Non Perturbative Quantum Field Theory
Quantum field theory provides an elegant and well-defined theoretical description of fundamental interactions. According to the Standard Model of elementary particle physics, strong, weak and electro-magnetic interactions are described by gauge field theories. It is a well-established fact that field theory is the language of theoretical particle physics at energies below the Plank scale.
At weak coupling, field theoretical calculations can be performed in Feynman perturbation theory. Unfortunately, this perturbative expansion is only suited to weak coupling. It is well-known that many interesting phenomena in quantum field theory are nonperturbative in nature, i.e. cannot be understood in perturbation theory. The known analytical non-perturbative methods are semi-classical in nature. They amount to evaluating the path integral by expanding the fields around non-trivial classical solutions -- instantons. It generally turns out that such an expansion is again valid only at weak coupling. However it does give contributions to the path integral which are non-perturbative, i.e. non-analytic functions of the coupling constant.
Remarkably, the last few years have seen a quantum leap in our understanding of strongly-coupled supersymmetric gauge theories through direct analytic non-perturbative calculations. The important feature of these results is that they describe the gauge dynamics in the strong-coupling regime and are valid in the domain where the standard perturbative and semi-classical methods fail.
The idea is to isolate a special class of Green's functions, which are protected by supersymmetric non-renormalization theorems. Then the results of non-perturbative semi-classical calculations, intrinsically valid at weak coupling, can be legitimately extrapolated to strong coupling. The goal is to develop and perform explicit semi-classical calculations in gauge theory around multi-instantons and/or monopoles, then to extrapolate the results to the strong coupling domain using appropriate non-renormalization theorems.