Boundary fluctuations and a reduction entropy

The boundary Weyl anomalies live on a codimension-1 boundary, $\partial \mathcal{M}$. The entanglement entropy originates from infinite correlations on both sides of a codimension-2 surface, $\mathrm{\Sigma }$. 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 $\partial \mathcal{M}$ to fluctuate. We discuss how a reduction might be performed so that the reduction entropy reproduces the entanglement structure.

Figure 1

The codimension-2 surface $\mathrm{\Sigma }$ (red line) is thickened to define a codimension-1 boundary $\partial \mathcal{M}$ (tube), where the boundary metric is allowed to fluctuate. A reduction is performed by sending configurations evaluated on $\partial \mathcal{M}$ back to $\mathrm{\Sigma }$.