N=2 supersymmetric spin foams in three dimensions

TL;DR

This paper develops a spin foam model for N=2 supergravity in three dimensions, addressing representation issues and identifying fermion propagation, contributing to quantum supergravity theories.

Contribution

It constructs a consistent state-sum model for N=2 supergravity using restricted osp(2|2) representations, enabling analysis of fermions and BPS states.

Findings

The state-sum is a topological invariant.

Restricted representations form a closed tensor category.

Fermions propagate with positive and negative charges.

Abstract

We construct the spin foam model for N=2 supergravity in three dimensions. Classically, it is a BF theory with gauge algebra osp(2|2). This algebra has representations which are not completely reducible. This complicates the procedure when building a state sum. Fortunately, one can and should excise these representations. We show that the restricted subset of representations form a subcategory closed under tensor product. The resulting state-sum is once again a topological invariant. Furthermore, within this framework one can identify positively and negatively charged fermions propagating on the spin foam. These results on osp(2|2) representations and intertwiners apply more generally to spin network states for N=2 loop quantum supergravity (in 3+1 dimensions) where it allows to define a notion of BPS states.