Vacuum currents for a scalar field in models with compact dimensions

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.


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 (see, e.g., [1]- [9]).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 observables 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 [10]- [14]), 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 [15]).
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 energymomentum 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 toroidally compactified spatial dimensions and in the presence of constant gauge fields have been investigated in [16,17].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, [18,19]).The corresponding quantum field theory lives in spaces with topologies R 1 × S 1 and S 1 × 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 [20,21] 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 [22] and [23,24] 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 [25]- [28].The corresponding applications to higher dimensional braneworld models with compact subspaces have been discussed.
The physical nature of the current densities considered in [16,17] is similar to that for persistent currents in mesoscopic rings [29]- [34].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., [35]- [50] and [51]- [59] 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 toroidal 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 curvatures 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.

General Formalism
Consider a charged scalar field ϕ(x) with the curvature coupling parameter ξ in background of (D +1)dimensional spacetime described by the line element ds 2 = g µν (x)dx µ dx ν .Here we use x = (x 0 = t, x 1 , . . ., x D ) for the notation of spacetime points.In the presence of a classical vector gauge field A µ (x) the corresponding action has the form where R is the Ricci scalar for the metric tensor g µν (x) and D µ = ∇ µ + ieA µ , with ∇ µ being the related covariant derivative operator.The most important special cases correspond to the fields with minimal (ξ = 0) and conformal (ξ = ξ D ≡ (D − 1)/(4D)) couplings.The field equation obtained from (1) by standard variational procedure reads We assume that a part of coordinates corresponding to the subspace (x p+1 , x p+2 , . . ., x D ) are compactified to circles of the lengths (L p+1 , L p+2 , . . ., L D ), respectively, and 0 ≤ x l ≤ L l , l = p + 1, . . ., D.
In addition, the metric tensor is periodic along the compact dimensions: g µν (t, x 1 , . . ., x p , . . ., x l + L l , . . ., x D ) = g µν (t, x 1 , . . ., x p , . . ., x l , . . ., x D ), where p < l ≤ D. For the scalar and gauge fields less trivial quasiperiodicity conditions are imposed ϕ(t, x 1 , . . ., x p , . . ., x l + L l , . . ., x D ) = e iα l (x) ϕ(t, x 1 , . . ., x p , . . ., x l , . . ., x D ), where the real functions α l (x), l = p + 1, . . ., 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 , . . ., 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., [60,61,62]).We are interested in the VEV of the current density 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 ′ ) defined as the VEV where , with |0 being the vacuum state, stands for the VEV.In particular, for the VEV of the current density we have with g ν ′ µ being the bivector of parallel displacement.We recall that the vector cµ = g ν ′ µ c ν ′ is obtained by the parallel transport of c ν from the point x ′ 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 {ϕ of the mode functions for the field specified by the set of quantum numbers σ.We will follow the second approach.For the field operator one has the expansion where a σ and b † σ are the annihilation and creation operators.The symbol σ stands for the summation over the discrete quantum numbers in the set σ and the integration over the continuous ones.Substituting in the expression (6) for the Hadamard function and by using the relations a σ |0 = b σ |0 = 0, we get the mode sum With this sum, the formal expression for the VEV of the current density takes the form 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 x q = (x p+1 , . . ., x D ) will be used in the compact subspace.For the classical gauge field we will assume a simple configuration A µ = const and in the periodicity conditions (4) the phases α l (x) = α l = const will be taken.Special cases α l = 2π and α l = π 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 (ϕ, A µ ) → ϕ ′ , A ′ µ given as with the function χ(x) = A µ x µ .For this function one gets A ′ µ = 0 and the field equation takes the form The periodicity conditions for the new scalar field read ϕ ′ (t, x 1 , . . ., x p , . . ., x l + L l , . . ., x D ) = e i αl ϕ ′ (t, x 1 , . . ., x p , . . ., x l , . . ., x D ), (13) with new phases αl = α l + eA l L l , for l = p + 1, . . ., D. The physics is invariant under the gauge transformation and we could expect that the components A µ , µ = 0, 1, . . ., 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 ϕ ′ , A ′ µ = 0 omitting the primes.

