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it is well known that different types of topological objects may have been formed in the early universe after planck time by the vacuum phase transition @xcite . depending on the topology of the vacuum manifold these are domain walls , strings , monopoles and textures . among them the cosmic strings are of special interest . although the recent observational data on the cosmic microwave background radiation have ruled out cosmic strings as the primary source for primordial density perturbations , they are still candidates for the generation of a number of interesting physical effects such as the generation of gravitational waves , gamma ray bursts and high - energy cosmic rays ( see , for instance , @xcite ) . recently , cosmic strings attract a renewed interest partly because a variant of their formation mechanism is proposed in the framework of brane inflation sara02 .    in the simplest theoretical model describing the infinite straight cosmic string the spacetime is locally flat except on the string where it has a delta shaped riemann curvature tensor . in quantum field theory the corresponding non - trivial topology induces non - zero vacuum expectation values for physical observables . explicit calculations for the geometry of a single cosmic string have been done for different fields @xcite - spin08 . vacuum polarization effects by higher - dimensional composite topological defects constituted by a cosmic string and global monopole are investigated in refs . @xcite for scalar and fermionic fields . another type of vacuum polarization arises when boundaries are present . the imposed boundary conditions on quantum fields alter the zero - point fluctuations spectrum and result in additional shifts in the vacuum expectation values of physical quantities . this is the well - known casimir effect ( for a review see @xcite ) . in ref . @xcite , we have studied both types of sources for the polarization of the scalar vacuum , namely , a cylindrical boundary and a cosmic string , assuming that the boundary is coaxial with the string and that on this surface the scalar field obeys robin boundary condition . for a massive scalar field with an arbitrary curvature coupling parameter we evaluated the wightman function and the vacuum expectation values of the field squared and the energy - momentum tensor . the polarization of the electromagnetic vacuum by a conducting cylindrical shell in the cosmic string spacetime is investigated in @xcite ( for a combination of topological and boundary induced quantum effects in the gravitational field of a global monopole see refs . @xcite . ) continuing in this line of investigation , in the present paper we analyze the polarization of the fermionic vacuum by a cylindrical shell coaxial with the cosmic string on which the field obeys mit bag boundary condition . we evaluate the fermionic condensate and vacuum expectation values of the energy - momentum tensor in both interior and exterior regions of the shell . the renormalized vacuum expectation value of the energy - momentum tensor for a fermionic field in the geometry of a cosmic string without boundaries is investigated in @xcite . in addition to describing the physical structure of the quantum field at a given point , the energy - momentum tensor acts as a source of gravity in the einstein equations and plays an important role in modelling a self - consistent dynamics involving the gravitational field . in the problem under consideration all calculations can be performed in a closed form and it constitutes an example in which the topological and boundary - induced polarizations of the vacuum can be separated in different contributions .    from the point of view of the physics in the region outside the string , the geometry considered in the present paper can be viewed as a simplified model for the non - trivial core . this model presents a framework in which the influence of the finite core effects on physical processes in the vicinity of the cosmic string can be investigated . in particular , it enables to specify conditions under which the idealized model with the core of zero thickness can be used . the corresponding results may shed light  upon features of finite core effects in more realistic models , including those used for string - like defects in crystals and superfluid helium . in addition , the problem considered here is of interest as an example with combined topological and boundary induced quantum effects in which the vacuum characteristics such as energy density and stresses can be found in closed analytic form . from the results of the present paper , as a special case , we obtain the fermionic casimir densities for a cylindrical shell with mit boundary conditions in the minkowski background ( for the combined effects of a magnetic fluxon and mit boundary conditions on the vacuum energy of a dirac field see refs . @xcite ) . note that , in addition to traditional problems of quantum field theory under the presence of material boundaries , the casimir effect for cylindrical geometries can also be important to the flux tube models of confinement @xcite and for determining the structure of the vacuum state in interacting field theories @xcite . we have organized the paper as follows . in the next section , the eigenspinors for the region inside the cylindrical boundary are constructed and the eigenvalues of the corresponding quantum numbers are specified . these eigenspinors are the basis for the anlysis of the casimir densities in the following sections . in section [ sec : fermc ] , by using a variant of the generalized abel - plana formula , we extract from the mode - sum of the fermionic condensate the part corresponding to the geometry of a cosmic string without the shell . the part induced by the shell is