## Periodic Reporting for period 1 - HIPSAM (HIgher Polylogarithms and String AMplitudes)

Période du rapport: 2020-09-01 au 2022-08-31

The main technical novelty of the project is the introduction of analogues of polylogarithms on higher-genus Riemann surfaces. Polylogarithms are important special functions which appear in several areas of mathematics, given by iterated integrals over configuration spaces of points on a Riemann sphere. Enriquez, Levin, Racinet, Brown and others introduced similar functions for genus-one Riemann surfaces, leading to the recent theory of elliptic polylogarithms, which found spectacular applications in high-energy physics. The next goal in this research area is to go beyond genus one, and its importance for this project stems from the expectation that genus-g polylogarithms are the mathematical tool needed to describe genus-g string amplitudes, an observation which has proved to be extremely useful at low genus.

Another important aspect of the project is to clarify the relation between closed string amplitudes and the newborn mathematical theory of single-valued periods, which would yield a deeper understanding of the relations between closed and open string amplitudes, and ultimately between gauge theories and gravity.

Conclusions of the action : we have achieved to construct a generalisation of polylogarithms to higher-genus Riemann surfaces, and we have characterised the space of functions that they generate. This is an important result in mathematics but also, potentially, in high-energy physics. Such construction is not yet suited to be applied to string amplitudes of genus higher than one, as one needs a more explicit formulation highlighting the dependence on the complex structure of the surface, which is currently under investigation. As for low-genus string amplitudes, and their relation with single-valued periods, we have clarified several aspects of such relations at genus-zero, and the analogous problem at genus-one is currently under investigation.

At the same time, we worked on low-genus string amplitudes and their relation with single-valued periods, with some variation with respect to the research lines which were originally planned. As a main result, together with Pierre Vanhove, and building on a previous unpublished joint work, we wrote an article ("Single-valued hyperlogarithms, correlation functions and closed string amplitudes"), which will soon be published by Advances in Theoretical and Mathematical Physics, where we have provided new interpretations of the relations between closed string theory amplitudes at genus zero and single-valued periods. For example, we have deduced the celebrated KLT formula by identifying closed string integrals with special values of single-valued correlation functions in two dimensional conformal field theory, and by obtaining their conformal block decomposition. Moreover, we have written the asymptotic expansion coefficients as multiple integrals over the complex plane of special functions known as single-valued hyperlogarithms, and used this fact to demonstrate that the asymptotic expansion coefficients belong to the ring of single-valued multiple zeta values.