# Vacuum Currents for a Scalar Field in Models
with Compact Dimensions

Aram A. Saharian
Institute of Physics, Yerevan State University,
1 Alex Manogian Street, 0025 Yerevan, Armenia

###### Abstract

This paper reviews the investigations on the vacuum expectation value of the
current density for a charged scalar field in spacetimes with toroidally
compactified spatial dimensions. As background geometries locally Minkowskian
(LM), locally de Sitter (LdS) and locally anti-de Sitter (LAdS) spacetimes are
considered. Along compact dimensions quasiperiodicity conditions are imposed
on the field operator and the presence of a constant gauge field is assumed.
The vacuum current has non-zero components only along compact dimensions.
Those components are periodic functions of the magnetic flux enclosed by
compact dimensions with the period equal to the flux quantum. For LdS and LAdS
geometries and for small values of the length of a compact dimension, compared
with the curvature radius, the leading term in the expansion of the the vacuum
current along that dimension coincides with that for LM bulk. In this limit
the dominant contribution to the mode sum for the current density comes from
the vacuum fluctuations with wavelength smaller than the curvature radius and
the influence of the gravitational field is weak. The effects of the
gravitational field are essential for lengths of compact dimensions larger
than the curvature radius. In particular, instead of the exponential
suppression of the current density in LM bulk one can have power law decay in
LdS and LAdS spacetimes.

Keywords: vacuum currents; nontrivial topology; Casimir effect; de Sitter
spacetime; anti-de Sitter spacetime

##  1 Introduction

In a number of physical models the dynamics of the system is formulated in
background geometries having compact dimensions. The examples include Kaluza-
Klein type models with extra dimensions, string theories with different types
of compactifications of 6-dimensional internal subspace, condensed matter
systems like fullerenes, cylindrical nanotubes and toroidal loops. The
periodicity conditions imposed on dynamical variables along compact dimensions
are sources of a number of interesting effects, such as the topological
generation of mass, various mechanisms for symmetry breaking and different
kinds of instabilities. In quantum field theory the nontrivial spatial
topology modifies the spectrum of zero-point fluctuations of fields and as a
consequence of that the vacuum expectation values (VEVs) of physical
obesrvables are shifted by an amount that depends on geometrical
characteristics of the compact space. This is the analog of the Casimir effect
(for reviews see [1]-[5]), where the conditions imposed on constraining
boundaries are replaced by periodicity conditions, and it is known under the
general name of the topological Casimir effect. In Kaluza-Klein models this
effect yields an effective potential for the lengths of compact dimensions and
can serve as a compactification and stabilization mechanism for internal
subspace (Candelas-Weinberg mechanism [6]).

The most popular quantity in the investigations of the zero temperature
Casimir effect is the vacuum energy. Been an important global characteristic
of the vacuum state, it also determines the vacuum forces acting on boundaries
constraining the quantization volume. More detailed information is contained
in the local characteristics such as the vacuum expectation value (VEV) of the
energy-momentum tensor. The latter is a source of the gravitational field in
semiclassical Einstein equations and determines the back-reaction of the
quantum effects on the geometry of the spacetime. That VEV of the energy-
momentum tensor in the Casimir effect, in general, does not obey the energy
conditions in Hawking-Penrose singularity theorems and, hence, is an
interesting source of singularity free solutions for the gravitational field.
For charged fields, another important local characteristic of the vacuum state
is the expectation value of the current density. It is a source of the
electromagnetic field in Maxwell equations and should be taken into account
when considering the self-consistent dynamics of the electromagnetic field.

Unlike to the energy-momentum tensor, in order to have nonzero vacuum currents
the parity symmetry of the model should be broken. That can be done by
introducing external fields or by imposing appropriate boundary or periodicity
conditions (in models with compact spatial dimensions). In particular, the
vacuum currents for charged scalar and fermionic fields in locally Minkowski
spacetime with totoidally compactified spatial dimensions and in the presence
of constant gauge fields have been investigated in [7, 8]. The parity symmetry
in the corresponding models is broken by an external gauge field or by
quasiperiodic conditions along compact dimensions with nontrivial phases. The
results for a fermionic field in (2+1)-dimensional spacetime have been applied
to carbon nanotubes and nanoloops. Those structures are obtained from planar
graphene sheet by appropriate identifications along a single or two spatial
dimensions. In the long wavelength approximation, the dynamics of electronic
subsystem in graphene is well described by an effective Dirac model with the
Fermi velocity of electrons appearing instead of the velocity of light (see,
for example, [9, 10]). The corresponding quantum field theory lives in spaces
with topologies $R^{1}\times S^{1}$ and $S^{1}\times S^{1}$ for nanotubes and
nanoloops respectively. The appearance of the vacuum currents along compact
dimensions can be understood as a kind of the topological Casimir effect. The
combined influence of the boundary-induced and topological Casimir effects on
the vacuum currents have been studied in [11, 12] for scalar and fermionic
fields on background of locally flat spacetime with toral dimensions and in
the presence of planar boundaries.

The boundary conditions imposed on quantum fields serve as simplified models
for external fields and the Casimir effect can be considered as a vacuum
polarization sourced by those conditions. Another type of vacuum polarization
is induced by external gravitational fields. The combined effects of those two
sources on the VEV of the current density have been investigated in [13] and
[14, 15] for locally de Sitter (dS) and anti-de Sitter (AdS) spacetimes with a
part of spatial dimensions compactified to a torus. The high symmetry of these
background geometries allowed to find exact expressions for the current
densities in scalar and fermionic vacua. The effects of additional boundaries
parallel to the AdS boundary were studied in [16]-[19]. The corresponding
applications to higher dimensional braneworld models with compact subspaces
have been discussed.

The physical nature of the current densities considered in [7, 8] is similar
to that for persistent currents in mesoscopic rings [20]-[25]. Those currents
are among the most interesting manifestations of the Aharonov-Bohm effect and
appear as a result of phase coherence of the charge carriers, extended over
the whole mesoscopic ring. The persistent currents have been studied
extensively in the literature for electronic subsystems in different condensed
matter systems and for bosonic and fermionic atoms by making use of discrete
or continuum models (see, e.g., [26]-[41] and [42]-[50] for theoretical and
experimental investigations, respectively, and references therein). Their
dependence on the geometry of ring is an interesting direction of those
investigations.

The present paper reviews the results of investigations of the vacuum current
densities for charged scalar fields in locally Minkowski, dS and AdS
spacetimes with toral subspaces. It is organized as follows. In Section 2 we
present the general formalism for the evaluation of the current density for a
charged scalar filed in models with compact dimensions. The application of the
formalism in the locally Minkowskian background geometry is considered in
Section 3. The vacuum currents for locally dS and AdS background spacetimes
are studied in Sections 4 and 5. The features of current densities and
comparative analysis for background geometries with zero, positive and
negative curvatues are discussed in Section 6. The main results are summarized
in Section 7. In appendix A we present the properties of the functions
appearing in the expressions of the current densities.

##  2 General Formalism

Consider a charged scalar field $\varphi(x)$ with the curvature coupling
parameter $\xi$ in background of $(D+1)$-dimensional spacetime described by
the line element $ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}$. Here we use
$x=(x^{0}=t,x^{1},\ldots,x^{D})$ for the notation of spacetime points. In the
presence of a classical vector gauge field $A_{\mu}(x)$ the corresponding
action has the form

$S\left[\varphi\right]=\frac{1}{2}\int
d^{D+1}x\,\sqrt{|g|}\left[g^{\mu\nu}\left(D_{\mu}\varphi\right)\left(D_{\nu}\varphi\right)^{\dagger}+\left(\xi
R+m^{2}\right)\varphi\varphi^{\dagger}\right].$ (1)

where $R$ is the Ricci scalar for the metric tensor $g_{\mu\nu}(x)$ and
$D_{\mu}=\nabla_{\mu}+ieA_{\mu}$, with $\nabla_{\mu}$ being the related
covariant derivative operator. The most important special cases correspond to
the fields with minimal ($\xi=0$) and conformal
($\xi=\xi_{D}\equiv(D-1)/(4D)$) couplings. The field equation obtained from
(1) by standard variational procedure reads

$\left(g^{\mu\nu}D_{\mu}D_{\nu}+\xi R+m^{2}\right)\varphi(x)=0.$ (2)

We assume that a part of coordinates corresponding to the subspace
$(x^{p+1},x^{p+2},\ldots,x^{D})$ are compactified to circles of the lengths
$(L_{p+1},L_{p+2},\ldots,L_{D})$, respectively, and $0\leq x^{l}\leq L_{l}$,
$l=p+1,\ldots,D$. In addition, the metric tensor is periodic along the compact
dimensions:

$g_{\mu\nu}(t,x^{1},\ldots,x^{p},\ldots,x^{l}+L_{l},\ldots,x^{D})=g_{\mu\nu}(t,x^{1},\ldots,x^{p},\ldots,x^{l},\ldots,x^{D}),$
(3)

where $p<l\leq D$. For the scalar and gauge fields less trivial
quasiperiodicity conditions are imposed

$\displaystyle\varphi(t,x^{1},\ldots,x^{p},\ldots,x^{l}+L_{l},\ldots,x^{D})$
$\displaystyle=$ $\displaystyle
e^{i\alpha_{l}(x)}\varphi(t,x^{1},\ldots,x^{p},\ldots,x^{l},\ldots,x^{D}),$
$\displaystyle A_{\mu}(t,x^{1},\ldots,x^{p},\ldots,x^{l}+L_{l},\ldots,x^{D})$
$\displaystyle=$ $\displaystyle
A_{\mu}(t,x^{1},\ldots,x^{p},\ldots,x^{l},\ldots,x^{D})-(1/e)\nabla_{\mu}\alpha_{l}(x),$
(4)

where the real functions $\alpha_{l}(x)$, $l=p+1,\ldots,D$, are periodic along
the compact dimensions. With these conditions, the Lagrangian density in (1)
is periodic in the subspace $(x^{p+1},x^{p+2},\ldots,x^{D})$. The gauge field
is periodic up to a gauge transformation and this type of quasiperiodic
conditions are referred as C-periodic boundary conditions (see, e.g., [51, 52,
53]).

We are interested in the VEV of the current density

$j_{\mu}=\frac{i}{2}e\left[\\{D_{\mu}\varphi,\varphi^{\dagger}\\}-\\{D_{\mu}\varphi,\varphi^{\dagger}\\}^{\dagger}\right],$
(5)

where the figure brackets stand for the anticommutator. The VEVs of the field
bilinear combinations are expressed in terms of the Hadamard function
$G(x,x^{\prime})$ defined as the VEV

$G(x,x^{\prime})=\left\langle\\{\varphi(x),\varphi^{\dagger}(x^{\prime})\\}\right\rangle,$
(6)

where $\left\langle\cdots\right\rangle=\left\langle
0\right|\cdots\left|0\right\rangle$, with $\left|0\right\rangle$ being the
vacuum state, stands for the VEV. In particular, for the VEV of the current
density we have

$\left\langle j_{\mu}(x)\right\rangle=\frac{ie}{2}\lim_{x^{\prime}\rightarrow
x}\left(D_{\mu}-g_{\mu}^{\quad\nu^{\prime}}D_{\nu^{\prime}}^{\ast}\right)G(x,x^{\prime}),$
(7)

with $g_{\mu}^{\quad\nu^{\prime}}$ being the bivector of parallel
displacement. We recall that the vector
$\tilde{c}_{\mu}=g_{\mu}^{\quad\nu^{\prime}}c_{\nu^{\prime}}$ is obtained by
the parallel transport of $c_{\nu}$ from the point $x^{\prime}$ to the point
$x$ along the geodesic connecting those points.

The Hadamard function in (7) can be evaluated in two different ways: by
solving the corresponding differential equation obtained from the field
equation or by using the complete set
$\\{\varphi_{\sigma}^{(+)}(x),\varphi_{\sigma}^{(-)}(x)\\}$ of the mode
functions for the field specified by the set of quantum numbers $\sigma$. We
will follow the second approach. For the field operator one has the expansion

$\varphi(x)=\sum_{\sigma}\left[a_{\sigma}\varphi_{\sigma}^{(+)}(x)+b_{\sigma}^{\dagger}\varphi_{\sigma}^{(-)}(x)\right],$
(8)

where $a_{\sigma}$ and $b_{\sigma}^{\dagger}$ are the annihilation and
creation operators. The symbol $\sum_{\sigma}$ stands for the summation over
the discrete quantum numbers in the set $\sigma$ and the integration over the
continuous ones. Substituting in the expression (6) for the Hadamard function
and by using the relations
$a_{\sigma}\left|0\right\rangle=b_{\sigma}\left|0\right\rangle=0$, we get the
mode sum

$G(x,x^{\prime})=\sum_{\sigma}\left[\varphi_{\sigma}^{(+)}(x)\varphi_{\sigma}^{(+)\ast}(x^{\prime})+\varphi_{\sigma}^{(-)}(x)\varphi_{\sigma}^{(-)\ast}(x^{\prime})\right].$
(9)

With this sum, the formal expression for the VEV of the current density takes
the form

$\left\langle
j_{\mu}\right\rangle=-e\sum_{\sigma}\mathrm{Im}\left(\varphi_{\sigma}^{(+)\ast}D_{\mu}\varphi_{\sigma}^{(+)}+\varphi_{\sigma}^{(-)\ast}D_{\mu}\varphi_{\sigma}^{(-)}\right).$
(10)

The expression on the right-hand side is divergent and a regularization with
the subsequent renormalization is required to obtain a finite result.