investigated in various asymptotic regions for the parameters . section [ sec : emt ] is devoted to the investigation of the boundary induced parts in the vacuum expectation value of the energy - momentum tensor inside the cylindrical shell . the vacuum densities in the region outside the cylindrical shell are discussed in section [ sec : exterior ] . the main results of the paper are summarized in section [ sec : conc ] . in appendix [ sec : appendix ] we give an alternative representation for the fermionic condensate and the expectation value of the energy - momentum tensor for a massive fermionic field in the geometry of a cosmic string without boundaries . we consider the background spacetime corresponding to an infinitely long straight cosmic string with the conical line element@xmath0where @xmath1 and the spatial points @xmath2 and @xmath3 are to be identified . the planar angle deficit is related to the mass per unit length of the string , @xmath4 , by @xmath5 , where @xmath6 is the newton gravitational constant . in the discussion below , in addition to the parameter @xmath7 , we will use the combination@xmath8    the dynamics of a massive fermionic field is governed by the dirac equation @xmath9with the covariant derivative operator defined as @xmath10here @xmath11 are the dirac matrices in curved spacetime and @xmath12 is the spin connection given in terms of the flat space dirac matrices @xmath13 by the relation @xmath14note that in this formula @xmath15 means the standard covariant derivative for vector fields . in the relations above @xmath16 is the tetrad basis satisfying @xmath17 , where @xmath18 is the minkowski spacetime metric tensor . in this paper we are interested in the change of the vacuum expectation values ( vevs ) of the fermionic condensate and the energy - momentum tensor for a fermionic field , induced by a cylindrical shell coaxial  with the string on which the field obeys mit bag boundary condition : @xmath19where @xmath20 is the cylinder radius and @xmath21 is the outward - pointing normal to the boundary . for the evaluation of the vevs we will use the direct mode summation procedure . in this approach we need to have the complete set of the eigenfunctions satisfying boundary condition ( [ bagcond ] ) .    in order to find these eigenfunctions , we will use the standard representation of the flat space dirac matrices:@xmath22with @xmath23 being the pauli matrices . we take the tetrad fields in the form used before in @xcite ( see also beze06,skar94):@xmath24where the index @xmath20 identifies the rows of the matrix . with this choice , the gamma matrices are given by@xmath25where we have introduced the @xmath26 matrices@xmath27for the spin connection and the combination appearing in the dirac equation we find@xmath28and the dirac equation takes the form@xmath29    for positive frequency solutions , assuming the time - dependence of the eigenfunctions in the form @xmath30 and decomposing the bispinor @xmath31 into the upper and lower components , denoted by @xmath32 and @xmath33 , respectively , we find the equations@xmath34substituting the function @xmath33 from the first equation into the second one , we obtain the second order differential equation for the function @xmath35:@xmath36\varphi = 0 ,   \label{phieq}\]]where @xmath37 . the same equation is obtained for the function @xmath33 .    because the above equation is in diagonal matrix form , we decompose the spinor @xmath32 into the upper , @xmath38 , and lower , @xmath39 , components . taking the eigenfunctions corresponding to these components in the form @xmath40 , @xmath41 , we can see that the solutions of the equations for the radial functions , regular on the string , are expressed in terms of the bessel function of the first kind : @xmath42 , where the order is defined by the relations@xmath43with @xmath44 , and@xmath45hence , the components of the upper spinor are given by the formula@xmath46 , \;l=1,2 . \label{phil}\]]having the upper spinor , we can find the components @xmath47 of the lower one by using the first equation in ( [ phixieq ] ) . from this equation we find the following relations@xmath48and@xmath49 , \;l=1,2 ,   \label{xil}\]]with the coefficients@xmath50we can see that the bispinor with the components defined by relations ( phil ) and ( [ xil ] ) is an eigenfunction of the projection of the total momentum along the cosmic string:@xmath51where@xmath52    for the further specification of the eigenfunctions we can impose an additional condition relating the constants @xmath53 and @xmath54 . as such a condition , following @xcite , we will require the following relations between the upper and lower components:@xmath55from the expressions for the spinor components we find the eigenvalues of the parameter @xmath56,@xmath57and the relation@xmath58for the coefficients in ( [ phil ] ) .    hence , the positive frequency solutions to the dirac equation , specified by the set of quantum numbers @xmath59 , has the form@xmath60 , \label{psi+}\]]where the orders of the bessel functions are defined in terms of @xmath61 as@xmath62 in ( [ psi+ ] ) and in the consideration below we use the notations@xmath63note that one has the relation @xmath64 . the eigenvalues of the radial quantum number @xmath65 are determined from the boundary condition ( [ bagcond ] ) imposed on the eigenspinor ( [ psi+ ] ) . for fixed values of @xmath61 and @xmath66 , this leads to the single equation@xmath67using the recurrence relations for the bessel functions , this equation may also be written in the form@xmath68here and in what follows we use the notation@xmath69 ,   \label{ftilde}\end{aligned}\]]with @xmath70 . we will denote the solutions of the equation ( lambvalue1 ) by @xmath71 , @xmath72 , assuming that they are arranged in ascending order . now the set of quantum numbers is specified by @xmath73 . the coefficient @xmath74 in ( [ psi+ ] ) is determined from the normalization condition @xmath75where @xmath76 is the determinant of the spatial metric and the integration goes over the region inside the cylindrical shell . the delta symbol on the right - hand side of eq . ( [ normcond ] ) is understood as the dirac delta function for continuous quantum numbers ( @xmath77 ) and the kronecker delta for discrete ones ( @xmath78 ) . substituting the eigenspinors ( [ psi+ ] ) into eq . ( [ normcond ] ) and using the value of the standard integral involving the square of the bessel function @xcite , we find@xmath79 ,   \label{csig+}\]]with the notation @xmath80 . the negative frequency eigenspinors are found in a way similar to that used above for the positive frequency ones and have the form@xmath81 , \label{psi-}\]]with the same notations as in ( [ psi+ ] ) . the boundary condition imposed on this eigenspinor leads to the same equation ( [ lambvalue ] ) for the eigenvalues of @xmath65 . the coefficient @xmath82 is found from the orthonormalization condition which is similar to ( [ normcond ] ) and has the form @xmath83 . fermionic condensate is evaluated by using the mode - sum formula @xmath84where @xmath85 is the dirac adjoint and@xmath86substituting the eigenspinor ( [ psi- ] ) into ( [ fcmodesum ] ) , we find @xmath87_{\lambda = \lambda _ { \beta , l}/a } , \label{fc1}\]]where @xmath88 is defined by expression ( [ kappas ] ) and @xmath89the fermionic condensate given by formula ( [ fc1 ] ) is divergent and some regularization procedure is necessary . we will assume that a cutoff function is introduced in formula ( [ fc1 ] ) without explicitly writing it . as the explicit form for @xmath90 is not known , formula ( fc1 ) is not convenient for the direct evaluation of the condensate . in addition , the separate terms in the mode - sum are highly oscillatory for large values of the quantum numbers . a convenient form can be obtained by applying to the series over @xmath91 the summation formula , previously derived in ref . @xcite ( see , also , @xcite ) . in @xcite by using the generalized abel - plana formula , it has been shown that for a function @xmath92 analytic in the half - plane @xmath93 and satisfying the condition @xmath94with @xmath95 and @xmath96 for @xmath97 , the following formula takes place @xmath98   \notag \\ & & { } -\frac{1}{\pi } \int_{0}^{\infty } dx\,\bigg[e^{-\beta \pi i}h(xe^{\pi i/2})\frac{k_{\beta } ^{(+)}(x)}{i_{\beta } ^{(+)}(x)}+e^{\beta \pi i}h(xe^{-\pi i/2})\frac{k_{\beta } ^{(-)}(x)}{i_{\beta } ^{(-)}(x)}\bigg]\ , \label{sumlamb}\end{aligned}\]]where @xmath99 , @xmath100 are the modified bessel functions , and @xmath101 is defined by the relation @xmath102\ .   \label{r1}\]]in formula ( [ sumlamb ] )  we used the notations @xmath103for a given function @xmath104 .    by taking into account the relation@xmath105we can write the mode - sum for the fermionic condensate in the form @xmath106with the notation@xmath107note that in ( [ fc2])@xmath108and we have the property@xmath109for @xmath110 . now we apply to the sum over @xmath91 in ( [ fc2 ] ) formula ( sumlamb ) taking @xmath111 . for this function the residue term in ( [ sumlamb ] ) the part in the fermionic condensate with the last integral on the right - hand side of ( [ sumlamb ] ) vanishes in the limit @xmath112 , whereas the part with the first integral does not depend on @xmath20 . from here it follows that the latter presents the fermionic condensate in the geometry of a cosmic string without boundaries . this can also be seen by direct evaluation . indeed , when the cylindrical boundary is absent , the positive and negative frequency eigenspinors are still given by formulae ( [ psi+ ] ) and ( psi- ) , where now the spectrum for @xmath65 is continuous , @xmath113 . the corresponding normalization coefficients are found from the condition ( [ normcond ] ) and have the form@xmath114substituting the eigenspinors into the mode - sum formula ( [ fcmodesum ] ) , for the fermionic condensate in the geometry of a cosmic string without boundaries we find@xmath115 . \label{fc0}\end{aligned}\]]this coincides with the result obtained from the first term on the right of formula ( [ sumlamb ] ) applied to mode - sum ( [ fc2 ] ) . formula ( [ fc0 ] ) is further simplified by taking into account the expression for @xmath116 and after the summation over @xmath66:@xmath117 . \label{fc0b}\]]here and in what follows@xmath118as it is seen from formula ( [ fc0b ] ) , for a massless field the fermionic condensate vanishes in the boundary - free cosmic string spacetime . since the geometry outside of the string is flat , the renormalization of the fermionic condensate given by ( [ fc0b ] ) is done by subtracting the corresponding quantity for the boundary - free minkowski spacetime . the latter is obtained from ( [ fc0b ] ) taking in this formula @xmath119 . note that for the minkowski case the summation over @xmath61 is explicitly done by using the formula @xmath120=2% \sideset{}{'}{\sum}_{n=0}^{\infty } j_{n}^{2}(\lambda r)=1 ,   \label{sumj2}\]]where the prime on the sign of summation means that the @xmath121 term should be halved . the renormalized fermionic condensate in the geometry of a cosmic string without boundaries is further investigated in appendix sec : appendix .    