Periodic Reporting for period 1 - HIPSAM (HIgher Polylogarithms and String AMplitudes)
Reporting period: 2020-09-01 to 2022-08-31
The main goal of this project is to develop the mathematical tools necessary to describe the perturbative expansion of string theory amplitudes, at least for low genus, re-interpreting and generalizing recent beautiful progress made at genus zero. Understanding the mathematical structure of perturbative string amplitudes would yield new information on string theory predictions for fundamental interactions, and point towards new directions in many fields of mathematics, such as mixed motives and moduli spaces of curves, opening new research lines completely inspired by physics. The techniques developed would also allow to attack similar nowadays intractable computations of amplitudes in quantum field theory, which can be compared with the experimental data produced by particle accelerators.
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.
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.
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.
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.
The main result obtained is the introduction of higher-genus analogues of polylogarithms, which is achieved in collaboration with Benjamin Enriquez, and which was announced as a main goal of this MSCA IF. We expect that this will have a big impact in the near future on the computation of scattering amplitudes, both in string theory and in quantum field theory, similarly to what happened with the introduction of elliptic analogues of polylogarithms, which is now the subject of yearly conferences within the amplitude community. Moreover, this should also have an impact in mathematics, as it gives the first explicit construction of periods of fundamental groups of curves beyond the classical periods of the curves.