Quantum gravity and black holes as Intermezzo

Note that

[a, b, c, d] = (a−b)(c−d)

−(a−d)(b−c)= (c−d)(a−b)

−(c−b)(d−a) = [c, d, a, b].

It is noticed that the result above seems to indicate that there is a choice of Schottky uniformization involved as an additional datum for Arakerov ge- ometry at arithmetic infinity. However, as remarked previously, at least in the case of archimedean primes that are real embeddings as the case of arithmetic infinity for Q, the Schottky uniformization is detemined by the real structure, by splittingX(C) along the real locusX(R), when the latter is non-trivial.

Green function and geodesics. By combining the basic formula by the oriented distance with the formula calculated above for the Arakelov Green function on a Riemann surface with Shottky uniformization, we can replace each termappearing in the formula with the corresponding termwhich computes the oriented geodesic length of a certain arc of geodesic inYΓ as

g((a)−(b),(c)−(d)) =−

h∈Γ

ord(a∗[hc, hd], b∗[hc, hd])

+ g

l=1

xl(a, b)

h∈S(γ_l)

ord(z⁺(h)∗[c, d], z⁻(h)∗[c, d]).

The coefficients xl(a, b) can also be expressed in terms of geodesics, using the system equations given above.

In this case, theMinkowski space time Mk^d ≈R^d ≈R×R^d−1 with the metric as the diagonal sum

G= (gμν) =−1⊕1d−1.

Note also that, withG=−12⊕13, in the anti-de Sitter space a-dS4, ds²=ds·ds=(−12⊕13)(du, dv, dx, dy, dz)^t, d(u, v, x, y, z).

The holographic principle postulates the existence of an explicit correspon- dence between gravity on a bulk space which is asymptotically a-dS^∼_d+1, like a space obtained as a global quotient of a-dS^∼_d+1 by a discrete group of isometries, and field theory on its conformal boundary at infinity.

The above given, relation between the log of the cross ratio as [a, b, c, d]

fora, b, c, d ∈P¹(C) and the oriented distance ord(·,·) in terms of geodesics in H³, which identifies the Green function g((a)−(b),(c)−(d)) on P¹(C) with the oriented length of a geodesic arc in H³, can be thought of as an instance of the holography principle, when we interpret one side as geodesic propaga- tor on the bulk space in a semi-classical approximation, and the other side as the Green function as the two-point correlation function of the boundary field theory. Note that, because of the prescribed behavior of the Green function at the singularities given by the points of the divisor, the four-point invariant as g((a)−(b),(c)−(d)), when aand b converge toc, d respectively, gives the two-point correlator with a logarithmic divergence which is intrinsic and does not depend on a choice of cut-off functions, unlike the way in regularization is often used in the physics literature.

It is shown in [111] that the above given, (another) relation between Arakelov Green functions and configurations of geodesics in the hyperbolic handle-body YΓ, proved by Manin in [106], provides in fact precisely the correspondence pre- scribed by the holography principle, for a class of (2 + 1)-dimensional space-time known as Euclidean Krasnov black holes. These as holes include the Ba˜nados- Teitelboim-Zanelli (BTZ) black holds, as an important class of space-times in the context of (2 + 1)-dimensional quantumgravity.

The Ba˜nados-Teitelboim-Zanelli black hole. Consider the case of a hy- perbolic handle-body asYΓ of genus one, as a solid (open) torus (as a donut), with conformal boundary at infinity given by an elliptic curve.

Recall that elliptic curves can be described via the Jacobi uniformization, as already encountered in the previous sections, in the context of noncommutative elliptic curves. LetXq(C) =C^∗/q^Z be such a description of an elliptic curve, whereqis a hyperbolic element ofP SL2(C) with fixed points as{0,∞}on the 2-sphere asP¹(C) at infinity, forH³, so thatq∈C^∗ with|q|<1. The action of qonP¹(C) extends to an action onH³=H³∪P¹(C) by

q·(z, y) = (qz,|q|y)∈C×(0,∞)≈H³.

It follows that the quotient space Yq =H³/q^Z by this action is a solid (open) torus as a topological space, which is compactified at infinity by the conformal boundaryXq(C) as∂Yq (as a torus).

The spaceYqis well known in the physics literature as the Euclidean Ba˜nados- Teitelboim-Zanelli black hole, where the parameter q∈ C^∗ is written as the form

q= exp

2π(i|r₋| −r+) l

with r_±² =1 2

M l±

M²l²+J , where M and J are the mass and angular momentum of the rotating black hole, and −_l¹2 is the cosmological constant. The corresponding black hole in Minkowskian signaturemay be illustrated, but no figure provided (as in [3]).

In the case of the elliptic curve asXq(C) =C^∗/q^Z, the formula of Alvarez- Gaum´e, Moore, and Vafa [2] gives the operator product expansion of the path integral for bosonic field theory as

g(z,1) = log

|q|^{2−1B2(loglog|z||q|)}|1−z|Π^∞n=1|1−qⁿz||1−qⁿz⁻¹| .

This is in fact the Arakelov Green function onXq(C). In terms of geodesics in the Euclidean BTZ black hole (BH), that becomes

g(z,1) =−1 2l(γ0)B2

lγ₀(z,1) l(γ0)

+

n≥0

lγ₁(0, zn) +

n≥1

lγ₁(0, z^∼_n),

where B2(v) =v²−v+¹₆ is the second Bernoulli polynomial, and we use the notation as that x = x∗[0,∞], zn = qⁿz∗[1,∞), and z_n^∼ = qⁿz⁻¹∗[1,∞) in H³ as well asXq, as in [106]. (No respective figures.) These terms describe gravitational properties of the Euclidean BTZBH. For instance,l(γ0)measures the black hole entropy. The whole expression is a combination of geodesic propagators.

Krasnov black holes. The problemof computing the bosonic field propagator on an algebraic curveX_C (such asX(C) = Γ\ΩΓ) can be solved by providing differentials of the third kind with purely imaginary periods as

ω(a)−(b)=ν(a)−(b)− g l=1

xl(a, b)ωγ_l,

and thus it can be related directly to the problem of computing the Arakelov Green function.

Differentials as above then determine all the higher correlation functions as G(z1,· · ·, zm, w1,· · ·, wl) =

m j=1

l k=1

qkϕ(zk, zk)ϕ(wj, wj)q_j,

for qk a system of charges at positions zk interacting with charges q_j at posi- tions wj from the basic two-point correlator Gμ(a−b, z) given by the Green function expressed in terms of the differentialsω(a)−(b)(z). By using a Schottky uniformization, obtained are the differentialsω(a)−(b)as given above.

The bulk space corresponding to the conformal boundary X(C) is given by the hyperbolic handle-body YΓ. As in the case of the BTZ black hole, it is possible to interpret these real hyperbolic 3-manifolds as analytic continuations to Euclidean signature of Minkowskian black holes that are global quotients of a-dS2+1. This is not just the effect of the usual rotation from Minkowskian to Euclidean signature, but amore refined form of analytic continuation which is adapted to the action of a Schtokky group, and which is introduced by Kirill Krasnov [92], [93] in order to deal with this class of space-times.

The above given formula for the Green functiong(·,·) in terms of oriented distances of geodesics as inYΓ gives the explicit bulk and boundary correspon- dence of the holography principle for this class of space times, so that each term in the Bosonic field propagator forX_Cis expressed in terms of geodesics in the EuclideanKrasnov black hole asYΓ =H³/Γ.