Quantum deformation of two four-dimensional spin foam models

TL;DR

This paper develops q-deformed four-dimensional spin foam models, extending the Euclidean and Lorentzian EPRL models using quantum group theory, ensuring their convergence and analyzing their algebraic properties.

Contribution

It introduces the first q-deformed versions of the Euclidean and Lorentzian EPRL spin foam models based on quantum group representations.

Findings

Both models are proven to be convergent.

Quantum EPRL intertwiners are explicitly constructed.

The models exhibit well-defined braiding properties.

Abstract

We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the EPRL model. The q-deformed models are based on the representation theory of two copies of U_q(su(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent.