Great effort was devoted on identifying analogues of polylogarithms for general Riemann surfaces. This is the most mathematical component, and main cornerstone, of the original project, and it was undertaken in collaboration with Benjamin Enriquez, who works at the University of Strasbourg. We have worked on three main research lines. First of all, we wanted to explicitly develop the algebraic de Rham theory of the fundamental group of configuration spaces of curves, following ideas of Hain. We have succeeded in writing down general homotopy invariant iterated integrals of rational functions on one curve (previously known only up to length two), and we are now left with generalizing this to configuration spaces. An article about this should be written up in the near future. The second research line consisted in constructing a single-valued flat connection over the configuration space of genus-g Riemann surfaces. We have succeeded in our goal by modifying a multi-valued flat connection previously constructed by Enriquez. We uploaded to the Arxiv in October 2021 a preprint ("Construction of Maurer-Cartan elements over configuration spaces of curves") which contains such result. Combining these two research lines leads to explicitly construct higher-genus analogues of polylogarithms, which was the main expected mathematical milestone of this project. A third research line consisted in studying the associated space of functions, and we obtained spectacular results in the case of affine curves. More specifically, we have constructed in three different ways a natural candidate for the space of hyperlogarithms (i.e. multiple polylogarithms with all but one variable fixed) on a general punctured Riemann surface, studied its algebraic structure, and identified a basis for such function spaces, whose elements constitute higher-genus analogues of classical functions first considered by Poincaré. These results have already been written up, and should appear in a preprint at the end 2022. Another research direction, currently under investigation and crucially important for applying such results to the computation of string amplitudes, is the study of the dependence on the complex structure of the Riemann surface, which is known only at genus one. Several of the results described above were announced and explained at invited seminar talks (in Dijon, Durham, Montpellier, Oxford and Zurich), as well as through two events which were planned for this MSCA IF, jointly organised with Pierre Vanhove: the (online) seminar "Motives and periods integrals in quantum field theory and string theory", and the special session "Mathematical Physics of Gravity" of the AMS-EMS-SMF joint meeting held in Grenoble in July 2022.
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.
