2-Group Representations for Spin Foams

TL;DR

This paper explores 2-group representations, specifically the Euclidean 2-group, to develop state sum models for 4d quantum topology, linking higher category theory with quantum gravity and field theory.

Contribution

It introduces the infinite-dimensional unitary representations of the Euclidean 2-group and constructs a new model with geometric and metric data for 4d quantum topology.

Findings

Developed a model based on 2-group representations with geometric content.

Connected the model to background-independent quantum field theory.

Provided explicit metric data on triangulation edges.

Abstract

Just as 3d state sum models, including 3d quantum gravity, can be built using categories of group representations, "2-categories of 2-group representations" may provide interesting state sum models for 4d quantum topology, if not quantum gravity. Here we focus on the "Euclidean 2-group", built from the rotation group SO(4) and its action on the group of translations of 4d Euclidean space. We explain its infinite-dimensional unitary representations, and construct a model based on the resulting representation 2-category. This model, with clear geometric content and explicit "metric data" on triangulation edges, shows up naturally in an attempt to write the amplitudes of ordinary quantum field theory in a background independent way.