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^{3}
[111]

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].

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.

Living Rev. Relativity 14, (2011), 8
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