Locally Minkowski Spacetime with Toral Dimensions
We start the consideration with LM spacetime having the spatial topology R p × T q , q = D − p, where the q-dimensional torus T q corresponds to the subspace with the compact coordinates x q = (x p+1 , x p+2 , . . ., x D ).For this geometry the metric in the Cartesian coordinates is expressed as g µν = η µν = diag (1, −1, . . ., −1).With this metric tensor one has ∇ µ = ∂ µ .The topological Casimir effect in flat spacetimes with toral dimensions has been widely considered in the literature (see [10,14], [63]- [78] 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 [16,17,20,21,79] 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 k = (k 1 , k 2 , . . ., k D ) and are given by where x p = (x 1 , . . ., x D ) stands for the set of coordinates in the uncompact subspace, ω k = √ 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, k p = (k 1 , . . ., k p ), we have k l ∈ (−∞, +∞), l = 1, . . ., p, whereas the eigenvalues along the compact dimensions, k q = (k p+1 , . . ., k D ), are discretized by the periodicity conditions (13): with l = p + 1, ..., D. The integer part of the ratio αl /(2π) in the expressions of the VEVs is absorbed by shifting the integer number n l in the corresponding summation and, hence, we can take |α l | ≤ π without loss of generality.Note that one has and hence for αl = nπ, with n being an integer, the parity (P-) symmetry with respect to the reflection x l → −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 lth 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 ∆t = t−t ′ , ∆x p = x p −x ′ p , and ∆x q = x q − x ′ q .Inserting the modes (15) in (9), one gets where n q = (n p+1 , . . ., n D ).In the gauge under consideration and for the geometry at hand the formula (7) for the current density takes the form For µ = 0, 1, . . ., p the derivatives ∂ µ G(x, x ′ ) and ∂ µ ′ G(x, x ′ ) are odd functions of ∆x µ and the charge density and the components of the current density along uncompact dimensions vanish, j µ = 0 for µ = 0, 1, . . ., p.Of course, we could expect that from the problem symmetry under the reflections x µ → −x µ along respective coordinates.In order to find the current density along the rth 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 [70,80] 2π with the functions g(z) = e iz∆x r and f (z . By making use of the expansion 1/(e y − 1) = ∞ l=1 e −ly , the integrals over z and then over k p are expressed in terms of the Macdonald function where The n r = 0 term here corresponds to the Hadamard function in the geometry with spatial topology R p+1 × T q−1 , where the rth dimension is decompactified.The divergences in the coincidence limit x ′ → 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 rth compact dimension, for the corresponding contravariant component one finds The specific features of the vacuum current will be discussed below in Section 6.As it has been already mentioned, for αr = 0, π the problem is symmetric with respect to the inversion x r → −x r and, as expected, the VEV j r 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 αr = 0 there is also a zero mode with k r = 0 which does not contribute to the current density along the rth dimension.
In the model with a single compact dimension x D one has p = D − 1, ω n q−1 = m and the formula ( 22) is reduced to In particular, for a massless field we get For odd values of D the series is expressed in terms of the Bernoulli polynomials B n (x) (see, e.g., [81]) and we get for 0 < αD < 2π.In particular, for D = 1 and D = 3 one finds For D ≥ 2 the current density is a continuous function of αD , whereas for D = 1 the current density for a massless field is discontinuous at αD = 2πn with integer n.
We could directly start from the mode sum formula (10) with D µ = ∂ µ .The substitution of the mode functions (15) leads to the expression Introducing the generalized zeta function ζ(s) in accordance with the current density is written as 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, [14,82,83]).In order to realize the analytic continuation we first integrate over the momentum in the uncompact subspace with the result The application of the generalized Chowla-Selberg formula [84] to the multiple series in (30) gives where the prime on the summation sign means that the term with n q = (0, 0, . . ., 0) should be excluded.
Here we have introduced the q-component vector αq = (α p+1 , αp+2 , . . ., αD ) and the notation 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 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 The representation with this replacement is given in [16].The equivalence of two representations ( 22) and ( 33) follows from the relation (zg(L q , n q )), (35) with s = p + 1 and z = m.This relation is proved in [70] 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 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.