In the following sections we apply the presented general formalism for special
background geometries. Three cases will be considered: locally Minkowski
spacetime (LM), locally dS spacetime (LdS) and locally AdS spacetime (LAdS).
For all these geometries, the planar coordinates
$\mathbf{x}_{q}=(x_{p+1},\ldots,x_{D})$ will be used in the compact subspace.
For the classical gauge field we will assume a simple configuration
$A_{\mu}=\mathrm{const}$ and in the periodicity conditions (4) the phases
$\alpha_{l}(x)=\alpha_{l}=\mathrm{const}$ will be taken. Special cases
$\alpha_{l}=2\pi$ and $\alpha_{l}=\pi$ correspond to periodic and antiperiodic
fields and have been widely considered in the literature. The constant gauge
field is removed from the field equation for the scalar field by the gauge
transformation
$\left(\varphi,A_{\mu}\right)\rightarrow\left(\varphi^{\prime},A_{\mu}^{\prime}\right)$
given as

$\varphi^{\prime}(x)=\varphi(x)e^{ie\chi(x)},\;A_{\mu}^{\prime}=A_{\mu}-\partial_{\mu}\chi(x),$
(11)

with the function $\chi(x)=A_{\mu}x^{\mu}$. For this function one gets
$A_{\mu}^{\prime}=0$ and the field equation takes the form

$\left(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}+\xi
R+m^{2}\right)\varphi^{\prime}(x)=0.$ (12)

The periodicity conditions for the new scalar field read

$\varphi^{\prime}(t,x^{1},\ldots,x^{p},\ldots,x^{l}+L_{l},\ldots,x^{D})=e^{i\tilde{\alpha}_{l}}\varphi^{\prime}(t,x^{1},\ldots,x^{p},\ldots,x^{l},\ldots,x^{D}),$
(13)

with new phases

$\tilde{\alpha}_{l}=\alpha_{l}+eA_{l}L_{l},$ (14)

for $l=p+1,\ldots,D$. The physics is invariant under the gauge transformation
and we could expect that the components $A_{\mu}$, $\mu=0,1,\ldots,p$, will
not appear in the expressions for physical quantities. That is not the case
for the components of the vector potential along the compact dimensions. They
will appear in the VEVs through the new phases (14). This is an Aharonov-Bohm
type effect for a constant gauge field in topologically nontrivial spaces. The
further consideration will be presented in terms of new fields
$\left(\varphi^{\prime},A_{\mu}^{\prime}=0\right)$ omitting the primes.

##  3 Locally Minkowski Spacetime with Toral Dimensions

We start the consideration with LM spacetime having the spatial topology
$R^{p}\times T^{q}$, $q=D-p$, where the $q$-dimensional torus $T^{q}$
corresponds to the subspace with the compact coordinates
$\mathbf{x}_{q}=(x^{p+1},x^{p+2},\ldots,x^{D})$. For this geometry the metric
in the Cartesian coordinates is expressed as
$g_{\mu\nu}=\eta_{\mu\nu}=\mathrm{diag}\left(1,-1,\ldots,-1\right)$. With this
metric tensor one has $\nabla_{\mu}=\partial_{\mu}$. The topological Casimir
effect in flat spacetimes with toral dimensions has been widely considered in
the literature (see [1, 5], [54]-[69] and references therein). As
characteristics of the ground state for quantum fields, the expectation values
of the energy density and the stresses were studied. The expectation value of
the current density for scalar and fermionic fields have been investigated in
[7, 8, 11, 12, 70] at zero and finite temperatures. In this section we review
the results for the VEV of the current density for a charged scalar field.

For the geometry under consideration the normalized scalar mode functions are
specified by the momentum $\mathbf{k}=\left(k_{1},k_{2},\ldots,k_{D}\right)$
and are given by

$\varphi_{\mathbf{k}}^{(\pm)}(x)=(2^{p+1}\pi^{p}V_{q}\omega_{\mathbf{k}})^{-1/2}e^{\mp
i\omega_{\mathbf{k}}t+\mathbf{k}_{p}\cdot\mathbf{x}_{p}+\mathbf{k}_{q}\cdot\mathbf{x}_{q}},$
(15)

where $\mathbf{x}_{p}=(x^{1},\ldots,x^{D})$ stands for the set of coordinates
in the uncompact subspace, $\omega_{\mathbf{k}}=\sqrt{\mathbf{k}^{2}+m^{2}}$
is the respective energy and $V_{q}=L_{p+1}....L_{D}$. For the components of
the momentum along uncompact dimensions,
$\mathbf{k}_{p}=(k_{1},\ldots,k_{p})$, we have $k_{l}\in(-\infty,+\infty)$,
$l=1,\ldots,p$, whereas the eigenvalues along the compact dimensions,
$\mathbf{k}_{q}=(k_{p+1},\ldots,k_{D})$, are discretized by the periodicity
conditions (13):

$k_{l}=\frac{2\pi n_{l}+\tilde{\alpha}_{l}}{L_{l}},\quad n_{l}=0,\pm 1,\pm
2,\ldots.,$ (16)

with $l=p+1,...,D$. The integer part of the ratio $\tilde{\alpha}_{l}/(2\pi)$
in the expressions of the VEVs is absorbed by shifting the integer number
$n_{l}$ in the corresponding summation and, hence, we can take
$|\tilde{\alpha}_{l}|\leq\pi$ without loss of generality. Note that one has

$\varphi_{\mathbf{k}}^{(\pm)}(t,x^{1},\ldots,x^{p},\ldots,x^{l}-L_{l},\ldots,x^{D})=e^{-i\tilde{\alpha}_{l}}\varphi_{\mathbf{k}}^{(\pm)}(t,x^{1},\ldots,x^{p},\ldots,x^{l},\ldots,x^{D}),$
(17)

and hence for $\tilde{\alpha}_{l}\neq n\pi$, with $n$ being an integer, the
parity (P-) symmetry with respect to the reflection $x^{l}\rightarrow-x^{l}$
is broken by the corresponding quasiperiodicity condition. As it will be seen
below, the breaking of this inversion symmetry results in a non-zero component
of the current density along the $l$th compact dimension.

The Hadamard function is obtained from (9) with the modes (15). Because of the
spacetime homogeneity the dependence on the coordinates appears in the form of
differences $\Delta t=t-t^{\prime}$,
$\Delta\mathbf{x}_{p}\mathbf{=x}_{p}-\mathbf{x}_{p}^{\prime}$, and
$\Delta\mathbf{x}_{q}\mathbf{=x}_{q}-\mathbf{x}_{q}^{\prime}$. Inserting the
modes (15) in (9), one gets

$G(x,x^{\prime})=\frac{1}{V_{q}}\int\frac{d\mathbf{k}_{p}}{(2\pi)^{p}}\sum_{\mathbf{n}_{q}}e^{i\mathbf{k}_{p}\cdot\Delta\mathbf{x}_{p}+i\mathbf{k}_{q}\cdot\Delta\mathbf{x}_{q}}\frac{\cos(\omega_{\mathbf{k}}\Delta
t)}{\omega_{\mathbf{k}}},$ (18)

where $\mathbf{n}_{q}=(n_{p+1},\ldots,n_{D})$. In the gauge under
consideration and for the geometry at hand the formula (7) for the current
density takes the form

$\left\langle j_{\mu}\right\rangle=\frac{ie}{2}\lim_{x^{\prime}\rightarrow
x}\left(\partial_{\mu}-\partial_{\mu^{\prime}}\right)G(x,x^{\prime}).$ (19)

For $\mu=0,1,\ldots,p$ the derivatives $\partial_{\mu}G(x,x^{\prime})$ and
$\partial_{\mu^{\prime}}G(x,x^{\prime})$ are odd functions of $\Delta x^{\mu}$
and the charge density and the components of the current density along
uncompact dimensions vanish, $\left\langle j_{\mu}\right\rangle=0$ for
$\mu=0,1,\ldots,p$. Of course, we could expect that from the problem symmetry
under the reflections $x^{\mu}\rightarrow-x^{\mu}$ along respective
coordinates. In order to find the current density along the $r$th compact
dimension it is convenient to transform the corresponding summation over
$n_{r}$ in (18). In order to do that we use a variant of the Abel-Plana
formula [61, 71]

$\frac{2\pi}{L_{r}}\sum_{n_{r}=-\infty}^{\infty}g(k_{r})f(\left|k_{r}\right|)=\sum_{\lambda=\pm
1}\left[\int_{0}^{\infty}dzg(\lambda z)f(z)+i\int_{0}^{\infty}dz\,g(i\lambda
z)\frac{f(iz)-f(-iz)}{e^{zL_{r}+i\lambda\tilde{\alpha}_{r}}-1}\right],$ (20)

with the functions $g(z)=e^{iz\Delta x^{r}}$ and $f(z)=\cos\left(\Delta
t\sqrt{z^{2}+\omega_{p,q-1}^{2}}\right)/\sqrt{z^{2}+\omega_{p,q-1}^{2}}$,
where $\omega_{p,q-1}=\sqrt{\mathbf{k}_{p}^{2}+\mathbf{k}_{q-1}^{2}+m^{2}}$
and $\mathbf{k}_{q-1}=(k_{p+1},\ldots,k_{r-1},k_{r+1},\ldots,k_{D})$. By
making use of the expansion $1/(e^{y}-1)=\sum_{l=1}^{\infty}e^{-ly}$, the
integrals over $z$ and then over $\mathbf{k}_{p}$ are expressed in terms of
the Macdonald function $K_{\nu}(x)$. Introducing the notation
$f_{\nu}(x)=x^{-\nu}K_{\nu}(x)$, we get

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2L_{r}V_{q}^{-1}}{(2\pi)^{p/2+1}}\sum_{n_{r}=-\infty}^{+\infty}\sum_{\mathbf{n}_{q-1}}e^{in_{r}\tilde{\alpha}_{r}+i\mathbf{k}_{q-1}\cdot\Delta\mathbf{x}_{q-1}}\omega_{\mathbf{n}_{q-1}}^{p}$
(21) $\displaystyle\times
f_{\frac{p}{2}}\left(\omega_{\mathbf{n}_{q-1}}\sqrt{|\Delta\mathbf{x}_{p}|^{2}+\left(\Delta
x^{r}-n_{r}L_{r}\right)^{2}-\left(\Delta t\right)^{2}}\right),$

where $\mathbf{n}_{q-1}=(n_{p+1},\ldots,n_{r-1},n_{r+1},\ldots,n_{D})$ and
$\omega_{\mathbf{n}_{q-1}}^{2}=\mathbf{k}_{q-1}^{2}+m^{2}$. The $n_{r}=0$ term
here corresponds to the Hadamard function in the geometry with spatial
topology $R^{p+1}\times T^{q-1}$, where the $r$th dimension is decompactified.
The divergences in the coincidence limit $x^{\prime}\rightarrow x$ are
contained in that term only. The remaining part is induced by the
compactification of the coordinate $x^{r}$ and it is finite in the coincidence
limit. The latter property is related to the fact that the compactification to
a circle does not change the local geometry and, hence, the structure of local
divergences as well.

Plugging the Hadamard function (21) in (19) and noting that the term $n_{r}=0$
does not contribute to the component of the current density along the $r$th
compact dimension, for the corresponding contravariant component one finds

$\left\langle
j^{r}\right\rangle=\frac{2^{1-p/2}eL_{r}^{2}}{\pi^{p/2+1}V_{q}}\sum_{n_{r}=1}^{\infty}n_{r}\sin(n_{r}\tilde{\alpha}_{r})\sum_{\mathbf{n}_{q-1}}\omega_{\mathbf{n}_{q-1}}^{p+2}f_{p/2+1}(n_{r}L_{r}\omega_{\mathbf{n}_{q-1}}).$
(22)

The specific features of the vacuum current will be discussed below in Section
6. As it has been already mentioned, for $\tilde{\alpha}_{r}=0,\pi$ the
problem is symmetric with respect to the inversion $x^{r}\rightarrow-x^{r}$
and, as expected, the VEV $\left\langle j^{r}\right\rangle$ vanishes. For
those special values the contribution from the right moving vacuum
fluctuations with $k_{r}>0$ is cancelled by the contribution coming from the
left moving modes with $k_{r}<0$. For $\tilde{\alpha}_{r}=0$ there is also a
zero mode with $k_{r}=0$ which does not contribute to the current density
along the $r$th dimension.

