Cosmic string and brane induced effects on the fermionic vacuum in AdS spacetime

We investigate the combined effects of a magnetic flux-carrying cosmic string and a brane on the fermionic condensate (FC) and on the vacuum expectation value (VEV) of the energy-momentum tensor for a massive charged fermionic field in background of 5-dimensional anti-de Sitter (AdS) spacetime. The brane is parallel to the AdS boundary and it divides the space into two regions with distinct properties of the fermionic vacuum. For two types of boundary conditions on the field operator and for the fields realizing two inequivalent representations of the Clifford algebra, the brane-induced contributions in VEVs are explicitly separated. The VEVs are even periodic functions of the magnetic flux, confined in the core, with the period of flux quantum. Near the horizon the FC and the vacuum energy-momentum tensor are dominated by the brane-free contribution, whereas the brane-induced part dominates in the region near the brane. Both the contributions vanish on the AdS boundary. At large distances from the cosmic string, the topological contributions in the VEVs, as functions of the proper distance, exhibit an inverse power-law decrease in the region between the brane and AdS horizon and an exponential decrease in the region between the brane and AdS boundary. We show that the FC and the vacuum energy density can be either positive or negative, depending on the distance from the brane. Applications are discussed in fermionic models invariant under the charge conjugation and parity transformation and also in $Z_{2}$% -symmetric braneworld models. By the limiting transition we derive the expressions of the FC and the vacuum energy-momentum tensor for a cosmic string on 5-dimensional Minkowski bulk in the presence of a boundary perpendicular to the string.


Introduction
The vacuum polarization is among the most interesting manifestations of the nontrivial properties of quantum vacuum. Various types of sources for the polarization of vacuum have been considered in the literature. They include external electromagnetic and gravitational fields or boundary conditions on quantum fields induced by the presence of boundaries having different physical nature. Examples of the latter are the material boundaries in quantum electrodynamics, interfaces separating different phases of physical systems, horizons, etc. The spectrum of vacuum fluctuations is influenced by the boundary conditions and, as a consequence, the expectation values of physical observables are shifted. Those modifications depend on the bulk and boundary geometries, on the boundary conditions imposed and are known as the Casimir effect (for reviews see [1]). Another interesting source for the vacuum polarization is the nontrivial topology induced by compactification of spatial dimensions or by the presence of topological defects. The corresponding effects play an important role in both high-energy quantum field theoretical models and in effective models describing the condensed matter systems. In the present paper we consider an exactly solvable problem for the polarization of the fermionic vacuum under the combined influence of several sources: background gravitational field, a topological defect of the cosmic string type and a boundary.
As to the background geometry we will take a locally anti-de Sitter (AdS) spacetime. There are several motivations for that. The AdS spacetime is the maximally symmetric solution of the Einstein equations with the negative cosmological constant as the only source for the gravitational field. The high degree of symmetry allows to obtain closed analytic expressions for local characteristics of the vacuum state. That is important for elucidating the features of the influence of the gravitational field on the properties of quantum field-theoretical systems in more complicated geometries. Of course, the results obtained for the AdS bulk are also of separate interest, due to its important role as the ground state in supergravity and superstring theories and also in two exciting developments of the contemporary theoretical physics, namely, in braneworld models with large extra dimensions [2] and in the AdS/conformal field theory (CFT) correspondence [3]. The braneworlds provide a geometrical solution of the hierarchy problem for the energy scales in particle physics and have been extensively used in cosmological and astrophysical contexts. The AdS/CFT correspondence is a realization of the holographic type duality between supergravity or string theories on background of the AdS spacetime and CFT located on its boundary. It is an interesting way to investigate non-perturbative effects in the theory by mapping them onto perturbative effects in the dual theory and has important applications in high-energy physics and in a number of condensed matter systems.
In the consideration below the nontrivial topology is induced by the presence of a two-dimensional topological defect, carrying a gauge field flux along its core. The investigation will be done within the framework of a simple model that reduces the influence of the defect on the geometry to the generation of a planar angle deficit in the spacetime line element. The image of the defect on the boundary of the AdS spacetime corresponds to the standard idealized cosmic string geometry in (3+1)-dimensional spacetime and we use the term cosmic string for the defect under consideration as well. The cosmic strings form a class of topological defects created as a result of symmetry breaking phase transitions during the expansion in the early Universe (see [4] for reviews). Among other types of topological defects (domain walls, monopoles, textures), they have been most frequently considered in the literature. That is related to interesting physical effects induced by the presence of cosmic strings, such as the generation of gamma ray bursts, high-energy cosmic rays, gravitational waves and scale invariant cosmological perturbations. The formation of macroscopic defects of cosmic string type has been also predicted within the framework of fundamental string theory [5]. An example is provided by the brane-inflation scenario where cosmic strings are produced towards the end of inflation. Cosmic string type solutions on AdS bulk and in braneworld models have been considered, for example, in [6,7]. The polarization of quantum vacuum around cosmic strings has been mainly considered for the background Minkowski geometry (see, for instance, references given in [8,9]). The combined effects of the gravitational field and of a cosmic string on the local characteristics of scalar, fermionic and electromagnetic vacua in the case of the de Sitter (dS) background were discussed in [10]. Another exactly solvable background with a cosmic string corresponds to the AdS bulk. The vacuum expectation values (VEVs) of the field squared and of the energy-momentum tensor induced by a straight cosmic string on that bulk have been investigated in [11] for scalar and fermionic fields.
As an additional source of the vacuum polarization we will consider a brane parallel to the AdS boundary with two types of boundary conditions on the fermionic field operator. The first one corresponds to the bag boundary condition and the second one differs by the sign of the term containing the normal to the brane. In braneworld models of the Randall-Sundrum type those boundary conditions are dictated by the Z 2 -symmetry. The boundary conditions give rise to Casimir type contributions in the VEVs of physical observables. We investigate those contributions for the fermionic condensate (FC) and for the expectation value of the energy-momentum tensor for fields realizing two inequivalent irreducible representations of the Clifford algebra. The corresponding VEV of the current density for a charged fermionic field has been considered in [12]. Another type of conditions imposed on the operator of a quantum field appears in models with compact spatial dimensions. Similar to the case of boundary conditions, the periodicity conditions along compact dimensions give rise to topological Casimir contributions in the vacuum characteristics. In a locally AdS spacetime with compact dimensions the combined effects of branes and compactification on the vacuum energy and on the VEV of the energy-momentum tensor have been investigated in [13]. The corresponding vacuum currents for charged scalar and fermionic fields were discussed in [14,15]. An additional topological influence of a cosmic string on the vacuum currents has been considered in [16,17]. The FC and the VEV of the energy-momentum tensor for a charged massive fermionic field were investigated in [18,19]. The influence of the brane on those characteristics of the fermionic vacuum in the geometry of a cosmic string on a locally AdS bulk is the main concern in the present paper. The choice of the FC and the energy-momentum tensor is motivated by their important role in the self-consistent description of the dynamics of the system including the influence of the gravitational field. Being the VEVs of local operators, because of the global nature of the vacuum state, they also contain information on the global characteristics of the background geometry. In particular, the FC is the central quantity in the discussions of dynamical breaking of chiral symmetry (see, for example, [20] for the chiral symmetry breaking in Nambu-Jona-Lasino and Gross-Neveu models on background of a curved spacetime with non-trivial topology).
The paper is organized as follows. In Section 2 we describe the system under consideration and present a complete set of fermionic modes in both the regions of the spacetime separated by a brane. By using those modes, in Section 3 the FC is investigated. A general formula is provided with explicit separation of the brane-induced contribution. Various limiting cases and the behavior of the FC in asymptotic regions of the parameters are discussed. Similar analysis for the VEV of the energy-momentum tensor is presented in Section 4. The model under consideration lives on an odd dimensional spacetime and the corresponding Clifford algebra has two inequivalent irreducible representations. In Section 5 we discuss the relations of the fields realizing those representations with the field discussed in the previous sections. Depending on the boundary conditions imposed on the separate fields, in addition to the bag boundary condition, one needs to consider the boundary condition that differs by the sign of the term involving the normal to the brane. The corresponding vacuum densities are presented. We also discuss the VEVs in models invariant under the charge conjugation and parity transformation. Applications are given in Z 2 -symmetric braneworld models of the Randall-Sundrum type with a single brane and in the presence of a cosmic string. The main results of the paper are summarized in Section 6. In Appendix A we present the details of the separation of the brane-induced contributions in the FC and in the VEV of the energy-momentum tensor.

