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# Algebra and Combinatorics Seminar: Andrew Harder, Lehigh University, The geometry of 2-loop Feynman integrals

## April 5, 2023 | 12:45 pm - 1:45 pm EDT

Speaker’s webpage: https://math.cas.lehigh.edu/andrew-harder

In particle physics, many quantities of interest are expressed in terms of Feynman integrals. These integrals are attached to combinatorial objects called Feynman graphs, and can be expressed as integrals over (infinite) domains inside the real plane. In examples, one often finds that Feynman integrals are equal to special values of functions that are of interest to algebraic and arithmetic geometers. For instance, multiple zeta functions, elliptic dilogarithms and the like. One explanation for this comes from the work of Bloch-Esnault-Kreimer, and subsequent work of Brown, which shows that Feynman integrals can be interpreted as periods of smooth algebraic varieties in the sense of Kontsevich and Zagier. In the literature, almost all examples that have been worked out from this perspective belong to a rather restricted class of graphs called “primitively divergent” graphs.

I will talk about recent work with Doran and Vanhove which studies an infinite class of (non-primitively divergent) graphs with first betti number equal to 2. We show that in this case, the Feynman integrals which appear are constructed from algebraic functions and periods of hyperelliptic curves.

I will talk about recent work with Doran and Vanhove which studies an infinite class of (non-primitively divergent) graphs with first betti number equal to 2. We show that in this case, the Feynman integrals which appear are constructed from algebraic functions and periods of hyperelliptic curves.