In the model with a single compact dimension $x^{D}$ one has $p=D-1$,
$\omega_{\mathbf{n}_{q-1}}=m$ and the formula (22) is reduced to

$\left\langle
j^{D}\right\rangle=\frac{4em^{D+1}L_{D}}{\left(2\pi\right)^{\frac{D+1}{2}}}\sum_{n=1}^{\infty}n\sin(n\tilde{\alpha}_{D})f_{\frac{D+1}{2}}(mnL_{D}).$
(23)

In particular, for a massless field we get

$\left\langle
j^{D}\right\rangle=\frac{2e\Gamma\left(\frac{D+1}{2}\right)}{\pi^{\frac{D+1}{2}}L_{D}^{D}}\sum_{n=1}^{\infty}\frac{\sin(n\tilde{\alpha}_{D})}{n^{D}}.$
(24)

For odd values of $D$ the series is expressed in terms of the Bernoulli
polynomials $B_{n}(x)$ (see, e.g., [72]) and we get

$\left\langle
j^{D}\right\rangle=\frac{\left(-1\right)^{\frac{D+1}{2}}\pi^{\frac{D}{2}}e}{\Gamma\left(\frac{D}{2}+1\right)L_{D}^{D}}B_{D}\left(\frac{\tilde{\alpha}_{D}}{2\pi}\right),$
(25)

for $0<\tilde{\alpha}_{D}<2\pi$. In particular, for $D=1$ and $D=3$ one finds

$\displaystyle\left\langle j^{D}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{e}{L_{D}}\left(1-\frac{\tilde{\alpha}_{D}}{\pi}\right),\;D=1,$
$\displaystyle\left\langle j^{D}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{e\tilde{\alpha}_{D}}{6L_{D}^{3}}\left(1-\frac{\tilde{\alpha}_{D}}{\pi}\right)\left(2-\frac{\tilde{\alpha}_{D}}{\pi}\right),\;D=3.$
(26)

For $D\geq 2$ the current density is a continuous function of
$\tilde{\alpha}_{D}$, whereas for $D=1$ the current density for a massless
field is discontinuous at $\tilde{\alpha}_{D}=2\pi n$ with integer $n$.

We could directly start from the mode sum formula (10) with
$D_{\mu}=\partial_{\mu}$. The substitution of the mode functions (15) leads to
the expression

$\left\langle
j^{r}\right\rangle=\frac{e}{V_{q}}\int\frac{d\mathbf{k}_{p}}{(2\pi)^{p}}\sum_{\mathbf{n}_{q}}\frac{k_{r}}{\omega_{\mathbf{k}}}=\frac{eL_{r}}{V_{q}}\frac{\partial}{\partial\tilde{\alpha}_{r}}\int\frac{d\mathbf{k}_{p}}{(2\pi)^{p}}\sum_{\mathbf{n}_{q}}\omega_{\mathbf{k}}.$
(27)

Introducing the generalized zeta function $\zeta(s)$ in accordance with

$\zeta(s)=\frac{1}{V_{q}}\int\frac{d\mathbf{k}_{p}}{(2\pi)^{p}}\sum_{\mathbf{n}_{q}}\omega_{\mathbf{k}}^{-2s}=\frac{1}{V_{q}}\int\frac{d\mathbf{k}_{p}}{(2\pi)^{p}}\sum_{\mathbf{n}_{q}}\left(\mathbf{k}_{p}^{2}+\mathbf{k}_{q}^{2}+m^{2}\right)^{-s},$
(28)

the current density is written as

$\left\langle
j^{r}\right\rangle=eL_{r}\left.\frac{\partial}{\partial\tilde{\alpha}_{r}}\zeta(s)\right|_{s=-1/2},$
(29)

where $|_{s=-1/2}$ is understood in the sense of the analytical continuation
of the representation (28) (for applications of the zeta function technique in
the investigations of the Casimir effect see, for example, [5, 73, 74]). In
order to realize the analytic continuation we first integrate over the
momentum in the uncompact subspace with the result

$\zeta(s)=\frac{\Gamma(s-p/2)}{2^{p}\pi^{p/2}V_{q}\Gamma(s)}\sum_{\mathbf{n}_{q}}\left(\mathbf{k}_{q}^{2}+m^{2}\right)^{p/2-s}.$
(30)

The application of the generalized Chowla-Selberg formula [75] to the multiple
series in (30) gives

$\zeta(s)=\frac{m^{D-2s}}{\left(4\pi\right)^{\frac{D}{2}}\Gamma(s)}\left[\Gamma\left(s-\frac{D}{2}\right)+2^{\frac{D}{2}+1-s}\sum_{\mathbf{n}_{q}}^{\prime}\cos\left(\mathbf{n}_{q}\cdot\mathbf{\tilde{\alpha}}_{q}\right)f_{\frac{D}{2}-s}(mg(\mathbf{L}_{q},\mathbf{n}_{q}))\right],$
(31)

where the prime on the summation sign means that the term with
$\mathbf{n}_{q}=(0,0,\ldots,0)$ should be excluded. Here we have introduced
the $q$-component vector
$\mathbf{\tilde{\alpha}}_{q}=(\tilde{\alpha}_{p+1},\tilde{\alpha}_{p+2},\ldots,\tilde{\alpha}_{D})$
and the notation

$g(\mathbf{L}_{q},\mathbf{n}_{q})=\left(\sum_{i=p+1}^{D}n_{i}^{2}L_{i}^{2}\right)^{1/2}.$
(32)

The first term in the right-hand side of (31) corresponds to the geometry
without compact dimensions and it does not contribute to the current density.
The last term in (31) is finite at the physical point and can be directly used
in (29) to obtain the following expression for the current density

$\left\langle
j^{r}\right\rangle=\frac{2em^{D+1}L_{r}}{\left(2\pi\right)^{\frac{D+1}{2}}}\sum_{\mathbf{n}_{q}}^{\prime}n_{r}\sin\left(\mathbf{n}_{q}\cdot\mathbf{\tilde{\alpha}}_{q}\right)f_{\frac{D+1}{2}}(mg(\mathbf{L}_{q},\mathbf{n}_{q})).$
(33)

In the special case of a single compact dimension $x^{D}$ this result
coincides with (23). Note that in the representation (33) we can make the
replacement

$n_{r}\sin\left(\mathbf{n}_{q}\cdot\mathbf{\tilde{\alpha}}_{q}\right)\rightarrow
n_{r}\sin(n_{r}\tilde{\alpha}_{r})\cos\left(\mathbf{n}_{q-1}\cdot\mathbf{\tilde{\alpha}}_{q-1}\right).$
(34)

The representation with this replacement is given in [7]. The equivalence of
two representations (22) and (33) follows from the relation

$\sum_{\mathbf{n}_{q-1}}\left(z^{2}+\mathbf{k}_{q-1}^{2}\right)^{\frac{s+1}{2}}f_{\frac{s+1}{2}}(n_{r}L_{r}\sqrt{z^{2}+\mathbf{k}_{q-1}^{2}})=\frac{V_{q}z^{s+q}}{(2\pi)^{\frac{q-1}{2}}L_{r}}\sum_{\mathbf{n}_{q-1}}\cos(\mathbf{n}_{q-1}\cdot\boldsymbol{\tilde{\alpha}}_{q-1})f_{\frac{s+q}{2}}(zg(\mathbf{L}_{q},\mathbf{n}_{q})),$
(35)

with $s=p+1$ and $z=m$. This relation is proved in [61] by using the Poisson’s
resummation formula. For a massless field, by using the asymptotic for the
modified Bessel function for small argument, one finds

$\left\langle
j^{r}\right\rangle=\frac{\Gamma\left(\frac{D+1}{2}\right)}{\pi^{\frac{D+1}{2}}}eL_{r}\sum_{\mathbf{n}_{q}}^{\prime}\frac{n_{r}\sin\left(\mathbf{n}_{q}\cdot\mathbf{\tilde{\alpha}}_{q}\right)}{g^{D+1}(\mathbf{L}_{q},\mathbf{n}_{q})}.$
(36)

For a single compact dimension this formula coincides with (24). Properties of
the current density described by (22) and (36) will be discussed in Section 6
below.

##  4 Current Density in Locally dS Spacetime with Compact Dimensions

In this section we consider $(D+1)$-dimensional locally dS spacetime with a
part of spatial dimensions compactified to $q$-dimensional torus in planar
(inflationary) coordinates. It is the solution of the Einstein field equations
with the positive cosmological constant $\Lambda$ as the only source of the
$(D+1)$-dimensional gravitation. The standard dS spacetime (without
compactification) is maximally symmetric and is one of the most popular
background geometries in gravity and in field theories. This has several
motivations. First of all, the high degree of symmetry allows to have a
relatively large number of exactly solvable problems. The corresponding
results shed light on the influence of the gravitational field on various
physical processes in more complicated curved backgrounds. In accordance of
the inflationary scenario, the dS spacetime approximates the geometry of the
early Universe and the investigation of the respective effects is an important
step to understand the dynamics of the Universe in the postinflationary stage.
In particular, the quantum fluctuations of fields in the early dS phase of the
expansion serve as seeds for large-scale structure formation in the Universe.
This is currently the most popular mechanism for the formation of large-scale
structures. Another motivation for the importance of dS spacetime is
conditioned by its role in the $\Lambda$CDM model for the cosmological
expansion. In that model the accelerated expansion of the Universe at recent
epoch is sourced by a positive cosmological constant and the dS spacetime
appears as the future attractor of the Universe expansion.

The explicit way to see the symmetries of the dS spacetime is its embedding as
a hyperboloid

$\left(Z^{0}\right)^{2}-\left(Z^{1}\right)^{2}-\cdots-\left(Z^{D+1}\right)^{2}=-a^{2},$
(37)

in $(D+2)$-dimensional Minkowski spacetime with the line element
$ds_{D+2}^{\mathrm{(M)}2}=\left(dZ^{0}\right)^{2}-\left(dZ^{1}\right)^{2}-\cdots-\left(dZ^{D+1}\right)^{2}$.
The parameter $a$ in (37) determines the curvature radius of dS spacetime.
Different coordinate systems have been used to exclude an additional degree of
freedom by using the relation (37). For the following discussion of the
current density we will use the planar coordinates
$(\tau,x^{1},x^{2},\ldots,x^{D})$ which are connected to the coordinates in
the embedding spacetime by the relations (see, for example, [76] in the case
$D=3$)

$\displaystyle Z^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2\tau}\left[a^{2}+(-1)^{i}\left(\sum_{l=1}^{D}(x^{l})^{2}-\tau^{2}\right)\right],\;i=0,1,$
$\displaystyle Z^{l}$ $\displaystyle=$
$\displaystyle\frac{a}{\tau}x^{l-1},\;l=2,\ldots,D+1.$ (38)

For the dS line element this gives
$ds_{D+1}^{\mathrm{(dS)}2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}$, with the metric
tensor

$g_{\mu\nu}(x)=\left(a/\tau\right)^{2}\eta_{\mu\nu}.$ (39)

In inflationary models a part of dS spacetime with the conformal time
coordinate in the range $-\infty<\tau<0$ is employed. For the corresponding
synchronous time coordinate $t$, $-\infty<t<+\infty$, one has
$t=-a\ln|\tau|/a$ and the line element is expressed as

$ds_{D+1}^{\mathrm{(dS)}2}=dt^{2}-\exp\left(\frac{2t}{\alpha}\right)\sum_{i=1}^{D}(dx^{i})^{2}.$
(40)

The metric (39) is the solution of Einstein’s equations with positive
cosmological constant $\Lambda=\frac{D(D-1)}{2a^{2}}$ as the only source of
the gravitational field.

The hyperbolic embedding (37) in the locally Minkowski spacetime with the line
element $ds_{D+2}^{\mathrm{(M)}2}$ works equally well for LdS spacetime with a
$q$-dimensional toroidal subspace covered by the coordinates
$(x^{p+1},x^{p+2},\ldots,x^{D})$. The coordinate $x^{l}$, $l=p+1,\ldots,D$,
varies in the range $0\leq x^{l}\leq L_{l}$. In this case, the subspace with
the coordinates $(Z^{p+2},\ldots,Z^{D+2})$ is compactified to a torus. The
length of the compact dimension $Z^{l+1}$, $l=p+1,\ldots,D$, is given by
$L_{\mathrm{(p)}l}=aL_{l}/\eta=e^{t/a}L_{l}$, with $\eta=|\tau|$. Note that
$L_{\mathrm{(p)}l}$ is the proper length of the compact dimension $x^{l}$ in
the LdS spacetime.

For LdS spacetime with the metric tensor (39) and compact subspace
$\mathbf{x}_{q}=(x^{p+1},x^{p+2},\ldots,x^{D})$, the scalar mode functions can
be presented in the form
$\varphi_{\sigma}^{(+)}(x)=\eta^{D/2}g_{\nu}(k\eta)e^{i\mathbf{k}_{p}\cdot\mathbf{x}_{p}\mathbf{+ik}_{q}\cdot\mathbf{x}_{q}}$,
where $g_{\nu}(y)$ is a cylinder function of the order

$\nu=\left[\frac{D^{2}}{4}-D(D+1)\xi-a^{2}m^{2}\right]^{1/2}.$ (41)

The eigenvalues of the momentum components along compact dimensions are given
by (16). Different choices of the function $g_{\nu}(y)$ correspond to
different vacuum states for a scalar field in dS spacetime. Here we will
investigate the current density in the Bunch-Davies vacuum state [77].