from the discussion above it follows that , after the application of the summation formula ( [ sumlamb ] ) , the part in the fermionic condensate with the second integral on the right - hand side of this formula corresponds to the vev induced by the presence of the cylindrical shell . by using property ( [ fprop ] ) and noting that under the change @xmath122 we have @xmath123 , @xmath124 , the fermionic condensate in the geometry with the cylindrical shell is presented in the decomposed form @xmath125here the second term on the right - hand side,@xmath126 ,   \label{fc3}\end{aligned}\]]is the part in the fermionic condensate induced by the cylindrical boundary . the number of the integrations in this formula is reduced by using the relation@xmath127further , redefining @xmath122 in the part of the summation over the negative values @xmath61 , it can be seen that the negative and positive values of @xmath61 give the same contributions to the fermionic condensate . finally , we arrive to the following formula @xmath128 , \label{fccyl}\]]where @xmath129and the function @xmath130 is given by the expression @xmath131 in ( [ fccyl ] ) , for the modified bessel functions we use the barred notations@xmath132with @xmath133 , and @xmath134 , @xmath135 . the boundary induced part ( [ fccyl ] ) is finite for points away from the cylindrical shell and the renormalization is necessary for the boundary - free part , @xmath136 , only . note that the ratio in the integrand of eq . ( [ fccyl ] ) may also be presented in the form@xmath137 + 2\mu xi_{\beta } ( x)i_{\beta + 1}(x ) } ,   \label{kbib}\]]where@xmath138 + 2\mu xi_{\beta } ( x)k_{\beta + 1}(x ) . \label{wbeta}\]]for a massless fermionic field , from ( [ fccyl ] ) and ( [ kbib ] ) we find@xmath139 in this case the boundary - free part vanishes and the fermionic condensate is always negative . now we turn to the investigation of the fermionic condensate given by eq . ( [ fccyl ] ) in the asymptotic regions of the parameters . for large values of the cylinder radius we use the asymptotic formulae for the modified bessel functions when the argument is large . by taking into account that the main contribution into the integral in ( [ fccyl ] ) comes from the lower limit of the integral , to the leading order we find@xmath140for @xmath141 . for a massless field , expanding the integrand in ( [ fcm0 ] ) we see that the main contribution is due to the term with @xmath142 and one has@xmath143for @xmath144 .    for points near the string , @xmath145 , the main contribution into the boundary - induced vev ( [ fccyl ] ) comes from the lowest mode @xmath142 and to the leading order we find@xmath146 . \label{fconaxis}\]]this quantity is non - zero in the case of the cylindrical boundary in the minkowski bulk and vanishes for the cosmic string geometry with @xmath147 . for a massless field formula ( [ fconaxis ] ) is reduced to eq . ( [ fclargeam0 ] ) . for points near the boundary the main contribution comes from large values of @xmath61 and in this case we can use the uniform asymptotic expansions for the modified bessel functions for large values of the order ( see , for instance , @xcite ) . in this way , to the leading order we have@xmath148for @xmath149 . as we see , the leading term does not depend on the planar angle deficit and corresponds to the same one obtained for a cylindrical boundary in the minkowski bulk . now we consider the limit when the parameter @xmath150 is large which corresponds to small values of @xmath7 and , hence , to a large planar angle deficit . in this limit the order of the modified bessel functions in ( [ fccyl ] ) is large and we replace these functions by their uniform asymptotic expansions . assuming fixed value of the ratio @xmath151 , the integral is estimated by the laplace method and to the leading order we have@xmath152}{2\pi ^{2}r^{3}[1-(r / a)^{2}]}\left ( \frac{r}{a}% \right ) ^{q},\;q\gg 1 . \label{fclargeq}\]]hence , for large values of the angle deficit the fermionic condensate is exponentially suppressed . in this section we consider the vev of the energy - momentum tensor of the fermionic field inside a cylindrical shell on which the field satisfies the boundary condition ( [ bagcond ] ) . this vev can be evaluated by making use of the mode - sum formula @xmath153\ ,   \label{modesum}\]]with the negative frequency eigenspinors given by ( [ psi- ] ) . we can see that the vacuum energy - momentum tensor is diagonal and the separate components are given by the formulae ( no summation over @xmath154 ) @xmath155where the following notations were introduced@xmath156 ,   \notag \\ f_{\beta } ^{(1)}(x , y ) & = & j_{\beta } ^{2}(y)+j_{\beta + \epsilon _ { j}}^{2}(y)-% \frac{2\beta + \epsilon _ { j}}{y}j_{\beta } ( y)j_{\beta + \epsilon _ { j}}(y ) , \label{fbetamu } \\ f_{\beta } ^{(2)}(x , y ) & = & \frac{2\beta + \epsilon _ { j}}{y}j_{\beta } ( y)j_{\beta + \epsilon _ { j}}(y),\;\,f_{\beta } ^{(3)}(x , y)=(k^{2}/\omega ^{2})f_{\beta } ^{(0)}(x , y ) . \notag\end{aligned}\]]the other notations in ( [ emt ] ) are the same as in ( [ fc2 ] ) . the vev given by ( [ emt ] ) is divergent and , as in the case of the fermionic condensate , it is assumed that a cutoff function is present . now , we can explicitly verify that the vevs ( [ emt ] ) satisfy the trace relation @xmath157    in order to extract from the vev of the energy - momentum tensor the part corresponding to the geometry of a string without boundaries , we apply to the series over @xmath91 in ( [ emt ] ) the summation formula ( [ sumlamb ] ) with@xmath158 in a way similar to that used in the case of the fermionic condensate , the vev can be written in the decomposed form:@xmath159where @xmath160 is the fermionic energy - momentum tensor in the geometry of a cosmic string when the cylindrical shell is absent . the second term on the right - hand side of formula ( [ emt1 ] ) is induced by the cylindrical shell and is given by the formula ( no summation over @xmath154)@xmath161 ,   \label{emtcyl}\]]where @xmath162 is defined by eq . ( [ betaj ] ) . in this formula we have introduced the notations@xmath163and @xmath164 . note that the function for the radial component is also presented in the form@xmath165as we see , the vacuum stress along the axis of the string is equal to the energy density . of course , this property is a direct consequence of the bust invariance along this axis .    