3.7 Useful mathematical tools
3.7.1 Curvature of space with a conical singularity
Consider a space
, which is an
-fold covering of a smooth manifold
along the Killing vector
, generating an abelian isometry. Let surface
be a stationary point of this isometry so that near
the space
looks like a direct product,
, of the surface
and a two-dimensional cone
with angle deficit
. Outside the singular surface
the space
has the same
geometry as a smooth manifold
. In particular, their curvature tensors coincide. However, the conical
singularity at the surface
produces a singular (delta-function like) contribution to the curvatures. This
was first demonstrated by Sokolov and Starobinsky [195] in the two-dimensional case by using
topological arguments. These arguments were generalized to higher dimensions in [7
]. One
way to extract the singular contribution is to use some regularization procedure, replacing the
singular space
by a sequence of regular manifolds
. This procedure was developed by
Fursaev and Solodukhin in [111
]. In the limit
one obtains the following results [111
]:
where
is the delta-function,
;
are two orthonormal vectors
orthogonal to the surface
,
and the quantities
,
and
are
computed in the regular points
by the standard method.
These formulas can be used to define the integral
expressions
[111
]
where
and
. We use a shorthand notation for the surface integral
.
The terms proportional to
in Eqs. (56) – (59) are defined on the regular space
. The terms
in Eqs. (57) – (59) are something like a square of the
-function. They are not well-defined
and depend on the way the singular limit
is taken. However, those terms are not important in
the calculation of the entropy since they are of higher order in
. However, there are certain
invariants, polynomial in the Riemann tensor, in which the terms
do not appear at all. Thus,
these invariants are well defined on the manifolds with conical singularity. Below we consider two examples
of such invariants [111
].
Topological Euler number.
The topological Euler number of a
-dimensional smooth manifold
is given by the
integral
Suppose that
has several singular surfaces (of dimension
)
, each with conical deficit
, then the Euler characteristic of this manifold is [111
]
A special case is when
possesses a continuous abelian isometry. The singular surfaces
are the
fixed points of this isometry so that all surfaces have the same angle deficit
. The Euler number in
this case is [111
]
An interesting consequence of this formula is worth mentioning. Since the introduction of a conical
singularity can be considered as the limit of certain smooth deformation, under which the topological
number does not change, one has
. Then one obtains an interesting formula reducing the
number
of a manifold
to that of the fixed points set of its abelian isometry [111
]
A simple check shows that Eq. (63) gives the correct result for the Euler number of the sphere
.
Indeed, the fixed points of 2-sphere
are its “north” and “south” poles. Each of these points has
and one gets from Eq. (63):
. On the other hand, the singular surface of
(
) is
and from Eq. (63) the known identity
follows. Note that Eq. (63) is
valid for spaces with continuous abelian isometry and it may be violated for an orbifold with conical
singularities.
Lovelock gravitational action.
The general Lovelock gravitational action is introduced on a d-dimensional
Riemannian manifold as the following polynomial [166]
where
is the totally antisymmetrized product of the Kronecker symbols and
is
(or
) for even (odd) dimension
. If the dimension of spacetime is
, the action
reduces
to the Euler number (60) and is thus topological. In other dimensions the action (64) is not topological,
although it has some nice properties, which make it interesting. In particular, the field equations, which
follow from Eq. (64), are quadratic in derivatives even though the action itself is polynomial in
curvature.
On a conical manifold
, the Lovelock action is the sum of volume and surface parts [111
]
where the first term is the action computed at the regular points. As in the case of the topological Euler
number, all terms quadratic in
mutually cancel in Eq. (65). The surface term in Eq. (65) takes
the form of the Lovelock action on the singular surface
. It should be stressed that integrals
are defined completely in terms of the intrinsic Riemann curvature
of
and
. Eq. (65) allows us to compute the entropy in the Lovelock gravity by applying the replica
formula. In [145] this entropy was derived in the Hamiltonian approach, whereas arguments based
on the dimensional continuation of the Euler characteristics have been used for its derivation
in [7].
3.7.2 The heat kernel expansion on a space with a conical singularity
The useful tool to compute the effective action on a space with a conical singularity is the heat kernel
method already discussed in Section 2.8. In Section 2.9 we have shown how, in flat space, using the
Sommerfeld formula (22), to compute the contribution to the heat kernel due to the singular surface
.
This calculation can be generalized to an arbitrary curved space
that possesses, at least locally, an
abelian isometry with a fixed point. To be more specific we consider a scalar field operator
, where
is some scalar function. Then, the trace of the heat kernel
has
the following small
expansion
where the coefficients in the expansion decompose into bulk (regular) and surface (singular) parts
The regular coefficients are the same as for a smooth space. The first few coefficients are
The coefficients due to the singular surface
(the stationary point of the isometry) are
The form of the regular coefficients (69) in the heat kernel expansion has been well studied in physics and
mathematics literature (for a review see [219
]). The surface coefficient
in Eq. (70) was calculated by
the mathematicians McKean and Singer [174] (see also [42]). In physics literature this term
has appeared in the work of Dowker [69]. (In the context of cosmic strings one has focused
more on the Green’s function rather on the heat kernel [3, 100].) The coefficient
was
first obtained by Fursaev [101
] although in some special cases it was known before in works of
Donnelly [64, 65].
It should be noted that due to the fact that the surface
is a fixed point of the abelian isometry, all
components of the extrinsic curvature of the surface
vanish. This explains why the extrinsic curvature
does not appear in the surface terms (70) in the heat kernel expansion.