###  4.1 Hadamard Function

For the Bunch-Davies vacuum state the normalized scalar mode functions are
specified by $\sigma=\mathbf{k}=(\mathbf{k}_{p},\mathbf{k}_{q})$ and are
expressed as

$\left\\{\begin{array}[]{l}\varphi_{\mathbf{k}}^{(+)}(x)\\\
\varphi_{\mathbf{k}}^{(-)}(x)\end{array}\right\\}=\left(\frac{a^{1-D}e^{i(\nu-\nu^{\ast})\pi/2}}{2^{p+2}\pi^{p-1}V_{q}}\right)^{\frac{1}{2}}e^{i\mathbf{k}_{p}\cdot\mathbf{x}_{p}\mathbf{+ik}_{q}\cdot\mathbf{x}_{q}}\eta^{D/2}\left\\{\begin{array}[]{l}H_{\nu}^{(1)}(k\eta)\\\
H_{\nu^{\ast}}^{(2)}(k\eta)\end{array}\right\\},$ (42)

where $H_{\nu}^{(1,2)}(y)$ are the Hankel functions and the star stands for
the complex conjugate. Note that, depending on the curvature coupling
parameter and on the mass of the field quanta, the order of the Hankel
functions can be either nonnegative real or purely imaginary. With the modes
(42), the mode sum for the Hadamard function reads

$G(x,x^{\prime})=\frac{\left(\eta\eta^{\prime}\right)^{\frac{D}{2}}e^{i\frac{\pi}{2}(\nu-\nu^{\ast})}}{2^{p+2}\pi^{p-1}V_{q}a^{D-1}}\int
d\mathbf{k}_{p}\,\sum_{\mathbf{n}_{q}}e^{i\mathbf{k}_{p}\cdot\Delta\mathbf{x}_{p}+i\mathbf{k}_{q}\cdot\Delta\mathbf{x}_{q}}[H_{\nu}^{(1)}(k\eta)H_{\nu^{\ast}}^{(2)}(k\eta^{\prime})+H_{\nu}^{(1)}(k\eta^{\prime})H_{\nu^{\ast}}^{(2)}(k\eta)].$
(43)

Applying to the sum over $n_{r}$ the summation formula (20) and assuming that
${\mathrm{Re}\,\nu<1}$, one finds the representation

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{4\left(\eta\eta^{\prime}\right)^{\frac{D}{2}}L_{r}}{\left(2\pi\right)^{\frac{p+3}{2}}V_{q}a^{D-1}}\int_{0}^{\infty}d\lambda\,\lambda\left[I_{-\nu}(\eta\lambda)K_{\nu}(\eta^{\prime}\lambda)+K_{\nu}(\eta\lambda)I_{\nu}(\eta^{\prime}\lambda)\right]\sum_{\mathbf{n}_{q}}e^{in_{r}\alpha_{r}}$
(44) $\displaystyle\times
e^{i\mathbf{k}_{q-1}\cdot\Delta\mathbf{x}_{q-1}}\left(\lambda^{2}+\mathbf{k}_{q-1}^{2}\right)^{\frac{p-1}{2}}f_{\frac{p-1}{2}}(\sqrt{\lambda^{2}+\mathbf{k}_{q-1}^{2}}\sqrt{|\Delta\mathbf{x}_{p}|^{2}+\left(\Delta
x^{r}-n_{r}L_{r}\right)^{2}}),$

where $I_{\nu}(y)$ and $K_{\nu}(y)$ are the modified Bessel functions. Similar
to the case of the Minkowski bulk, the contribution coming from the term
$n_{r}=0$ gives the Hadamard function for the geometry where the $r$th
coordinate is decompactified (spatial topology $R^{p+1}\times T^{q-1}$). The
effects of the compactification of that coordinate are included in the part
with $n_{r}\neq 0$.

For a conformally coupled massless field we have $\nu=1/2$ and

$I_{-\nu}(\eta\lambda)K_{\nu}(\eta^{\prime}\lambda)+K_{\nu}(\eta\lambda)I_{\nu}(\eta^{\prime}\lambda)=\frac{\cosh\left(\lambda\Delta\eta\right)}{\lambda\sqrt{\eta\eta^{\prime}}}.$
(45)

With this function, the integral over $z$ in (44) is evaluated by using the
formula from [78] and we get

$G(x,x^{\prime})=\left(\frac{\eta\eta^{\prime}}{a^{2}}\right)^{\frac{D-1}{2}}G_{\mathrm{M}}(x,x^{\prime}),$
(46)

where the Minkowskian Hadamard function $G_{\mathrm{M}}(x,x^{\prime})$ is
given by (21) with $m=0$. For a conformally coupled massless field this is the
standard relation between two conformally related geometries.

###  4.2 Vacuum Current

For the components of the current density along uncompact dimensions
$x^{\mu}$, $\mu=1,2,\ldots,p$, one has $\partial_{\mu}G(x,x^{\prime})\propto
g_{\mu\alpha}\Delta x^{\alpha}$ and the corresponding expectation values
vanish. By using the properties of the modified Bessel functions we can see
that

$\lim_{x^{\prime}\rightarrow
x}(\partial_{0}-\partial_{0}^{\prime})\left\\{\left(\eta\eta^{\prime}\right)^{\frac{D}{2}}\left[I_{-\nu}(\eta\lambda)K_{\nu}(\eta^{\prime}\lambda)+K_{\nu}(\eta\lambda)I_{\nu}(\eta^{\prime}\lambda)\right]\right\\}=0,$
(47)

and, hence, the charge density vanishes as well. In order to find the
component of the current density along the $r$th compact dimension we use the
representation (44) for the Hadamard function in combination with (7) where
$D_{\mu}=\partial_{\mu}$. In (44), the derivative of the term $n_{r}=0$ with
respect to $x^{r}$ is an odd function of $\Delta x^{r}$ and vanishes in the
coincidence limit. As it has been mentioned before, that term corresponds to
the Hadamard function in the geometry with uncompactified $x^{r}$ and the
corresponding current density vanishes by the symmetry. The part in (44)
induced by the compactification of the direction $x^{r}$ (the terms with
$n_{r}\neq 0$) is finite in the coincidence limit and that limit can be
directly taken in the expression for the VEV. This gives the following
expression for the contravariant component [13]:

$\displaystyle\left\langle j^{r}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{8ea(\eta/a)^{D+2}}{\left(2\pi\right)^{\frac{p+3}{2}}V_{q}}L_{r}^{2}\int_{0}^{\infty}d\lambda\,\lambda\left[I_{-\nu}(\eta\lambda)+I_{\nu}(\eta\lambda)\right]K_{\nu}(\eta\lambda)$
(48)
$\displaystyle\times\sum_{n_{r}=1}^{\infty}n_{r}\sin(n_{r}\tilde{\alpha}_{r})\sum_{\mathbf{n}_{q-1}}\left(\lambda^{2}+\mathbf{k}_{q-1}^{2}\right)^{\frac{p+1}{2}}f_{\frac{p+1}{2}}(n_{r}L_{r}\sqrt{\lambda^{2}+\mathbf{k}_{q-1}^{2}}).$

An alternative representation for the current density is obtained by applying
the formula (35) with $s=p$. This gives

$\displaystyle\left\langle j^{r}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{4eL_{r}}{\left(2\pi\right)^{\frac{D}{2}+1}a^{D+1}}\int_{0}^{\infty}du\,u^{D+1}\left[I_{-\nu}(u)+I_{\nu}(u)\right]K_{\nu}(u)$
(49)
$\displaystyle\times\sum_{\mathbf{n}_{q}}n_{r}\sin(\mathbf{n}_{q}\cdot\boldsymbol{\tilde{\alpha}}_{q})f_{\frac{D}{2}}(ug(\mathbf{L}_{q},\mathbf{n}_{q})/\eta).$

The integral in the right-hand side is evaluated by using the formula

$\int_{0}^{\infty}dz\,z^{\frac{D}{2}+1}\left[I_{\nu}(z)+I_{-\nu}(z)\right]K_{\nu}(z)K_{\frac{D}{2}}(bz)=\frac{\sqrt{2\pi}}{4}b^{\frac{D}{2}}p_{\nu-\frac{1}{2}}^{-\frac{D+1}{2}}\left(\frac{b^{2}}{2}-1\right),$
(50)

where we use the notation

$p_{\alpha}^{-\mu}(u)=\Gamma(\mu-\alpha)\Gamma\left(\mu+\alpha+1\right)\frac{P_{\alpha}^{-\mu}(u)}{\left(u^{2}-1\right)^{\frac{\mu}{2}}},$
(51)

with $P_{\nu}^{\mu}(u)$ being the associated Legendre function of the first
kind. The expression of the function $p_{\alpha}^{-\mu}(u)$ in terms of the
hypergeometric function is given in Appendix A. The result (50) is obtained
from the integral involving the product
$I_{\nu}(z)K_{\nu}(z)K_{\frac{D}{2}}(bz)$ and given in [78]. That integral is
expressed in terms of the sum of two hypergeometric functions. The
contribution of the second function is canceled in evaluating the integral
(50). Then, we express the hypergeometric function in terms of the associated
Legendre functions. By taking into account (50) in (49) the current density is
expressed as

$\left\langle
j^{r}\right\rangle=\frac{eL_{r}}{\left(2\pi\right)^{\frac{D+1}{2}}a^{D+1}}\sum_{\mathbf{n}_{q}}n_{r}\sin(\mathbf{n}_{q}\cdot\boldsymbol{\tilde{\alpha}}_{q})p_{\nu-\frac{1}{2}}^{-\frac{D+1}{2}}\left(\frac{g^{2}(\mathbf{L}_{q},\mathbf{n}_{q})}{2\eta^{2}}-1\right).$
(52)

In both the formulas (49) and (52) we can make the replacement (34). Note that
one has the property
$p_{\nu-\frac{1}{2}}^{-\frac{D+1}{2}}\left(u\right)=p_{-\nu-\frac{1}{2}}^{-\frac{D+1}{2}}\left(u\right)$
and the expression on the right-hand side is real for both real and purely
imaginary values of $\nu$.

##  5 AdS Spacetime with Compact Dimensions

Now we turn to the LAdS spacetime with a part of spatial dimensions
compactified to a torus. It obeys the $(D+1)$-dimensional Einstein equations
with the negative cosmological constant $\Lambda$. The usual AdS spacetime is
maximally symmetric and appears as the ground state in supergravity and in
string theories. That was the reason for the early interest in the AdS
physics. The interest to those investigations was further increased related to
two exciting developments in modern theoretical physics. The first one, dubbed
as AdS/CFT correspondence (see, e.g., [79]-[82]), establishes duality between
the supergravity and string theory on the AdS bulk and conformal field theory
(CFT) on its boundary. This duality is a unique way to investigate strong
coupling effect in one theory by mapping it on the dual theory. A number of
examples can be found in the literature, including those with applications in
condensed matter physics. The second development, with the AdS spacetime as a
background geometry, corresponds to braneworld models of the Randall-Sundrum
type [83] with large extra dimensions. In the corresponding setup the standard
model fields are localized in a 4-dimensional hypersurface (brane) on
background of higher dimensional AdS spacetime. The braneworld models provide
a geometrical solution to the hierarchy problem between the electroweak and
Planck energy scales and naturally arise in the string/M theory context. They
present a novel setting in considerations of various phenomenological and
cosmological issues, in particular, the generation of cosmological constant
localized on the brane.

By analogy of the dS bulk, it is convenient to visualize the AdS spacetime as
a hyperboloid

$\left(Z^{0}\right)^{2}-\sum_{i=1}^{D}\left(Z^{i}\right)^{2}+\left(Z^{D+1}\right)^{2}=-a^{2},$
(53)

where the line element of the embedding $(D+2)$-dimensional flat spacetime is
given by
$ds_{D+2}^{2}=\left(dZ^{0}\right)^{2}-\sum_{i=1}^{D}\left(dZ^{i}\right)^{2}+\left(dZ^{D+1}\right)^{2}$.
The Poincaré coordinates $(t,x^{1}=z,x^{2},\ldots,x^{D})$ are introduced by
the relations

$\displaystyle Z^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2z}\left[a^{2}+(-1)^{i}\left(\sum_{l=1}^{D}(x^{l})^{2}-t^{2}\right)\right],\;i=0,1,$
$\displaystyle Z^{l}$ $\displaystyle=$
$\displaystyle\frac{a}{z}x^{l},\;Z^{D+1}=\frac{a}{z}t,\;l=2,\ldots,D.$ (54)

In those coordinates the metric tensor of the AdS spacetime is expressed as

$g_{\mu\nu}=\frac{a^{2}}{z^{2}}\eta_{\mu\nu},$ (55)

with $0\leq z<\infty$. The hypersurfaces $z=0$ and $z=\infty$ present the AdS
boundary and the horizon, respectively. The proper distance along the
direction $x^{1}$ is measured by the coordinate $y=a\ln(z/a)$,
$-\infty<y<+\infty$, in terms of which the line element is written as

$ds^{2}=\exp\left(-\frac{2y}{a}\right)\left[dt^{2}-\sum_{l=2}^{D}(dx^{l})^{2}\right]-dy^{2}.$
(56)

Here we consider the LAdS geometry with the coordinates
$(x^{p+1},x^{p+2},\ldots,x^{D})$ compactified to a torus as described in
section 2. It can be embedded as a hyperboloid (53) in the spacetime with
coordinates $(Z^{0},Z^{1},\ldots,Z^{D+1})$, where the coordinate $Z^{l}$,
$l=p+1,\ldots,D$, is compactified to a circle with the length
$L_{\mathrm{(p)}l}=aL_{l}/z=e^{-y/a}L_{l}$. The latter is the proper length
for the compact dimension in LAdS. It is exponentially small near the horizon.