by using eq . ( [ kbib ] ) , we may write the expressions for the components of the energy - momentum tensor in more explicit form given by @xmath166 - 2\mu i_{\beta _ { j}}^{2}(xr / a)}{i_{\beta _ { j}+1}^{2}(x)+i_{\beta _ { j}}^{2}(x)+2(\mu /x)i_{\beta _ { j}}(x)i_{\beta _ { j}+1}(x ) } ,   \label{t00int}\end{aligned}\]]for the energy density and by ( no summation over @xmath154)@xmath167f_{\beta _ { j}}^{(\nu ) } ( x , xr / a)}{i_{\beta _ { j}+1}^{2}(x)+i_{\beta _ { j}}^{2}(x)+2(\mu /x)i_{\beta _ { j}}(x)i_{\beta _ { j}+1}(x ) } ,   \label{tnuint}\]]for the radial and azimuthal stresses , @xmath168 . the vev of the fermionic energy - momentum tensor in the geometry of a cosmic string when the cylindrical shell is absent , corresponds to the first term on the right - hand side of summation formula ( [ sumlamb ] ) . for this vev we have the mode - sum ( no summation over @xmath154)@xmath169the summation over @xmath66 in this formula is done explicitly and noting that the negative and positive values of @xmath61 give the same contributions , we find@xmath170where now@xmath171 ,   \notag \\ g_{\beta _ { j}}^{(1)}(\lambda , y ) & = & j_{\beta _ { j}}^{2}(y)+j_{\beta _ { j}+1}^{2}(y)-\frac{2qj}{y}j_{\beta _ { j}}(y)j_{\beta _ { j}+1}(y ) , \label{gmu } \\ g_{\beta _ { j}}^{(2)}(\lambda , y ) & = & \frac{2qj}{y}j_{\beta _ { j}}(y)j_{\beta _ { j}+1}(y),\;g_{\beta _ { j}}^{(3)}(\lambda , y)=(k^{2}/\omega ^{2})g_{\beta _ { j}}^{(0)}(\lambda , y ) . \notag\end{aligned}\]]as in the case of the fermionic condensate , the vev ( [ emts1 ] ) is renormalized by subtracting the corresponding vev in the minkowski spacetime without boundaries . the boundary - free renormalized vev  of the energy - momentum tensor for a massive fermionic field is further investigated in appendix [ sec : appendix ] . it can be explicitly verified that the boundary - induced parts satisfy the trace relation @xmath172 and the covariant conservation equation @xmath173 . for the geometry under consideration the latter is reduced to the single equation@xmath174=\left\langle t_{2}^{2}\right\rangle _ { \mathrm{cyl}}.   \label{conteq}\]]this equation is easily checked by using the relation @xmath175 between the radial and azimuthal functions in ( [ fbetamu ] ) . in the case of a massless fermionic field , for the vev induced by the cylindrical boundary from ( [ t00int ] ) and ( [ tnuint ] ) we have ( no summation over @xmath154 ) @xmath176where @xmath177  for @xmath168 , and the corresponding function for the energy density is defined as@xmath178 . \label{fbeta0m0}\]]in the special case with @xmath119 , from the formulae given above we obtain the fermionic casimir densities for a cylindrical boundary in the minkowski spacetime .  on the left panel of figutre [ fig1 ] we have plotted the corresponding vevs for the energy density and radial stress as functions of the radial coordinate for a massless fermionic field . on the right panel the boundary induced parts are presented in the geometry of a cosmic string with the parameter @xmath179 .    [ cols="^,^ " , ] in this section we consider the fermionic condensate and the vev of the energy - momentum tensor in the region outside the cylindrical shell . as in the interior case this can be done by the direct mode summation . the corresponding eigenspinors have the form ( [ psi+ ] ) and ( [ psi- ] ) with the difference that now , instead of the bessel functions @xmath180 , the linear combinations of the functions @xmath180 and @xmath181 should be taken , with @xmath182 being the neumann function . the ratio of the coefficients in this combination is determined from the boundary condition ( [ bagcond ] ) imposed on the cylindrical surface . in this way for the positive and negative frequency eigenspinors we have the expressions @xmath183 , \label{psiext}\]]where the function @xmath184 is defined by the formula@xmath185and@xmath186with @xmath187 given by relations ( [ bet12a ] ) . here the notation @xmath188 is defined by ( [ ftilde ] ) with the replacement @xmath189 . the eigenspinors are orthonormalized by condition ( [ normcond ] ) , where now the radial integration goes over the exterior region . the eigenvalues for @xmath65 are continuous and on the right - hand side of the normalization condition we have @xmath190 . since the radial integral diverges for @xmath191 , the main contribution to this integral comes from large values @xmath192 and we cane replace the bessel and neumann functions with the arguments @xmath193 , by the corresponding asymptotic expressions . in this way , for the normalization coefficients in ( [ psiext ] ) we find:@xmath194 ,   \label{csigext}\]]where @xmath88 and @xmath195 are defined by eqs . ( [ kappas ] ) and ( [ bsplmin ] ) .      substituting the eigenspinors ( [ psiext ] ) into the mode - sum formula ( fcmodesum ) , for the fermionic condensate in the region outside the cylindrical shell we obtain the formula@xmath196 }   \notag \\ & & \times \lbrack ( -m+s\sqrt{\lambda ^{2}+m^{2}})z_{\beta } ^{2}(\lambda a,\lambda r)-(m+s\sqrt{\lambda ^{2}+m^{2}})z_{\beta + \sigma _ { j}}^{2}(\lambda a,\lambda r ) ] ,   \label{fcext}\end{aligned}\]]where@xmath197 in order to extract from the vev ( [ fcext ] ) the part induced by the cylindrical shell , we subtract the fermionic condensate for the geometry of a string without the shell . as it has been shown before , the latter is given by formula ( [ fc0 ] ) . for further evaluation of the difference , we use the identities@xmath198where @xmath199 or @xmath200 , and @xmath201 are the hankel functions . as a result for the fermionic condensate we obtain @xmath202 . \label{fcext1}\end{aligned}\ ] ]    now , in the complex plane @xmath65 we rotate the integration contour in the integral over @xmath65 on the right - hand side of formula ( [ fcext1 ] ) by the angle @xmath203 for @xmath204 term and by the angle @xmath205 for @xmath206 term . by using the symmetry properties of the integrands , it can be seen that the parts of the integrals over @xmath207 and @xmath208 are cancelled . the number of the remained integrations is reduced by using formula ( [ intformula ] ) . in this way , introducing the modified bessel functions , we present the fermionic condensate in the decomposed form ( [ fc4 ] ) , where the part induced by the cylindrical boundary in the region @xmath209 is given by the expression@xmath210 ,   \label{fccylext}\]]where @xmath162 is defined by eq . ( [ betaj ] ) and we are using notation@xmath211note that the ratio of the combinations of the modified bessel functions in ( [ fccylext ] ) can be written in the form@xmath212 - 2\mu xk_{\beta } ( x)k_{\beta + 1}(x ) } ,   \label{ibkb}\]]where the function @xmath213 is defined by eq . ( [ wbeta ] ) . for a massless fermionic field one has the formula@xmath214and the fermionic condensate is positive . let us consider the behavior of the fermionic condensate in asymptotic regions of the parameter . in the limit @xmath215 with fixed values @xmath192 , we introduce in ( [ fccylext ] ) a new integration variable @xmath216 and expand the integrand in powers of @xmath20 . the main contribution comes from the mode @xmath142 and we have the leading term given below@xmath217 . \label{fcsmalla}\]]for a massless fermionic field , by using the result from @xcite for the integral involving the square of the macdonald function , from here we find@xmath218    at large distances from the cylindrical boundary , for a massive field under the condition @xmath219 the main contribution into the integral in ( fccylext ) comes from the lower limit of the integration and to the leading order we find@xmath220 . \label{fclarger}\]]here the imaginary part is easily taken by using eq . ( [ ibkb ] ) . as we could expect , in this limit the vev is exponentially suppressed . at large distances and for a massless field the behavior of the fermionic condensate is described by eq . ( [ fclargerm0 ] ) . note that the decreasing of the fermionic condensate at large distances is stronger than in the case when the string is absent . for points near the boundary , by using the uniform asymptotic expansions for the modified bessel functions , we can see that the leading term in the asymptotic expansion of the fermionic condensate over the distance from the boundary is given by the same expression ( [ fcnear ] ) as in the interior region . the vev for the energy - momentum tensor in the exterior region is found in the way similar to that for the fermionic condensate . here we omit the details of the calculations and give the final result . the vev is decomposed into the sum of boundary - free and boundary - induced parts in the form given by ( [ emt1 ] ) . in the region outside the cylindrical shell the boundary - induced part is ( no summation over @xmath154)@xmath221 . \label{emtext}\]]in this formula we introduced the notations@xmath222and @xmath223 . as an additional check we can see that these vevs satisfy the trace relation and the covariant conservation equation . by using formula ( [ ibkb ] ) , we can write the vacuum densities in the form@xmath224 + 2\mu k_{\beta _ { j}+1}^{2}(xr / a)}{k_{\beta _ { j}+1}^{2}(x)+k_{\beta _ { j}}^{2}(x)-2(\mu /x)k_{\beta _ { j}}(x)k_{\beta _ { j}+1}(x ) } ,   \label{t00ext}\end{aligned}\]]for the energy density and in the form ( no summation over @xmath154 )    @xmath167f_{\beta _ { j}}^{\mathrm{(ex)}(\nu ) } ( x , xr / a)}{k_{\beta _ { j}+1}^{2}(x)+k_{\beta _ { j}}^{2}(x)-2(\mu /x)k_{\beta _ { j}}(x)k_{\beta _ { j}+1}(x ) } ,   \label{tnuext}\ ] ]    for the radial and azimuthal stresses , @xmath168 . we recall that the summation over @xmath61 in these formulae goes in accordance with eq . ( sumjnot ) .    in the case of a massless fermionic field from ( [ emtext ] ) we find the following expressions ( no summation over @xmath154)@xmath225where@xmath226 ,   \label{f0ext}\]]and for the corresponding functions for the radial and azimuthal stresses we have @xmath227 , @xmath168 . now we turn to the investigation of the vev in the energy - momentum tensor induced by the cylindrical shell in the exterior region in limiting cases . first let us consider the limit @xmath215 for fixed values @xmath192 . expanding the integrands in powers of @xmath20 , we can see that the main contribution comes from the terms with @xmath142 , and the leading terms are given by the expressions ( no summation over @xmath154)@xmath228with @xmath168 . for a massless field these terms vanish . in this case from eq . ( [ emtextm0 ] ) we find the following leading behavior ( no summation over @xmath154):@xmath229with the coefficients@xmath230for @xmath119 the vevs behave like @xmath231 .    at large distances from the cylinder and for a massive field the main contribution comes from the lower limit of the integral in ( [ emtext ] ) . by using the asymptotic formulae for the macdonald function for large values of the argument , we find@xmath232 ,   \label{emtextlargr}\end{aligned}\]]for @xmath219 . as we see , in this limit @xmath233 . at large distances from the cylinder and for a massless field the asymptotic behavior of the boundary induced parts is given by formula ( [ emtextlargm0 ] ) . the asymptotic behavior of the vev  for the energy - momentum tensor near the cylindrical shell is found in a way similar to that for the interior region and the leading terms are given by the formulae ( [ t22nearcyl ] ) , ( t00nearcyl ) . hence , near the boundary the energy density and the azimuthal stress in the interior and exterior regions have opposite signs , whereas the radial stresses have the same sign . the boundary induced parts in the vevs of the energy density and the radial stress for the exterior region are plotted