# Research Fields

**Our current understanding of the basic constituents of matter and interactions among them** is based on quantum field theory and gauge symmetries. All known forces in Nature are mediated by gauge fields. The gauge interactions are well under theoretical control in their weak coupling regime, where perturbative computations based on Feynman diagrams reach remarkable levels of precision. These include the weak and electromagnetic interactions, and strong interactions at asymptotically high energies. However, gauge theories such as quantum chromodynamics (QCD) are in general strongly coupled, and the full understanding of their dynamics requires non-perturbative methods.

**Exactly solvable, or integrable, models** play a distinguished role in physics, helping us to understand the behavior of strongly correlated, strongly coupled systems in regimes where other methods fail. They have many applications in quantum theory ranging from condensed-matter to high energy physics. For a long time, exact solvability was associated only with (1+1)-dimensional quantum field theories and statistical systems. Powerful methods of integrability provide tools to understand the dynamics of these systems even in the strongly-coupled, non-perturbative regime.

**The application of similar techniques directly to four dimensional gauge theories has been a long-standing dream, which has begun to be realised in a quite unexpected manner**. The major step in this direction was the discovery of the AdS/CFT (Anti-de Sitter / Conformal Field Theory) duality, which established a precise relationship between gauge fields and strings. It was then realized that the two-dimensional dynamics on the string worldsheet is in many cases integrable, paving the way for application of non-perturbative, integrability-based methods to four-dimensional gauge theories.

The best studied example of this sort concerns N = 4 super-Yang-Mills (SYM) theory in the limit of large numbers of colours (‘large-N’). Feynman diagrams that contribute in this limit have planar topology and it has long been expected that the large-N limit is described by some kind of string theory. The AdS/CFT correspondence is the first case in which the relationship to string theory has been made precise. Moreover, planar diagrams in N = 4 SYM can be summed exactly with the help of integrability. The spectrum of this theory is described by the Hamiltonian of an integrable spin chain, as was first established by explicit perturbative computations.

On the other side of the AdS/CFT duality, the string worldsheet is described by an integrable two-dimensional sigma-model. By using integrability it was possible to find a set of Bethe ansatz equations that diagonalize the exact S-matrix of elementary magnon excitations of the spin chain, or equivalently, the soliton spectrum of the sigma-model.

The EC funded GATIS network (2013-2016) has had major impact on the development of these topics, by advancing and exploiting novel integrability based techniques. For the first time ever, teams of the network were able to compute the scaling exponents in a 4-dimensional conformal gauge theory exactly, including all perturbative and non-perturbative corrections. They also developed open codes to make these results available for a wider scientific community. The breakthrough employs a new tool in the theory of integrable systems, the so-called quantum spectral curve, which has found applications in other research areas already, most notably in studies of the Hubbard model.

A second key success could be achieved in the context of scattering amplitudes where GATIS teams solved a central challenge of gauge theory that had been around for half a century: To compute amplitudes entirely in terms of the flux tube, i.e. from a 1-dimensional quantum system. This important development, which was based on the so-called Wilson loop OPE, has spurred numerous spin-offs. For example, it resulted in a complete and easy-to-use solution of planar scattering amplitudes involving up to six gluons in multi-Regge kinematics to arbitrary logarithmic order. In another direction, the key strategies of the Wilson loop OPE could also be extended to the case of correlation functions.

These and other developments are fueling intense exchange with neighboring scientific communities, in a healthy competition to advance the understanding of gauge theories, our most successful framework for the description of nature.