The scalar mode functions in the coordinates corresponding to the metric
tensor (55) and obeying the periodicity conditions (13) are written in the
form $\varphi_{\sigma}^{(\pm)}(x)=e^{\mp i\omega
t+i\mathbf{k}_{p-1}\mathbf{x}_{p-1}+\mathbf{k}_{q}\mathbf{x}_{q}}f(z)$, with
$\mathbf{x}_{p-1}=(x^{2},\ldots,x^{p})$,
$\mathbf{k}_{p-1}=(k_{2},\ldots,k_{p})$. The equation for the function $f(z)$
is obtained from the field equation. The corresponding solution is presented
as $z^{D/2}\left[c_{1}J_{\nu_{+}}(\lambda z)+c_{2}Y_{\nu_{+}}(\lambda
z)\right]$, where $J_{\nu}(\lambda z)$ and $Y_{\nu}(\lambda z)$ are the Bessel
and Neumann functions,
$\lambda^{2}=\omega^{2}-\mathbf{k}_{p-1}^{2}-\mathbf{k}_{q}^{2}$, and

$\nu_{+}=\left[\frac{D^{2}}{4}-D(D+1)\xi+a^{2}m^{2}\right]^{1/2}.$ (57)

For the stability of the Poincaré vacuum the parameter $\nu_{+}$ should be
real [84, 85, 86]. For $\nu_{+}\geq 1$ from the normalizability condition it
follows that $c_{2}=0$ in the linear combination of the cylinder functions.
For $0\leq\nu_{+}<1$ the modes with $c_{2}\neq 0$ are normalizable. In this
case one of the coefficients in the linear combination is determined from the
normalization condition and the second one is fixed by the boundary condition
on the AdS boundary. The general class of allowed boundary condition has been
discussed in [87, 88]. Here we will consider the special case of Dirichlet
boundary condition for which $c_{2}=0$ and
$f(z)=c_{1}z^{D/2}J_{\nu_{+}}(\lambda z)$. The normalized mode functions are
expressed as

$\varphi_{\sigma}^{(\pm)}(x)=\left(\frac{\pi^{-p}a^{1-D}\lambda}{2^{p+1}\omega
V_{q}}\right)^{\frac{1}{2}}z^{\frac{D}{2}}e^{i\mathbf{k}_{p-1}\mathbf{x}_{p-1}+i\mathbf{k}_{q}\mathbf{x}_{q}\mp
i\omega t}J_{\nu_{+}}(\lambda z).$ (58)

The modes are specified by the set
$\sigma=(\lambda,\mathbf{k}_{p-1},\mathbf{k}_{q})$ with $0\leq\lambda<\infty$
and the energy is given by
$\omega=\sqrt{\lambda^{2}+\mathbf{k}_{p-1}^{2}+\mathbf{k}_{q}^{2}}$.

With the mode functions (58), the Hadamard function takes the form

$G(x,x^{\prime})=\frac{a^{1-D}(zz^{\prime})^{\frac{D}{2}}}{(2\pi)^{p-1}V_{q}}\sum_{\mathbf{n}_{q}}\int
d\mathbf{k}_{p-1}\,e^{i\mathbf{k}_{p-1}\cdot\Delta\mathbf{x}_{p-1}+i\mathbf{k}_{q}\cdot\Delta\mathbf{x}_{q}}\int_{0}^{\infty}d\lambda\,\frac{\lambda}{\omega}J_{\nu_{+}}(\lambda
z)J_{\nu_{+}}(\lambda z^{\prime})\cos(\omega\Delta t).$ (59)

Similar to the cases of the locally Minkowski and dS geometries, we aplly to
the series over $n_{r}$ the summation formula (20) to see the representation

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2(zz^{\prime})^{\frac{D}{2}}L_{r}}{(2\pi)^{\frac{p+1}{2}}V_{q}a^{D-1}}\sum_{\mathbf{n}_{q}}e^{i\mathbf{k}_{q-1}\cdot\Delta\mathbf{x}_{q-1}+in_{r}\tilde{\alpha}_{r}}\int_{0}^{\infty}d\lambda\,\lambda
J_{\nu_{+}}(\lambda z)J_{\nu_{+}}(\lambda z^{\prime})$ (60)
$\displaystyle\times\left(\lambda^{2}+\mathbf{k}_{q-1}^{2}\right)^{\frac{p-1}{2}}f_{\frac{p-1}{2}}\left(\sqrt{\lambda^{2}+\mathbf{k}_{q-1}^{2}}\sqrt{|\Delta\mathbf{x}_{p-1}|^{2}+\left(\Delta
x^{r}-n_{r}L_{r}\right)^{2}-\left(\Delta t\right)^{2}}\right).$

The term with $n_{r}=0$ in this representation corresponds to the Hadamard
function in the geometry where the $r$th dimension is decompactified.

Another representation for the function (59) is obtained in [14] by using an
integral representation for the ratio $\cos(\omega\Delta t)/\omega$. The
integral over $\lambda$ is expressed in terms of the modified Bessel function.
Integrating over the components of the momentum along uncompact dimensions and
applying to the series the Poisson resummation formula the Hadamard function
is expressed as

$G(x,x^{\prime})=\frac{a^{1-D}}{(2\pi)^{D/2}}\sum_{\mathbf{n}_{q}}e^{i\tilde{\mathbf{\alpha}}\cdot\mathbf{n}_{q}}\int_{0}^{\infty}dx\,x^{D/2-1}I_{\nu}(x)e^{-v_{\mathbf{n}_{q}}x},$
(61)

where

$v_{\mathbf{n}_{q}}=1+\frac{1}{2zz^{\prime}}\left[(\Delta
z)^{2}+(\Delta\mathbf{x}_{p-1})^{2}+\sum_{i=p+1}^{D}\left(\Delta
x^{i}-L_{i}n_{i}\right)^{2}-(\Delta t)^{2}\right],$ (62)

and $\Delta z=z-z^{\prime}$. By using the result from [78] for the integral in
(61), the following representation is obtained:

$G(x,x^{\prime})=\frac{2a^{1-D}}{(2\pi)^{\frac{D+1}{2}}}\sum_{\mathbf{n}_{q}}e^{i\tilde{\mathbf{\alpha}}\cdot\mathbf{n}_{q}}q_{\nu_{+}-\frac{1}{2}}^{\frac{D-1}{2}}(v_{\mathbf{n}_{q}}),$
(63)

where the function $q_{\alpha}^{\mu}(x)$ is expressed in terms of the
associated Legendre function of the second kind, $Q_{\alpha}^{\mu}(x)$ (for
the expression in terms of the hypergeometric function see Appendix A):

$q_{\nu-\frac{1}{2}}^{\mu}(x)=\frac{e^{-i\pi\mu}Q_{\nu-\frac{1}{2}}^{\mu}(x)}{(x^{2}-1)^{\frac{\mu}{2}}}.$
(64)

The contribution in (63) corresponding to the term $\mathbf{n}_{q}=0$ presents
the Hadamard function in AdS spacetime with Poincaré coordinates
$-\infty<x^{\mu}<+\infty$ for $\mu=2,3,\ldots,D$. The divergences in the
coincidence limit are contained in that part. The topological contribution
with $\mathbf{n}_{q}\neq 0$ is finite in that limit and can be directly used
in evaluating the current density.

As before, the charge density and the components of the current density along
uncompact dimensions vanish: $\left\langle j^{\mu}\right\rangle=0$ for
$\mu=0,1,\ldots,p$. Combining the formulas (7) and (60), for the component
along the $r$th dimension we find

$\displaystyle\langle j^{r}\rangle$ $\displaystyle=$
$\displaystyle\frac{4ez^{D+2}L_{r}^{2}}{(2\pi)^{\frac{p+1}{2}}a^{D+1}V_{q}}\sum_{n_{r}=1}^{\infty}n_{r}\sin\left(n_{r}\tilde{\alpha}_{r}\right)\sum_{\mathbf{n}_{q-1}}\int_{0}^{\infty}d\lambda\,\lambda
J_{\nu_{+}}^{2}(\lambda z)$ (65)
$\displaystyle\times\left(\lambda^{2}+\mathbf{k}_{q-1}^{2}\right)^{\frac{p+1}{2}}f_{\frac{p+1}{2}}\left(n_{r}L_{r}\sqrt{\lambda^{2}+\mathbf{k}_{q-1}^{2}}\right).$

By applying the formula (35) with $s=p$ to the series over $\mathbf{n}_{q-1}$
one gets

$\langle
j^{r}\rangle=\frac{4ez^{D+2}L_{r}}{(2\pi)^{\frac{D}{2}}a^{D+1}}\sum_{n_{r}=1}^{\infty}n_{r}\sin\left(n_{r}\tilde{\alpha}_{r}\right)\sum_{\mathbf{n}_{q-1}}\cos(\mathbf{n}_{q-1}\cdot\boldsymbol{\tilde{\alpha}}_{q-1})\int_{0}^{\infty}d\lambda\,\lambda^{D+1}J_{\nu_{+}}^{2}(\lambda
z)f_{\frac{D}{2}}(\lambda g(\mathbf{L}_{q},\mathbf{n}_{q})).$ (66)

The integral in (66) is expressed in terms of the function (64) [89] (note
that there is a misprint in the similar integral given in [78]) and the
current density is presented in the form [14]

$\langle
j^{r}\rangle=\frac{4eL_{r}}{(2\pi)^{\frac{D+1}{2}}a^{D+1}}\sum_{n_{r}=1}^{\infty}n_{r}\sin(\tilde{\alpha}_{r}n_{r})\sum_{\mathbf{n}_{q-1}}\,\cos(\tilde{\mathbf{\alpha}}_{q-1}\cdot\mathbf{n}_{q-1})q_{\nu_{+}-\frac{1}{2}}^{\frac{D+1}{2}}\left(1+\frac{g^{2}(\mathbf{L}_{q},\mathbf{n}_{q})}{2z^{2}}\right).$
(67)

This representation could be directly obtained by using the Hadamard function
in the form (63) and the relation
$\partial_{x}q_{\alpha}^{\mu}(x)=-q_{\alpha}^{\mu+1}(x)$ for the function
(64).

##  6 Features of the Current Density

###  6.1 General Features

The physical component of the charge density is given by $\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle=\sqrt{|g_{rr}|}\left\langle
j^{r}\right\rangle$. It determines the charge flux through the spatial
hypersurface $x^{r}=\mathrm{const}$, expressed as $n_{r}\left\langle
j^{r}\right\rangle$, with $n_{r}=\sqrt{|g_{rr}|}$ being the corresponding
normal. The expressions obtained above for the $r$th component of the current
density can be combined as

$\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle=\frac{2eL_{\mathrm{(p)}r}}{\left(2\pi\right)^{\frac{D+1}{2}}a^{D+1}}\sum_{\mathbf{n}_{q}}n_{r}\sin\left(\mathbf{n}_{q}\cdot\mathbf{\tilde{\alpha}}_{q}\right)F_{D}(am,g(\mathbf{L}_{\mathrm{(p)}q}/a,\mathbf{n}_{q})),$
(68)

where we have defined the function

$\displaystyle F_{D}(am,x)$ $\displaystyle=$
$\displaystyle\left(am\right)^{D+1}f_{\frac{D+1}{2}}(amx),\mathrm{\;for\;LM},$
$\displaystyle F_{D}(am,x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}du\,u^{D+1}Z(u)f_{\frac{D}{2}}(ux),\mathrm{\;for\;LdS,LAdS},$
(69)

with

$\displaystyle Z(u)$ $\displaystyle=$
$\displaystyle\left[I_{-\nu}(u)+I_{\nu}(u)\right]K_{\nu}(u),\mathrm{\;for\;LdS},$
$\displaystyle Z(u)$ $\displaystyle=$ $\displaystyle\pi
J_{\nu_{+}}^{2}(u),\mathrm{\;for\;LAdS}.$ (70)

Alternative expressions for locally dS and AdS geometries are obtained from
(52) and (67):

$\displaystyle F_{D}(am,x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}p_{\nu-\frac{1}{2}}^{-\frac{D+1}{2}}\left(x^{2}/2-1\right),\mathrm{\;for\;LdS},$
$\displaystyle F_{D}(am,x)$ $\displaystyle=$ $\displaystyle
q_{\nu_{+}-\frac{1}{2}}^{\frac{D+1}{2}}\left(x^{2}/2+1\right),\mathrm{\;for\;LAdS}.$
(71)

For even values of the spatial dimension $D$ the functions (71) are expressed
in terms of elementary functions. The corresponding representations are given
by the formulas (99) and (100). In odd number of spatial dimensions the
expressions for the functions (71) in terms of the Legendre functions
$P_{\nu-\frac{1}{2}}(u)$ and $Q_{\nu_{+}-\frac{1}{2}}(u)$ are given by (101).

In asymptotic analysis for some limiting cases it is more convenient to use
the representations (22), (48), and (65). For LdS and LAdS geometries the
corresponding formulas can be combined by using (70):

$\displaystyle\left\langle j_{\mathrm{(p)}}^{r}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{8ea^{-p-2}L_{\mathrm{(p)}r}^{2}}{\left(2\pi\right)^{\frac{p+3}{2}}V_{q}^{\mathrm{(p)}}}\sum_{n_{r}=1}^{\infty}n_{r}\sin(n_{r}\tilde{\alpha}_{r})\sum_{\mathbf{n}_{q-1}}\int_{0}^{\infty}du\,uZ(u)$
(72)
$\displaystyle\times\left(u^{2}+a^{2}\mathbf{k}_{q-1}^{\mathrm{(p)}2}\right)^{\frac{p+1}{2}}f_{\frac{p+1}{2}}(n_{r}L_{\mathrm{(p)}r}/a\sqrt{u^{2}+a^{2}\mathbf{k}_{q-1}^{\mathrm{(p)}2}}).$

with the notation

$\mathbf{k}_{q-1}^{\mathrm{(p)}2}=\sum_{i=p+1,\neq r}^{D}\left(\frac{2\pi
n_{i}+\tilde{\alpha}_{i}}{L_{\mathrm{(p)}i}}\right)^{2}.$ (73)

Note that the vector $\mathbf{k}_{q-1}^{\mathrm{(p)}}$ is the physical
momentum in the compact subspace with the set of coordinates
$(x^{p+1},\ldots,x^{r-1},x^{r+1},\ldots,x^{D})$.

