New geometry in classical and quantum gravity
In his thesis, Kevin van Helden studies variants on the theory of general relativity. In order to do this in a mathematically rigorous way, he introduces the theories of pseudo-Riemannian geometry, Cartan geometry and quantum Riemmannian geometry. Next, Van Helden proves that the same concepts from differential geometry in those different theories, such as a metric, connection, torsion, curvature an a action, are equivalent.
After that, Van Helden uses Cartan geometry, Spencer cohomology in particular, to describe the cokernel of the Spencer differential for Galilean and Carrollian p-branes and to give geometric criteria for the different cases for which the intrinsic torsion of a given spacetime reside in one to the subrepresentations. He subsequently repeats those results in the language of pseudo-Riemannian geometry (with indices) and gives a number of examples of Galilean limits of general relativity, for which it turns out that the intrinsic torsion can reside in those subrepresentations. Van Helden also does this for Newton-Cartan geometry and string Newton-Cartan geometry, obtaining comparable results.
Furthermore, Van Helden introduces quantum Riemannian geometry to study Euclidean discrete models. He gives the equations which a *-compatible quantum Levi-Civita connection of such a model must satisfy, and he gives a solution for metrics that are invariant in transversal directions. At last, Van Helden classifies 2-term L∞-algebras in terms of a Lie algebra, a representation of that Lie algebra, a cohomology class of the corresponding Lie algebra cohomology and a vector space.