Integrability in Quantum Field Theories on Non-commutative Geometries
In this project we investigate possible integrable structures in quantum field theories on non-commutative geometries. They are characterized by a recursive system of equations which should be solved by rational functions. The goal is to determine the generating function of the equation system and relate its rational coefficients with intersection numbers of tautological characteristic classes on suitable modul spaces. Details
Group field theory is a generalization of matrix field theory to higher rank and is a candidate for a quantum theory of gravity. Is it possible to generalize also the non-perturbative solutions as recently found by the mathematical physics group for matrix field theory? In this research project we address this challenge using the algebraic structure of renormalization on the level of Dyson-Schwinger equations. Quantum symmetries related both to the tensorial structure as well as the gauge invariance of the theory allow to simplify these equations. In this way we want to find under which conditions group field theory can be solved non-perturbatively and derive solutions. Control over the non-perturbative regime is an open issue of huge physical interest, in particular for the limit to continuum space-time in such a theory of quantum gravity.