Background geometry and fermionic modes
In this section we present the background geometry of a 5-dimensional locally AdS spacetime in the presence of an idealized cosmic string and the complete set of the positive and negative energy fermionic modes, being the solutions of the Dirac equation subject to the bag boundary condition on a brane parallel to the AdS boundary.
We consider a (4+1)-dimensional spacetime covered by the spatial coordinates (r, φ, y, z), with r ≥ 0, 0 ≤ φ ≤ 2π/q, −∞ < y, z < +∞. The local geometry is described by the line element where t ∈ (−∞, ∞). In the special case q = 1, it corresponds to 5-dimensional AdS spacetime generated by the cosmological constant Λ = −6/a 2 . For q = 1, the line element (2.1) describes an idealized topological defect with the core localized on the two-dimensional spatial hypersurface r = 0. The geometry of the core is given by the element ds 2 core = e −2y/a dt 2 − dz 2 − dy 2 . Note that the presence of the defect does not change the local characteristics of the geometry outside the core and they coincide with those for (4+1)-dimensional AdS spacetime. Two projections of the element (2.1) describe idealized cosmic strings with the planar angle deficit 2π(1 − 1/q). The first one, corresponding to a hypersurface y = const, presents a cosmic string in (3+1)-dimensional Minkowski spacetime. In particular, that is the case for the geometry of the conformal field theory in the context of the AdS/CFT correspondence. The second projection with z = const corresponds to a cosmic string in background of (3+1)-dimensional AdS spacetime. In the discussion below it will be convenient to work in the coordinates where the geometry outside the core r = 0 is conformally flat. Introducing a new coordinate w in accordance with w = ae y/a , the line element is written in the form conformally related to a 5-dimensional Minkowski spacetime: r, φ, w, z). This explicitly shows the conformal connection to the geometry of a cosmic string in background of (4+1)-dimensional Minkowski spacetime with the line element where, as before, 0 ≤ φ ≤ 2π/q. The new coordinate is defined in the interval 0 ≤ w < ∞ and the endpoints w = 0 and w = ∞ correspond to the AdS boundary and horizon, respectively. Having specified the background geometry we turn to the field content. A quantum fermionic field ψ will be considered that realizes an irreducible representation of the Clifford algebra for the Dirac matrices γ µ . In a 5-dimensional spacetime it is presented by a 4-component spinor. The Dirac matrices are expressed in terms of the corresponding flat spacetime matrices γ (b) as γ µ = e µ (b) γ (b) , where e µ (b) are the vielbein fields. The dynamics of the field is described by the Lagrangian density with the covariant derivative operator D µ = ∂ µ + Γ µ + ieA µ and the Dirac adjointψ = ψ † γ (0) . Here, Γ µ is the spin connection, A µ is the vector potential for a gauge field and ±e are the charges of the field quanta. In odd dimensional spacetimes the Clifford algebra has two inequivalent irreducible representations and the values s = 1 and s = −1 of the parameter s distinguish two fields realizing those representations (see the discussion in Section 5). Taking the vielbein fields e µ (b) = (w/a)δ µ b , we use the following representation for the 4 × 4 Dirac matrices: where the 2 × 2 Pauli matrices σ l , l = 1, 2, 3, corresponding to the coordinates (r, φ, w), are given by 6) and The equation of motion corresponding to the Lagrangian density (2.4) reads In the discussion below we assume a special configuration of a classical gauge field with the vector potential A µ = (0, 0, A, 0, 0) having the only nonzero constant covariant component A 2 = A. This corresponds to a magnetic flux Φ = −2πA/q, running along the core of the defect. Outside the core the field tensor F µν = ∂ µ A ν − ∂ ν A µ vanishes and the effect of the magnetic flux on the local characteristics of the fermionic vacuum in the region r > 0 is purely topological. Here we are interested in the effects of a codimension one brane, parallel to the AdS boundary, on the vacuum FC and on the VEV of the energy-momentum tensor. Assuming that the brane is located at w = w 0 , the MIT bag boundary condition will be imposed on the field operator: 8) where n µ is the normal to the brane. The latter is given as n µ = δ 3 µ a/w in the region 0 ≤ w ≤ w 0 (referred to as the L(left)-region) and as n µ = −δ 3 µ a/w for the region w 0 ≤ w < ∞ (R(right)-region). The value of the y-coordintae corresponding to the location of the brane we will denote by y 0 .
The analysis of the FC and of the mean energy-momentum tensor on the pure AdS spacetime considering the presence of branes has been developed in [21]. As to the investigation of those VEVs on the AdS background in the presence of a magnetic flux-carrying cosmic string, it was considered in [18,19]. Here in this paper, we want to investigate the influence of the gravitational field, the nontrivial spatial topology and the presence of the brane on the local properties of the FC and the VEV of the energy-momentum tensor.
The VEVs of physical observables bilinear in the field operator can be expressed in terms of the mode sum over the complete set of solutions of the field equation. For the system under consideration, those solutions were obtained in [12]. They are specified by the set of quantum numbers σ = (λ, j, p, k z , η), where 0 ≤ λ, p < ∞, j = ±1/2, ±3/2, . . ., −∞ < k z < +∞, and η = ±1. The energy is expressed as The positive (upper sign) and negative (lower sign) energy fermionic mode functions are presented as where J β j (λr) is the Bessel function [22] of the order 11) and the parameter α is expressed in terms of the magnetic flux as α = −Φ/Φ 0 , being Φ 0 = 2π/e the quantum flux. In the coefficients of the spinor components we have introduced the notations The dependence of the fermionic modes on the coordinate w enters in the form of the functions w 5/2 W ν 1 (pw) and w 5/2 W ν 2 (pw), where is a linear combination of the Bessel and Neumann functions and ν l = ma + (−1) l s/2. (2.14) The coefficients in the linear combination depend on the region of the space under consideration. In the L-region we take (C 1 , C 2 ) = (1, 0) and in the R-region the ratio of the coefficients is determined by the boundary condition (2.8) on the brane: From the boundary condition (2.8) for the modes in the L-region it follows that the allowed values of the quantum number p are the positive roots of the equation The roots with respect to the argument of the Bessel function will be denoted by p i = pw 0 , i = 1, 2, 3, ..., assuming that p i+1 > p i . The remaining normalization coefficient C 18) where δ σσ ′ is understood as the Dirac delta function for continuous quantum numbers in the set σ and the Kronecker delta for discrete ones. The integration over w in (2.18) goes over [0, w 0 ] in the L-region and over [w 0 , ∞) in the R-region. We can show that 19) where J = L, R specifies the spatial region and . (2.20) In the L-region one has p = p i /w 0 . Here, a comment related to the choice of the coefficients (C 1 , C 2 ) = (1, 0) in the L-region is in order. In the range of the mass ma ≥ 1/2 that choice is dictated by the normalizability condition for the fermionic modes used in the second quantization procedure. In the range 0 ≤ ma < 1/2, the mode functions with C 2 = 0 are normalizable and for a unique specification of the ratio C 2 /C 1 an additional boundary condition is required on the AdS boundary. Our choice of the modes corresponds to the setup with two branes located at w = ε < w 0 and w = w 0 , where the bag boundary condition is imposed for w = ε and then the limiting transition ε → 0 is taken.