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 Λ 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 ΛCDM 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 in (D + 2)-dimensional Minkowski spacetime with the line element ds 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 (τ, x 1 , x 2 , . . ., x D ) which are connected to the coordinates in the embedding spacetime by the relations (see, for example, [85] in the case D = 3) For the dS line element this gives ds (dS)2 D+1 = g µν (x)dx µ dx ν , with the metric tensor In inflationary models a part of dS spacetime with the conformal time coordinate in the range −∞ < τ < 0 is employed.For the corresponding synchronous time coordinate t, −∞ < t < +∞, one has t = −a ln |τ |/a and the line element is expressed as The metric (39) is the solution of Einstein's equations with positive cosmological constant Λ = D(D−1) as the only source of the gravitational field.The hyperbolic embedding (37) in the locally Minkowski spacetime with the line element ds works equally well for LdS spacetime with a q-dimensional toroidal subspace covered by the coordinates (x p+1 , x p+2 , . . ., x D ).The coordinate x l , l = p + 1, . . ., D, varies in the range 0 ≤ x l ≤ L l .In this case, the subspace with the coordinates (Z p+2 , . . ., Z D+2 ) is compactified to a torus.The length of the compact dimension Z l+1 , l = p + 1, . . ., D, is given by L (p)l = aL l /η = e t/a L l , with η = |τ |.Note that L (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 x q = (x p+1 , x p+2 , . . ., x D ), the scalar mode functions can be presented in the form ϕ The eigenvalues of the momentum components along compact dimensions are given by (16).Different choices of the function g ν (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 [86].

Hadamard Function
For the Bunch-Davies vacuum state the normalized scalar mode functions are specified by σ = k = (k p , k q ) and are expressed as where H (1,2) ν (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 ν * (kη ′ ) + H (1)  ν (kη ′ )H (2) Applying to the sum over n r the summation formula (20) and assuming that Re ν < 1, one finds the representation where I ν (y) and K ν (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 rth coordinate is decompactified (spatial topology R p+1 × T q−1 ).The effects of the compactification of that coordinate are included in the part with n r = 0.For a conformally coupled massless field we have ν = 1/2 and With this function, the integral over z in ( 44) is evaluated by using the formula from [87] and we get where the Minkowskian Hadamard function G M (x, x ′ ) is given by ( 21) with m = 0.For a conformally coupled massless field this is the standard relation between two conformally related geometries.

Vacuum Current
For the components of the current density along uncompact dimensions x µ , µ = 1, 2, . . ., p, one has ∂ µ G(x, x ′ ) ∝ g µα ∆x α and the corresponding expectation values vanish.By using the properties of the modified Bessel functions we can see that lim and, hence, the charge density vanishes as well.In order to find the component of the current density along the rth compact dimension we use the representation (44) for the Hadamard function in combination with (7) where D µ = ∂ µ .In (44), the derivative of the term n r = 0 with respect to x r is an odd function of ∆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 = 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 [22]: n r sin(n r αr ) An alternative representation for the current density is obtained by applying the formula (35) with s = p.This gives The integral in the right-hand side is evaluated by using the formula where we use the notation with P µ ν (u) being the associated Legendre function of the first kind.The expression of the function p −µ α (u) in terms of the hypergeometric function is given in Appendix A. The result ( 50) is obtained from the integral involving the product I ν (z)K ν (z)K D 2 (bz) and given in [87].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 In both the formulas ( 49) and ( 52) we can make the replacement (34).Note that one has the property p (u) and the expression on the right-hand side is real for both real and purely imaginary values of ν.

