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Quantum field theory is the framework describing the known elementary particles and their interactions. Though very successful as a tool for calculating the quantities observed in experiment, mathematically the theory is incomplete in many ways. In turn, this has led to a very fruitful interaction between physics and mathematics as quantum field theory motivates interesting new mathematical questions together with heuristic ideas for their solution as well as mathematics providing a rigorous understanding why the field-theoric toolbox works. Combinatorially non-local field theory is a particularly interesting example of this interaction. Such non-local interactions give rise to a perturbative expansion in graphs encoding topological manifolds. This might be the crucial additional structure to define perturbative field theory in a rigorous way, but these specific combinatorics also relate the theory to random geometries and possibly quantum gravity.
In this lecture I will give a systematic introduction to field theories with combinatorially non-local interactions from their diagrammatic combinatorics and perturbative renormalization of amplitudes to non-perturbative effects and properties of their renormalization group flow.
Topics:
- Large-N expansions in local QFT
- Matrix fields from non-commutativity
- Generalization to tensor fields
- Feynman graphs of non-local fields
- Perturbative renormalization
- Renormalizable tensorial theories
- Renormalization Hopf algebra
- Dyson-Schwinger equations
- Renormalization group flow
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