in figures [ fig1]-[fig3 ] as functions of the radial coordinate and the parameters @xmath150 and @xmath234 . in this paper the vacuum polarization effects are investigated for a fermionic field in the geometry of a cosmic string with a coaxial cylindrical shell . we have assumed that on the shell the field obeys the mit bag boundary condition . in order to evaluate the fermionic condensate and the vev  of the energy - momentum tensor one needs the complete set of normalized eigenspinors satisfying the boundary condition . this set for the region inside the cylindrical shell is considered in section sec : eigspinor . the corresponding mode - sums for both fermionic condensate and the energy - momentum tensor contain series over the zeros of the combination ( [ ftilde ] ) of the bessel function of the first kind and its derivative . for the summation of these series we used a variant of the generalized abel - plana formula previously derived in ref . this formula allows us to extract from the respective vevs the parts corresponding to the cosmic string geometry without a cylindrical shell and to present the part induced by the shell in terms of exponentially convergent integrals for points away from the boundary . in this way the renormalization procedure for the fermionic condensate and the energy - momentum tensor is reduced to the renormalization of the corresponding quantities in the geometry of the boundary - free cosmic string . the renormalized vev of the energy - momentum tensor for a fermionic field in the boundary - free geometry is well investigated in literature . in appendix [ sec : appendix ] , by using the abel - plana summation formula , we give alternative integral representations for both fermionic condensate and the energy - momentum tensor in the case of a massive field .    in the region inside the shell , the parts in the vevs induced by the presence of the cylindrical boundary are given by formula ( [ fccyl ] ) for the fermionic condensate and by eq . ( [ t00int ] ) , ( [ tnuint ] ) for the vacuum energy densities and stresses . these formulae are further simplified for a massless fermionic field with the vacuum densities given by eqs . ( fcm0 ) and ( [ emtm0 ] ) . for points near the cylindrical shell the boundary induced parts in the vevs dominate over the boundary - free parts and diverge on the cylindrical shell . these type of divergences are well known in quantum field theory with boundaries and are investigated for various bulk and boundary geometries . in the problem under consideration , the leading terms in the asymptotic expansions in powers of the distance from the boundary are given by eq . ( [ fcnear ] ) for the fermionic condensate and by eqs . ( [ t22nearcyl ] ) and ( [ t00nearcyl ] ) for the components of the energy - momentum tensor . these leading terms do not depend on the planar angle deficit and are the same as for a cylindrical boundary in the minkowski bulk . the boundary induced parts in the vevs vanish on the string axis for @xmath147 and are non - zero in the case of a cylindrical boundary in the minkowski bulk . since the boundary - free part diverges on the string axis , for points near the string it dominates . for large values of the parameter @xmath235 , which corresponds to a large planar angle deficit , the boundary - induced vevs are suppressed by the factor @xmath236 . the boundary induced vevs have non - trivial dependence on the mass of the field and , as it is illustrated by figure [ fig3 ] , for a massive field the polarization effects can be stronger than for a massless one . fermionic vacuum densities in the region outside a cylindrical shell with mit bag boundary condition are investigated in section [ sec : exterior ] . subtracting from the mode - sums the parts corresponding to the geometry of a string without boundaries and by making use of a complex rotation , we have derived explicit expressions for the boundary induced vevs . the corresponding parts in the fermionic condensate and the energy - momentum tensor are given by eqs . ( [ fccylext ] ) , ( [ t00ext ] ) and ( [ tnuext ] ) . when the cylinder radius goes to zero , for a fixed value of the radial distance , the boundary induced part in the vev  of the energy - momentum tensor vanishes as @xmath237 for a massive field and as @xmath238 for a massless one . at large distances from the cylindrical shell this part is exponentially suppressed for a massive field and decay as @xmath239 in the case of a massless field . note that in the latter case the boundary - free part behaves as @xmath240 and it dominates at large distances . for points near the cylindrical shell the leading terms in the asymptotic expansions in powers of the distance from the boundary are given by the same formulae as for the interior region . in this limit the total vev is dominated by the boundary induced part . in dependence of the mass , the vacuum stresses can be either positive or negative , whereas the energy density is positive . in the special case @xmath119 , from the formulae derived in the present paper we obtain the fermionic casimir densities for a cylindrical boundary in the minkowski spacetime . we have considered the idealized geometry of a cosmic string with zero thickness . a realistic model for cosmic string has a non - trivial structure on a length scale defined by the phase transition at which it is formed . as it has been shown in refs . @xcite , the internal structure of the string may have non - negligible effects even at large distances . here we note that when the cylindrical boundary is present , the vevs of the physical quantities in the exterior region are uniquely defined by the boundary conditions and the bulk geometry . this means that if we consider a non - trivial core model with finite thickness @xmath241 and with the line element ( [ ds21 ] ) in the region @xmath242 , the results in the region outside the cylindrical shell will not be changed . as regards to the interior region , the formulae given in this paper are the first stage of the evaluation of the vevs and other effects could be present in a realistic cosmic string . note that from the point of view of the physics in the exterior region the cylindrical surface with mit bag boundary condition can be considered as a simple model of non - trivial string core . e.r.b.m . and v.b.b . thank conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) , fapesq - pb / cnpq ( pronex ) and fapes - es / cnpq ( pronex ) for partial financial support . a.a.s . was supported by the armenian ministry of education and science grant no . 