Arithmetic Geometry of Feynman Integrals

This will be a hybridized lecture course + seminar series on Feynman integrals.

The course will be taught in the Winter Semester of 2024-25 at the Max Planck Institute for Mathematics in the Sciences, Leipzig.

To audit the course online, please send me an email at kidambi[AT]duck[dot]com

Organizational details

Website: Click here

Course schedule

All lectures will take place at 9:15 AM German time. All lectures will take place in person at the Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig. Any deviation from the schedule will be announced.

You can find a calendar under Teaching.

Course Outline

The goal is to build enough background such that one can study the Broadhurst-Kreimer or Brown-Schnetz papers. Meaning that we will focus more on the appearance of multiple zeta zalues in Feynman integrals, periods and their arithmetic geometry.

However, the arithmetic geometric structure of Feynman integrals goes beyond MZVs, as is recently seen in the Banana graphs where periods of Calabi-Yau motives appear.

Lecture 1: 21.10.2024: Motivating Feynman integrals and their arithemtic structure
Lecture 2: 28.10.2024: Graph theory
Lecture 3: 4.11.2024: Homology of graphs, Symanzik polynomials, Feynman Graphs
11.11.2024: No Lectures
18.11.2024: No Lectures
Lecture 4: 25.11.2024: Parametrizations of Feynman integrals from Feynman Graphs, graph hypersurfaces
Lecture 5: 02.12.2024: Periods and Multiple Zeta Values (MZV’s)
Lecture 7: 09.12.2024: More on MZVs, Broadhurst-Kreimer conjecture, Zagier conjecture
Lecture 6: 16.12.2024: Singular, Betti and algebraic de Rham cohomology of smooth affine varieties
Lecture 8: 06.01.2025: Algebraic de Rham cohomology of smooth non-affine varieties, Relative cohomology, comparison isomorphism
Lecture 9: 13.01.2025: Proof of comparison isomorphism, GAGA theorem, periods revisited
Lecture 10: 20.01.2025: MZVs as periods, mixed Hodge structures and Mixed Tate Motives over \(\mathbb Z\)
Lecture 11: 27.01.2025: MZVs as periods, mixed Hodge structures and Mixed Tate Motives over \(\mathbb Z\)
Lecture 12: 03.02.2025: The Kontsevich conjecture and disproof (Belkale-Brosnan, Brown-Schnetz)

Literature

There is no literature source in particular that one should consult. The most comprehensive source is the book by Stefan Weinzierl, which is available on the arxiv: Feynman Integrals - Stefan Weinzierl. I will base the course roughly around expanding Section 1 of Feynman integrals and motives by Matilde Marcolli with a significant amount of arithmetic geometry and number theory.

If you are interested in learning how other special functions and period integrals appear in Feynman integrals, Feynman itegrals, toric geometry and mirror symmetry by Pierre Vanhove is a good resource.

Course Notes

Still being typeset!