# Fermionic vacuum stresses in models with toroidal compact dimensions

A. A. Saharian1,2, R. M. Avagyan1,2, G. H. Harutyunyan1, G. H. Nikoghosyan1

1Institute of Physics, Yerevan State University,
1 Alex Manoogian Street, 0025 Yerevan, Armenia
2Institute of Applied Problems of Physics NAS RA,
25 Hrachya Nersissyan Street, 0014 Yerevan, Armenia E-mail<EMAIL_ADDRESS>
###### Abstract

We investigate vacuum expectation value of the energy-momentum tensor for a
massive Dirac field in flat spacetime with a toroidal subspace of a general
dimension. Quasiperiodicity conditions with arbitrary phases are imposed on
the field operator along compact dimensions. These phases are interpreted in
terms of magnetic fluxes enclosed by compact dimensions. The equation of state
in the uncompact subspace is of the cosmological constant type. It is shown
that, in addition to the diagonal components, the vacuum energy-momentum
tensor has nonzero off-diagonal components. In special cases of twisted
(antiperiodic) and untwisted (periodic) fields the off diagonal components
vanish. For untwisted fields the vacuum energy density is positive and the
energy-momentum tensor obeys the strong energy condition. For general values
of the phases in the periodicity conditions the energy density and stresses
can be either positive or negative. The numerical results are given for a
Kaluza-Klein type model with two extra dimensions.

Keywords: Topological Casimir effect, Dirac field, toroidal compactification

##  1 Introduction

The field theoretical models in background spacetimes with compact dimensions
appear in a number of theories in fundamental physics like string theories,
supergravities and Kaluza-Klein theories. The quantum creation of universe
with a compact space has been considered in [1]-[3]. In this type of models
the probability of inflation in the early stages of the universe expansion is
not exponentially small. The effects caused by the non-trivial topology of the
universe on cosmological scales are discussed, for example, in [4, 5]. They
include the ghost images of galaxies and quasars, cosmological magnetic fields
and observable effects on cosmic microwave background. Physical models
formulated on background geometries with nontrivial topology also appear in a
number of condensed matter physics systems. Examples are topological
structures of graphene, like carbon nanotubes and nanoloops. The long
wavelength excitations of the electronic subsystem in those structures are
described by an effective field theory (Dirac model, see [6, 7]) with
2-dimensional spatial topologies $R^{1}\times S^{1}$ and $T^{2}=S^{1}\times
S^{1}$, respectively.

In quantum field theory the nontrivial spatial topology is a source of a
number of interesting effects. In particular, the periodicity conditions along
compact dimensions modify the spectrum of quantum fluctuations of fields and,
as a consequence, the expectation values of the physical characteristics are
shifted by an amount that depends on the geometry and topology of the compact
subspace. This general phenomenon is known as the topological Casimir effect
(see [8]-[12]). The vacuum energy in the topological Casimir effect depends on
the size of compact dimensions and this provides a stabilization mechanism for
the corresponding moduli fields. The topological Casimir energy may also
appear as the source of the accelerated expansion of uncompact subspace
playing the role of the dark energy at recent epoch of the Universe expansion
(see, for example, [13]-[18]).

An important physical characteristic for charged fields is the expectation
values of the current density. For a relatively simple model of toroidal
compactification in flat spacetime (for quantum field theory in models with
toroidal spatial dimensions see, for example, [19]), in references [20]-[23]
it has been shown that the nontrivial phases in the periodicity conditions
along compact dimensions give rise to nonzero currents along those dimensions.
The phases can be interpreted in terms of magnetic fluxes enclosed by those
dimensions. The currents in the compact subspace are sources of magnetic
fields having components along uncompactified dimensions. The dependence of
the vacuum energy density and diagonal stresses for a massive fermionic field
in the same model of flat spacetime with a part of spatial dimensions
compactified on a torus has been studied in [24]. In the present paper we show
that, in addition to the diagonal components, the vacuum expectation value
(VEV) of the energy-momentum tensor may have nonzero off-diagonal components
(vacuum stresses) in the compact subspace.

The paper is organized as follows. In the next section present the problem
setup and the eigenmodes for a Dirac field obeying the quasiperiodicity
conditions along compact dimensions. The general formulas for the vacuum
energy density and stresses are obtained in section 3. The asymptotic and
numerical analysis of the VEVs in a model with two compact dimensions is
presented in section 4. The main results are summarized in section 5.

##  2 Problem setup

The background geometry we are going to consider is a flat spacetime with
topology $M^{p+1}\times T^{q}$, where $M^{p+1}$ is $(p+1)$-dimensional
Minkowski spacetime covered by the Cartesian coordinates
$(z^{0}=t,\mathbf{z}_{p})=(z^{0},z^{1},\ldots,z^{p})$, and $T^{q}=(S^{1})^{q}$
is a $q$-dimensional torus with the coordinates
$\mathbf{z}_{q}=(z^{p+1},\ldots,z^{D})$. The length of the $l$th compact
dimension will be denoted by $L_{l}$ and, hence, $0\leq z^{l}\leq L_{l}$,
$l=p+1,\ldots,D$. The volume of the compact subspace is expressed as
$V_{q}=L_{p+1}\cdots L_{D}$. For the uncompact dimensions, as usual, we have
$z^{l}\in(-\infty,+\infty)$. The line element has the standard Minkowskian
form

$ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=dt^{2}-d\mathbf{z}^{2},\;\mathbf{z}=(\mathbf{z}_{p},\mathbf{z}_{q}),$
(1)

and $\eta_{\mu\nu}$ is the Minkowski metric tensor in Cartesian coordinates.