First of all, we see that the current density along the $r$th dimension is an
even periodic function of the parameters $\tilde{\alpha}_{i}$, $i\neq r$, with
the period $2\pi$ and an odd periodic function of $\tilde{\alpha}_{r}$ with
the same period. This corresponds to the periodicity with respect to the
magnetic flux with the period equal to the flux quantum. From the formula (68)
it follows that the physical component $n_{r}\left\langle j^{r}\right\rangle$
depends on the lengths of compact dimensions and on the coordinates through
the ratios $L_{\mathrm{(p)}i}/a$, $i=p+1,\ldots,D$. They present the proper
lengths of compact dimensions measured in units of the curvature radius. This
feature is related to the maximal symmetry of the dS and AdS spacetimes.

The numerical examples below will be given for models with a single compact
dimension $x^{D}$ having the length $L=L_{D}$. In Figure 1 we present the
dependence of the respective current density, multiplied by
$L_{\mathrm{(p)}}^{D}/e$, as a function of the parameter
$\tilde{\alpha}_{D}/(2\pi)$ and of the proper length of the compact dimension
$L_{\mathrm{(p)}}=L_{\mathrm{(p)}D}$ in the LM spacetime with $D=4$. In the
numerical evaluation we have taken $ma=0.5$. In the LM bulk the current
density does not depend on the curvature coupling parameter and
$L_{\mathrm{(p)}}=L$. As it follows from (24), for a massless field the
dimensionless combination $L_{\mathrm{(p)}}^{D}\left\langle
j_{\mathrm{(p)}}^{D}\right\rangle/e$ does not depend on $L_{\mathrm{(p)}}$.

Figure 1: The current density in the $D=4$ LM spacetime versus the parameter
$\tilde{\alpha}_{D}/(2\pi)$ and the proper length of the compact dimension (in
units of $a$). The graph is plotted for $ma=0.5$.

The current densities for the $D=4$ LdS and LAdS background geometries and for
$ma=0.5$ are plotted in Figures 2 and 3. The left and right panels on both
figures correspond to conformally and minimally coupled fields.

|
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Figure 2: The current density in the $D=4$ LdS spacetime, multiplied by $L_{\mathrm{(p)}}^{D}/e$, versus the parameter $\tilde{\alpha}_{D}/(2\pi)$ and the proper length of the compact dimension for $ma=0.5$. The left and right panels correspond to conformally and minimally coupled fields, respectively.  |
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Figure 3: The same as in Figure 2 for the LAdS bulk.

###  6.2 Conformal Coupling and Minkowskian Limit

Let us consider special cases of general formulas. For a conformally coupled
massless field one has $\xi=\xi_{D}$ and $\nu=\nu_{+}=1/2$. The current
density for the Minkowskian case does not depend on the curvature coupling
parameter and from (33) we get

$\left\langle
j^{r}\right\rangle_{\mathrm{LM}}=eL_{r}\frac{\Gamma\left(\frac{D+1}{2}\right)}{\pi^{\frac{D+1}{2}}}\sum_{\mathbf{n}_{q}}\frac{n_{r}\sin\left(\mathbf{n}_{q}\cdot\mathbf{\tilde{\alpha}}_{q}\right)}{g^{D+1}(\mathbf{L}_{q},\mathbf{n}_{q})}.$
(74)

The corresponding functions $F_{D}(am,x)$ for dS and AdS geometries are
obtained from (69) by taking into account that
$\left[I_{-\nu}(u)+I_{\nu}(u)\right]K_{\nu}(u)=1/u$ for $\nu=1/2$ and
$J_{\nu_{+}}(u)=\sqrt{2/\pi u}\sin u$ for $\nu_{+}=1/2$. The integrals are
evaluated by using the formulas from [78] and we get

$\displaystyle\left\langle j^{r}\right\rangle_{\mathrm{LdS}}$ $\displaystyle=$
$\displaystyle\left(\frac{\eta}{a}\right)^{D+1}\left\langle
j^{r}\right\rangle_{\mathrm{LM}},$ $\displaystyle\left\langle
j^{r}\right\rangle_{\mathrm{LAdS}}$ $\displaystyle=$
$\displaystyle\left(\frac{z}{a}\right)^{D+1}\left\langle
j^{r}\right\rangle_{\mathrm{LM}}^{(1)},$ (75)

where

$\left\langle
j^{r}\right\rangle_{\mathrm{LM}}^{(1)}=eL_{r}\frac{\Gamma\left(\frac{D+1}{2}\right)}{\pi^{\frac{D+1}{2}}}\sum_{\mathbf{n}_{q}}n_{r}\sin\left(\mathbf{n}_{q}\cdot\mathbf{\tilde{\alpha}}_{q}\right)\left[\frac{1}{g^{D+1}(\mathbf{L}_{q},\mathbf{n}_{q})}-\frac{1}{\left(4z^{2}+g^{2}(\mathbf{L}_{q},\mathbf{n}_{q})\right)^{\frac{D+1}{2}}}\right].$
(76)

The result (75) for the dS bulk was expected from the conformal relation
between the problems in the Minkowski and dS geometries with the same range of
the spatial coordinates and between the Bunch-Davies and Minkowski vacua. For
the AdS bulk the contribution of the first term in the square brackets of (76)
will give the Minkowskian current density multiplied by the conformal factor
$(z/a)^{D+1}$. The presence of the part coming from the second term in the
square brackets is related to the boundary condition on the AdS boundary at
$z=0$. Because of that condition the problem on the AdS bulk for a conformally
coupled massless field is conformally related to the corresponding problem in
the Minkowski bulk with an additional boundary at $z=0$ with Dirichlet
boundary condition for the field. The VEV (76) is the current density for a
conformally coupled massless field in the region $0<z<\infty$ of the locally
Minkowski bulk with Dirichlet boundary at $z=0$.

In Figure 4, for a massless field, we have plotted the dependence of the
ratios $\left\langle j^{D}\right\rangle_{\mathrm{LAdS}}/\left\langle
j^{D}\right\rangle_{\mathrm{LM}}$ and $\left\langle
j^{D}\right\rangle_{\mathrm{LdS}}/\left\langle
j^{D}\right\rangle_{\mathrm{LM}}$ on the proper length
$L_{\mathrm{(p)}}=L_{\mathrm{(p)}D}$ of a single compact dimension (in units
of the curvature radius $a$). It is assumed that the compact dimension has the
same proper length in LAdS, LdS and LM spacetimes. The left and right panels
correspond to conformally and minimally coupled fields, respectively, and the
numbers near the curves present the values of the respective spatial
dimension. For a conformally coupled field $\left\langle
j^{D}\right\rangle_{\mathrm{LdS}}/\left\langle
j^{D}\right\rangle_{\mathrm{LM}}=1$ and only the case of the LAdS bulk is
depicted on the left panel. The full and dashed curves on the right panel
correspond to the LAdS and LdS geometries, respectively. As seen from the
graphs, for massless fields the decay of the current density, as a function of
the proper length of the compact dimension, is stronger in the LAdS spacetime
(compared to the case of the LM bulk). For a minimally coupled field in the
LdS geometry the fall-off of the current density is stronger in the LM
spacetime. For small values of the proper length compared to the curvature
radius the effect of the gravitational field is weak and the ratio
$\left\langle j^{D}\right\rangle/\left\langle
j^{D}\right\rangle_{\mathrm{LM}}$ tends to 1. All these features will be
confirmed below by asymptotic analysis.

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Figure 4: The left panel presents the ratio of the current densities for a
conformally coupled massless scalar field in LAdS and LM spacetimes, with a
single compact dimension of the proper length
$L_{\mathrm{(p)}}=L_{\mathrm{(p)}D}$, versus the ratio $L_{\mathrm{(p)}}/a$.
On the right panel the ratio $\left\langle j^{D}\right\rangle/\left\langle
j^{D}\right\rangle_{\mathrm{LM}}$ is plotted for a minimally coupled massless
scalar field in LAdS (full curves, $\left\langle
j^{D}\right\rangle=\left\langle j^{D}\right\rangle_{\mathrm{LAdS}}$) and LdS
(dashed curves, $\left\langle j^{D}\right\rangle=\left\langle
j^{D}\right\rangle_{\mathrm{LdS}}$). The numbers bear the curves present the
corresponding spatial dimension.

Now let us check the Minkowskian limit for dS and AdS geometries. As seen from
(40) and (56), it is obtained taking $a\rightarrow\infty$ for fixed spacetime
coordinates $(t,x^{1},\ldots,x^{D})$ and $(t,y,x^{2},\ldots,x^{D})$ for the dS
and AdS cases, respectively. For large curvature radius one has $\nu\approx
iam$, $\eta\approx a-t$ in LdS bulk and $\nu_{+}\approx am$, $z\approx a+y$
for LAdS. In the case of LAdS we need the asymptotic expression of the
function $q_{\nu_{+}-1/2}^{(D+1)/2}\left(u^{2}/2+1\right)$ for $\nu_{+}\gg 1$
and $u\ll 1$. That expression is obtained by using the uniform asymptotic
expression of the associated Legendre function of the second kind for large
degree and fixed order, given in [90]. With that asymptotic, it can be checked
that in the limit under consideration

$q_{\nu_{+}-1/2}^{\frac{D+1}{2}}\left(1+\frac{g^{2}(\mathbf{L}_{\mathrm{(p)}q},\mathbf{n}_{q})}{2a^{2}}\right)\approx\left(am\right)^{D+1}f_{\frac{D+1}{2}}\left(mg(\mathbf{L}_{q},\mathbf{n}_{q})\right),$
(77)

confirming the transition to the Minkowskian result. In the case of LdS it is
convenient to use the representation (69) for the function $F_{D}(am,x)$. In
the limit at hand $\nu\approx iam$, $am\gg 1$. The corresponding uniform
asymptotic expansions for the functions $I_{\pm\nu}(u)$ and $K_{\nu}(u)$ can
be found, for example, in [91, 92]. From those expansions it can be seen that
the dominant contribution to the integral in the expression for $F_{D}(am,x)$
comes from the region $u>am$, where

$\left[I_{\nu}(u)+I_{-\nu}(u)\right]K_{\nu}(u)\sim\frac{1}{\sqrt{u^{2}-a^{2}m^{2}}}.$
(78)

with $\nu\approx iam$. The respective integral is evaluated by using the
formula

$\int_{0}^{\infty}du\,\left(u^{2}+b^{2}\right)^{\mu}f_{\mu}(c\sqrt{u^{2}+b^{2}})=\sqrt{\frac{\pi}{2}}b^{2\mu+1}f_{\mu+\frac{1}{2}}(cb),$
(79)

and we see that to the leading order

$F_{D}(am,x)\approx\left(ma\right)^{D+1}f_{\frac{D+1}{2}}(max),$ (80)

which coincides with the Minkowskian result.

###  6.3 Large and Small Proper Lengths of Compact Dimensions

For small values of the proper length of the $r$th compact dimension,
$L_{\mathrm{(p)}r}\ll a,1/m$, we first consider the contribution in (68) of
the terms for which at least one of $n_{i}$, $i\neq r$, is different from
zero. For that part the dominant contribution to the series over $n_{r}$ comes
from the terms with large values $|n_{r}|$ and we replace the corresponding
summation by the integration. The corresponding integral involving the product
of the sin and Macdonald functions is evaluated by using the formula from [89]
and is expressed in terms of the Macdonald function with large argument. By
using the corresponding asymptotic we see that the contribution of the term
for which at least one of $n_{i}$, $i\neq r$, is not zero is suppressed by the
factor
$\exp[-g(\mathbf{L}_{\mathrm{(p)}q-1}/a,\mathbf{n}_{q-1})\tilde{\alpha}_{r}a/L_{\mathrm{(p)}r}]$,
where

$g^{2}(\mathbf{L}_{\mathrm{(p)}q-1}/a,\mathbf{n}_{q-1})=\sum_{i=p+1,\neq
r}^{D}n_{i}^{2}\frac{L_{i}^{2}}{a^{2}}.$ (81)

For the contribution of the terms with $n_{i}=0$, $i\neq r$, in (68) the
argument $x$ of the function $F_{D}(am,x)$ is small. By using the
corresponding asymptotic for the Macdonald function we see that in the case of
LM bulk

$F_{D}(am,x)\approx\frac{2^{\frac{D-1}{2}}}{x^{D+1}}\Gamma\left(\frac{D+1}{2}\right),\;x\ll
1.$ (82)

In the cases of the LdS and LAdS geometries we note that the main contribution
to the integrals in (69) comes from the region with large values of $u$. By
using the respective approximations for the functions
$\left[I_{-\nu}(u)+I_{\nu}(u)\right]K_{\nu}(u)$ and $J_{\nu_{+}}^{2}(u)$ and
evaluating the integrals, we can see that the corresponding asymptotics are
given by the same expression (82). Hence, in the limit $L_{\mathrm{(p)}r}\ll
a,1/m$ the dominant contribution to the current density comes from the modes
with $n_{i}=0$, $i\neq r$, and to the leading order

$\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle\approx\frac{2e\Gamma\left(\frac{D+1}{2}\right)}{\pi^{\frac{D+1}{2}}L_{\mathrm{(p)}r}^{D}}\sum_{n_{r}=1}^{\infty}\frac{\sin\left(n_{r}\tilde{\alpha}_{r}\right)}{n_{r}^{D}}.$
(83)

The expression in the right-hand side presents the current density for a
massless scalar field in the LM spacetime with spatial topology $R^{D-1}\times
S^{1}$ with a single compact dimension $x^{r}$ with length
$L_{r}=L_{\mathrm{(p)}r}$. For small values of $L_{\mathrm{(p)}r}$ the
dominant contribution to the VEVs comes from the vacuum fluctuations with
small values of the wavelength (compared to the curvature radius) and the
effects of gravity are weak.

