I report on work in progress with Erik Panzer (University of Oxford) on applications of Picard-Lefschetz theory to the study of Feynman integrals in the (Schwinger-)parametric representation. These are parameter-dependent integrals over relative cycles in the complement of a subspace arrangement in a compact complex manifold, defining multivalued analytic functions of the parameters. To understand their analytic structure is an old (and difficult) problem of great interest both in theoretical and "practical" particle physics. Picard-Lefschetz theory allows to study it using methods from differential topology.
Although known since the 60s (by the work of Pham et al, based on earlier work by Leray, Thom and Whitney), this approach has not received much attention from physics; the main reason being that general Feynman integrals are just outside of the scope of the mathematical theory. In particular, the parametric representation, which has uncovered many fascinating connections between particle physics and algebraic geometry and number theory, does not fit into the framework of classical Picard-Lefschetz theory.
In this talk I will give a quick overview of the problem, then discuss a relative version of Picard-Lefschetz theory. Although it is slightly more technical than its non-relative counterpart, we will see that these technicalities turn out to be features rather than obstructions. I finish by discussing how these features can be exploited to study the analytic structure of Feynman integrals (and other problems).
Although known since the 60s (by the work of Pham et al, based on earlier work by Leray, Thom and Whitney), this approach has not received much attention from physics; the main reason being that general Feynman integrals are just outside of the scope of the mathematical theory. In particular, the parametric representation, which has uncovered many fascinating connections between particle physics and algebraic geometry and number theory, does not fit into the framework of classical Picard-Lefschetz theory.
In this talk I will give a quick overview of the problem, then discuss a relative version of Picard-Lefschetz theory. Although it is slightly more technical than its non-relative counterpart, we will see that these technicalities turn out to be features rather than obstructions. I finish by discussing how these features can be exploited to study the analytic structure of Feynman integrals (and other problems).
- Arrangør: Centre for Quantum Mathematics
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