We are interested in the vacuum stresses for a massive Dirac field $\psi(x)$,
$x=(t,\mathbf{z})$, induced by compactification of a part of spatial
dimensions. The field equation reads

$\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi(x)=0,$ (2)

where $\gamma^{\mu}$, $\mu=0,1,\ldots,D$, are $N\times N$ Dirac matrices with
$N$ given by $N=2^{\left[\frac{D+1}{2}\right]}$ and $[a]$ stands for the
integer part of $a$. The background geoemtry has non-trivial topology and in
order to fix the dynamics uniquely the periodicity conditions along the
compact dimensions have to be specified for the field operator. We impose
quasiperiodicity conditions

$\psi(t,\mathbf{z}_{p},\ldots,z^{l}+L_{l},\ldots,z^{D})=e^{i\alpha_{l}}\psi(t,\mathbf{z}_{p},\ldots,z^{l},\ldots,z^{D}),$
(3)

with phases $\alpha_{l}=\mathrm{const}$ and $l=p+1,\ldots,D$. The special
cases of periodic and antiperiodic fields correspond to $\alpha_{l}=0$ and
$\alpha_{l}=\pi$ (untwisted and twisted fields, respectively).

The VEV of the energy-momentum tensor $T_{\mu\nu}$ for the fermionic field is
expressed in terms of the mode sum over a complete set of normal modes
$\psi_{\beta}^{(\pm)}(x)$, where $\beta$ is the set of quantum numbers
specifying the solutions to the field equation and upper/lower signs
correspond to the positive/negative energy modes. Denoting the VEV by
$\left\langle T_{\mu\nu}\right\rangle=\left\langle
0\right|T_{\mu\nu}\left|0\right\rangle$, with $\left|0\right\rangle$ being the
vacuum state, the mode sum is expressed as

$\langle
T_{\mu\nu}\rangle=-\frac{i}{4}\sum_{\beta}\sum_{j=+,-}j[\bar{\psi}_{\beta}^{(j)}(x)\gamma_{(\mu}\partial_{\nu)}\psi_{\beta}^{(j)}(x)-(\partial_{(\mu}\bar{\psi}_{\beta}^{(j)}(x))\gamma_{\nu)}\psi_{\beta}^{(j)}(x)]\
,$ (4)

where $\gamma_{\mu}=\eta_{\mu\nu}\gamma^{\nu}$,
$\bar{\psi}(x)=\psi^{\dagger}(x)\gamma^{0}$ is the Dirac adjoint and the
parentheses including the indices mean symmetrization over those indices. The
symbolic notation $\sum_{\beta}$ stands for the summation over discrete
components of the collective index $\beta$ and the integration over the
continuous ones. The problem under consideration has planar symmetry and it is
natural to take the normal modes corresponding to plane waves.

Taking the chiral representation of the Dirac matrices,

$\gamma^{0}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),\;\gamma^{l}=\left(\begin{array}[]{cc}0&\sigma_{l}\\\
-\sigma_{l}^{\dagger}&0\end{array}\right),$ (5)

the positive and negative energy wave functions with momentum
$\mathbf{k}=(k_{1},\ldots,k_{D})$ and energy
$\varepsilon_{\mathbf{k}}=\sqrt{k^{2}+m^{2}}$ have the form [22]

$\psi_{\beta}^{(\pm)}(x)=\left[\frac{1+m/\varepsilon_{\mathbf{k}}}{2(2\pi)^{p}V_{q}}\right]^{\frac{1}{2}}e^{i\mathbf{k}\cdot\mathbf{z}\mp
i\varepsilon_{\mathbf{k}}t}\left(\begin{array}[]{c}\left(-\frac{\mathbf{k}\cdot\boldsymbol{\sigma}}{\varepsilon_{\mathbf{k}}+m}\right)^{\frac{1\mp
1}{2}}w_{\chi}^{(\pm)}\\\
\left(\frac{\mathbf{k}\cdot\boldsymbol{\sigma}^{\dagger}}{\varepsilon_{\mathbf{k}}+m}\right)^{\frac{1\pm
1}{2}}w_{\chi}^{(\pm)}\end{array}\right),$ (6)

where $\mathbf{\sigma}=(\sigma_{1},\ldots,\sigma_{D})$. Here, the quantum
number $\chi=1,2,\ldots,N/2$ enumerates the polarization degrees of freedom,
$w_{\chi}^{(\pm)}$ are one-column matrices with $N/2$ rows and $l$th element
$w_{\chi l}^{(\pm)}=\delta_{\chi l}$. We will decompose the momentum into two
parts, $\mathbf{k}=(\mathbf{k}_{p},\mathbf{k}_{q})$, where
$\mathbf{k}_{p}=(k_{1},\ldots,k_{p})$ and
$\mathbf{k}_{q}=(k_{p+1},\ldots,k_{D})$ are the parts in uncompact and compact
subspaces. For the components $k_{l}$, $l=1,2,\ldots,p$, one has
$-\infty<k_{l}<+\infty$. The eigenvalues of the components $k_{l}$ along
compact dimensions are quantized by the periodicity conditions (3):

$k_{l}=\frac{2\pi n_{l}+\alpha_{l}}{L_{l}},\;n_{l}=0,\pm 1,\pm 2,\ldots,$ (7)

for $l=p+1,\ldots,D$.

