Gauge/string theory duality and integrable systems

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In quantum field theory (QFT) in four dimensions calculations can often only be done by perturbation theory w.r.t the coupling constant. In the case of electromagnetism this is the electric charge which is a very small parameter so that the leading terms of the perturbative expansion yield a good approximation of experimental data. In other realistic models - e.g.quantum chromodynamics (QCD) at low energy - the coupling constant is not small and thus the series expansion is hardly helpful; a non-perturbative treatment would be needed.On the other hand, many two-dimensional systems are integrable, i.e. they can be solved exactly. The theory of integrable systems is mathematically very rich, but so far it has proven hard to import the relevant concepts into four-dimensional physics.This project is aimed at the development of non-perturbative methods in four-dimensional QFT. 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 of physical theories; e.g. the perturbation theory of the model reproduces a part of that of QCD itself. For instance, the calculation of probability amplitudes for particle scattering in QCD may be partially obtained from the N=4 model, while the remainder is the structurally simpler part.Due to its high symmetry, the N=4 SYM theory has a number of interesting properties; most prominently the strong coupling limit of the model is described by a string theory in a certain curved space.For an important set of quantities in this gauge/string theory system, an integrable model has recently been constructed which is powerful enough to interpolate between the opposite regimes of strong and weak coupling. We wish to further develop this picture, and thereby to learn simultaneously about QFT at intermediate and strong coupling, the quantisation of string theory on curved spaces, and integrable systems related to four-dimensional physics.