General formula
We start our investigation of local VEVs from the FC. It is defined as the VEV 0|ψψ|0 ≡ ψ ψ , where |0 stands for the vacuum state. Having the complete set of the fermionic modes, the FC can be evaluated by the following mode sum formula: The details of the evaluation procedure for the brane-induced effects do not depend on the regularization of the divergent expression in the right-hand side of (3.1) and we will not specify the corresponding procedure. For example, we could regularize by the point-splitting technique or by introducing a cutoff function. Substituting the mode functions (2.10) in (3.1), after the summation over the quantum number η, we can see that where j = j=±1/2,±3/2,··· , the functions W ν (pw) and U (J) ν 2 (pw 0 ) in the L-and R-regions are defined by (2.15) and (2.20), p = p i /w 0 in the L-region and The parameter α, codifying the effects of the magnetic flux, enters in (3.2) through the combination j + α. If we present it in the form α = N + α 0 , with N being an integer and |α 0 | ≤ 1/2, then the integer part is absorbed by the redefinition j + N → j in the summation over j. From here it follows that the FC depends on the fractional part α 0 only and in (3.2) we can replace α by α 0 .
In order to extract explicitly the contributions induced by the cosmic staring and by the brane, we substitute in the right-hand side of (3.2) the representation The evaluation of the integral over k z is elementary and the λ-integral is evaluated by the formula where x = r 2 /(2v 2 ) and I ν (x) is the modified Bessel function [22]. Passing to a new integration variable x in the integral over v, we obtain where For the further transformation of the expression in the right-hand side of (3.6) we employ the representation [23] J (q, α 0 , x) = 2 q e x + 4 q where [q/2] represents the integer part of q/2, c k = cos(πk/q), and the function h(q, α 0 , x) is defined as h(q, α 0 , x) = l=±1 cos[πq(1/2 + lα 0 )] sinh[(1/2 − lα 0 )qx]. (3.9) In the case 1 ≤ q < 2, the term with the summation on the right-hand side of (3.8) must be omitted. Note that in the absence of the cosmic string and magnetic flux one has q = 1, α 0 = 0 and J (q, α 0 , x) = 2e x . We will denote the corresponding FC by ψ ψ J . Now we see that the part in the FC (3.6) coming from the first term in the right-hand side of (3.8) coincides with ψ ψ (0) J . In the remaining part the integral over x is expressed in terms of the modified Bessel function K ν (z) and one gets with the notation s k = sin(πk/q). In order to extract the brane-induced contribution, let us consider the FC in the brane-free geometry. The corresponding mode functions are given by (2.10) with W ν (pw) = J ν (pw) and the respective normalization constant is obtained from (2.19) The FC in the brane-free geometry, ψ ψ AdS cs , is obtained from (3.10) with the same replacements and with (p) from (3.3) for the R-region. It is decomposed as ψ ψ AdS cs = ψ ψ AdS + ψ ψ cs , (3.11) where ψ ψ AdS is the FC in pure AdS spacetime and the part is the contribution of the cosmic string in the brane-free geometry (see [18]). The integral over p in this expression is expressed in terms of the associated Legendre function of the second kind and the final expression for ψ ψ cs can be found in [18]. As seen from (3.12), one has ψ ψ cs | s=−1 = − ψ ψ cs | s=+1 . The FC (3.10) is presented as 13) where the expressions for the functions f (J) (w 0 , w, γ), with J = R, L, are given in Appendix A. In J − ψ ψ AdS is the contribution in the FC induced by the brane in the geometry where the cosmic string is absent. The latter has been investigated in [21] in general number of spatial dimensions D for the case s = 1. In the special case of (4+1)-dimensional spacetime with D = 4, the corresponding result for s = ±1 have the form with the functions where ν 1 and ν 2 are defined by (2.14).
The further transformation of the functions f (J) (w 0 , w, γ) in the last term of (3.13) is presented in Appendix A. Substituting (3.14), (A.3) and (A.6) in (3.13), the FC is decomposed as 16) where the contribution induced by the brane is given by the formula In the expression of the right-hand side we have introduced the notation (3.18) Here and below, the prime on the sign of summation means that the term k = 0 should be taken with additional coefficient 1/2. That term is reduced to 2 −n−1 /Γ(n + 1). Note that in (3.17) the contribution of the term with k = 0 presents the FC ψ ψ (0) b,J , given by (3.14). For points away from the defect core and outside of the brane (r = 0, w = w 0 ) the renormalization in (3.16) is required for the pure AdS part ψ ψ AdS only. In the absence of the defect and brane the background geometry is maximally symmetric and the renormalized FC ψ ψ AdS does not depend on the spacetime point. Comparing the results for the R-and L-regions we see that the formula for the brane-induced FC in the L-region is obtained from the formula for the R-region by the replacements I ⇄ K, ν 1,2 → ν 2,1 (see (3.15)). As seen from (3.17), the brane-induced FC depends on the coordinates w, w 0 and r through the ratios w/w 0 and r/w. For the first one we have w/w 0 = e (y−y 0 )/a and it determines the physical distance of the observation point from the brane: The proper distance from the cosmic string core is given by r p = ar/w and the ratio r/w presents the proper distance measured in units of the curvature radius a. The general expression (3.17) is further simplified in two special cases. In the absence of the magnetic flux one has α 0 = 0 and h(q, 0, 2u) = 2 cos (πq/2) sinh (qu) . (3.20) The corresponding expression for the function (3.18) is reduced to The second special case corresponds to the absence of the planar angle deficit with q = 1. In this case 22) and the expression for the function (3.18) takes the form For general values of the parameters, both the brane-free and brane-induced contributions in the FC are even periodic functions of the magnetic flux Φ with the period equal to the flux quantum. In Figure 1 we have presented the dependence of the brane-induced FC on the fractional part of the ratio of the magnetic flux to the flux quantum, codified in the parameter α 0 . The left and right panels correspond to the R-and L-regions with w/w 0 = 1.5 and w/w 0 = 0.75, respectively, and the numbers near the curves are the values of the parameter q. The full and dashed curves present the condensate for the fields with s = +1 and s = −1. For the remaining parameters we have taken ma = 1 and r/w 0 = 0.25. As seen, the dependence on the magnetic flux is stronger for larger values of the parameter q.

Minkowski bulk
First we consider the limit of the Minkowski bulk corresponding to a → ∞ with fixed values of y and y 0 . For the coordinate w one has w ≈ a + y. Both the orders and the arguments of the modified Bessel functions in the expressions for F (J) (pw 0 , pw) are large and we use the corresponding uniform asymptotic expansions [22]. To the leading order this gives (3.24) In this case the FC in the R-and L-regions are symmetric. Substituting (3.24) in (3.17) we see that where the boundary-induced contribution in the Minkowski bulk is given by ψ ψ (3.25) with the function H 1 (q, α 0 , x) defined in (3.18). The integral is expressed in terms of the modified Bessel function K ν (x) (see [24]) and Here, we have introduced the functions 27) and Note that the function f ν (x) obeys the relation f ′ ν (x) = −xf ν+1 (x). The expression for the FC induced by the cosmic string in the boundary-free (4+1)-dimensional Minkowski spacetime can be found in [18]. For a massless field the latter vanishes. In the massless limit the boundary-induced contribution (3.26) We recall that the k = 0 term gives the boundary-induced contribution in the absence of the cosmic string. The divergence of the FC on the boundary comes from that term alone.

Massless field
In the case of a massless field on the AdS bulk the contribution in the FC induced by a cosmic string in the absence of the boundary is given by [18] ψ Comparing with (3.29), we see that in the right-hand side is the FC in the locally Minkowski bulk with the line element (2.3), induced by the boundary at w = 0. It is given by (3.29) with y 0 = 0 and with the replacement y → w. We could expect the relation (3.31), by taking into account that for a massless fermionic field the boundary-free problem on the AdS bulk is conformally related to the problem in the Minkowski bulk with a single boundary at w = 0. The latter is the conformal image of the AdS boundary.
For a massless field on the AdS bulk one has ν 1 = −s/2, ν 2 = s/2 and the modified Bessel functions in (3.15) are expressed in terms of the elementary functions. This gives: . (3.32) In the expression (3.17) for the R-region the integral is evaluated by using the formula from [25]: The contribution to the brane-induced part ψ ψ b,R coming from the second term in the figure brackets of (3.33) with k = 0 is cancelled by the brane-free term ψ ψ cs , given by (3.30), and the total condensate in the R-region is presented as Comparing with (3.29) (with the replacements y, y 0 → w, w 0 ) we see that the last term in (3.34) is conformally related to the FC in the Minkowski bulk (2.3) induced by the boundary at w = w 0 . In the massless case, the expression for the FC in the L-region is obtained substituting (3.32) in (3.17). Compared to the R-region, the corresponding expression is more complicated. That is related to the fact that the probem in the L-region is conformally related to the problem in the Minkowski with two planar boundaries located at w = 0 and w = w 0 . The former is the conformal image of the AdS boundary.

