Scattering amplitudes, renormalization and integrability
The IMPRS will try to deepen the mathematical analysis of QFTs in the directions of scattering amplitudes, renormalization and integrability. In the context of integrability, particular emphasis is put on the study of exact solutions in certain non-abelian QFTs, most notably the N=4 maximally supersymmetric Yang-Mills model. The latter plays a central role in better appreciating the connections between string theory and quantum gravity, especially in the context of the celebrated AdS/CFT correspondence. The integrability of the N=4 model was largely explored and developed at AEI and HU and has led to exact results for previously uncalculable quantities. The methodology is interdisciplinary, with strong connections to condensed matter topics such as integrable spin chains and cutting-edge mathematical topics.
The study of exactly solvable models has led to completely new insights into the integrable structure of Yangian invariant scattering amplitudes. Scattering amplitudes are the key observables linking QFT with collider experiments. In a variety of QFTs and perturbative approaches to supergravity, scattering amplitudes were recently found to exhibit rich symmetries and beautiful mathematical structures. Most of these unexpected features go far beyond standard textbook prescriptions based on Feynman diagrams, but use modern on-shell methods as well as superspace techniques in various dimensions. String theory has proven to be a fruitful tool to reveal the hidden simplicity of scattering amplitudes in its point particle limit, in particular the striking connection between amplitudes of gauge theories and perturbative gravity. Moreover, string corrections to field theories provide a laboratory to encounter number theoretic aspects of field theory amplitudes such as multiple zeta values, polylogarithms and their Hopf algebra structure within a simpler context.
The treatment of locally interacting quantum fields in four space-time dimensions requires the use of renormalization techniques in order to render physical observables finite. The mathematical structure of renormalization together with the arithmetic properties of the periods that appear in the computation of renormalization group functions will be another key focus in this research area. It leads to manifold contact with algebraic and arithmetic geometry.