Discretization of 4d Poincar\'e BF theory: from groups to 2-groups

TL;DR

This paper explores the discretization of 4d Poincaré BF theory, relating continuum fields to discrete variables, and examines how boundary terms influence the symmetry structure and quantum state selection.

Contribution

It establishes a connection between continuum fields and discrete variables in BFCG theory, revealing complex dependencies and the impact of boundary terms on discretized symmetry structures.

Findings

Discretized variables depend on multiple continuum fields.

Relation between continuum fields and discrete variables is more complex than in BF theory.

Boundary terms influence the symmetry structure and quantum states.

Abstract

We study the discretization of a Poincar\'e/Euclidean BF theory. Upon the addition of a boundary term, this theory is equivalent to the BFCG theory defined in terms of the Poincar\'e/Euclidean 2-group. At an intermediate step in the discretization, we note that there are multiple options for how to proceed. One option brings us back to recovering the discrete variables and phase space of BF theory. Another option allows us to rediscover the phase space related to the G-networks given in [2]. Indeed, our main result is that we are now able to relate the continuum fields with the discrete variables in [2]. This relation is important to determine how to implement the simplicity constraints to recover gravity using the BFCG action. In fact, we show that such relation is not as simple as in the BF discretization: the discretized variables on the triangles actually depend on several of the continuum fields instead of solely the continuum B-field. We also compare and contrast the discretized BF and BFCG models as pairs of dual 2-groups. This work highlights (again) how the choice of boundary term influences the resulting symmetry structure of the \textit{discretized theory} -- and hence ultimately the choice of quantum states.