A major research focus of the group “Mathematical Physics of Space, Time and Matter” is on a currently intensively-studied model that has been termed by some the ”hydrogen atom of the 21st century”. The basic idea and goal is to construct a mathematically exact solution of an, admittedly idealized, quantum field theory of the general type as occurs in the description of the forces between our universe’s elementary particles, with the notable exception of the gravitational force.

Yang-Mills gauge theory is named for its inventors, Chen Ning Yang and Robert Mills. The word gauge refers to the fact that at the heart of these theories lies a certain built-in redundancy in its mathematical description hard to eliminate, while apparently necessary in order to properly record and understand the rules of the game. It is a bit like deciding on proper time zones for the earth: We require them in order to be able to communicate, their need is ultimately due to the spherical nature of Earth as well as its rotation around its axis, but the precise location of the zones is completely man-made. The idealized system at the focus of our group is called N=4 super Yang-Mills gauge theory, where “super” does not mean super interesting (even though this is certainly true!), but rather refers to a beautiful symmetry not yet discovered in nature, which is called supersymmetry. It stipulates that in addition to our standard continuous (“bosonic”) spacetime dimensions, certain hidden discrete (“fermionic”) dimensions exist. The number N=4 refers to the fact that this model has not just one but as many as four such curious symmetries. In fact, this is the maximum number that mathematical consistency allows for a quantum field theory in four spacetime dimensions.

So in this sense, the N=4 gauge model is the most beautiful and simple Yang-Mills theory one can come up with, even though it certainly does not directly appear in nature. It is also a deeply mysterious model, as it has become clear in recent years that it possesses further hidden symmetries as well as seemingly contradictory, alternative descriptions, which promise to allow for a complete solution of the model, at least for certain quantities and in certain limits.

One such amazing, alternative description was hypothesized in 1997 by Juan Maldacena (IAS, Princeton). He suggested that our N=4 model can also be described by a system of vibrating chords moving and wiggling in ten spacetime dimensions, a string theory. Not approximately, but exactly. Confusingly, in this description gravitational forces are included, while there is no trace of them in the gauge theory, as we have already mentioned above. So it appears that the N=4 Yang-Mills theory is somehow hiding the gravitational force. This explains much of the fascination the model has for today’s theoretical physicist: It promises to give insight into all forces between elementary particles, including the quantum mechanically elusive gravitational attraction.

A detailed qualitative and quantitative understanding of this – at first sight rather crazy sounding - marvellous duality between a gauge and a string theory is not easy to gain, however. Which is where the above-mentioned further hidden symmetries come in. They are of a type first discovered by the physicist Hans Bethe in the early 1930s, just before his forced emigration to the United States. They lead to the so-called complete integrability of the model. This means that it is possible to write down a system of equations that encode various quantities of interest in the model in an exact fashion.

A simpler analog of such a secret symmetry allowing for a mathematically exact treatment is the so-called Laplace-Runge-Lenz vector, which, while unbeknownst to him, allowed Isaac Newton to find the exact solution for Johannes Kepler’s classical two-body problem of planetary motion, and Erwin Schrodinger and Wolfgang Pauli to determine the exact solution of the quantum mechanical hydrogen atom. This analogy is the chief reason why the N=4 gauge theory has been called the hydrogen atom of the 21st century. However, while physicists are convinced that the analog of the Laplace-Runge-Lenz vector and the corresponding exact solution of the model also exist in the case of gauge theory, finding it requires prolonged and intense work. Our research is meant to be an important stepping stone in this direction.