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 Λ.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., [88]- [91]), 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 [92] 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 where the line element of the embedding (D + 2)-dimensional flat spacetime is given by ds 2 D+2 = dZ 0 2 − D i=1 dZ i 2 + dZ D+1 2 .The Poincaré coordinates (t, x 1 = z, x 2 , . . ., x D ) are introduced by the relations In those coordinates the metric tensor of the AdS spacetime is expressed as with 0 ≤ z < ∞.The hypersurfaces z = 0 and z = ∞ 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), −∞ < y < +∞, in terms of which the line element is written as Here we consider the LAdS geometry with the coordinates (x p+1 , x p+2 , . . ., 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 , . . ., Z D+1 ), where the coordinate Z l , l = p + 1, . . ., D, is compactified to a circle with the length L (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 ϕ (±) σ (x) = e ∓iωt+ik p−1 x p−1 +kqxq f (z), with x p−1 = (x 2 , . . ., x p ), k p−1 = (k 2 , . . ., k p ).The equation for the function f (z) is obtained from the field equation.The corresponding solution is presented as z D/2 c 1 J ν + (λz) + c 2 Y ν + (λz) , where J ν (λz) and Y ν (λz) are the Bessel and Neumann functions, For the stability of the Poincaré vacuum the parameter ν + should be real [93,94,95].For ν + ≥ 1 from the normalizability condition it follows that c 2 = 0 in the linear combination of the cylinder functions.For 0 ≤ ν + < 1 the modes with c 2 = 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 [96,97].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 ν + (λz).The normalized mode functions are expressed as The modes are specified by the set σ = (λ, k p−1 , k q ) with 0 ≤ λ < ∞ and the energy is given by With the mode functions (58), the Hadamard function takes the form 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 The term with n r = 0 in this representation corresponds to the Hadamard function in the geometry where the rth dimension is decompactified.
Another representation for the function ( 59) is obtained in [23] by using an integral representation for the ratio cos(ω∆t)/ω.The integral over λ 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 where and ∆z = z − z ′ .By using the result from [87] for the integral in (61), the following representation is obtained: where the function q µ α (x) is expressed in terms of the associated Legendre function of the second kind, Q µ α (x) (for the expression in terms of the hypergeometric function see Appendix A): The contribution in (63) corresponding to the term n q = 0 presents the Hadamard function in AdS spacetime with Poincaré coordinates −∞ < x µ < +∞ for µ = 2, 3, . . ., D. The divergences in the coincidence limit are contained in that part.The topological contribution with n q = 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: j µ = 0 for µ = 0, 1, . . ., p. Combining the formulas ( 7) and ( 60), for the component along the rth dimension we find By applying the formula (35) with s = p to the series over n q−1 one gets n r sin (n r αr ) (λg(L q , n q )). ( 66) The integral in ( 66) is expressed in terms of the function ( 64) [98] (note that there is a misprint in the similar integral given in [87]) and the current density is presented in the form [23] This representation could be directly obtained by using the Hadamard function in the form ( 63) and the relation ∂ x q µ α (x) = −q µ+1 α (x) for the function (64).
6 Features of the Current Density

General Features
The physical component of the charge density is given by j r a D+1 nq n r sin (n q • αq ) F D (am, g(L (p)q /a, n q )), (68) where we have defined the function (amx), for LM, with Alternative expressions for locally dS and AdS geometries are obtained from ( 52) and ( 67): 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 ( 102) and (103).In odd number of spatial dimensions the expressions for the functions (71) in terms of the Legendre functions (u) are given by (104).
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): n r sin(n r αr ) with the notation Note that the vector k q−1 is the physical momentum in the compact subspace with the set of coordinates (x p+1 , . . ., x r−1 , x r+1 , . . ., x D ).
First of all, we see that the current density along the rth dimension is an even periodic function of the parameters αi , i = r, with the period 2π and an odd periodic function of αr 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 j r depends on the lengths of compact dimensions and on the coordinates through the ratios L (p)i /a, i = p + 1, . . ., 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

Conformal Coupling and Minkowskian Limit
Let us consider special cases of general formulas.For a conformally coupled massless field one has ξ = ξ D and ν = ν + = 1/2.The current density for the Minkowskian case does not depend on the curvature coupling parameter and from (33) we get The corresponding functions F D (am, x) for dS and AdS geometries are obtained from ( 69) by taking into account that [ The integrals are evaluated by using the formulas from [87] and we get where 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 < ∞ 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 j D LAdS / j D LM and j D LdS / j D LM on the proper length L (p) = L (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 j D LdS / j D 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 j D / j D LM tends to 1.All these features will be confirmed below by asymptotic analysis.Now let us check the Minkowskian limit for dS and AdS geometries.As seen from ( 40) and ( 56), it is obtained taking a → ∞ for fixed spacetime coordinates (t, x 1 , . . ., x D ) and (t, y, x 2 , . . ., x D ) for the dS and AdS cases, respectively.For large curvature radius one has ν ≈ iam, η ≈ a − t in LdS bulk and ν + ≈ am, z ≈ a + y for LAdS.In the case of LAdS we need the asymptotic expression of the function q (D+1)/2 ν + −1/2 u 2 /2 + 1 for ν + ≫ 1 and u ≪ 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 [99].With that asymptotic, it can be checked that in the limit under consideration 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 ν ≈ iam, am ≫ 1.The corresponding uniform asymptotic expansions for the functions I ±ν (u) and K ν (u) can be found, for LM is plotted for a minimally coupled massless scalar field in LAdS (full curves, j D = j D LAdS ) and LdS (dashed curves, j D = j D LdS ).The numbers bear the curves present the corresponding spatial dimension.The different behaviour for LAdS and LdS geometries in the region of large compact dimensions will be clarified below by the asymptotic analysis.example, in [100,101].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 with ν ≈ iam.The respective integral is evaluated by using the formula and we see that to the leading order which coincides with the Minkowskian result.