The phases in the conditions (3) can be interpreted in terms of the magnetic
flux for a vector gauge field enclosed by compact dimensions. The
representation described above corresponds to the gauge with zero vector
potential, $(\psi,A_{\mu})=(\psi,0)$. Let us pass to a new gauge with the
fields $(\psi^{\prime},A_{\mu}^{\prime})$, where $A_{\mu}^{\prime}$ has
nonzero constant components along compact dimensions: $A_{\mu}^{\prime}=0$,
$\mu=0,1,\ldots,p$ and $A_{l}^{\prime}=\mathrm{const}$ for $l=p+1,\ldots,D$.
The gauge transformation has the form

$\psi^{\prime}=\psi
e^{-ie\lambda},\;A_{\mu}^{\prime}=A_{\mu}+\partial_{\mu}\lambda,\;\lambda=b_{\mu}x^{\mu},$

with constant $b_{\mu}$ and $e$ being the charge of the Dirac field. Choosing
$b_{\mu}=0$, $\mu=0,1,\ldots,p$ and $b_{l}=A_{l}^{\prime}$ for
$l=p+1,\ldots,D$, for $A_{\mu}=0$ in the new gauge we get
$(\psi^{\prime},A_{\mu}^{\prime})=(\psi
e^{-ieA_{\mu}^{\prime}x^{\mu}},A_{\mu}^{\prime})$. Taking
$A_{l}^{\prime}=\alpha_{l}/(eL_{l})$, we see that the new field obeys the
periodicity condition with $\alpha_{l}^{\prime}=0$. Hence, the initial problem
with a zero gauge field and quasiperiodicity condition (3) is transformed to a
gauge with periodic boundary conditions for the field and with a gauge field
having constant components along compact dimensions. In this interpretation
the phases can be expressed as $\alpha_{l}=-2\pi\Phi_{l}/\Phi_{0}$, where
$\Phi_{0}=2\pi/e$ is the flux quantum and $\Phi_{l}=-A_{l}^{\prime}L_{l}$ is
the formal magnetic flux enclosed by the compact dimension $x^{l}$. That flux
takes on real meaning in models where the space under consideration is
embedded in a space of higher dimension. Examples are the braneworld models
and carbon nanotubes. In the latter case the Dirac field describing the
electronic subsystem of graphene lives in a 2-dimensional space with topology
$R^{1}\times S^{1}$ and that space is embedded in a 3-dimensional Euclidean
space. The magnetic flux is located inside the tube.

##  3 Vacuum energy density and stresses

With the mode functions (6), the VEV of the energy-momentum tensor is
evaluated by using the formula (4). The mode sums for the energy density and
vacuum stresses are transformed to

$\displaystyle\langle T_{00}\rangle$ $\displaystyle=$
$\displaystyle-\frac{N}{2V_{q}}\int\frac{d\mathbf{k}_{p}}{\left(2\pi\right)^{p}}\sum_{\mathbf{n}_{q}\in\mathbf{Z}_{q}}\varepsilon_{\mathbf{k}},$
$\displaystyle\langle T_{\mu\nu}\rangle$ $\displaystyle=$
$\displaystyle-\frac{N}{2V_{q}}\int\frac{d\mathbf{k}_{p}}{(2\pi)^{p}}\sum_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}\frac{k_{\mu}k_{\nu}}{\varepsilon_{\mathbf{k}}},$
(8)

for $\mu,\nu=1,2,\ldots,D$, and $\langle T_{0\nu}\rangle=0$. The component
$\langle T_{00}\rangle$ corresponds to the energy density and it is presented
as the sum of the zero-point energies for elementary oscillators. The
expressions (8) for the diagonal components have been considered in [24]. The
expressions in (8) are divergent and in [24] two different methods have been
used in order to find the expressions for the renormalized VEVs. The first one
is based on the Abel-Plana summation formula and the second one uses the zeta
function technique [9, 25]. We will follow the second approach.

For the regularization of the mode sums we introduce the zeta function of a
complex variable $s$ as

$\zeta(s)=\frac{1}{V_{q}}\int\frac{d\mathbf{k}_{p}}{\left(2\pi\right)^{p}}\sum_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}\varepsilon_{\mathbf{k}}^{-2s},$
(9)

where the term $\mathbf{n}_{q}=0$ has to be excluded from the sum in the
special case $\alpha_{l}=0$, $l=p+1,\ldots,D$. After integration over
$\mathbf{k}_{p}$ and by using the generalized Chowla-Selberg formula [26, 27]
for the resulting series, the zeta function is decomposed as [24]
$\zeta(s)=\zeta_{\mathrm{M}}(s)+\zeta_{\mathrm{t}}(s)$, where
$\zeta_{\mathrm{M}}(s)$ is the corresponding function for the Minkowski
spacetime with trivial topology and the contribution $\zeta_{\mathrm{t}}(s)$
is induced by nontrivial topology. Introducing the vectors
$\boldsymbol{\alpha}_{q}=(\alpha_{p+1},\ldots,\alpha_{D})$ and
$\mathbf{L}_{q}=(L_{p+1},\ldots,L_{D})$ in the compact subspace, the
topological part is expressed as

