HIPSAM · HIgher Polylogarithms and String AMplitudes

HIgher Polylogarithms and String AMplitudes

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.

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