Large and Small Proper Lengths of Compact Dimensions
For small values of the proper length of the rth compact dimension, L (p)r ≪ a, 1/m, we first consider the contribution in (68) of the terms for which at least one of n i , i = 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 [98] 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 = r, is not zero is suppressed by the factor exp[−g(L (p)q−1 /a, n q−1 )α r a/L (p)r ], where For the contribution of the terms with n i = 0, i = 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 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 [I −ν (u) + I ν (u)] K ν (u) and J 2 ν + (u) and evaluating the integrals, we can see that the corresponding asymptotics are given by the same expression (82).Hence, in the limit L (p)r ≪ a, 1/m the dominant contribution to the current density comes from the modes with n i = 0, i = r, and to the leading order 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 ×S 1 with a single compact dimension x r with length L r = L (p)r .
For small values of L (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 = r, for which ω n q−1 = ω 0r = k (0)2 The beahvior of the current density is essentially different depending whether ω 0r is zero or not.In the first case the leading term in the current density is given as sin(n r αr ) 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 × S 1 having a single compact dimension x r .For ω 0r = 0 the dominant contribution to the current density is induced by the mode with n r = 1 and, to the leading order, In particular, for the model with a single compact dimension x D one has p = D − 1 and the asymptotic (86) takes the form where mL D ≫ 1.
For LdS and LAdS geometries and for large values of the proper length L (p)r it is more convenient to use the representations (48) and (65).The dominant contribution comes from the term in the summation with n q−1 = 0 (n l = 0 for l = r) and from the integration region near the lower limit.Two cases should be considered separately.The first one corresponds to the phases αi = 0, i = r.With these values and for LAdS bulk and for LdS bulk in the case ν > 0 the leading order term is expressed as sin (n r αr ) where and For LdS geometry and for imaginary values of ν, ν = i|ν|, by similar calculations the leading term is presented as where the coefficient C(ma) > 0 and the phase φ 0 are defined by the relation In this case the current density exhibits an oscillatory behavior with the amplitude decaying as 1/L p+1 (p)r .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 (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 (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 (p)r asymptotics for LdS and LAdS from that in the LM geometry, in Figures 5 and 6 we present the ratios j D / j D LM for D = 4 LdS and LAdS spacetimes, with a single compact dimension of the proper length L (p) = L (p)D , as functions of ma and L (p) /a for fixed αD = 2π/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.
If at least one of the phases αi , i = r, is different from zero and the proper length L (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 ν the leading contribution to the series over n r comes from the term n r = 1 and we get where, as before, µ = −ν and µ = ν + for LdS and LAdS.In the case of LdS bulk and imaginary ν the leading order term takes the form Figure 6: The same as in Figure 5 for the LAdS spacetime.
In the case of LdS the different asymptotic behavior for positive and imaginary values of ν is related to different asymptotics of the function [I ν (x) + I −ν (x)] K ν (x) for small arguments.For the LdS geometry and ν = 0 for the leading contribution we get Now let us consider the asymptotics with respect to the length of the lth dimension with l = r.For large values L l compared with the other length scales and for L (p)l /a ≫ 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 lth dimension is decompactified.The corrections induced by the respective compactification are suppressed by the factor e −mL l (1/L D+1 l for a massless field) in the LM bulk and by the factor 1/L D+2+2µ l for LdS and LAdS geometries, where µ 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 αl = 0 and αl = 0.In the first case the dominant contribution to the summation over n q−1 comes from the modes with n l = 0.
For the LM bulk the leading term in the expansion of L (p)l j r (p) coincides with the current density in D-dimensional LM spacetime which is obtained from the intital (D + 1)-dimensional spacetime excluding the lth dimension.The same is the case for LdS and LAdS bulks with the difference that in the leading terms of the expansion for L (p)l j r (p) the parameters ν and ν + are defined for (D + 1)-dimensional spacetime whereas in the formula for D-dimensional current density j r (p) the corresponding expressions for ν and ν + are obtained from ( 41) and ( 57) by the replacement D → D−1.For small values of L l and αl = 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 = 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 j r (p) is suppressed by the factor exp(−L r |α l |/L l ).

