3.7 Useful mathematical tools

3.7.1 Curvature of space with a conical singularity

Consider a space E α, which is an α-fold covering of a smooth manifold E along the Killing vector ∂φ, generating an abelian isometry. Let surface Σ be a stationary point of this isometry so that near Σ the space E α looks like a direct product, Σ × š’ž α, of the surface Σ and a two-dimensional cone š’žα with angle deficit δ = 2π (1 − α ). Outside the singular surface Σ the space E α has the same geometry as a smooth manifold E. In particular, their curvature tensors coincide. However, the conical singularity at the surface Σ 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 [7Jump To The Next Citation Point]. One way to extract the singular contribution is to use some regularization procedure, replacing the singular space Eα by a sequence of regular manifolds E&tidle;α. This procedure was developed by Fursaev and Solodukhin in [111Jump To The Next Citation Point]. In the limit E&tidle;α → E α one obtains the following results [111Jump To The Next Citation Point]:

R μν = R¯μν + 2 π(1 − α)((n μn )(nνn ) − (nμn )(nνn )) δ , αβμ μ αβ μ α β β α Σ R ν = R¯ ν + 2π(1 − α)(n n ν)δΣ, R = R¯ + 4π(1 − α )δΣ , (55 )
where δΣ is the delta-function, ∫ ∫ ā„³ f δΣ = Σf; μ nk = nk∂μ , k = 1, 2 are two orthonormal vectors orthogonal to the surface Σ, ∑ (nμnν) = 2k=1 nkμnkν and the quantities ¯R μναβ, ¯R μν and ¯R are computed in the regular points E αāˆ•Σ by the standard method.

These formulas can be used to define the integral expressions3 [111Jump To The Next Citation Point]

∫ ∫ ¯ ∫ Eα R = α E R + 4π (1 − α ) Σ 1, (56 ) ∫ ∫ ∫ R2 = α ¯R2 + 8π(1 − α ) ¯R + O ((1 − α)2), (57 ) ∫ Eα ∫E Σ ∫ RμνR μν = α ¯RμνR¯μν + 4π(1 − α) R¯ii + O ((1 − α )2), (58 ) ∫ Eα ∫E Σ ∫ μνλρ ¯μνλρ ¯ ¯ 2 E α R Rμνλρ = α E R R μνλρ + 8π (1 − α) Σ Rijij + O ((1 − α ) ), (59 )
where ¯ ¯ μ ν Rii = Rμνn i ni and ¯ ¯ μ λ ν ρ Rijij = Rμνλρni n injn j. We use a shorthand notation for the surface integral ∫ ∫ √ -- Σ ≡ Σ γdd−2šœƒ.

The terms proportional to α in Eqs. (56View Equation) – (59View Equation) are defined on the regular space E. The terms O ((1 − α)2) in Eqs. (57View Equation) – (59View Equation) are something like a square of the δ-function. They are not well-defined and depend on the way the singular limit &tidle; E β → ā„°β is taken. However, those terms are not important in the calculation of the entropy since they are of higher order in (1 − α). However, there are certain invariants, polynomial in the Riemann tensor, in which the terms O((1 − α )2) 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 [111Jump To The Next Citation Point].

Topological Euler number.
The topological Euler number of a 2p-dimensional smooth manifold ā„° is given by the integral4

∫ √ -- 2p χ = ā„°2p gd x , E ā„°2p = cpšœ–μ μ ...μ μ šœ–ν1ν2...ν2p−1ν2pR μ1μ2 ...R μ2p−1μ2p , cp = ---1--- . (60 ) 1 2 2p−1 2p ν1ν2 ν2p−1ν2p 23pπpp!
Suppose that E α has several singular surfaces (of dimension 2(p − 1)) Σ i, each with conical deficit 2π (1 − αi), then the Euler characteristic of this manifold is [111Jump To The Next Citation Point]
∫ ∑ χ[E α] = ā„°2p + (1 − αi)χ [Σi ]. (61 ) E αāˆ•Σ i
A special case is when Eα possesses a continuous abelian isometry. The singular surfaces Σi are the fixed points of this isometry so that all surfaces have the same angle deficit αi = α. The Euler number in this case is [111Jump To The Next Citation Point]
∑ χ[Eα ] = α χ[Eα=1] + (1 − α) χ [Σi ]. (62 ) i
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 χ [E α] = χ[Eα=1 ]. Then one obtains an interesting formula reducing the number χ of a manifold E to that of the fixed points set of its abelian isometry [111Jump To The Next Citation Point]
∑ χ [E α=1] = χ[Σi]. (63 ) i
A simple check shows that Eq. (63View Equation) gives the correct result for the Euler number of the sphere Sdα. Indeed, the fixed points of 2-sphere S2α are its “north” and “south” poles. Each of these points has χ = 1 and one gets from Eq. (63View Equation): χ[S2] = 1 + 1 = 2. On the other hand, the singular surface of Sd α (d ≥ 3) is d−2 S and from Eq. (63View Equation) the known identity d d− 2 χ[S ] = χ[S ] follows. Note that Eq. (63View Equation) 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]

