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Finding suitable diffeomorphism-invariant observables to probe gravity at the Planck scale is essential in quantum gravity. The Wilson loop of the 4-dimensional Christoffel connection is a potentially interesting ingredient for the construction of such an observable. We have investigated to what extent and what form of curvature information of the underlying spacetime may be extracted from Wilson loops through a Stokes’ theorem-like relation. We present an expression for the conservation of geometric flux as the quantity related to the gravitational Wilson loop. This expression is surface-independent and it holds for a certain class of manifolds with global symmetries.

Complements offer a separating device which proves useful for renormalisation purposes. A set and its set complement are disjoint, a vector space and its orthogonal complement have trivial intersection. Inspired by J. Pommersheim and S. Garoufalidis, we define a class of complement maps which give rise to a class of binary relations that generalise the disjointness of sets and the orthogonality of vector spaces. We discuss how these reflect locality in quantum field theory and how they can be used for renormalisation purposes. This talk is based on joint work with Pierre Clavier, Li Guo and Bin Zhang.

Causal Dynamical Triangulations (CDT) is a candidate theory for quantum gravity, formulated nonperturbatively as the scaling limit of a lattice theory in terms of triangulated spacetimes. An important feature of this approach is its elegant resolution of the problem of diffeomorphism symmetry in the full, background-free quantum theory. This has enabled the concrete computation of geometric observables in a highly nonperturbative, Planckian regime, an important step in putting quantum gravity on a quantitative footing, and understanding the structure of quantum spacetime. While the need to find quantum observables describing this regime is common to all approaches, CDT provides a concrete testing ground for implementation and measurements. In particular, a new notion of quantum Ricci curvature has opened a new window on the counterintuitive properties of quantum geometry.