Asymptotics with respect to the distance from the brane
First let us consider the FC in the R-region at large distances from the brane, w ≫ w 0 , that corresponds to (y − y 0 ) ≫ a. The dominant contribution to the integral over p in (3.17) comes from the region where pw 0 ≪ 1. Expanding the corresponding modified Bessel functions one gets If, in addition, w ≫ r, further simplification is made by expanding the integrand with respect to r/w (for the near string asymptotic see below). From (3.35) we see that the leading term in the brane-induced FC ψ ψ b,R behaves as (w 0 /w) ν 2 +|ν 2 | . This shows that for s = 1 the brane-induced FC decays as (w 0 /w) 2ma+1 . For the field with s = −1, depending on the mass, two qualitatively different cases are realized. The first one corresponds to the range of the mass ma > 1/2 and in this case the condensate ψ ψ b,R tends to zero like (w 0 /w) 2ma−1 . For the masses in the range ma < 1/2 and for fixed r/w the brane-induced contribution tends to a finite nonzero value in the limit w 0 /w → 0. Note that the asymptotic (3.35) also determines the behavior of the FC when the location of the brane tends to the AdS boundary, w 0 → 0, for fixed values of w and r.
In the L-region, at large distances from the brane one has w ≪ w 0 . For fixed values of w 0 and r this corresponds to points near the AdS boundary. In this limit, in the expression (3.17) for the L-region we introduce a new integration variable u = pw 0 and expand the modified Bessel functions with the arguments uw/w 0 . Keeping the leading order terms we get (3.36) and the brane-induced FC vanishes on the AdS boundary like (w/w 0 ) 2ma+5 . Note that near the AdS boundary the brane-free contribution behaves as [18] ψ ψ cs ≈ − Hence, in the limit w → 0 both the contributions tend to zero like w 5+2ma . Now we turn to the near brane asymptotic. The brane-induced FC diverges for points on the brane, w = w 0 . As it is seen from the representation (3.10), the divergence comes from the k = 0 term that corresponds to the FC in the geometry where the cosmic string is absent. The remaining contributions are finite on the brane due to the presence of the modified Bessel function K 1 (x). Moreover, from (3.10) it follows that for r > 0 the difference ψ ψ J − ψ ψ (0) J vanishes on the brane. Note that in the representation (3.16), with the brane-induced contribution from (3.17), for the evaluation of that difference on the brane the direct substitution w = w 0 in the corresponding integrand is not allowed. For points near the brane the dominant contribution to the integral for the k = 0 term in (3.17) comes from the integration range where the arguments of the modified Bessel functions are large. By making use of the corresponding asymptotic expressions [22], we can see that to the leading order F (J) (pw 0 , pw) ≈ e −2p|w−w 0 | /(2pw). After evaluating the integral we get the leading behavior: Note that for points near the brane one has |y − y 0 | ≪ a and 1 − w 0 /w ≈ (y − y 0 )/a. The leading term is written as ψ ψ b,J ≈ 3 (y − y 0 ) −4 /(32π 2 ). It coincides with the corresponding result in the Minkowski bulk and the effects of the spacetime curvature are weak. That is related to the fact that near the brane the main contribution to the VEVs comes from fluctuations with small wavelengths and the influence of gravity on those fluctuations is weak. The leading term (3.39) does not depend on the planar angle deficit and in the region near the brane the effects of the cosmic string are subdominant. Figure 2 presents the dependence of the FC, induced by the cosmic string and brane, as a function of the ratio w/w 0 that determines the distance from the brane (see (3.19)). The left and right panels correspond to the R-and L-regions and the full and dashed curves correspond to the fields with s = +1 and s = −1. The graphs are plotted for ma = 1, r/w = 0.25, α 0 = 0.3 and the numbers near the curves are the values of the parameter q. The graphs on Figure 2 confirm the features clarified by the asymptotic analysis: near the brane the FC is dominated by the brane-induced contribution and the dependence on q is weak, whereas at large distances and near the horizon the brane-free contribution dominates. As seen, depending on the distance from the brane, the FC may change the sign.

Small and large distances from the cosmic string
We finish the qualitative description of the behavior for the FC considering small and large distance asymptotics with respect to the cosmic string core. At small proper distances r p from the cosmic string core, compared with the curvature radius, one has r/w ≪ 1, and the brane-free contribution for a massive field behaves as [18] ψ ψ cs ≈ − mh 3 (q, α 0 ) 8π 2 (ar/w) 3 .
For 2|α 0 | > 1 − 1/q the brane-free part ψ ψ cs for a massless field diverges on the string like (w/r) 1−(1−2|α 0 |)q . For 2|α 0 | < 1 − 1/q the limiting value for the brane-induced contribution ψ ψ b,J on the string is obtained directly from (3.17) putting in the integrand r = 0: Comparing with the FC in the geometry where the cosmic string is absent, given by (3.14), the following relation is seen: b,J . From (3.8) for the factor in this expression one has Now, by taking into account (3.7), we see that for 2|α 0 | < 1 − 1/q the right-hand side of (3.42) tends to zero as x q(1/2−|α 0 |)−1/2 . Hence, we conclude that ψ ψ b,J | r=0 = 0 for 2|α 0 | < 1 − 1/q. In the range 2|α 0 | > 1 − 1/q the contribution ψ ψ b,J diverges on the string. The divergence comes from the integral over u in (3.17). For points close to the string the dominant contribution to that integral comes from large values of u. By using the corresponding asymptotic, after evaluating the integral, to the leading order we get where α q = (1/2 − |α 0 |)q. Note that under the condition 2|α 0 | > 1 − 1/q one has α q < 1/2 and the condensate (3.43) is negative near the string. From (3.42) it follows that 1 + 2h 0 (q, α 0 ) = q/2 for α q = 1/2. The limiting value of the FC on the cosmic string for this special case is obtained from (3.41) with that replacement. The same result is also obtained from (3.43) in the limit α q → 1/2. In the discussion of the large distance asymptotic it is more convenient to use the representation (3.10), where the last term is induced by the presence of the cosmic string. The behavior of the FC in that region is qualitatively different for the L-and R-regions and we consider them separately. In the L-region the spectrum of the quantum number p is discrete and at large distances the main contribution to the string-induced part comes from the term with the lowest mode p = p 1 /w 0 . For q > 2 the term k = 1 dominates and the cosmic string induced effects, given by ψ ψ L − ψ ψ (0) L , are suppressed by the factor e −2p 1 s 1 r/w 0 . For 1 ≤ q < 2 the sum over k is absent in (3.10) and we need to estimate the integral. In the region under consideration the integral is dominated by the integration range near the lower limit. By expanding the integrand for small values of u, we can see that ψ ψ L − ψ ψ L ∝ e −2(r/w 0 )p 1 and in this case the suppression is stronger. In the R-region the spectrum of the quantum number is continuous and (p) in (3.10) is defined in accordance with (3.3). At large distances from the cosmic string the dominant contribution to the corresponding integral over p comes from the region near the lower limit of the integration and we can use the asymptotic formulas for the Bessel and Neumann functions for small arguments [22]. By using the integral [25] ∞ we can see that for the cases s = 1 and s = −1, ma < 1/2. In the case s = −1, ma > 1/2 the asymptotic behavior is described by the expression These estimates show that, unlike the L-region, the decay of the topological contributions in the R-region, as functions of the proper distance from the cosmic string, exhibit an inverse power-law decrease at large distances.
In Figure 3 we have displayed the brane-induced contribution to the FC as a function of the ratio r/w 0 for the R-and L-regions (left and right panels, respectively). The full and dashed curves correspond to the fields with s = 1 and s = −1. The graphs are plotted for ma = 1, α 0 = 0.3 and the numbers near the curves correspond to the values of the parameter q. The graphs on the left and right panels are plotted for w/w 0 = 1.5 and w/w 0 = 0.75, respectively. Note that the case q = 2.5 corresponds to the critical value α q = 1/2 for which the brane-induced FC takes a finite nonzero value on the cosmic string. In accordance with (3.43), for q = 1 the condensate ψ ψ b,J diverges on the string. For q = 3 the brane-induced condensate tends to zero in the limit r → 0. At large distances from the cosmic string one has ψ ψ J → ψ ψ (0) J . The results obtained above can be used to estimate the effects of the nonzero FC in models with self-interacting fermions or with fermions interacting with other quantum fields. Examples are the fermionic fields with Nambu-Jona-Lasinio type self-interaction (described by the Lagrangian density proportional to ψ ψ 2 ) and the fermions interacting with a scalar fields ϕ through the Lagrangian density proportional toψψϕ 2 . The formation of the FC gives rise to mass terms in the field equations proportional to ψ ψ ψ and ψ ψ ϕ, for the fermionic and scalar fields, respectively. To the leading order with respect to the interactions, the FC in those terms is given by the quantity evaluated within the framework of the free field theory. For the geometry under consideration that quantity is determined by the expressions given in this section. Note that, depending on the sign of the FC, the generation of additional mass terms may induce instabilities in the respective field theories.

