## 6 Logarithmic Term in the Entropy of Generic Conformal Field Theory

As we have already seen, in even dimensions, there typically appears a logarithmic term in the
entanglement entropy. This term is universal in the sense that it does not depend on the scheme, which is
used to regularize the UV divergences. In conformal field theories the logarithmic terms in
the entropy are closely related to the conformal anomaly. In this section we discuss in detail
this aspect and formulate precisely the relation between entanglement entropy and conformal
anomalies.

Consider a conformal field theory in spacetime dimensions. As we have discussed throughout this
review the most efficient way to calculate the corresponding entanglement entropy for a non-extremal black
hole is to introduce a small angle deficit at the horizon surface , compute the effective
action on a manifold with a singular surface and then apply the replica formula
and obtain from it the entanglement entropy. In dimensions the effective
action has the general structure

The logarithmic term in this expansion appears only if dimension is even. Thus, only even will be
considered in this section. The terms , ,…representing the power UV divergences, are not
universal, while the term is universal and is determined by the integrated conformal anomaly. The
term is the UV finite part of the effective action. Under a global rescaling of the metric on ,
, one has
An important property of the expansion (269) for a quantum field theory, which classically is
conformally invariant, is that the logarithmic term is conformally invariant (see [219] and references
therein),

On a
manifold with a conical singularity at the surface , the coefficients have a bulk part and a
surface part. To first order in one finds that
For the coefficients and are the integrated bulk and the surface conformal
anomalies respectively. The bulk and surface terms are independently invariant under conformal
transformation ,
Applying the replica formula one obtains the entanglement entropy
For a regular manifold each term is the integral of a polynomial of degree in the Riemann
curvature. Respectively, the surface terms are the integral over the singular surface of
polynomial of degree in the Riemann curvature and its projections onto the subspace
orthogonal to surface . The concrete structure of the polynomials depends on the dimension
.