### 3.5 The role of the rotational symmetry

We emphasize that the presence of the rotational symmetry with respect to the Killing vector ,
which generates rotations in the 2-plane orthogonal to the entangling surface , plays an important role
in our construction. Indeed, without such a symmetry it would be impossible to interpret for an
arbitrary as a partition function in some gravitational background. In general, two points are important
for this interpretation:
i) that the spacetime possesses, at least locally near the entangling surface, a rotational symmetry so that,
after the identification , we get a well-defined spacetime , with no more
than just a conical singularity; this holds automatically if the surface in question is a Killing
horizon;

ii) and that the field operator is invariant under the “rotations”, ; this is automatic if the field
operator is a covariant operator.

In particular, point ii) allows us to use the Sommerfeld formula (more precisely its generalization to
a curved spacetime) in order to define the Green’s function or the heat kernel on the space
. As is shown in [184] (see also discussion in Section 2.13) in the case of the non-Lorentz
invariant field operators in flat Minkowski spacetime, the lack of the symmetry ii) makes the
whole “conical space” approach rather obscure. On the other hand, in the absence of rotational
symmetry i) there may appear terms in the entropy that are “missing” in the naively applied
conical space approach: the extrinsic curvature contributions [204] or even some curvature
terms [134].

In what follows we consider the entanglement entropy of the Killing horizons and deal with the covariant
operators so that we do not have to worry about i) or ii).