It is expected that the effects of gravity on the vacuum currents will be
essential for proper lengths of compact dimensions of the order or larger than
the curvature radius. We start the consideration for large values of the
lengths with the LM case, assuming that $L_{r}$ is much larger than the other
length scales of the model. From the formula (22) it follows that the dominant
contribution comes from the modes with $n_{i}=0$, $i\neq r$, for which
$\omega_{\mathbf{n}_{q-1}}=\omega_{0r}=\sqrt{k_{\mathbf{n}_{q-1}}^{(0)2}+m^{2}}$
with

$k_{\mathbf{n}_{q-1}}^{(0)2}=\sum\nolimits_{l=p+1,\neq
r}^{D}\tilde{\alpha}_{l}^{2}/L_{l}^{2}.$ (84)

The beahvior of the current density is essentially different depending whether
$\omega_{0r}$ is zero or not. In the first case the leading term in the
current density is given as

$\left\langle
j^{r}\right\rangle_{\mathrm{LM}}\approx\frac{2e\Gamma(\frac{p}{2}+1)}{\pi^{\frac{p}{2}+1}L_{r}^{p}V_{q}}\sum_{n_{r}=1}^{\infty}\frac{\sin(n_{r}\tilde{\alpha}_{r})}{n_{r}^{p+1}}.$
(85)

Comparing with (83) we see that the right-hand side of (85), multiplied by
$V_{q-1}=V_{q}/L_{r}$ presents the current density for a massless scalar field
in $(p+2)$-dimensional LM spacetime with spatial topology $R^{p}\times S^{1}$
having a single compact dimension $x^{r}$. For $\omega_{0r}\neq 0$ the
dominant contribution to the current density is induced by the mode with
$n_{r}=1$ and, to the leading order,

$\left\langle
j^{r}\right\rangle_{\mathrm{LM}}\approx\frac{2e\sin(\tilde{\alpha}_{r})\omega_{0r}^{\frac{p+1}{2}}}{(2\pi)^{\frac{p+1}{2}}L_{r}^{\frac{p-1}{2}}V_{q}}e^{-L_{r}\omega_{0r}}.$
(86)

In particular, for the model with a single compact dimension $x^{D}$ one has
$p=D-1$ and the asymptotic (86) takes the form

$\left\langle
j^{D}\right\rangle_{\mathrm{LM}}\approx\frac{2e\sin(\tilde{\alpha}_{D})m^{\frac{D}{2}}}{(2\pi
L_{D})^{\frac{D}{2}}}e^{-mL_{D}},$ (87)

where $mL_{D}\gg 1$.

For LdS and LAdS geometries and for large values of the proper length
$L_{\mathrm{(p)}r}$ it is more convenient to use the representations (48) and
(65). The dominant contribution comes from the term in the summation with
$\mathbf{n}_{q-1}=0$ ($n_{l}=0$ for $l\neq r$) and from the integration region
near the lower limit. Two cases should be considered separately. The first one
corresponds to the phases $\tilde{\alpha}_{i}=0$, $i\neq r$. With these values
and for LAdS bulk and for LdS bulk in the case $\nu>0$ the leading order term
is expressed as

$\langle
j_{\mathrm{(p)}}^{r}\rangle\approx\frac{4ea^{1+2\mu}B_{D}(am)}{\pi^{\frac{p+1}{2}}V_{q}^{\mathrm{(p)}}L_{\mathrm{(p)}r}^{p+1+2\mu}}\Gamma\left(\frac{p+3}{2}+\mu\right)\sum_{n_{r}=1}^{\infty}\frac{\sin\left(n_{r}\tilde{\alpha}_{r}\right)}{n_{r}^{p+2+2\mu}},$
(88)

where

$\mu=\left\\{\begin{array}[]{cc}-\nu,&\mathrm{for\;LdS}\\\
\nu_{+},&\mathrm{for\;LAdS}\end{array}\right.,$ (89)

and

$B_{D}(am)=\left\\{\begin{array}[]{cc}\Gamma\left(\nu\right)/(2\pi),&\mathrm{for\;LdS}\\\
1/\Gamma\left(\nu_{+}+1\right),&\mathrm{for\;LAdS}\end{array}\right..$ (90)

For LdS geometry and for imaginary values of $\nu$, $\nu=i|\nu|$, by similar
calculations the leading term is presented as

$\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle\approx\frac{4eC(ma)\eta^{D+1}}{\pi^{\frac{p+3}{2}}V_{q}a^{D}L_{r}^{p+1}}\sum_{n_{r}=1}^{\infty}\frac{\sin(n_{r}\tilde{\alpha}_{r})}{n_{r}^{p+2}}\cos\left[2|\nu|\ln\left(n_{r}L_{r}/\eta\right)+\phi_{0}\right],$
(91)

where the coefficient $C(ma)>0$ and the phase $\phi_{0}$ are defined by the
relation

$\Gamma(i|\nu|)\Gamma\left(\frac{p+3}{2}-i|\nu|\right)=C(ma)e^{i\phi_{0}}.$
(92)

In this case the current density exhibits an oscillatory behavior with the
amplitude decaying as $1/L_{\mathrm{(p)}r}^{p+1}$. Comparing (88) and (91)
with (86) we see that the gravitational field essentially modifies the
asymptotic behavior of the current density for large values of the proper
length $L_{\mathrm{(p)}r}$: one has a power law decay in LdS and LAdS
geometries instead of exponential suppression for the LM bulk.

In particular, the formulas (88) and (91) with $p=D-1$ describe the behavior
of the current density for large values of $L_{\mathrm{(p)}D}$ in models with
a single compact dimension $x^{D}$. In that special case the asymptotic of the
Minkowskian current density for a massive field is described by (87). In order
to display the the essential difference of the large $L_{\mathrm{(p)}r}$
asymptotics for LdS and LAdS from that in the LM geometry, in Figures 5 and 6
we present the ratios $\left\langle j^{D}\right\rangle/\left\langle
j^{D}\right\rangle_{\mathrm{LM}}$ for $D=4$ LdS and LAdS spacetimes, with a
single compact dimension of the proper length
$L_{\mathrm{(p)}}=L_{\mathrm{(p)}D}$, as functions of $ma$ and
$L_{\mathrm{(p)}}/a$ for fixed $\tilde{\alpha}_{D}=2\pi/5$. The ratios are
evaluated for the same values of the proper lengths in the LM, LdS and LAdS
spacetimes and all the quantities are measured in units of $a$.

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Figure 5: The ratio of the current densities in the $D=4$ LdS and LM spacetimes with the same proper lengths of the single compact dimension versus the mass and the proper length (in units of $a$). The left and right panels correspond to conformally and minimally coupled fields and the graphs are plotted for $\tilde{\alpha}_{D}=2\pi/5$.  |
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Figure 6: The same as in Figure 5 for the LAdS spacetime.

If at least one of the phases $\tilde{\alpha}_{i}$, $i\neq r$, is different
from zero and the proper length $L_{\mathrm{(p)}r}$ is large, we use the
asymptotic expression of the Macdonald function for large argument. For the
LAdS geometry and LdS geometry with positive values of $\nu$ the leading
contribution to the series over $n_{r}$ comes from the term $n_{r}=1$ and we
get

$\langle
j_{\mathrm{(p)}}^{r}\rangle\approx\frac{2ea^{1+2\mu}B_{D}(am)\sin\left(\tilde{\alpha}_{r}\right)}{2^{\frac{p}{2}+\mu}\pi^{\frac{p}{2}}V_{q}^{\mathrm{(p)}}L_{\mathrm{(p)}r}^{p+1+2\mu}}(L_{r}|\mathbf{k}_{q-1}^{(0)}|)^{\frac{p}{2}+1+\mu}e^{-L_{r}|\mathbf{k}_{q-1}^{(0)}|},$
(93)

where, as before, $\mu=-\nu$ and $\mu=\nu_{+}$ for LdS and LAdS. In the case
of LdS bulk and imaginary $\nu$ the leading order term takes the form

$\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle\approx\frac{4e\eta^{D+1}\sin(\tilde{\alpha}_{r})}{\left(2\pi\right)^{\frac{p+1}{2}}a^{D}V_{q}L_{r}^{p+1}}\frac{(L_{r}|\mathbf{k}_{q-1}^{(0)}|)^{\frac{p}{2}+1}e^{-|\mathbf{k}_{q-1}^{(0)}|L_{r}}}{\sqrt{2|\nu|\sinh\left(\pi|\nu|\right)}}\cos\left[|\nu|\ln\left(|\mathbf{k}_{q-1}^{(0)}|\frac{L_{r}}{2}\right)-\mathrm{arg}(\Gamma(i|\nu|))\right].$
(94)

In the case of LdS the different asymptotic behavior for positive and
imaginary values of $\nu$ is related to different asymptotics of the function
$\left[I_{\nu}(x)+I_{-\nu}(x)\right]K_{\nu}(x)$ for small arguments. For the
LdS geometry and $\nu=0$ for the leading contribution we get

$\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle\approx\frac{4e\sin(\tilde{\alpha}_{r})\eta^{D+1}}{\left(2\pi\right)^{\frac{p}{2}+1}a^{D}V_{q}L_{r}^{p+1}}(|\mathbf{k}_{q-1}^{(0)}|L_{r})^{\frac{p}{2}+1}e^{-|\mathbf{k}_{q-1}^{(0)}|L_{r}}\ln\left(|\mathbf{k}_{q-1}^{(0)}|L_{r}\right).$
(95)

Now let us consider the asymptotics with respect to the length of the $l$th
dimension with $l\neq r$. For large values $L_{l}$ compared with the other
length scales and for $L_{\mathrm{(p)}l}/a\gg 1$ the leading contribution to
(68) comes from the term with $n_{l}=0$. As expected, this leading term
coincides with the current density in the geometry where the $l$th dimension
is decompactified. The corrections induced by the respective compactification
are suppressed by the factor $e^{-mL_{l}}$ ($1/L_{l}^{D+1}$ for a massless
field) in the LM bulk and by the factor $1/L_{l}^{D+2+2\mu}$ for LdS and LAdS
geometries, where $\mu$ is given by (89).

In the opposite limit of small values of $L_{l}$ it is more convenient to use
the representations (22) and (72). The behavior of the current density is
essentially different for the cases $\tilde{\alpha}_{l}=0$ and
$\tilde{\alpha}_{l}\neq 0$. In the first case the dominant contribution to the
summation over $\mathbf{n}_{q-1}$ comes from the modes with $n_{l}=0$. For the
LM bulk the leading term in the expansion of $L_{\mathrm{(p)}l}\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle$ coincides with the current density in
$D$-dimensional LM spacetime which is obtained from the intital
$(D+1)$-dimensional spacetime excluding the $l$th dimension. The same is the
case for LdS and LAdS bulks with the difference that in the leading terms of
the expansion for $L_{\mathrm{(p)}l}\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle$ the parameters $\nu$ and $\nu_{+}$ are
defined for $(D+1)$-dimensional spacetime whereas in the formula for
$D$-dimensional current density $\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle$ the corresponding expressions for $\nu$ and
$\nu_{+}$ are obtained from (41) and (57) by the replacement $D\rightarrow
D-1$. For small values of $L_{l}$ and $\tilde{\alpha}_{l}\neq 0$ the
contribution of the modes with $n_{r}=1$ and $n_{l}=0$ dominates in (22) and
(72). In the remaining summations over $n_{i}$, $i\neq r,l$, the main
contribution comes from large values $n_{i}$ and we replace the corresponding
series by integrations. In this way it can be seen that the current density
$\left\langle j_{\mathrm{(p)}}^{r}\right\rangle$ is suppressed by the factor
$\exp(-L_{r}|\tilde{\alpha}_{l}|/L_{l})$.

##  7 Conclusion

In the present paper we have discussed the features of the vacuum currents in
field-theoretical models formulated in background of spacetimes with compact
dimensions. Three cases of background geometries are considered: LM, LdS and
LAdS. In the decompactification limit they correspond to maximally symmetric
solutions of the Einstein field equations in $(D+1)$-dimensional spacetime
with zero, positive and negative cosmological constants, respectively. The
toroidal compactification of a part of spatial dimensions does not change the
local geometrical characteristics and the high symmetry allows to find closed
analytic expressions for the vacuum currents along compact dimensions. For an
external gauge field we have taken the simplest configuration with a constant
gauge field. Though the corresponding magnetic field is zero, because of the
nontrivial topology respective vector potential gives rise to Aharonov-Bohm
like effect on the vacuum characteristics. By a gauge transformation the gauge
field potential is reinterpreted in terms of the phases in the periodicity
conditions on the field operator along compact dimensions. The
quasiperiodicity conditions with nontrivial phases break the reflection
symmetry along the respective direction and, as a consequence, the
contributions of the left and right moving modes of the vacuum fluctuations of
the quantum field do not compensate each other. As a result a net current
appears that is the analog of the persistent currents in mesoscopic metallic
rings.

