Feynman Periods in Classical and Quantum Field Theory

• Julio PARRA-MARTINEZ (IHES)

Room: Salle Yvette Cauchois In this talk I will review the appearance of periods as Feynman integral associated to classical and quantum scattering. Using some specific examples, I will give a brief survey of the state of the art of the subject and describe in which physical quantities they arise.

Numerical computations of periods and monodromy representations

• Eric PICHON-PHARABOD (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany)

Room: Salle Yvette Cauchois The period matrix of a smooth complex projective variety encodes the isomorphism between its singular homology and its algebraic De Rham cohomology. Numerical approximations with sufficient precision of the entries of the period matrix allow to recover some algebraic invariants of the variety. Such approximations can be obtained from an effective description of the homology of the variety, which itself can be obtained from the monodromy representation associated to a generic fibration. I will describe these methods to several hundred digits, and showcase implementations and applications, in particular to computation of the Picard rank of certain K3 surfaces.

Thimble decomposition and Wall Crossing Structure for Physical Integrals

• Roberta ANGIUS (Institute for Theoretical Physics, University of Hamburg, Germany)

Room: Salle Yvette Cauchois A growing body of evidence suggests that the complexity of physical integrals is most naturally understood through geometry. Recent mathematical developments by Kontsevich and Soibelman [arXiv:2402.07343] have illuminated the role of exponential integrals as periods of twisted de Rham cocycles over Betti cycles, offering a structured approach to address this problem in a wide range of settings. In this talk, I will first introduce the key tools underlying this structure and then apply them to show how families of physically relevant integrals, ranging from holomorphic exponentials to logarithmic multivalued functions, can be reformulated within this language. For holomorphic exponents, I will present an explicit decomposition of a family of integrals into thimbles expansion together with a detailed analysis of the wall-crossing structure behind the analytic continuation of its relevant parameter. Finally, I will discuss the generalization to multivalued functions, which provides the appropriate framework for describing Feynman integrals in special representations. In this context, the thimble decomposition is expected to match the decomposition into Master Integrals, while the study of the wall-crossing structure yields a precise count of independent Master Integrals (or periods), circumventing complications arising from Stokes phenomena