My recent research has been focused on the development of Shape Dynamics, a canonical metric theory with refoliation invariance replaced by spatial conformal symmetry. Shape Dynamics can be shown to be equivalent to classical General Relativity (work with Henrique Gomes and Sean Gryp). This equivalence can be shown using a linking gauge theory, which admits both General Relativity and Shape Dynamics as partial gauge fixings (work with Henrique Gomes). Linking gauge theories are similar but not equivalent to parent gauge theories used in string theory dualities. It turns out that the large volume behavior of Shape Dynamics is universal, leading to a classical correspondence between large volume gravity and asymptotic conformal theories (work with Henrique Gomes, Sean Gryb and Flavio Mercati). To explore Shape Dynamics in detail, we considered gravity in 2+1 dimensions, where we where able to perform a full metric Dirac quantization of Shape Dynamics and found evidence of a quantum gravity-CFT correspondence (work with Timothy Budd). My latest conjecture about linking gauge theories is that they may guide the way towards fixed points of the renormalization group, because they have enhanced gauge symmetry, thus satisfy more Ward identities.
There are several ongoing projects involving Shape Dynamics: (1) It turns out that coupling to the standard model is possible. (2) There exist several perturbative constructions of the Shape Dynamics Hamiltonian. (3) Shape Dynamics can be cast in Ashtekar-variables and some of the loop quantization programme can be carried out explicitly.
Further topics in my recent work where the relation of Loop Quantum Cosmology and Loop Quantum Gravity (work with Johannes Brunnemann), application of the functional renormalization group to matrix models (work with Alessandro Sfondrini and ongoing collaboration with Astrid Eichhorn and Ilya Vilensky) as well as re-interpretation of the Connes-Chameseddine model using the functional renormalization group (ongoing collaboration with Christoph Stephan).