The key interests of our group are the symmetries and integrable structures arising in different contexts of quantum field theory. In the following we give a brief overview of how these interests are embedded into the landscape of theoretical high energy physics.
Elementary Particles and Forces
Elementary particle physics aims at understanding the most fundamental building blocks of our world. Physicists have found a number of different particles which are believed to be elementary. These particles interact via four different forces: the gravitational, the electromagnetic, the weak, and the strong force. In the context of quantum theory, also the forces are understood in terms of the exchange of force particles: the graviton, the photon, the W- and Z-boson, and the gluons. While the last century came with great insights into the fundamental laws of physics, the underlying mathematical framework and explicit calculations still pose major challenges.
Quantum Field Theory
Three of the four fundamental forces of nature are described by the so-called Standard Model of particle physics. This particular quantum field theory model captures all essential features of the weak, strong and electromagnetic force and makes predictions for scattering experiments at particle colliders. Quantum field theory was developed during the last century and unifies the theory of special relativity with quantum mechanics. An open, and perhaps the biggest open problem in elementary particle physics is to include gravity into a consistent quantum field theory. On large scales all observations on the gravitational force are very well described by Enstein's theory of general relativity.
Scattering Amplitudes
The computation of scattering amplitudes within the framework of quantum field theory allows to make predictions for collisions at particle colliders like the Large Hardron Collider (LHC) at CERN close to Geneva. On the other hand, these scattering amplitudes are generically hard to compute and fascinating objects on their own with a rich mathematical structure. This motivates to study their properties in more generality and beyond theories like the Standard Model of particle physics.
Gauge/Gravity Duality
The most cited scientific paper in theoretical high energy physics conjectures the duality between string theory on a ten-dimensional spacetime background and a four-dimensional (gauge) quantum field theory on its boundary. Remarkably, it connects gravity, which is included into the string theory and typically understood via Einstein's theory of general relativity, with forces well described by gauge quantum field theory. Moreover, the gauge/gravity or AdS/CFT correspondence relates regimes of the gauge theory that are hard to access (strong coupling), to regimes of the string theory that are easily captured (weak coupling) and vice versa. In a certain limit, which consists of taking an infinite number of color charges for the quantum field theory particles, this duality becomes particularly beautiful due to the emergence of integrable structures.
Integrability
Theoretical models with a large amount of symmetry are ubiquitous in physics and often key to developing efficient methods for complex problems. If the number of symmetries surpasses a critical threshold, a system is called integrable with a prime example being the Kepler problem of planetary motion. While integrability typically comes with a rich spectrum of mathematical methods, it is often hard to identify the underlying symmetries. Examples of integrable models range from propagating water waves to the above mentioned models for elementary particle physics. Often integrable structures go along with sophisticated solution techniques. In the case of the above gauge/gravity duality, integrability makes the goal of solving a four-dimensional quantum field theory seem reachable.
Black Hole Dynamics
The observation of gravitational waves in 2015, followed by the Nobel prize in 2016, has led to an increased interest in efficient techniques for calculating the dynamics of interacting black holes that emit these waves. As it turns out, the quantum field theory toolbox developed for the computation of scattering amplitudes is extremely useful for this endeavour and has already contributed to the cutting edge predictions. Along with our interest in the symmetries of graviton scattering, we have recently entered this terrain, which connects quantum field theory and gravitational wave physics. In particular, we studied the interactions of three black holes coupled by Einstein gravity. Notably, the gravitational three-body problem is in general chaotic and exact solutions only exist for finetuned parameter values. The two-body problem on the other hand is generically integrable in the above sense (at least at leading orders in the bodies' velocity) and thus completely solvable.