Higher Gauge Theory and Discrete Geometry

Abstract

In four dimensions, gravity can be seen as a constrained topological model. This provides a natural way to construct quantum gravity models, since topological models are relatively straightforward to quantize. Difficulty arises in the implementation of the constraints at the quantum level. Different procedures have generated so-called spin foam models.

Following the dimensional/categorical ladder, the natural structure to quantize 4d topological models are 2-categories, augmenting the gauge group symmetries of the model into 2-group symmetries. One can study these models classically by examining their phase space. At the quantum level, one attempts to construct a partition function. As there are no local degrees of freedom in topological theories, it is convenient to characterize its phase space in terms of a discretization, providing insights to the quantum theory. A key question is understanding how these topological models defined in terms of 2-categories can be related to gravity.

A first hint that 2-categories are relevant to describe quantum gravity models comes when we introduce a cosmological constant in the theory. As we will recall, this can be done at the classical (discrete) level in a consistent manner, only if we use 2-group symmetries.

This thesis focuses on understanding the symmetry aspects, the different possible discretizations and the quantization of four dimensional topological theories for some skeletal 2-group symmetries. When we discuss the quantum aspects using some field theory techniques to generate the quantum amplitudes, we extend the construction to non-skeletal 2-groups.