Non-perturbative Group field theory from combinatorial Dyson-Schwinger equations and their algebraic structure II

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One of the greatest theoretical challenges in fundamental physics is to combine general relativity and quantum field theory to a quantum theory of gravity. Quantum field theories on non-commutative geometry have recently been found to be solvable non-perturbatively in a matrix-theory representation. Group field theory is a generalization of such matrix field theory to higher rank and is a candidate for a quantum theory of gravity. It is therefore an important question to what extent non-perturbative solutions can be obtained in group field theory as well. In this research project we address this challenge making use of 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 will 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 since the limit to continuum space-time coincides with the limit to critical loci in such a theory of quantum gravity.