dr Anna Pachoł, Queen Mary University of London

Digital quantum geometries

Noncommutative geometry, as the generalised notion of geometry, allows us to model the quantum gravity effects in an effective description without full knowledge of quantum gravity itself. On a curved space one must use the methods of Riemannian geometry - but in their quantum version, including quantum differentials, quantum metrics and quantum connections. The brief introduction to the general framework involving noncommutative differential graded algebra and construction of quantum Riemannian geometry elements will be provided. This framework has been applied to classification of all possible noncommutative Riemannian geometries in small dimensions (including finding explicit forms for quantum Levi-Civita connections and Riemann, Ricci and Einstein tensors), working over the field F_2 of 2 elements and with coordinate algebras up to dimension n<=3. We have found a rich moduli of examples for n=3 and top form degree 2 (providing a landscape of all reasonable up to 2D quantum geometries), including many which are not flat. Their coordinate algebras are commutative but their differentials are not. The choice of the finite field in this framework proposes a new kind of 'discretisation scheme', which we called the 'digital geometry'.