Simplicity condition and boundary-bulk duality

Understand

In the first-order formulation, general relativity could be formally viewed as the topological $BF$ theory with a specific constraint, the Plebanski constraint.

• $BF$ theory is expected to be the classical limit of the Crane-Yetter~(CY) topological state sum.
• In the Euclidean case, the Plebanski constraint could be lifted in an elegant way to a quantum version in the CY state sum, called the simplicity condition.
• The constrained state sum is known as the Barrett-Crane~(BC) model.