Higher-point functions and integrability

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In quantum field theory (QFT) in four dimensions, calculations can often only be done by perturbation theory in the coupling constant. In the case of quantum electrodynamics, this is the electric charge. It is a very small parameter, and hence the leading terms of the perturbative expansion yield a good approximation of experimental data. In other physical models - e.g. quantum chromodynamics (QCD) at low energy - the coupling is not small, and thus the series expansion is hardly helpful. A non-perturbative treatment is needed.On the other hand, many two-dimensional systems possessing a coupling constant are integrable, i.e. can be solved exactly at generic coupling. The theory of integrable systems is mathematically very rich. However, until recently it has proven hard to import the relevant concepts into four-dimensional physics.This agenda is aimed at the development of non-perturbative methods in four-dimensional QFT utilising integrability. We propose studying the maximally supersymmetric gauge theory in four dimensions (N=4 SYM). Although this model is not itself phenomenologically relevant, it shares some generic features with physical theories. For instance, the perturbation theory of the model reproduces a part of that of QCD. In particular, the most complicated part of the probability amplitudes for high energy particle scattering in QCD may be obtained from the N=4 model.Due to its high symmetry, N=4 SYM has a number of interesting properties. Most prominently, in the strong coupling limit it is described by a string theory in a certain curved space. This is the famous AdS/CFT duality conjecture, frequently called ``AdS/CFT correspondence''. The spectral problem of this gauge/string theory system, i.e. the calculation of the string energy levels or, equivalently, of the anomalous dimensions in field theory, is described by an integrable model that powerfully interpolates between the opposite regimes of strong and weak coupling.Current work in the field is concerned with integrable structures controlling higher-point objects, where the functional kinematical dependence plays an important role. The main incentive of the present proposal is to develop the idea of tiling n-point correlation functions by elementary patches fixed by symmetry. We aim at the success of integrability-based methods in this new arena, and ultimately once more at the regime of strong and even intermediate coupling.The project offers a fresh perspective on perturbative field theory. Potentially there will be spin-offs also in the field of integrable systems, because the quantities studied and the methods employed are new.