$\zeta_{\mathrm{t}}(s)=\frac{2^{1-s}m^{D-2s}}{(2\pi)^{\frac{D}{2}}\Gamma(s)}\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}\cos(\mathbf{n}_{q}\cdot\boldsymbol{\alpha}_{q})f_{\frac{D}{2}-s}(mg(\mathbf{L}_{q},\mathbf{n}_{q})),$
(10)

where the prime on the summation sign means that the term $\mathbf{n}_{q}=0$
should be excluded and we have introduced the functions

$f_{\nu}(x)=\frac{K_{\nu}(x)}{x^{\nu}},\;g(\mathbf{L}_{q},\mathbf{n}_{q})=\left(\sum_{i=p+1}^{D}L_{i}^{2}n_{i}^{2}\right)^{1/2},$
(11)

with $K_{\nu}(x)$ being the modified Bessel function of the second kind.

An alternative representation for the zeta function, convenient in the
asymptotic analysis of the off-diagonal stress, is obtained from (10) by using
the formula (2.40) from [22]. It is given by the formula

$\displaystyle\zeta_{\mathrm{t}}(s)$ $\displaystyle=$
$\displaystyle\zeta_{p+2,q-2}(s)+\frac{2^{1-s}}{(2\pi)^{\frac{p}{2}+1}\Gamma(s)V_{q-2}}\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}\cos\left(n_{\mu}\alpha_{\mu}\right)$
(12)
$\displaystyle\times\cos\left(n_{\nu}\alpha_{\nu}\right)\varepsilon_{\mathbf{n}_{q-2}}^{p-2s+2}f_{\frac{p}{2}-s+1}\left(\sqrt{n_{\mu}^{2}L_{\mu}^{2}+n_{\nu}^{2}L_{\nu}^{2}}\varepsilon_{\mathbf{n}_{q-2}}\right),$

where $\zeta_{p+2,q-2}(s)$ is the zeta function in the model of topology
$M^{p+3}\times T^{q-2}$ with decompactified dimensions $x^{\mu}$ and
$x^{\nu}$. Here, $M^{p+3}$ stands for $(p+3)$-dimensional Minkowski spacetime
with trivial topology. The prime on the summation sign in (12) means that the
term $n_{\mu}=n_{\nu}=0$ is excluded from the summation and we have defined

$\varepsilon_{\mathbf{n}_{q-2}}=\left(\sum_{l=p+1,\neq\mu,\nu}^{D}k_{l}^{2}+m^{2}\right)^{1/2}.$
(13)

The function $\zeta_{p+2,q-2}(s)$ is given by a formula similar to (10) with
the summation over $\mathbf{n}_{q-2}\in\mathbf{Z}^{q-2}$.

The background geometry is flat and the renormalization is reduced to the
subtraction of the Minkowskian VEV. The renormalized energy density is
expressed as $\langle
T_{0}^{0}\rangle_{\mathrm{t}}=-N\zeta_{\mathrm{t}}(-1/2)/2$. For the diagonal
components of the vacuum stresses along uncompact dimensions one gets (no
summation over $\mu$) $\langle T_{\mu}^{\mu}\rangle_{\mathrm{t}}=\langle
T_{0}^{0}\rangle_{\mathrm{t}}$, $\mu=1,2,\ldots,p$. The diagonal vacuum
stresses in the compact subspace are found by using the relation (no summation
over $\mu$) $\langle
T_{\mu}^{\mu}\rangle_{\mathrm{t}}=\left(L_{\mu}/V_{q}\right)\partial_{L_{\mu}}\left(V_{q}\langle
T_{0}^{0}\rangle_{\mathrm{t}}\right)$, $\mu=p+1,\ldots,D$. In this way, from
(10) for the diagonal component we find [24] (no summation over $\mu$)

$\langle
T_{\mu}^{\mu}\rangle_{\mathrm{t}}=\frac{Nm^{D+1}}{(2\pi)^{\frac{D+1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}\cos(\mathbf{n}_{q}\cdot\boldsymbol{\alpha}_{q})F_{(\mu)}(mg(\mathbf{L}_{q},\mathbf{n}_{q})),$
(14)

with the functions

$F_{(\mu)}(x)=\left\\{\begin{array}[]{cc}f_{\frac{D+1}{2}}(x),&\mu=0,1,2,\ldots,p\\\
f_{\frac{D+1}{2}}(x)-m^{2}L_{\mu}^{2}n_{\mu}^{2}f_{\frac{D+3}{2}}(x),&\mu=p+1,\ldots,D\end{array}\right..$
(15)

The corresponding expressions in the case of periodic conditions,
$\alpha_{l}=0$, $l=p+1,\ldots,D$, are obtained from (14) with
$\cos(\mathbf{n}_{q}\cdot\boldsymbol{\alpha}_{q})=1$. In this case the vacuum
energy density is positive and for the diagonal stresses one has (no summation
over $l$) $\langle T_{l}^{l}\rangle_{\mathrm{t}}<\langle
T_{0}^{0}\rangle_{\mathrm{t}}$, $l=p+1,\ldots,D$. As it will be shown below,
for periodic conditions the off-diagonal components vanish. By using the
relation

$x^{2}f_{\nu+1}(x)=f_{\nu-1}(x)+2\nu f_{\nu}(x),$ (16)

it can be seen that $\sum_{l=1}^{D}\langle
T_{l}^{l}\rangle_{\mathrm{t}}<\langle T_{0}^{0}\rangle_{\mathrm{t}}$. Hence,
the vacuum energy-momentum tensor for a fermionic field with periodic
conditions obey the strong energy condition. For twisted fields with
$\alpha_{l}=\pi$, $l=p+1,\ldots,D$, one has
$\cos(\mathbf{n}_{q}\cdot\boldsymbol{\alpha}_{q})=(-1)^{n_{p+1}+\cdots+n_{D}}$.