Vacuum expectation value of the energy-momentum tensor
In this section we investigate another important characteristic of the fermionic vacuum, the VEV of the energy-momentum tensor 0|T µν |0 ≡ T µν . It is evaluated by using the mode sum formula where the covariant derivative operator acting on the Dirac adjoint spinor is given by D µψ = ∂ µψ − ieA µψ −ψΓ µ and the brackets in the index expression mean the symmetrization over the enclosed indices.

General expressions
Inserting the mode functions (2.10), using the relation {γ µ , Γ µ } = 0, and summing over χ and η, we can see that the VEVs for the off-diagonal components vanish. The VEVs for the diagonal components are presented in the combined form (no summation over µ) where, as before, J = R, L for the R-and L-regions, respectively. The functions for the separate components are defined by the expressions and The remaining notations are the same as those in (3.2). Note that we have the relations It can be checked that the components (4.2) obey the trace relation The further transformations of (4.2) are similar to those for the FC. By using the integral representation (3.4) and integrating over k z we get (no summation over µ) For the components with µ = 0, 3, 4 the integrals over λ are evaluated by the formula (3.5). For the component µ = 2 we use the formula 8) and the relation with x = r 2 /(2v 2 ). The remaining integral over λ for µ = 1 is obtained by using the relation (4.5). After integrating by parts the energy density, the VEV is presented as (no summation over µ) (2x∂ x + 1) δ 2µ e −x J (q, α 0 , x), (4.10) with the notation (3.8). From here it follows that and Based on this relations, we continue our discussion for the components µ = 0, 3. For the factor J (q, α 0 , x) in the integrand of (4.10) we have the representation (3.8). The part coming from the first term in the right-hand side gives the VEV in the geometry without a cosmic string, denoted here by T µ µ (0) J . In the part induced by the presence of the string the integral over x is expressed in terms of the modified Bessel function and we get (no summation over µ = 0, 3) In order to discuss the effects induced by the brane, it is useful to have the VEV in the brane-free geometry. We will denote it by T µ µ AdS cs . The corresponding expression is obtained from (4.13) with W ν (pw) = J ν (pw), U (J) ν 2 (pw 0 ) → 1 and (p) = w 2 0 ∞ 0 dp p. It is decomposed as (4.14) where T µ µ AdS is the VEV in pure AdS spacetime and the string-induced contribution is given by [18] (no summation over µ = 0, 3) where . (4.16) Note that this contribution is the same for s = +1 and s = −1.
The VEV of the energy-momentum tensor is decomposed as (4.17) where the functions f AdS is the contribution of the brane in the geometry where the cosmic string is absent. The expression for that contribution in the case s = 1 has been obtained in [21] for general spatial dimension (the VEV of the energy-momentum tensor for scalar and vector fields in both single and two-brane geometries were investigated in [26]). Specifying for the case D = 4 and generalizing for s = −1, the corresponding formulas read (no summation over µ) with the functions (R) (x, y) = 4.19) in the R-region and (4.20) in the L-region. By making use of the representations for the functions f (µ) (J) (w 0 , w, γ), given in Appendix A, the final expression for the VEV of the energy-momentum tensor is decomposed as (no summation over µ) T (4.21) where the contribution induced by the brane is given by the formula Here, the functions F (µ) (J) (pw 0 , pw) in the R-and L-regions (J = R and J = L, respectively) are given by (4.19), (4.20) and we have defined the functions (4.23) with H n (q, α 0 , x) from (3.18). The contribution of the k = 0 term in (3.18) to (4.22) presents the VEV induced by the brane in the geometry where the cosmic string is absent. It is given by the expression (4.18). For points outside the defect core and the brane, the last two contributions are finite and the renormalization is needed for the part T µ µ AdS only. From the maximal symmetry of the background geometry for that part it follows that for the renormalized VEV we have T ν µ AdS = const · δ ν µ . Comparing the expressions for the energy-momentum tensor in the R-and L-regions, we can see that the brane-induced contribution in the L-region is obtained from the corresponding one in the R-region by the replacements K → I, I → K, of the modified Bessel functions, and ν 1,2 → ν 2,1 in their orders. The special cases of the general formula (4.22) in the absence of the magnetic flux or planar angle deficit are obtained by using the corresponding expressions (3.21) and (3.23) in (4.23).
The parts T ν µ cs and T ν µ b,J separately obey the trace relations T µ µ cs = sm ψ ψ cs and T µ µ b,J = sm ψ ψ b,J . In particular, these tensors are traceless for a massless fermionic field. Moreover, it can be verified that the covariant conservation equation, ∇ µ T µν = 0, is also satisfied by the separate terms in (4.21). In the problem at hand, it is reduced to the following two relations (4.24) between the separate components. By taking into account that T 1 1 = T 0 0 , we see that the first relation in (4.24) coincides with (4.12).
The VEV of the energy-momentum tensor is an even periodic function of the magnetic flux with the period equal to the flux quantum. That is expressed in terms of the dependence on the parameter α 0 . That dependence for the brane-induced vacuum energy density in the R-region (for w/w 0 = 1.5, left panel) and L-region (for w/w 0 = 0.75, right panel) is displayed in Figure 4 for different values of the parameter q (the numbers near the curves). The full and dashed curves correspond to the fields with s = +1 and s = −1. The graphs are plotted for ma = 1 and r/w 0 = 0.25. Similar to the case of the FC, we see that the dependence on the magnetic flux is stronger for higher values of the planar angle angle deficit.

Asymptotic analysis and numerical results
In order to clarify the behavior of the vacuum energy-momentum tensor we consider special cases and asymptotics.

Minkowskian limit
For large values of the curvature radius a the orders of the modified Bessel functions in the integrand of (4.18) are large and we use the corresponding uniform asymptotic expansions. It can be seen that in the leading order we get 1, 2, 4, (4.25) where the asymptotic expression for F (J) (pw 0 , pw) is given by (3.24). Substituting the asymptotic expression (4.25) in (4.22), we see that the leading term in the expansion of T µ µ b,J does not depend on a and it coincides with the corresponding VEV induced by a planar boundary in the (4+1)-dimensional Minkowski spacetime with a cosmic string. Denoting the latter by T ν µ (M) b , for the components with µ = 3 one obtains (no summation over µ)

Massless field
For a massless fermionic field one has ν 1 = −ν 2 = −s/2, and by taking into account the property K ν (z) = K −ν (z) for the Macdonald function, we see that the brane-induced VEV of the energymomentum tensor vanishes in the R-region. This result could be directly obtained based on the conformal relation with the problem of a single boundary on the Minkowski bulk with a cosmic string. 1 For the L-region one has F (µ) (L) (x, y) = −F (3) (L) (x, y) = 2/[y(se 2x + 1)], µ = 3, and the VEV is presented as (no summation over µ) (4.27) This result is conformally related to the corresponding formula for a cosmic string in the Minkowski bulk with the line element (2.3) and with two planar boundaries located at w = 0 and w = w 0 . In the case of a massless field for the brane-free part one has [18] (no summation over µ) (4.28) for µ = 0, 1, 3, 4 and T 2 2 cs = −4 T 0 0 cs .

