Related papers: Iterated integrals related to Feynman integrals as…
A systematic algorithm for obtaining recurrence relations for dimensionally regularized Feynman integrals w.r.t. the space-time dimension $d$ is proposed. The relation between $d$ and $d-2$ dimensional integrals is given in terms of a…
Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic…
The Eynard-Orantin invariants of a plane curve are multilinear differentials on the curve. For a particular class of genus zero plane curves these invariants can be equivalently expressed in terms of simpler expressions given by polynomials…
We present a general method for analytically factorizing the n-fold form factor integrals $f^{(n)}_{N,N}(t)$ for the correlation functions of the Ising model on the diagonal in terms of the hypergeometric functions…
Using lower bounds for linear forms in elliptic logarithms we determine the integral points of the modular curve associated to the normalizer of a non-split Cartan group of level 11. As an application we obtain a new solution of the class…
A space of modular iterated integrals sits inside the space of real analytic modular forms. We present a theorem for producing length three modular iterated integrals which are not simply combinations of real analytic Eisenstein series;…
We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with…
We consider the problem of obtaining higher order in regularization parameter $\epsilon$ analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct…
The Feynman Path Integral is extended in order to capture all solutions of a quantum field theory. This is done via a choice of appropriate integration cycles, parametrized by M in SL(2,C), i.e., the space of allowed integration cycles is…
In this paper we construct parameterizations of elliptic curves over the rationals which have many consecutive integral multiples. Using these parameterizations, we perform searches in GMP and Magma to find curves with points of small…
We find that all Feynman integrals (FIs), having any number of loops, can be completely determined once linear relations between FIs are provided. Therefore, FIs computation is conceptually changed to a linear algebraic problem. Examples up…
The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$…
A genus one curve of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our…
In modern quantum field theory, one of the most important tasks is the calculation of loop integrals. Loop integrals appear when evaluating the Feynman diagrams with one or more loops by integrating over the internal momenta. Even though…
This talk summarizes recent developments in the evaluation of Feynman integrals using hyperlogarithms. We discuss extensions of the original method, new results that were obtained with this approach and point out current problems and future…
We revisit the idea of numerically integrating the differential form of Feynman integrals. With a novel approach for the treatment of branch cuts, we develop an integrator capable of evaluating a basis of master integrals in double and…
We consider elliptic curves defined by an equation of the form $y^2=x^3+f(t)$, where $f\in k[t]$ has coefficients in a perfect field $k$ of characteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain, under suitable…
It is by now well established that, by means of the integration by part identities, all the integrals occurring in the evaluation of a Feynman graph of given topology can be expressed in terms of a few independent master integrals. It is…
We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications of…
We introduce a type of graph integrals which are holomorphic analogs of configuration space integrals. We prove their (ultraviolet) finiteness by considering a compactification of the moduli space of graphs with metrics, and study their…