### 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.