The result (14) shows that the vacuum stresses in the uncompact subspace are
equal to the energy density, $\langle
T_{\mu}^{\mu}\rangle_{\mathrm{t}}=\langle T_{0}^{0}\rangle_{\mathrm{t}}$ ,
$\mu=1,\ldots,p$ (no summation over $\mu$). Of course, this is a consequence
of the Lorentz invariance in that subspace. By taking into account that for
the vacuum effective pressure along the direction $x^{\mu}$ one has
$P_{\mu}=-\langle T_{\mu}^{\mu}\rangle_{\mathrm{t}}$, we see that the equation
of state for the vacuum in the uncompact subspace is of the cosmological
constant type. The models with the topological Casimir energy as the source of
the accelerated expansion are based on this property.

For a massless fermionic field the general result (14) is reduced to (no
summation over $\mu$)

$\left\langle
T_{\mu}^{\mu}\right\rangle_{\mathrm{t}}=\frac{N}{2\pi^{\frac{D+1}{2}}}\Gamma\left(\frac{D+1}{2}\right)\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}\frac{\cos(\mathbf{n}_{q}\cdot\boldsymbol{\alpha}_{q})}{g^{D+1}(\mathbf{L}_{q},\mathbf{n}_{q})}F_{(\mu)}^{(0)}(\mathbf{L}_{q},\mathbf{n}_{q}),$
(17)

where $F_{(\mu)}^{(0)}(\mathbf{L}_{q},\mathbf{n}_{q})=1$ for
$\mu=0,1,2,\ldots,p$, and

$F_{(\mu)}^{(0)}(\mathbf{L}_{q},\mathbf{n}_{q})=1-\frac{\left(D+1\right)n_{\mu}^{2}L_{\mu}^{2}}{g^{2}(\mathbf{L}_{q},\mathbf{n}_{q})},$
(18)

for $\mu=p+1,\ldots,D$. In this special case the vacuum energy-momentum tensor
is traceless, $\left\langle T_{\mu}^{\mu}\right\rangle_{\mathrm{t}}=0$.

Here we are interested in the off-diagonal components. For
$\mu=0,1,2,\ldots,p$ and $\nu\neq\mu$ one gets $\langle
T_{\mu\nu}\rangle_{\mathrm{t}}=0$. The possible nonzero components $\langle
T_{\mu\nu}\rangle_{\mathrm{t}}$ correspond to $\mu,\nu=p+1,\ldots,D$. In order
to use the zeta function, we note that the following relation takes place

$\frac{k_{\nu}k_{\mu}}{\varepsilon_{\mathbf{k}}}=\frac{L_{\mu}L_{\nu}}{3\left(2\pi\right)^{2}}\partial_{\alpha_{\mu}}\partial_{\alpha_{\nu}}\varepsilon_{\mathbf{k}}^{3}.$
(19)

This allows to write the off-diagonal components in the form

$\langle
T_{\mu\nu}\rangle=-\frac{NL_{\mu}L_{\nu}}{6\left(2\pi\right)^{2}}\partial_{\alpha_{\mu}}\partial_{\alpha_{\nu}}\frac{1}{V_{q}}\int\frac{d\mathbf{k}_{p}}{(2\pi)^{p}}\sum_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}\varepsilon_{\mathbf{k}}^{3}=-\frac{NL_{\mu}L_{\nu}}{6\left(2\pi\right)^{2}}\partial_{\alpha_{\mu}}\partial_{\alpha_{\nu}}\zeta\left(-\frac{3}{2}\right).$
(20)

By using the formula (10), for the topological part we get

$\langle
T_{\mu\nu}\rangle_{\mathrm{t}}=\frac{m^{D+3}NL_{\mu}L_{\nu}}{(2\pi)^{\frac{D+1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}n_{\mu}n_{\nu}\cos(\mathbf{n}_{q}\cdot\boldsymbol{\alpha}_{q})f_{\frac{D+3}{2}}(mg(\mathbf{L}_{q},\mathbf{n}_{q})).$
(21)

