Mathematical Physics and String Theory

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(16.05.) Kronecker forms at higher genera: Schottky parametrizations for higher polylogarithms

Johannes Brödel (ETH Zürich)

16.05.2024 at 16:15

Integrating rational functions on Riemann surfaces leads to polylogarithms, which have evolved into a canonical language for the expression of scattering amplitudes in quantum field theory and string theory. Polylogarithms on a genus-zero Riemann surface are very well understood. For Riemann surfaces of genus one various flavours of so-called elliptic polylogarithms are known, which can be represented as iterated integrals over integration kernels, which in turn are generated by the so-called Kronecker form.
In this talk I will discuss the so-called Schottky parametrization of Riemann surfaces, which allows the generalization of the Kronecker form to arbitrary genus. I will construct those higher-genus Kronecker function and show that its associated connection matches the connection of Enriquez. Using integration kernels generated by a genus-two Kronecker form, I will construct hyperelliptic polylogarithms and point out the stunning advantages of the Schottky parametrization for their numerical evaluation.