The combined expression for the current density along the $r$th compact
dimension, valid for all three background geometries, is given by the formula
(68). The information on specific geometry is encoded in the function
$F_{D}(am,x)$ defined by (69). The component of the current density
$\left\langle j_{\mathrm{(p)}}^{r}\right\rangle$ is an odd periodic function
of the phase $\tilde{\alpha}_{r}$ with the period $2\pi$ and an even periodic
function of the remaining phases $\tilde{\alpha}_{l}$, $l\neq r$, with the
same period. This periodicity is also interpreted as periodicity in terms of
the magnetic flux enclosed by compact dimensions. In this interpretation the
period is equal to the flux quantum. For curved backgrounds the current
density depends on the lengths of the compact dimensions and on the
coordinates (temporal $\tau$ and spatial $z$ coordinates for LdS and LAdS,
respectively) in the form of the proper lengths $L_{\mathrm{(p)}l}$. This
feature is a consequence of the maximal symmetry of dS and AdS spacetimes. For
a conformally coupled massless scalar field the current densities in LM and
LdS spacetimes are connected by the standard relation (75). For LAdS geometry
one has a conformal relation with the current density in LM spacetime (given
by (76)) with an additions planar boundary perpendicular to one of the
uncompact dimensions. The boundary in LM spacetime with Dirichlet boundary
condition on the scalar field operator is the conformal image of the AdS
boundary.

For LdS and LAdS bulks and for small values of the length of compact dimension
the mode sum of the component of the current density along that dimension is
dominated by the contribution of the vacuum fluctuations with wavelengths
smaller than the curvature radius. The influence of the gravitational field on
those modes is weak and the leading term in the respective expansion, given by
(83), coincides with that for LM bulk with the length of the compact dimension
replaced by the proper lengths for LdS and LAdS geometries. The leading term
presents the current density for a massless field in $(D+1)$-dimensional LM
spacetime with a single compact dimension of the length $L_{\mathrm{(p)}r}$.
For small values of the length of the $l$th compact dimension, the behavior of
the current density along the $r$th dimension, $r\neq l$, essentially differs
for zero and nonzero values of the phase $|\tilde{\alpha}_{l}|\leq 1/2$. In
the first case, $\tilde{\alpha}_{l}=0$, the dominant contribution to the
current density $\left\langle j_{\mathrm{(p)}}^{r}\right\rangle$ comes from
the zero mode $n_{l}=0$ and the leading term in the expansion of the product
$L_{\mathrm{(p)}l}\left\langle j_{\mathrm{(p)}}^{r}\right\rangle$ coincides
with the corresponding current density in $D$-dimensional spacetime with
spatial coordinates $(x^{1},\ldots,x^{l-1},x^{l+1},\ldots,x^{D})$. For
$\tilde{\alpha}_{l}\neq 0$ the zero mode with respect to the $l$th dimension
is absent and the component $\left\langle j_{\mathrm{(p)}}^{r}\right\rangle$
decays like $\exp(-|\tilde{\alpha}_{l}|L_{r}/L_{l})$.

The effect of the spacetime curvature on the current density is essential for
lengths of compact dimensions of the order or larger compared with the
curvature radius. For large values of the length of the $r$th compact
dimension the asymptotic of the current $\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle$ is completely different for the cases
$\omega_{0r}=0$ and $\omega_{0r}\neq 0$ with
$\omega_{0r}=\sqrt{k_{\mathbf{n}_{q-1}}^{(0)2}+m^{2}}$ and
$k_{\mathbf{n}_{q-1}}^{(0)2}$ defined by (84). For $\omega_{0r}=0$,
corresponding to a massless field with zero phases $\tilde{\alpha}_{l}$,
$l\neq r$, the leading term for the LM bulk is given by (85). Multiplied by
$V_{q}/L_{r}$, that expression gives the current density for a massless field
in $(p+2)$-dimensional LM spacetime with a single compact dimension $x^{r}$.
For $\omega_{0r}\neq 0$ the large $L_{r}$ asymptotic is described by (86) and
the current density in LM bulk is exponentially suppressed. For LdS and LAdS
background geometries and for $\tilde{\alpha}_{l}=0$, $l\neq r$, the leading
term in the large $L_{\mathrm{(p)}r}$ asymptotic is given by the right-hand
side of (88) with $\nu>0$ for LdS bulk. This shows that the gravitational
field essentially changes the behavior of the current density for large
lengths of compact dimensions: instead of the exponential suppression in LM
bulk for a massive field, for LdS and LAdS geometries the fall-off of the
current density follows a power law. For LdS background and for imaginary
values of the parameter $\nu$ the behavior of the current density is described
by (94). In this case the decay with respect to $L_{r}$ is oscillatory with
the amplitude decreasing as $1/L_{r}^{p+1}$. In the case when at least one of
the phases $\tilde{\alpha}_{l}=0$, $l\neq r$, differs from zero, the
asymptotic behavior of the current density $\left\langle
j_{\mathrm{(p)}}^{r}\right\rangle$ is given by (93) for LAdS and for LdS in
the range $\nu>0$ with an exponential decay. For LdS bulk and imaginary $\nu$
the decay is oscillatory (see (94)).

## Acknowledgments

I am grateful to Stefano Bellucci, Eugenio Ramos Bezerra de Mello and Valeri
Vardanyan for fruitful collaboration. The work was supported by the grant No.
21AG-1C047 of the Higher Education and Science Committee of the Ministry of
Education, Science, Culture and Sport RA and by the ANSEF grant 23AN:PS-
hepth-2889.

##  Appendix A Properties of the Functions in the Expressions for the Currents

We have seen that the vacuum currents along compact dimensions in LdS and LAdS
geometries are expressed in terms of the functions (51) and (64). Here the
properties of those functions are considered. First of all we note that they
are expressed in terms of the hypergeometric function as

$\displaystyle p_{\nu-\frac{1}{2}}^{-\mu}(u)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\mu+\frac{1}{2}-\nu)\Gamma(\mu+\frac{1}{2}+\nu)}{2^{\mu}\Gamma\left(\mu+1\right)}F\left(\mu+\frac{1}{2}-\nu,\mu+\frac{1}{2}+\nu;\mu+1;\frac{1-u}{2}\right),$
$\displaystyle q_{\nu-\frac{1}{2}}^{\mu}(u)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\pi}\Gamma(\mu+\nu+\frac{1}{2})}{2^{\nu+\frac{1}{2}}\Gamma(\nu+1)u^{\mu+\nu+\frac{1}{2}}}F\left(\frac{\mu+\nu+\frac{3}{2}}{2},\frac{\mu+\nu+\frac{1}{2}}{2};\nu+1;\frac{1}{u^{2}}\right).$
(96)

By using the recurrence relations for the associated Legendre functions one
can see that $\partial_{u}p_{\alpha}^{-\mu}(u)=-p_{\alpha}^{-\mu-1}(u)$ and
$\partial_{u}q_{\alpha}^{\mu}(u)=-q_{\alpha}^{\mu+1}(u)$. From here we get the
relations

$p_{\alpha}^{-\mu-n}(u)=(-1)^{n}\partial_{u}^{n}p_{\alpha}^{-\mu}(u),\;q_{\alpha}^{\mu+n}(u)=(-1)^{n}\partial_{u}^{n}q_{\alpha}^{\mu}(u),$
(97)

with $n=1,2,\ldots$.

In the physical problems under consideration, depending on the spatial
dimension, the order $\mu$ for the functions $p_{\alpha}^{-\mu}(u)$ and
$q_{\alpha}^{\mu}(u)$ is a positive integer or half-integer. By making use of
(97) these functions are expressed in terms of $p_{\alpha}^{0}(u)$ and
$q_{\alpha}^{0}(u)$ or $p_{\alpha}^{-1/2}(u)$ and $q_{\alpha}^{1/2}(u)$.
Employing the corresponding expressions for the functions
$P_{\alpha}^{-1/2}(u)$ and $Q_{\alpha}^{1/2}(u)$ from [72] one obtains

$\displaystyle p_{\nu-1/2}^{-1/2}(\cosh\zeta)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2\pi}\sinh\left(\nu\zeta\right)}{\sin\left(\pi\nu\right)\sinh\zeta},$
$\displaystyle p_{\nu-1/2}^{-1/2}(\cos\theta)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2\pi}\sin\left(\nu\theta\right)}{\sin\left(\pi\nu\right)\sin\theta},$
$\displaystyle q_{\nu-1/2}^{1/2}(\cosh\zeta)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{2}}\frac{e^{-\nu\zeta}}{\sinh\zeta}.$ (98)

Combining these expressions with the relations (97) for even values of the
spatial dimension $D$ we get

$F_{D}(am,x)=\sqrt{\frac{\pi}{2}}(-1)^{\frac{D}{2}}\left(\frac{\partial_{\xi}}{\sinh\xi}\right)^{\frac{D}{2}}\frac{e^{-\nu_{+}\xi}}{\sinh\xi},\;x=2\sinh\left(\xi/2\right),$
(99)

in LAdS geometry and

$\displaystyle F_{D}(am,x)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\frac{D}{2}}\sqrt{\pi/2}}{\sin\left(\pi\nu\right)}\left(\frac{\partial_{\xi}}{\sinh\xi}\right)^{\frac{D}{2}}\frac{\sinh\left(\nu\xi\right)}{\sinh\xi},\;x=2\cosh\left(\xi/2\right),$
$\displaystyle F_{D}(am,x)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\pi/2}}{\sin\left(\pi\nu\right)}\left(\frac{\partial_{\theta}}{\sin\theta}\right)^{\frac{D}{2}}\frac{\sin\left(\nu\theta\right)}{\sin\theta},\;x=2\cos\left(\theta/2\right),$
(100)

for LdS bulk. For odd values of $D$ one has

$\displaystyle F_{D}(am,x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-1)^{\frac{D+1}{2}}\Gamma\left(\frac{1}{2}-\nu\right)\Gamma\left(\nu+\frac{1}{2}\right)\partial_{u}^{\frac{D+1}{2}}P_{\nu-\frac{1}{2}}(u),\mathrm{\;for\;LdS},$
$\displaystyle F_{D}(am,x)$ $\displaystyle=$
$\displaystyle(-1)^{\frac{D+1}{2}}\partial_{u}^{\frac{D+1}{2}}Q_{\nu_{+}-\frac{1}{2}}(u),\mathrm{\;for\;LAdS},$
(101)

where $u=x^{2}/2-1$ for LdS and $u=x^{2}/2+1$ for LAdS.

In order to find the behavior of the functions $p_{\nu-1/2}^{-\mu}(u)$ and
$q_{\nu-1/2}^{\mu}(u)$ for large values of $u$ we use the corresponding
asymptotics for the associated Legendre functions (see, for example, [90]).
The leading order terms read

$\displaystyle p_{\nu-\frac{1}{2}}^{-\mu}(u)$ $\displaystyle\sim$
$\displaystyle\frac{2^{\nu-\frac{1}{2}}}{\sqrt{\pi}}\frac{\Gamma\left(\nu\right)\Gamma\left(\mu+\frac{1}{2}-\nu\right)}{u^{\mu-\nu+\frac{1}{2}}},\;\mathrm{Re}\,\nu>0,\;\mu+\nu\neq-1,-2,\ldots,$
$\displaystyle p_{-\frac{1}{2}}^{-\mu}(u)$ $\displaystyle\sim$
$\displaystyle\sqrt{\frac{2}{\pi}}\Gamma\left(\mu+\frac{1}{2}\right)\frac{\ln
u}{u^{\mu+\frac{1}{2}}},\;\mu\neq-\frac{1}{2},-\frac{3}{2},\ldots,$
$\displaystyle q_{\nu-\frac{1}{2}}^{\mu}(u)$ $\displaystyle\sim$
$\displaystyle\frac{\sqrt{\pi}\Gamma\left(\mu+\nu+\frac{1}{2}\right)}{2^{\nu+\frac{1}{2}}\Gamma\left(\nu+1\right)u^{\mu+\nu+\frac{1}{2}}},\;\nu\neq-\frac{3}{2},-\frac{5}{2},\ldots$
(102)

The asymptotic for $p_{\nu-\frac{1}{2}}^{-\mu}(u)$ in the case of purely
imaginary values of $\nu$, $\nu=i|\nu|$, is obtained by using the relation
[90]

$P_{i|\nu|-\frac{1}{2}}^{-\mu}(u)=ie^{-i\mu\pi
i}\frac{Q_{i|\nu|-\frac{1}{2}}^{\mu}(u)-Q_{-i|\nu|-\frac{1}{2}}^{\mu}(u)}{\sinh\left(|\nu|\pi\right)|\Gamma\left(\mu+\frac{1}{2}+i|\nu|\right)|^{2}},$
(103)

and the corresponding asymptotic for $Q_{i|\nu|-\frac{1}{2}}^{\mu}(u)$. This
gives

$p_{i|\nu|-\frac{1}{2}}^{-\mu}(u)\sim\frac{\sqrt{2/\pi}}{u^{\mu+\frac{1}{2}}}\,\mathrm{Re}\left[\Gamma\left(\mu+\frac{1}{2}-i|\nu|\right)\frac{\Gamma\left(i|\nu|\right)}{\left(2u\right)^{i|\nu|}}\right].$
(104)

These asymptotic formulas have been used in the main text to study the
behavior of the current density in asymptotic regions of the parameters.

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