Maïté Dupuis, Florian Girelli, Oleksandra Hrytseniak, Wolfgang Wieland

Four-dimensional gravity admits many equivalent formulations - metric, Einstein-Cartan, teleparallel, McDowell-Mansouri, among others - each offering distinct advantages, particularly, in view of quantization. We propose a new formulation based on Weyl spinor-valued 1-forms, ultimately encoding the frame-field data. Starting from a topological field theory with a global $\mathrm{SL}(2,\mathbb{C})$ symmetry, we show that promoting this symmetry to a local gauge symmetry leads to the emergence of gravity. We analyze the covariant phase space of this theory, its symmetries and charge structure and explore the role of admissible corner terms together with their impact on boundary charges and their algebra. We study several extensions of this framework, including the incorporation of a cosmological constant and a novel $ G \rightarrow 0 $ scaling limit obtained from this model. The presence of the frame field already at the topological level allows point particles to be coupled uniformly in both the topological and gravitational theories. We perform a detailed Hamiltonian analysis of the theory and clarify the implementation of the reality conditions. We argue that this formulation provides structural features that make it particularly well suited for both discretization and quantization.