Asymptotics with respect to the brane location
Here we consider the asymptotic behavior of the vacuum energy-momentum tensor with respect to the ratio w/w 0 . In the R-region and for large values of this ratio, the dominant contribution to the integral (4.22) comes from the region where pw 1. In that region pw 0 is small and we expand the corresponding modified Bessel functions. To the leading order this gives (no summation over µ) For fixed values of w and r and for small w 0 this asymptotic describes the situation where the brane is close to the AdS boundary. Similar to the case of the FC, for fields with s = 1 and s = −1, ma > 1/2 the brane-induced VEV vanishes as w 2ma+s 0 . In the case s = −1, ma < 1/2 the VEV T µ µ b,R tends to a nonzero limiting value in the limit w 0 → 0. Fixing w 0 and r and for large values of w, we have the situation where the observation point is close to the horizon. In this case we can additionally expand the function H (µ) (q, α 0 , 2xr/w) for small values of the ratio r/w. The latter corresponds to small proper distances from the string compared with the AdS curvature radius. The corresponding asymptotic is discussed below.
In the L-region and for small values of the ratio w/w 0 we introduce a new integration variable x = pw and expand the integrand with respect to w/w 0 . The leading term is expressed as Again, we can consider two situations. The first one corresponds to large values of w 0 for fixed w and r (the brane is close to horizon) and we can further expand the function H (µ) (q, α 0 , 2xr/w 0 ) for small r/w 0 . The second situation corresponds to small values of w for fixed w 0 and r (the observation point is close to the AdS boundary). As seen, for fixed r/w 0 > 0, the brane-induced VEV T µ µ b,L vanishes on the AdS boundary like w 5+2ma . A similar behavior is exhibited by the brane-free part as well (see [18]). Now let us consider the brane-induced VEV (4.22) near the brane. All the terms in (4.22), except the k = 0 term in the summation over k, are finite on the brane. The k = 0 term corresponds to the brane-induced VEV in the problem where the cosmic string is absent and for a massive field it diverges on the brane. This means that near the brane the VEV (4.22) is dominated by that term. For points near the brane the main contribution to the corresponding integral comes from the region where p is large. By using the asymptotic expressions for the modified Bessel function with large arguments [22], we can see that F (µ) (R) (pw 0 , pw) ≈ smae −2p|w−w 0 | /(pw) 2 for µ = 3. Hence, to the leading order, for the components with µ = 3 one gets (no summation over µ) (4.31) The asymptotic expression for the stress T 3 3 J is most conveniently obtained by using the second equation in (4.24): (4.32) As seen, the leading term is symmetric with respect to the brane for the components with µ = 3 and has opposite signs for the normal stress T 3 3 J . As it has been mentioned above, near the brane one has 1 − w 0 /w ≈ (y − y 0 )/a and the leading term in (4.31) coincides with the corresponding result for a planar boundary in the Minkowski bulk with a cosmic string where the distance from the boundary is given by |y − y 0 |. Note that in the Minkowski bulk the normal stress T 3 3 J is zero and the nonzero effect in the right-hand side of (4.32) is an effect induced by the background curvature. For a massless field the brane-induced VEV T µ µ b,J is finite on the brane and the same is the case for the total VEV T µ µ J . In this case the brane-induced part is zero in the R-region and the limiting value T µ µ b,L | w=w 0 for the L-region is directly obtained from (4.27) putting in the integrand w = w 0 .
In Figure 5 we present the combined effects of the cosmic string and brane in the VEV of the vacuum energy density, considered as a function of the ratio w/w 0 . The left and right panels are plotted for the R-and L-regions and the numbers near the curves are the values of the parameter q. The full and dashed curves correspond to the fields with s = +1 and s = −1, respectively, and the numbers near the curves are the values of q. For the values of the parameters we have taken ma = 1, α 0 = 0.3, r/w 0 = 0.75. The vacuum energy density is dominated by the brane-induced contribution near the brane and by the brane-free part near the horizon. It can be either positive or negative, depending on the distance from the brane.

Small and large distances from the cosmic string
It remains to consider the asymptotics near the cosmic string and at large distances from it. For points away from the brane and for 2|α 0 | < 1 − 1/q, the brane-induced contribution is finite on the string (r = 0) and we can directly put r = 0 in the integrand of (4.22). This leads to the result (no summation over µ) where the brane-induced contribution T µ µ (0) b,J in the geometry without cosmic string is given by (4.18). As it has been already explained before, 1 + 2h 0 (q, α 0 ) = 0 in the range 2|α 0 | < 1 − 1/q and in that range of the parameters the brane-induced term T µ µ b,J vanishes on the string. In the special case 2|α 0 | = 1 − 1/q in (4.33) we have 1 + 2h 0 (q, α 0 ) = q/2. For 2|α 0 | > 1 − 1/q and near the cosmic string the main contribution in (4.22) comes from the term containing the integral over u. The integral is dominated by the region with large values of u. By using the corresponding asymptotic expressions, in the way similar to that for the FC, we get (no summation over µ) with the notations In the limit α q → 1/2 the result (4.34) is reduced to (4.33). As regards the brane-free contribution T µ µ cs , near the string the effect of the mass is weak and to the leading order it coincides with that for a massless field. In that region the influence of the background gravitational field on the string induced effects is weak and the VEV T µ µ cs behaves as (w/r) 5 (see (4.28)). In the discussion of the asymptotic for the cosmic string-induced contribution at large distances from the core it is more convenient to use the representation (4.13). In the R-region one has (p) = w 2 0 ∞ 0 dp p and the dominant contribution to the integral in (4.13) comes from the integration range p 1/r. In that range the arguments of the Bessel and Neumann functions, w 0 p and wp, are small in the region under consideration. By employing the asymptotics for those functions [22], the integral over p is evaluated by making use of the formula (3.44). For s = 1 and s = −1, ma < 1/2, to the leading order, for the components with µ = 0, 1, 4 one finds (no summation over µ) π 2 a 5 (2r/w) 5+2sma h 5+2sma (q, α 0 ). (4.36) For the contributions in the remaining components, induced by the cosmic string, we have In the case s = −1, ma > 1/2 the asymptotic at large distances has the form (no summation over µ) for µ = 0, 1, 4 and for the remaining stresses. As we see in the case s = −1 the decay of the topological contributions is more slowly.
In the L-region the spectrum of the quantum number p is discrete and the decrease of the stringinduced contributions is exponential, like e −2rp 1 /w 0 for 1 ≤ q < 2 and as e −2rp 1 s 1 /w 0 for q ≥ 2, where p 1 /w 0 is the lowest positive eigenvalue for p. Figure 6 displays the dependence of the brane-induced part in the energy density, as a function of the distance from the cosmic string, in the R-region for w/w 0 = 1.5. The left and right panels correspond to the fields with s = +1 and s = −1, respectively, and the numbers near the curves are the values of q. For the remaining parameters we have taken ma = 1 and α 0 = 0.3. The same graphs for w/w 0 = 0.75 (L-region) are presented in Figure 7.  5 Vacuum densities for a field realizing the second representation and applications in braneworlds

Fields realizing the two representations
We have considered a fermionic field in (4+1)-dimensional spacetime. In odd number of spacetime dimensions the Clifford algebra for gamma matrices has two inequivalent irreducible representations. In this section they will be specified by s = +1 and s = −1 and the corresponding fields will be denoted by ψ (s) . As we will see, the parameter s is identified with the parameter s in the Lagrangian density (2.4). In (4+1)-dimensional flat spacetime the two sets of Dirac matrices (in the coordinates (t, r, φ, w, z)) can be taken as γ (1) , γ (2) , γ (3) , sγ (4) }, where the matrices γ (b) are related to the matrices (2.5) as γ (b) = (a/w)δ b µ γ µ and for the matrix γ (4) one has γ (4) = −rγ (0) γ (1) γ (2) γ (3) . Introducing the corresponding curved spacetime Dirac matrices γ µ (s) = (w/a)δ µ b γ (b) and the related spin connection Γ (s) µ , the Lagrangian density for the free field ψ (s) is presented as In order to see the relation of this Lagrangian density to (2.4), we introduce new fields ψ ′ (s) in accordance with ψ ′ (s) = − γ (4) 1+δ 1s ψ (s) . In terms of these fields the Lagrangian density is presented as +1) and D µ is the same as in (2.4). This shows that the fields ψ ′ (s) correspond to the fields with s = +1 and s = −1 in the discussion of the previous sections.
Let us compare the boundary conditions for the fields ψ (s) and ψ ′ (s) . We start the discussion with the case when the fields ψ (s) obey the bag boundary condition on the brane: for w = w 0 . Transforming to the fields ψ ′ (s) , we can see that they obey the condition on w = w 0 . This shows that the VEVs for the field ψ (+1) are given by the formulas presented in Sections 3 and 4 for the case s = 1. The transformed field ψ ′ (−1) obeys the equation (2.7), however, the corresponding boundary condition (5.2) differs from the condition (2.8) by the sign of the term that contains the normal to the brane. The corresponding VEVs are obtained in the way similar to the ones in the previous sections and are discussed in the next subsection.

