Abstracts

Anna Pachol

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 mathematical framework behind it is the noncommutative differential graded algebra. After presenting the general framework, I will discuss some recent results on noncommutative Riemannian geometries in small dimensions, working over the field F_2 of 2 elements and with coordinate algebras up to dimension n<=3 (arXiv:1807.08492). We have found a rich moduli of examples for n=3 and top form degree 2, including 9 that are Ricci flat but not flat. The choice of the finite field in this framework proposes a new kind of 'discretisation scheme', which we called the 'digital geometry'.