The entropic boundary law in BF theory

We compute the entropy of a closed bounded region of space for pure 3d Riemannian gravity formulated as a topological BF theory for the gauge group SU(2)SU(2) and show its holographic behavior. More precisely, we consider a fixed graph embedded in space and study the flat connection spin network state without and with particle-like topological defects. We regularize and compute exactly the entanglement for a bipartite splitting of the graph and show it scales at leading order with the number of vertices on the boundary (or equivalently with the number of loops crossing the boundary). More generally these results apply to BF theory with any compact gauge group in any space–time dimension.