VEVs for the second type of boundary conditions
Let us consider the VEVs for a fermionic field ψ obeying the field equation (2.7) and the boundary condition (the condition (5.2) with s = −1) on the brane located at w = w 0 . The complete set of mode functions still has the form (2.10), where now W ν (pw) = J ν (pw) in the L-region and W ν (pw) = G ν 1 ,ν (pw 0 , pw) in the R-region. These functions obey the condition (5.3) in the R-region and from the boundary condition in the L-region it follows that the eigenvalues of the quantum number p in that region are roots of the equation J ν 2 (pw) = 0 (compare with (2.17)). The normalization coefficients are obtained from (2.19) by the replacement ν 2 → ν 1 or s → −s. Following the same steps as in Section 3, we can see that the FC is presented in the decomposed form (3.16), where the contributions ψ ψ AdS and ψ ψ cs are given by the same expressions and the brane-induced FC is given by with the function H 1 (q, α 0 , x) from (3.18). Here, the functions in the integrand are defined as (5.5) with ν 1 and ν 2 from (2.14). Comparing with (3.15) and (3.17), we see that, for a given s, the braneinduced contribution for the field obeying the equation (2.7) and the condition (5.3) differs from the FC for the field, obeying the field equation (2.7) with s replaced by −s and the condition (2.8), only in the sign. The same property is the case for the terms ψ ψ AdS and ψ ψ cs : they change the signs under the replacement s → −s.
In a similar way we can see that the VEV of the energy-momentum tensor for the field obeying the equation (2.7) and the boundary condition (5.3) is presented as (4.21) with the brane-induced contribution (no summation over µ) (5.6) where the functions H (µ) (q, α 0 , x) are given by (4.23). In the R-region the functions in the integrand are defined as and for the L-region (5.8) Comparison of this result with (4.19), (4.20) and (4.22) shows that the brane-induced VEV of the energy-momentum tensor for the field with a given s and obeying the boundary condition (5.3) coincides with the corresponding quantity for the field with s replaced by −s in the field equation (2.7) and obeying the condition (2.8). We recall that the brane-free contribution in the VEV of the energy-momentum tensor does not depend on s. Equivalently, the obtained features can be formulated as follows: The FC for the field obeying the field equation (2.7) and the boundary condition (1 + siγ µ n µ )ψ = 0 on the brane w = w 0 is an odd function of s, whereas the VEV of the energy-momentum tensor is an even function.

VEVs for the second representation and applications in RSII braneworld
For the representation of the Clifford algebra corresponding to s = −1 the transformed field ψ ′ (−1) obeys the field equation (2.7) with s = −1 and the boundary condition (5.3). Hence, in accordance with the result formulated in the previous subsection, we conclude that the corresponding FC is expressed as where ψ ψ s=+1 is given by the expressions in Section 3 with s = +1. In order to find the FC for the initial field ψ (−1) we make the inverse transformation of the field. It is easy to see that 1) . Combining with (5.9) we conclude that the FCs coincide for the fields realizing two inequivalent irreducible representations of the Clifford algebra if they obey the boundary condition (5.1): ψ (+1) ψ (+1) = ψ (−1) ψ (−1) . The corresponding expressions are given by the formulas in Section 3 with s = +1. The same is the case for the fields ψ (s) obeying the boundary condition (1 − iγ µ (s) n µ )ψ (s) = 0 (5.10) for w = w 0 . The corresponding FC is given by the formula (5.4) with s = +1 (or by the formulas in Section 3 with s = −1 and with the change of the sign in front of all the terms). In a similar way, we can see that the VEVs of the energy-momentum tensor for the fields ψ (±1) obeying the boundary condition (5.1) coincide with the VEVs for the fields ψ ′ (±1) obeying the condition (5.2). They are given by the formulas in Section 4 with s = +1 for the field ψ (s) obeying the boundary condition (5.1) and by the formula (5.6) with s = +1 (or equivalently by (4.22) with s = −1) for the boundary condition (5.10).
The mass term in the Lagrangian density L (s) is not invariant under the charge conjugation (C) and the parity transformation (P ). In the absence of an external gauge field, we can combine two fields ψ (+1) and ψ (−1) for the construction of fermionic models with the Lagrangian density L = s=±1 L (s) invariant under those transformations. The total FC and the VEV of the energy-momentum tensor are obtained by summing the corresponding VEVs for separate fields. They are given by the formulas in Sections 3 and 4, with an additional coefficient 2, for s = +1 in the case of the boundary conditions (5.1) and for s = −1 in the case of conditions (5.10) (with an additional change in the sign of the FC for (5.10)).
As a realization of the model under consideration we can consider the Randall-Sundrum model with a single brane (RSII model) [28] in the presence of a topological defect. The brane in the corresponding setup is located at y = 0 and the background geometry contains two copies of the R-region identified by the Z 2 -symmetry. The line element is obtained from (2.1) by the replacement e −2y/a → e −2|y|/a . The fields in the regions −∞ < y < 0 and 0 < y < +∞ are related by the Z 2 -symmetry of the model. The 4 × 4 matrix M in the relation ψ(t, r, φ, −y, z) = M ψ(t, r, φ, y, z) is determined by the requirement of the invariance of the action under the Z 2 identification (see [15,29]). The following conditions are obtained on the matrix M : {γ (0) , M } = 0, {γ (0) γ (3) , M } = 0, and [γ (0) γ (b) , M ] = 0 with b = 1, 2, 4. The corresponding solution with an additional constraint M 2 = 1 is given as M = ±diag(σ 3 , −σ 3 ) = ±iγ (3) . Two types of fermionic fields in braneworld models correspond to the upper and lower signs in the expression for the matrix M . The boundary conditions on the brane for those fields are obtained by taking y = 0 in the relation mapping two copies of the R-region. It is reduced to 1 ∓ iγ (3) ψ(x) = 0. By taking into account that in the R-region we had n µ = −δ 3 µ a/w, we see that for the field with M = iγ (3) the boundary condition dictated by the Z 2 -symmetry is reduced to (2.8). For the field corresponding to M = −iγ (3) we get the boundary condition (5.3). The FC and the VEV of the energy-momentum tensor in the RSII model with a topological defect (for quantum effects in higher-dimensional generalizations of RSII model see [30] and references therein) are expressed by the formulas given above for respective boundary conditions taking w 0 = a and with an additional coefficient 1/2. The latter is related to the fact that in Z 2symmetric braneworld model the integral over y in the normalization condition for the fermionic modes goes over the region −∞ < y < +∞ instead of the region y ∈ [y 0 = 0, ∞) in our consideration above for the R-region.
In the context of the braneworld scenario, for an observer located on the brane at y = 0 the induced line element takes the form ds 2 b = dt 2 − dr 2 − r 2 dφ 2 − dz 2 with 0 ≤ φ ≤ 2π/q. This is the line element for the geometry corresponding to a straight cosmic string in (3+1)-dimensional flat spacetime. From the point of view of physics on the brane it is of interest to compare the VEVs induced on the brane by the topological defect in the background AdS spacetime with the corresponding VEVs induced by a cosmic string on the Minkowski bulk with the line element ds 2 b . As it has been discussed above, the VEVs ψ ψ J and T µ µ J diverge on the brane. The divergences come from the purely brane-induced parts ψ ψ (0) J and T µ µ (0) J . They are absorbed by the renormalization of the on-brane FC and energymomentum tensor in the absence of the cosmic string. The corresponding renormalized VEVs do not depend on the characteristics of the cosmic string. The contributions in the VEVs induced by the cosmic string on the AdS bulk, given by ψ ψ J − ψ ψ (0) J and T µ µ J − T µ µ (0) J , are finite on the brane. Moreover, as it has been shown above, that contribution in the FC vanishes on the brane. For the components with µ = 0, 3, the cosmic string-induced contribution T µ µ J − T µ µ (0) J on the brane y = 0 is directly found from the representation (4.13) taking w = w 0 = a (with a coefficient 1/2 for RSII model).
For these values and for the R-region one has W (0) ν 1 (pa) = 4/ (πpa) 2 and W (3) ν 1 (pa) = −4/π 2 a 2 . The other components are found by using the relations (4.11) and (4.12). The corresponding expressions are further simplified for a massless field. In that case J 2 ν 2 (pa) + Y 2 ν 2 (pa) = 2/(πpa) and the integral over p in (4.13) for the R-region is evaluated by using the formula ∞ 0 dx x 2+δ µ3 K 2−δ µ3 (x) = 3π/2. For the cosmic string-induced part on the brane this gives (no summation over µ) (1, 1, −4, 1, 1). (5.11) Note that this result for a massless field does not depend on the curvature radius of the AdS spacetime. For a massive field, the curvature radius appears in the expression for the product in the form of dimensionless combinations ma and r/a. The expectation values induced on the brane by quantum fluctuations of bulk fields differ from those for a cosmic string in Minkowski spacetime with the line element ds 2 b . The VEVs for the latter geometry in the absence of magnetic flux have been considered in [31]. The corresponding FC is nonzero for a massive fermionic field and vanishes in the massless limit. The VEV of the energy-momentum tensor is different from zero for both massless and massive fields. For a massless field it is reduced to the result found in [32].

