Entanglement entropy is a useful tool which plays an important role in modern physics. The geometrical nature of entanglement entropy explains why it finds so many applications in various fields of physics, from black holes and holography to integrable models and quantum computers.

For conformal field theories, the entanglement entropy plays a special and important role since the logarithmic terms in the entropy are related to the conformal anomalies. Recently there has been some progress in understanding the conformal anomalies in the case where the spacetime is not infinite but has some boundaries. It is interesting that in the presence of boundaries the integrated anomaly is non-vanishing in odd spacetime dimensions, the relevant contribution being produced by the boundary terms only.

The boundary phenomenon in entanglement entropy is more general and is not restricted only to conformal field theories. Yet, the explicit calculations for arbitrary boundaries and surfaces are technically complicated, if even possible. Therefore, it is instructive to first analyze the problem in some simple cases, where the spacetime is flat and the boundary is composed by a collection of planes.

The first part of the talk provides a review of the basics of entanglement entropy and the standard replica method that uses the heat kernel and the conical singularity technology. The second part presents a number of explicit calculations, for a free massive scalar field, of entanglement entropy in the case where the entangling surface intersects orthogonally the boundary of d-dimensional Minkowski spacetime. The main focus is made on the role of the boundary conditions : Neumann, Dirichlet and Robin.

Finally, I consider the case where the entangling surface is bounded in every directions and show that IR divergences can cancel out by choosing appropriate boundary conditions.

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- Théorie