Note that in this representation we can make the replacement

$\cos(\mathbf{n}_{q}\cdot\boldsymbol{\alpha}_{q})\rightarrow\sin\left(n_{\mu}\alpha_{\mu}\right)\sin\left(n_{\nu}\alpha_{\nu}\right)\cos(\mathbf{n}_{q-2}\cdot\boldsymbol{\alpha}_{q-2}),$
(22)

where
$\mathbf{n}_{q-2}\cdot\boldsymbol{\alpha}_{q-2}=\sum_{l\neq\mu,\nu}n_{l}\alpha_{l}$.
This replacement explicitly shows that the off-diagonal component $\langle
T_{\mu\nu}\rangle_{\mathrm{t}}$ is an even periodic function of the phases
$\alpha_{l}$, $l\neq\mu,\nu$, with the period equal to $2\pi$, and an odd
periodic function of $\alpha_{\mu}$ and $\alpha_{\nu}$ with the same period.
Hence, without the loss of generality, we can assume that
$|\alpha_{\mu}|\leq\pi$. For a massless field, by taking into account that
$f_{\nu}(x)=2^{\nu-1}\Gamma(\nu)x^{-2\nu}$ for $x\ll 1$, one obtains

$\langle
T_{\mu\nu}\rangle_{\mathrm{t}}=\frac{NL_{\mu}L_{\nu}}{\pi^{\frac{D+1}{2}}}\Gamma\left(\frac{D+3}{2}\right)\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}n_{\mu}n_{\nu}\frac{\cos(\mathbf{n}_{q}\cdot\boldsymbol{\alpha}_{q})}{g^{D+3}(\mathbf{L}_{q},\mathbf{n}_{q})}.$
(23)

An equivalent expression for the off-diagonal stresses is obtained by using
the representation (12) for the zeta function in (20). The first term in the
right-hand side of (12) does not depend on $\alpha_{\mu}$ and $\alpha_{\nu}$
and, hence, it does not contribute to the stress. The following expression is
obtained:

$\displaystyle\langle T_{\mu\nu}\rangle_{\mathrm{t}}$ $\displaystyle=$
$\displaystyle-\frac{NL_{\mu}^{2}L_{\nu}^{2}}{(2\pi)^{\frac{p+3}{2}}V_{q}}\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}_{q}\in\mathbf{Z}^{q}}n_{\mu}n_{\nu}\sin\left(n_{\mu}\alpha_{\mu}\right)$
(24)
$\displaystyle\times\sin\left(n_{\nu}\alpha_{\nu}\right)\varepsilon_{\mathbf{n}_{q-2}}^{p+5}f_{\frac{p+5}{2}}\left(\sqrt{n_{\mu}^{2}L_{\mu}^{2}+n_{\nu}^{2}L_{\nu}^{2}}\varepsilon_{\mathbf{n}_{q-2}}\right).$

For $q=2$ ($p=D-2$) one has $\varepsilon_{\mathbf{n}_{q-2}}=m$ and the formula
is reduced to

$\langle
T_{\mu\nu}\rangle_{\mathrm{t}}=-\frac{Nm^{D+3}L_{\mu}L_{\nu}}{(2\pi)^{\frac{D+1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{n_{\mu},n_{\nu}=-\infty}^{+\infty}n_{\mu}n_{\nu}\sin\left(n_{\mu}\alpha_{\mu}\right)\sin\left(n_{\nu}\alpha_{\nu}\right)f_{\frac{D+3}{2}}\left(m\sqrt{n_{\mu}^{2}L_{\mu}^{2}+n_{\nu}^{2}L_{\nu}^{2}}\right).$
(25)

In the special case under consideration this coincides with (21).

The special case of the results corresponding to $D=2$ describes the
properties of the ground state for the electronic subsystem in graphene
nanotubes and nanoloops (toroidal nanotubes) described by the effective Dirac
model. For nanotubes one has $(p,q)=(1,1)$ and for nanoloops $(p,q)=(0,2)$.
For metallic nanotubes and in the absence of the threading magnetic flux the
phase along the periodic condition is zero, $\alpha_{l}=0$. Depending on the
chiral vector in semiconductor nanotubes two values of the phases are realized
with $\alpha_{l}=\pi/3$ and $\alpha_{l}=\pi/3$. The corresponding analysis for
the diagonal components of the ground state energy-momentum tensor can be
found in [24]. The off-diagonal component for nanoloops is obtained from (25)
with $D=2$ and $N=2$. In this special case one has
$f_{5/2}=\sqrt{\pi/2}x^{-5}e^{-x}(3+3x+x^{2})$.

##  4 Asymptotic analysis and numerical results

Let us consider some asymptotics of general formulae. For large values of the
length of the compact dimension $z^{l}$, $l\neq\mu,\nu$, $L_{l}\gg
L_{\mu},L_{\nu}$, the dominant contribution in (21) comes from the term with
$n_{l}=0$ and the leading order term coincides with the off-diagonal stress in
the model where the coordinate $z^{l}$ is decompactified. In the opposite
limit $L_{l}\ll L_{\mu},L_{\nu}$, it is more convenient to use the the
representation (24). The behavior of the stress is essentially different
depending on whether the phase $\alpha_{l}$ is zero or not. For $\alpha_{l}=0$
the main contribution to the VEV $\langle T_{\mu\nu}\rangle_{\mathrm{t}}$
comes from the modes with $n_{l}=0$. To the leading order we get

