class of discrete symmetries of families of Feynman integrals, defined as affine changes of variables that map different sectors of the family into each other. We show that these transformations…
The general structure of infrared divergences in the scattering of massive particles is captured by the soft anomalous dimension matrix. The latter can be computed from a correlation function of multiple Wilson lines. The state-of-the-art…
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…
We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent expansion in the dimensional regularisation parameter $\varepsilon$. This is done by transforming the…
Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…
A module $M$ is called an automorphism-invariant module if every isomorphism between two essential submodules of $M$ extends to an automorphism of $M$. This paper introduces the notion of dual of such modules. We call a module $M$ to be a…
We observe a property of orthogonality of the Mellin-Barnes transformation of the triangle one-loop diagrams, which follows from our previous papers [JHEP {\bf 0808} (2008) 106, JHEP {\bf 1003} (2010) 051, JMP {\bf 51} (2010) 052304]. In…
Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or…
Let $G$ be a finite subgroup of the linear group of a finite-dimensional complex vector $V$, $B={\operatorname S}(V)$ be the symmetric algebra, ${\mathcal D}=\mathcal D^G_B$ the ring of $G$-invariant differential operators, and ${\mathcal…
In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)-automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and…
Multiloop algebras determined by $n$ commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain necessary and…
We investigate the analytic structure of functions defined by integrals with integrands singular on a finite union of quadrics. The main motivation comes from Feynman integrals which belong to this class. Using isotopy techniques we derive…
Negative dimensional integration is a step further dimensional regularization ideas. In this approach, based on the principle of analytic continuation, Feynman integrals are polynomial ones and for this reason very simple to handle,…