Scattering amplitudes are traditionally assembled from Feynman diagrams, intuitive graphical representations for different ways how the process in question can happen. In spite of their success in matching with precision tests, Feynman diagrams are highly inefficient from a conceptual point of view: Each Feynman diagram evaluated separately is more complicated than the complete amplitude because it obscures crucial symmetries of the final answer. During the last 30 years, various instances of hidden simplicity and rich mathematical structures have been discovered in scattering amplitudes, and most of them are inaccessible to the Feynman diagrammatic approach. It is desirable to compute amplitudes within a framework which manifests these symmetries.
String theory was born from the study of scattering amplitudes among hadrons. The latter arise from vibration modes of open strings, but it became clear that string theory inevitably incorporates closed string excitations which describe gravity. Enforcing a unified description of gravitational and (sub-)atomic interactions, string theory is one of the most promising candidates for a unifying quantum theory of gravity. And independent on that, it opens up a new perspective on scattering amplitudes in QFT and perturbative gravity.
Our research focuses on scattering amplitudes in string theory and aims to better understand their structures and organizing principles. In string amplitudes, Feynman diagrams of QFTs and perturbative gravity are replaced by so-called world-sheets, two-dimensional surfaces which encode possible histories of splitting and joining strings. Once the string length is set to zero in the associated mathematical expressions, one recovers a multitude of Feynman diagrams from a single world-sheet. Hence, the point particle limit of string theory tames the Feynman approach to scattering amplitudes and helps to bypass its spurious complications.
Consistency of string theory requires supersymmetry, a widely studied symmetry between matter and forces which gives rise to particularly simple scattering amplitudes. The so-called pure spinor formalism allows to address string amplitudes in a manifestly supersymmetric way and to thereby identify their simplifying features from the outset. This not only facilitates the computation of world-sheet diagrams but also inspires a language to describe scattering amplitudes in supersymmetric QFT. The pure spinor setup singles out a toolbox of variables which allows to construct supersymmetric amplitudes from first principles. This program led to strikingly compact representations at classical level, and the list of successful extensions to quantum corrections is steadily growing.
Apart from their lessons for the point particle limit, string amplitudes provide a fruitful link to number theory through their dependence on the string length. String corrections to QFT and perturbative gravity are accompanied by multiple zeta values, a class of numbers of wide interest among mathematicians. Advanced mathematical tools were found to govern the string corrections to genus zero world-sheets, such as the Hopf algebra structure of multiple zeta values or their generating series known as the Drinfeld associator. The quantum corrections set by worldsheets with holes promise a laboratory for further number theoretic structures. Hence, string amplitudes furnish a profitable interdisciplinary bridge to mathematics.