Reading Seminar - Feynman Integrals and Moduli Spaces WS 2022/23

Organizer: Daniele Agostini.

Time: Friday, 9:00

Place: On Zoom. The link is here.

Announcements

• The last two meetings are moved to early 2023, date TBD. .
• The first session will be on Friday, November 4.

Content

The aim of this reading group is to understand the connection between Feynman integrals and amplitudes on one side and algebraic geometry, tropical geometry and D-modules on the other. These connections have appeared recently in different fields, such as enumerative geometry of real and complex curves, topology of moduli spaces of curves or numerical evaluations of Feynman integrals.

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References : At the beginning, we will be inspired by the paper:

• P. Tourkine, Tropical Amplitudes. Here all our actors make their appearance: amplitudes, Feynman integrals, Riemann surfaces, tropical curves and moduli spaces.
• There is a talk by Tourkine on this work here.

Depending on time, we might continue in different directions such as:

• F. Brown, Invariant Differential Forms on Complexes of Graphs and Feynman Integrals. Connections to the topology of the moduli space. Here is a video.
• J. Tevelev, Scattering amplitudes of stable curves. Connections to enumerative geometry.

Other useful references:

• D. Agostini, C. Fevola, A.-L. Sattelberger, S. Telen, Vector Spaces of Generalized Euler Integrals. An overview on Euler integrals, of which Feynman integrals are a subcase.
• E. Arbarello, M. Cornalba, P. A. Griffiths, Geometry of Algebraic Curves, Volume II. Very complete references on the moduli space of curves.
• L. Caporaso, Algebraic and tropical curves: comparing their moduli spaces. The two moduli spaces of curves that we care about.
• H. Markwig, Tropical curves and covers and their moduli spaces. Expository article on tropical curves, Feynman integrals make an appearance.
• D. Mumford, Tata Lectures on Theta, Volume I and Volume II. Very complete references on theta functions.
• S. Weinzierl, Feynman Integrals. Everything you wanted to know on the topic, and more.

Lectures and notes

1. 04.11.2022, Daniele: Introduction, Feynman diagrams, Feynman integrals. Notes.
2. 11.11.2022, Claudia and Dmitrii: Graph polynomials, trees and forests. Laplacian, deletion and contraction (Weinzierl, Chapter 3). Notes Claudia, Notes Dmitrii.
3. 18.11.2022, Javier: Riemann surfaces, stable curves and their moduli spaces. Examples in genus zero and one. Notes.
4. 25.11.2022, Lou-Jean: Tropical curves, linear systems and theta characteristics. Notes.
5. 02.12.2022, Daniele: Theta functions, abelian varieties and their moduli spaces. Jacobians, the Torelli map and the Schottky problem. Notes.
6. 09.12.2022, Victoria: Moduli spaces of tropical curves and tropical abelian varieties, the tropical Torelli map, and tropicalization. Notes.
7. 2023, Speaker: Classical and tropical prime form. Amplitudes.
8. 2023, Speaker: Amplitudes in the tropical limit.

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