The aim of the proposed research is to develop a new description of gravity in four spacetime dimensions. This will (i) serve as a new tool to investigate the conjecture that four-dimensional quantum gravity may be ultra-violet complete (asymptotically safe); (ii) provide new techniques for the problem of classification of geometric structures on four-manifolds. To this end we shall study a certain large class of diffeomorphism invariant SU(2) gauge theories. The low-energy physics of these theories is known to be indistinguishable from that of general relativity (GR). At high energies, they provide an interesting type of deformations of GR, with the key property that the number of propagating degrees of freedom is the same as in general relativity. To test the asymptotic safety conjecture we shall perform perturbative one-loop computations to determine how these theories are renormalized by quantum corrections and then study the resulting renormalization group flow. The same class of theories will also be used to solve some of fundamental conjectures about the geometric structures on four-manifolds. The most optimistic scenario results will prove the asymptotic safety in four-dimensional quantum gravity and explicitly describe the physics occurring around the ultra-violet fixed point. The impact of this on theoretical physics will be comparable to the impact of the 1973 discovery of asymptotic freedom on the high energy physics. The work on the proposal will involve some of the world leading scientists as collaborators and advisors. The project will be carried out in the School of Mathematical Sciences at the University of Nottingham, one of the major mathematics research centres in the UK. Regular long-term research visits to our group by leading scientists, as well as three meetings planned will establish the PI Nottingham research group as one of the leading European centres in the subject area.
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