Intersection Numbers: from Electrodynamics to Quantum Field Theory

by Mastrolia Pierpaolo (Padova)

Thursday 13 Jul 2023, 14:00 → 15:00 Europe/Berlin

HS2 (Physikdepartment)

I elaborate on Feynman integrals as elements of a vector space, whose algebraic and analytic properties have a deep geometric origin, which can be investigated by relating Stoke's theorem with Morse theory and de Rham theory to the integration of differential forms. Deriving the relations they fulfill, such as integration-by-parts identities, differential and difference equations, as well as quadratic relations, is therefore equivalent to the problem of decomposing a generic vector in a vector basis, where the "intersection number" is introduced to define a scalar product for differential forms. This novel formalism is found to be applicable to all integral functions admitting a Euler-Mellin integral representation, as well as to differential operators related to generalised hypergeometric systems. Beside its applications to Feynman integrals and GKZ systems, I show how the evaluation of matrix elements in Quantum Mechanics, Green-functions and tau-functions in QFT, and correlators in Cosmology can be remarkably addressed in the same way. With this seminar, I hope to offer a new point of view on the intertwinement between physics, geometry, and statistics, which exploits the analogy between fluxes, period integrals, and statistical moments, and which could suggest a universal, quantitative approach to be applied to several scientific disciplines.