$\langle T_{\mu\nu}\rangle_{\mathrm{t}}\approx\frac{N\langle
T_{\mu\nu}\rangle_{\mathrm{t}}^{(M^{p+1}\times T^{q-1})}}{N_{D-1}L_{l}},$ (26)

where $N_{D-1}=2^{\left[\frac{D}{2}\right]}$ and $\langle
T_{\mu\nu}\rangle_{\mathrm{t}}^{(M^{p+1}\times T^{q-1})}$ is the corresponding
VEV in $D$-dimensional spacetime with topology $M^{p+1}\times T^{q-1}$ which
is obtained from the geometry described by (1) excluding the compact dimension
$x^{l}$. For $\alpha_{l}\neq 0$ and assuming that $|\alpha_{l}|<\pi$, again,
the dominant contribution give the modes $n_{l}=0$. The argument of the
function $f_{(p+5)/2}(x)$ is large and we can use the corresponding asymptotic
of the modified Bessel function. This shows that in the limit under
consideration the off-diagonal stress $\langle T_{\mu\nu}\rangle_{\mathrm{t}}$
is suppressed by the factor
$\exp(-|\alpha_{l}|\sqrt{L_{\mu}^{2}+L_{\nu}^{2}}/L_{l})$.

For large values of the lengths $L_{\mu}$ and $L_{\nu}$ compared to the other
length scales $1/m$ and $L_{l}$, $l\neq\mu,\nu$, by using (24) we can see that
the topological contribution $\langle T_{\mu\nu}\rangle_{\mathrm{t}}$ is
exponentially suppressed by the factor
$\exp(-\varepsilon_{(0)}\sqrt{L_{\mu}^{2}+L_{\nu}^{2}})$, where
$\varepsilon_{(0)}=\varepsilon_{\mathbf{n}_{q-2}}|_{\mathbf{n}_{q-2}=0}$ and
$0<|\alpha_{l}|<\pi$. For small values of $L_{\mu}$ and $L_{\nu}$ the dominant
contribution in (24) comes from large values of $|n_{l}|$ and we can replace
the corresponding summations by the integration in accordance with

$\frac{L_{\mu}L_{\nu}}{V_{q}}\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}_{q-2}\in\mathbf{Z}^{q-2}}\varepsilon_{\mathbf{n}_{q-2}}^{p+5}f_{\frac{p+5}{2}}\left(b\varepsilon_{\mathbf{n}_{q-2}}\right)\rightarrow\int\frac{d^{q-2}\mathbf{y}}{\left(2\pi\right)^{q-2}}\left(\mathbf{y}^{2}+m^{2}\right)^{(p+5)/2}f_{\frac{p+5}{2}}\left(b\sqrt{\mathbf{y}^{2}+m^{2}}\right),$
(27)

where $b=\sqrt{n_{\mu}^{2}L_{\mu}^{2}+n_{\nu}^{2}L_{\nu}^{2}}$ and
$\mathbf{y}=(y_{1},\ldots,y_{q-2})$ with $-\infty<y_{l}<+\infty$. After
integration over the angular part, the integral over $|\mathbf{y}|$ is
evaluated by using the formula from [28]. In this way it can be seen that the
leading term in the expansion of $\langle T_{\mu\nu}\rangle_{\mathrm{t}}$
coincides with (25). Additionally assuming that
$m\sqrt{L_{\mu}^{2}+L_{\nu}^{2}}\ll 1$, in the leading approximation we get

$\langle
T_{\mu\nu}\rangle_{\mathrm{t}}\approx-\frac{NL_{\mu}L_{\nu}}{\pi^{\frac{D+1}{2}}}\Gamma\left(\frac{D+3}{2}\right)\sideset{}{{}^{\prime}}{\sum}_{n_{\mu},n_{\nu}=-\infty}^{+\infty}n_{\mu}n_{\nu}\frac{\sin\left(n_{\mu}\alpha_{\mu}\right)\sin\left(n_{\nu}\alpha_{\nu}\right)}{\left(n_{\mu}^{2}L_{\mu}^{2}+n_{\nu}^{2}L_{\nu}^{2}\right)^{\frac{D+3}{2}}}.$
(28)

Note that the right-hand side presents the off-diagonal component of the
vacuum energy-momentum tensor for a massless fermionic field in the model
$(p,q)=(D-2,2)$ with compact dimensions $x^{\mu}$ and $x^{\nu}$.

We will present the numerical analysis for the $D=5$ model with two compact
dimensions $x^{4}$ and $x^{5}$. This corresponds to the set $(p,q)=(3,2)$. By
taking into account that in [24] the numerical results for the energy density
and diagonal stresses are given for the model $D=4$ with a single compact
dimension, the analysis will be given for those quantities as well. We start
from the dependence of the expectation values on the phases $\alpha_{4}$ and
$\alpha_{5}$. Figure 1 presents that dependence of the energy density (left
panel) and off-diagonal stress $\langle T_{45}\rangle_{\mathrm{t}}$ (right
panel) for $mL_{4}=0.5$ and $mL_{6}=0.6$.

|
---|---
Figure 1: The expectation values of the vacuum energy density and off-diagonal
stress on the phases of the periodicity conditions in the model $(p,q)=(3,2)$
with $mL_{4}=0.5$, $mL_{5}=0.6$.