Conclusion
We have investigated the combined effects of a cosmic string and of a brane parallel to the AdS boundary on the local properties of the fermionic vacuum. As representatives of those properties the FC and the VEV of the energy-momentum tensor are considered. For the evaluation of the corresponding expectation values the direct summation over the complete set of the fermionic modes from [12] has been used. They are specified by the set of quantum numbers (λ, j, p, k z , η). The eigenvalues of the quantum number p are continuous in the R-region and discrete in the L-region and the properties of the vacuum are different in those regions. Both the FC and the vacuum energymomentum tensor are decomposed into three separate contributions. The first one corresponds to the pure AdS geometry when the cosmic string and the brane are absent and due to the maximal symmetry of the AdS spacetime and of the corresponding vacuum state it does not depend on a spacetime point. The second contribution to the VEVs presents the part that is induced by the cosmic string in the brane-free geometry. Those contribution for the FC and the energy-momentum tensor have been investigated in [18] and [19], respectively. Our main interest in the present paper is concentrated on the contributions in the FC and the energy-momentum tensor induced by the presence of the brane. In order to explicitly separate those contributions in the L-region we have used the generalized Abel-Plana formula for the summation of series over the zeros of the Bessel functions (related to the eigenvalues of the quantum number p). The corresponding representation for the R-region is obtained by an appropriate rotation of the integration contour in the integral over continuous eigenvalues of p. After those transformations the brane-induced contributions are given by (3.17) for the FC and by (4.22) for the energy-momentum tensor. They are even periodic functions of the magnetic flux with the period equal to the flux quantum.
In order to clarify the behavior of the brane-induced contributions in the VEVs we have considered limiting cases and asymptotic regions of the parameters. The general formulas are simplified in two special cases. The first one corresponds to the absence of the magnetic flux with α 0 = 0 and the second one corresponds to the absence of the planar angle deficit with q = 1. The respective expressions for the FC and the VEV of the energy-momentum tensor are obtained from the expressions (3.17) and (4.22) substituting the functions (3.18) from (3.21) and (3.23), respectively. In the limit a → ∞, with fixed values for y and y 0 , we have obtained the VEVs in the geometry of a cosmic string in background of (4+1)-dimensional Minkowski spacetime in the presence of a planar boundary on which the field obeys the bag boundary condition. For a massless fermionic field the problem under consideration is conformally related to the corresponding problem in the Minkowski bulk with a single boundary for the R-region and with two parallel boundaries in the L-region. The one of the boundaries in the latter case is the conformal image of the brane and the second boundary is the conformal image of the AdS boundary.
The brane-induced contributions in the VEVs are mainly located in the region near the brane. For the FC the leading term in the corresponding asymptotic expansion is given by (3.39). Near the brane the effects of the background curvature and of the mass are weak and the leading term (3.39) coincides with that for a boundary in the Minkowski bulk in the absence of cosmic string. For the R-region, the large values of the ratio w/w 0 correspond to large proper distances from the brane compared with the curvature radius of the background spacetime. For a given value of the ratio r/w, the brane-induced VEVs for the FC and the energy-momentum tensor behave like (w 0 /w) ν 2 +|ν 2 | . In particular, this shows that when the location of the brane tends to the AdS boundary the brane-induced contributions vanish as w 1+2ma 0 for the field with s = +1. For the field with s = −1 that contribution vanishes like w 2ma−1 0 in the range of the mass ma > 1/2 and tends to a nonzero finite value for ma < 1/2. In the L-region, the large proper distances from the brane correspond to small values of the ratio w/w 0 ≫ 1 and the brane-induced parts decay as (w/w 0 ) 5+2ma .
Near the cosmic string and for massive fields the VEVs are dominated by the contributions ψ ψ cs and T ν µ cs . They behave like (r/a) 3 and (r/a) 5 for the FC and energy-momentum tensor, respectively. For the energy-momentum tensor the leading behavior remains the same for a massless field, whereas for the FC the leading term vanishes and near the cosmic string the contributions ψ ψ cs and ψ ψ b,J are of the same order. The brane-induced contributions ψ ψ b,J and T µ µ b,J vanish on the cosmic string for 2|α 0 | < 1 − 1/q, take a finite limiting value for 2|α 0 | = 1 − 1/q and diverge as (r/w) (1−2|α 0 |)q−1 in the range 2|α 0 | > 1 − 1/q. The effects induced by the cosmic string in the FC and in the VEV of the energy-momentum tensor are described by the differences ψ ψ J − ψ ψ J . The behavior of those quantities at large distances from the string essentially differs in the R-and L-regions. For the R-region and in the cases s = +1 or s = −1, ma < 1/2, the cosmic string-induced effects decay as (r/w) 5+2sma , whereas for the case s = −1, ma > 1/2 the decay is slower, like (r/w) 3+2ma . It is of interest to note that, considered as a function of the proper distance from the string, the decay of the cosmic string-induced contributions in the VEVs at large distances follows a power-law for both the cases of massless and massive fields. This behavior is in contrast to the one for the Minkowski bulk with the exponential decay for massive fields. The eigenvalues of the quantum number p in the L-region are discrete and the cosmic string-induced VEVs at large distances are suppressed by the exponential factor e −2(r/w 0 )p 1 , where p 1 /w 0 is the lowest eigenvalue for p.
We have considered an odd dimensional background spacetime and the corresponding Clifford algebra for the gamma matrices has two inequivalent irreducible representations. The FC and the VEV of the energy-momentum tensor for the fields realizing those representations, denoted here as ψ (+1) and ψ (−1) , are obtained from the results we have provided in Sections 3 and 4. If the bag boundary condition is imposed on the brane for both the fields (see (5.1)) and they have equal massess, then the VEVs for the fields ψ (+1) and ψ (−1) coincide and they are given by the formulas in Sections 3 and 4 for s = +1. If the fields ψ (+1) and ψ (−1) obey the boundary condition (5.10), that differs from the bag boundary condition by the sign of the term containing the normal to the brane, then the corresponding VEVs for both the fields are obtained from (5.4) and (5.6) with s = +1. Equivalently, the VEVs are obtained from the formulae in Sections 3 and 4 with s = −1 additionally changing the sign for the FC. We have seen that two types of the considered boundary conditions naturally arise in Randall-Sundrum type braneworld models as a consequence of the Z 2 -symmetry with respect to the brane. The FC and the VEV of the energy-momentum tensor in RSII model in the presence of cosmic string are obtained directly from the formulas given above.