∑kd ∫ 1 [ν ν...ν ν ] ∑kd WL = λp -2p-δ[μ112μ2...2μp2p−−112μp2p]R μ1νμ12ν2...R μ2pν2−p1μ−21pν2p ≡ λpWp , (64 ) p=1 2 p! p=1
where [...] δ[...] is the totally antisymmetrized product of the Kronecker symbols and kd is (d − 2)āˆ•2 (or (d − 1)āˆ•2) for even (odd) dimension d. If the dimension of spacetime is 2p, the action Wp reduces to the Euler number (60View Equation) and is thus topological. In other dimensions the action (64View Equation) is not topological, although it has some nice properties, which make it interesting. In particular, the field equations, which follow from Eq. (64View Equation), are quadratic in derivatives even though the action itself is polynomial in curvature. On a conical manifold ā„³ α, the Lovelock action is the sum of volume and surface parts [111Jump To The Next Citation Point]
k∑d−1 WL [ā„³ α] = WL [ā„³ αāˆ•Σ ] + 2π(1 − α ) λp+1Wp [Σ ], (65 ) p=0
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 (1 − α) mutually cancel in Eq. (65View Equation). The surface term in Eq. (65View Equation) takes the form of the Lovelock action on the singular surface Σ. It should be stressed that integrals Wp [Σ ] are defined completely in terms of the intrinsic Riemann curvature Rij kn of Σ
∫ Wp [Σ ] = -1--- δ[i[j1......i2jp]]Ri1i2j j...Ri2p−j1i2pj (66 ) 22pp! Σ 1 2p 1 2 2p−1 2p
and W ≡ ∫ 0 Σ. Eq. (65View Equation) 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 (22View Equation), to compute the contribution to the heat kernel due to the singular surface Σ. This calculation can be generalized to an arbitrary curved space Eα that possesses, at least locally, an abelian isometry with a fixed point. To be more specific we consider a scalar field operator š’Ÿ = − (∇2 + X ), where X is some scalar function. Then, the trace of the heat kernel K = e− sš’Ÿ has the following small s expansion

TrK (s) = ---1---∑ a sn, (67 ) Eα (4πs )d2n=0 n
where the coefficients in the expansion decompose into bulk (regular) and surface (singular) parts
reg Σ an = an + an . (68 )
The regular coefficients are the same as for a smooth space. The first few coefficients are
reg ∫ reg ∫ 1-¯ a0 = Eα 1 , a1 = Eα(6R + X ), ∫ ( 1 1 1 1 1 1 ) ar2eg = ---R¯2μναβ − ---R¯2μν + -∇2 (X + -R¯) + -(X + --¯R)2 . (69 ) Eα 180 180 6 5 2 6
The coefficients due to the singular surface Σ (the stationary point of the isometry) are
∫ Σ Σ π-(1 −-α)(1 +-α)- a0 = 0; a 1 = 3 α Σ1 , (70 ) π (1 − α)(1 + α) ∫ 1 π (1 − α)(1 + α)(1 + α2)∫ aΣ2 = ----------------- (--¯R + X ) − ---------------3----------- (¯Rii − 2R¯ijij). 3 α Σ 6 180 α Σ
The form of the regular coefficients (69View Equation) in the heat kernel expansion has been well studied in physics and mathematics literature (for a review see [219Jump To The Next Citation Point]). The surface coefficient aΣ 1 in Eq. (70View Equation) 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 aΣ2 was first obtained by Fursaev [101Jump To The Next Citation Point] 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 Σ is a fixed point of the abelian isometry, all components of the extrinsic curvature of the surface Σ vanish. This explains why the extrinsic curvature does not appear in the surface terms (70View Equation) in the heat kernel expansion.

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