Topological Entanglement Entropy and the Coordinate Transformation

The entanglement entropies depend on how the system is bipartitioned. However, they remain invariant under a coordinate transformation when the bipartition also undergoes the same transformation. In the context of a topological quantum field theory (TQFT), these coordinate transformations reduce to representations of the mapping class group of the manifold on the Hilbert space. We employ this invariance property of the Rényi entropies under coordinate transformations in the study of TQFTs in (2+1) dimensions on various types of manifolds. Notably, for a torus, the equality obtained in the calculations using different bases under coordinate transformations yields a Verlinde-like formula. Furthermore, we can extend our analysis to compute entropies for more complex configurations and bipartitions within TQFTs.