Boundary fluctuations and a reduction entropy

The boundary Weyl anomalies live on a codimension-1 boundary, partial derivative M. The entanglement entropy originates from infinite correlations on both sides of a codimension-2 surface, Sigma. Motivated to have a further understanding of the boundary effects, we introduce a notion of reduction entropy, which, guided by thermodynamics, is a combination of the boundary effective action and the boundary stress tensor defined by allowing the metric on partial derivative M to fluctuate. We discuss how a reduction might be performed so that the reduction entropy reproduces the entanglement structure.