Fishnet Integrals from Integrability and Geometry

About

The computation of Feynman integrals still represents a bottle neck for precision calculations in quantum field theory. Recent progress goes hand in hand with the understanding of new function classes and their mathematical structure. In this talk we discuss the so-called fishnet integrals that represent an infinite family of scalar Feynman integrals featuring integrable structures. In particular we review the Yangian quantum group symmetry of these integrals which extends the conformal spacetime symmetry of the associated fishnet quantum field theories. We then focus on the specific case of fishnets in two spacetime dimensions and argue that these integrals compute the quantum volumes of Calabi-Yau varieties. Here the Yangian provides a convenient tool that yields the Picard-Fuchs differential equations for the Calabi-Yau periods, while the geometry dictates the particular linear combination of periods, which furnishes the Feynman integrals.