Fermionic currents
In the discussion above we have considered the current densities for a charged scalar field.Similar investigations for the massive Dirac field ψ(x) in general number of spatial dimensions, obeying the quasiperiodicity conditions ψ(t, x 1 , . . ., x p , . . ., x l + L l , . . ., x D ) = e iα l ψ(t, x 1 , . . ., x p , . . ., x l , . . ., x D ), (96) with constant phases α l , are presented in [17,22,24] for LM, LdS and LAdS geometries, respectively.The formulas from these references for the fermionic current density along the rth compact dimension are presented in the combined form a D+1 nq n r sin (n q • αq ) F (f) D (am, g(L (p)q /a, n q )), (97) with the same notations as in (68).Here, N = 2 [ D+1 2 ] ([x] stands for the integer part of x) is the number of spinor components for the Dirac field realizing the irreducible representation of the Clifford algebra.The functions F The replacement 2π αl → − αl in the expression for LM bulk, compared to the one given in [17], is related to different notations of the constants in the quasiperiodicity conditions (see also the comment in [24]).The applications of (97), with D = 2, in cylindrical nanotubes, described in terms of the effective Dirac theory, have been discussed in [17,24].As it is seen from ( 98), assuming the same masses and phases in the periodicity conditions for scalar and Dirac fields, the relation j l (f)LM = −(N/2) j l LM is obtained for the corresponding current densities in LM bulk.In particular, in supersymmetric models with the same number of scalar and spinor degrees of freedom the total vacuum current vanishes.That is not the case for LdS and LAdS geometries.In even number of spatial dimensions D, the Clifford algebra has two inequivalent representations with two different sets of the Dirac matrices.As it has been discussed in [24], the vacuum current densities coincide for the fields realizing those representations if the corresponding masses and the periodicity conditions are the same.More details of the properties for fermionic currents in LM, LdS and LAdS geometries with toroidal compact dimensions will be reviewed elsewhere.

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 rth 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 j r (p) is an odd periodic function of the phase αr with the period 2π and an even periodic function of the remaining phases αl , l = 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 τ and spatial z coordinates for LdS and LAdS, respectively) in the form of the proper lengths L (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

=
|g rr | j r .It determines the charge flux through the spatial hypersurface x r = const, expressed as n r j r , with n r = |g rr | being the corresponding normal.The expressions obtained above for the rth component of the current density can be combined as we present the dependence of the respective current density, multiplied by L D (p) /e, as a function of the parameter αD /(2π) and of the proper length of the compact dimension L (p) = L (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 (p) = L.As it follows from (24), for a massless field the dimensionless combination L D (p) j D (p) /e does not depend on L (p) .

Figure 1 :
Figure 1: The current density in the D = 4 LM spacetime versus the parameter αD /(2π) 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

Figure 2 :
Figure 2: The current density in the D = 4 LdS spacetime, multiplied by L D (p) /e, versus the parameter αD /(2π) 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.

Figure 3 :
Figure 3: The same as in Figure 2 for the LAdS bulk.

Figure 4 :
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 (p) = L (p)D , versus the ratio L (p) /a.On the right panel the ratio j D / j DLM is plotted for a minimally coupled massless scalar field in LAdS (full curves, j D = j D LAdS ) and LdS (dashed curves, j D = j D LdS ).The numbers bear the curves present the corresponding spatial dimension.The different behaviour for LAdS and LdS geometries in the region of large compact dimensions will be clarified below by the asymptotic analysis.

Figure 5 :
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 αD = 2π/5.