119 and by conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) . a.s.t . was supported in part by grant nfsat - crdf uc-06/07 . in this appendix we consider the renormalized fermionic condensate and the vev of the energy - momentum tensor in the cosmic string geometry when the cylindrical shell is absent . for a massless field the vacuum energy - momentum tensor was found in @xcite . fermionic propagators for a massive field are considered in refs . @xcite . in the case of a massive field , a representation of the vevs for the energy - momentum tensor in terms of contour integrals is given in @xcite . here an alternative integral formulae are given by applying to the corresponding mode - sums the abel - plana formula . we will do these calculations by using the mode - sum formulae ( [ fc0b ] ) and ( [ emts1 ] ) for the boundary - free vevs . first let us consider the fermionic condensate . the renormalization is done by subtracting the corresponding quantity for the minkowski background . the latter is obtained from ( [ fc0b ] ) putting @xmath119 . substituting in the corresponding formulae@xmath243integrating over @xmath77 and @xmath65 , and introducing a new integration variable @xmath244 , we find the following representation of the renormalized fermionic condensate:@xmath245 . \label{fc01}\end{aligned}\]]next , we apply to the series over @xmath61 the abel - plana summation formula in the form ( see , for example , @xcite ) @xmath246it is easily seen that for the summand in ( [ fc01 ] ) the first integral on the right - hand side of ( [ abel ] ) vanishes and , hence , the divergent parts are explicitly cancelled . introducing the macdonald function , we arrive to the following expression@xmath247 .   \label{fcren}\]]where the notation@xmath248 ,   \label{gqx}\]]is introduced .    in a similar way we can find the formula for the renormalized vev of the energy - momentum tensor . for the renormalized energy density and the azimuthal stress we have the representations given below:@xmath249 ,   \notag \\ \left\langle t_{2}^{2}\right\rangle _ { \mathrm{s , ren } } & = & \frac{r^{-4}}{\pi ^{2}}\int_{0}^{\infty } dy\,ye^{-m^{2}r^{2}/2y - y}\sum_{\delta = \pm 1}\delta \,\sum_{j}j\left [ q^{2}i_{qj-\delta /2}(y)-i_{j-\delta /2}(y)\right ] . \label{t00t22free}\end{aligned}\]]by making use of summation formula ( [ abel ] ) to the series over @xmath61 , we find the following formulae for the renormalized vevs:@xmath250 , \notag \\ \left\langle t_{2}^{2}\right\rangle _ { \mathrm{s , ren } } & = & \frac{4}{\pi ^{3}r^{4}}\int_{0}^{\infty } dy\,ye^{-m^{2}r^{2}/2y - y}\int_{0}^{\infty } dxxg(q , x){\mathrm{re}}\left [ k_{ix+1/2}(y)\right ] . \label{t00sren}\end{aligned}\]]the radial stress is found from ( [ fcren ] ) and ( [ t00sren ] ) by using the trace relation . formulae ( [ t00sren ] ) are further simplified for a massless fermionic field . the integration over @xmath251 is done by using the formula@xmath252substituting this into eq . ( [ t00sren ] ) and integrating over @xmath253 , we find @xmath254 in the massless case the radial stress is equal to the energy density . smith , in _ the formation and evolution of cosmic strings _ , proceedings of the cambridge workshop , cambridge , england , 1989 , edited by g.w . gibbons , s.w . hawking , and t. vachaspati ( cambridge university press , cambridge , england , 1990 ) . mostepanenko and n.n . trunov , _ the casimir effect and its applications _ ( oxford university press , oxford , 1997 ) ; m. bordag , u. mohidden , and v.m . mostepanenko , phys . rep . * 353 , * 1 ( 2001 ) ; k.a . the casimir effect : physical manifestation of zero - point energy _ ( world scientific , singapore , 2002 ) . s. leseduarte and a. romeo , commun . phys . * 193 * , 317 ( 1998 ) ; c.g . beneventano , m. de francia , k. kirsten , and e.m . santangelo , phys . d * 61 * , 085019 ( 2000 ) ; m. de francia and k. kirsten , phys . d * 64 * , 065021 ( 2001 ) .
the vacuum expectation values of the energy  momentum tensor and the fermionic condensate are analyzed for a massive spinor field obeying the mit bag boundary condition on a cylindrical shell in the cosmic string spacetime . 
 both regions inside and outside the shell are considered . by applying to the corresponding mode - sums a variant of the generalized abel 
 plana formula , we explicitly extract the parts in the expectation values corresponding to the cosmic string geometry without boundaries . in this way 
 the renormalization procedure is reduced to that for the boundary - free cosmic string spacetime . 
 the parts induced by the cylindrical shell are presented in terms of integrals rapidly convergent for points away from the boundary . 
 the behavior of the vacuum densities is investigated in various asymptotic regions of the parameters . in the limit of large values of the planar angle deficit , 
 the boundary - induced expectation values are exponentially suppressed . as a special case 
 , we discuss the fermionic vacuum densities for the cylindrical shell on the background of the minkowski spacetime . 
 pacs numbers : 03.70.+k , 98.80.cq , 11.27.+d