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Diffeomorphism Invariant Gauge Theories, Asymptotic Safety and Geometry

Final Report Summary - DIGT (Diffeomorphism Invariant Gauge Theories, Asymptotic Safety and Geometry)

One of the main outcomes of this project is the development of the chiral pure connection formalism for GR, together with other related chiral formulations. General Relativity in this formalism possesses some exceptional properties such as boundedness of the Euclidean signature action as well as the notable simplicity of the perturbative expansion. These properties are absent in the usual metric formulation. Our results make the class of chiral formulations of GR to be arguably the most computationally efficient and useful formalisms for GR available. The formalism developed gives a new framework for tackling problems in classical and quantum General Relativity. The simplicity and efficiency of this formalism make it ideally poised to become a tool in a wide range of applications of GR. Our results establish a new branch in the subject of General Relativity.

The other main outcome of the project is the discovery of an unexpected link between gravity theories in four spacetime dimensions and Hitchin-type theories of three-forms in seven dimensions. This suggests a completely new perspective on 4D gravity, whose full implications are still to be uncovered. This link also achieves a solution to the problem we set to solve in the framework of the grant, which is to find a principle that would select one of the theories from the "deformations of GR" class. These four-dimensional gravity theories are easiest to formulate in the chiral pure connection formalism. There were known before the start of the project, but their understanding has improved greatly due to the work performed. All these theories become indistinguishable from GR at low energies, and the physics side of the project was guided by the idea that the "correct" gravity theory at high energies is one of them. The embedding into a seven-dimensional theory discovered by this project provides a principle for selecting a specific theory from this class, thus achieving one of the original goals of this project. Another related result is the discovery of the new link between Schwarz-type topological theories of three-forms in seven dimensions and the topological theories of BF-type in four dimensions.

Yet another main outcome is the discovery of the relevance of the Lie algebra of diffeomorphisms in the usual Yang-Mills theory in flat space, with the Drinfeld double of the Lie algebra of diffeomorphisms being responsible for the off-shell version of the Yang-Mills color-kinematics duality at four points. The discovery of the relevance of the Drinfeld double of the Lie algebra of diffeomorphisms in Yang-Mills
theory suggests a new way to approach problems related to the color-kinematics duality in Yang-Mills theory and gravity equals gauge theory squared relation.

Two meetings were supported by the grant, a workshop on "Geometry of the Graviton Scattering Amplitudes" in July 2015 at the University of Nottingham, and a programme on "Gravity, Twistors and Amplitudes" in June-July 2016 at the Isaac Newton Institute for Mathematical Sciences (Cambridge). All talks given at these events were recorded and are publicly available either on the PI's youtube channel or via the Isaac Newton Institute website.