The corresponding results for the diagonal stresses along compact dimensions
are given in figure 2. As already mentioned above, the energy density and the
diagonal stresses are even periodic functions of $\alpha_{4}$ and
$\alpha_{5}$, whereas the off-diagonal component is an odd periodic function
of those phases. Depending on the specific values of the phases, all the
components can be either positive or negative. The energy density $\langle
T_{00}\rangle_{\mathrm{t}}$ and the vacuum pressures $\langle
T_{44}\rangle_{\mathrm{t}}$ and $\langle T_{55}\rangle_{\mathrm{t}}$ are
positive for the values of the phases near $(\alpha_{4},\alpha_{5})=(0,0)$
(periodic conditions) and negative near $(\alpha_{4},\alpha_{5})=(\pi,\pi)$
(antiperiodic conditions).

|
---|---
Figure 2: The same as in figure 1 for the stresses along the compact
dimensions $x^{4}$ (left panel) and $x^{5}$ (right panel).

The dependence of the VEVs of the components for the energy-momentum tensor on
the lengths of compact dimensions is presented in figures 3 (energy density
and off-diagonal component) and 4 (stresses along compact dimensions). For the
diagonal components we have taken the phases $\alpha_{4}=\pi/2$ and
$\alpha_{5}=0$ and the off-diagonal component is plotted for
$\alpha_{4}=\pi/2$, $\alpha_{5}=-0.6\pi$.

|
---|---
Figure 3: The expectation values of the vacuum energy density and off-diagonal stress on the length of compact dimensions in the model $(p,q)=(3,2)$. For the left panel we have taken $\alpha_{4}=\pi/2$, $\alpha_{5}=0$ and for the right panel $\alpha_{4}=\pi/2$, $\alpha_{5}=-0.6\pi$.  |
---|---
Figure 4: The vacuum stresses along the compact dimensions $x^{4}$ (left
panel) and $x^{5}$ (right panel) versus the lengths of those dimensions. The
graphs are plotted for $\alpha_{4}/2\pi=0.25$, $\alpha_{5}/2\pi=0$.

From the given graphs, one can get the impression that the energy density is a
monotonic function of the lengths of the compact dimensions. However, this is
not the case even for a massless field. In order to demonstrate that and by
taking into account that the VEVs for a massless field approximate the results
for massive fields in the limit of small values of the lengths of compact
dimensions, in figure 5 we have plotted the dimensionless quantity
$L_{4}^{6}\langle T_{00}\rangle_{\mathrm{t}}$ as a function of the ratio
$L_{5}/L_{4}$. The corresponding expression is given by the right-hand side of
(28). The graphs are plotted for $\alpha_{4}=\pi/2$ and the numbers near the
curves are the values of the ratio $\alpha_{5}/2\pi$. For large values of
$L_{5}/L_{4}$ all the curves tend to the corresponding result for the energy
density in the model where the direction $x^{5}$ is decompactified
($L_{5}\rightarrow\infty$).

Figure 5: The energy density for a massless fermionic field in the model
$(p,q)=(3,2)$ as a function of the ratio $L_{5}/L_{4}$ for fixed value
$\alpha_{4}=\pi/2$. The numbers near the graphs are the values of
$\alpha_{5}/2\pi$.

##  5 Conclusions

Continuing the investigations started in [24] we have studied the effects of
nontrivial topology on the local characteristics of the fermionic vacuum. A
toroidal compactification of a part of spatial dimensions in
$(D+1)$-dimensional flat spacetime is considered. In addition to the diagonal
components, studied in [24], the vacuum energy-momentum tensor has an off-
diagonal components having indices along compact dimensions. Those components
vanish for periodic ($\alpha_{l}=0$) and antiperiodic ($\alpha_{l}=\pi$)
conditions. In the first case the vacuum energy-momentum tensor for a
fermionic field obeys the strong energy condition. For general values of the
phases that is not the case. The phases in the periodicity conditions can be
interpreted in terms of magnetic fluxes enclosed by compact dimensions. The
VEVs are periodic functions of magnetic fluxes with the period of flux
quantum. The diagonal components are even functions of the phases
$\alpha_{l}$. The off-diagonal component $\langle
T_{\mu\nu}\rangle_{\mathrm{t}}$, $\mu\neq\nu$, $\mu,\nu=p+1,\ldots,D$, is an
even function of $\alpha_{l}$ with $l\neq\mu,\nu$, and odd function of the
phases $\alpha_{\mu}$ and $\alpha_{\nu}$. The vacuum stresses in the uncompact
subspace are isotropic and the corresponding equation of state is of the
cosmological constant type. Depending on the values of the phases the
components of the vacuum energy-momentum tensor can be either positive or
negative. For small values of the lengths $L_{\mu}$ and $L_{\nu}$, the off-
diagonal component is approximated by the corresponding result for a massless
field in the model with $q=2$ and compact subspace $(x^{\mu},x^{\nu})$ (see
(28)). The numerical analysis of the obtained results is presented for the
$D=5$ with $(p,q)=(3,2)$.

We have considered the effects of the nontrivial topology on the local
properties of the fermionic vacuum. In the presence of boundaries additional
contributions are induced in the VEVs of physical observables (the boundary-
induced Casimir effect). The effects of two planar boundaries with the bag
boundary conditions on the Dirac field in the geometry under consideration
have been discussed in [29, 30]. The results in the special case of
2-dimensional space are applied to finite length carbon nanotubes. The
fermionic condensate and the VEV of the energy-momentum tensor in toroidally
compactified de Sitter spacetime are studied in [31].

## Acknowledgments

The work was supported by the grant No. 21AG-1C047 of the Higher Education and
Science Committee of the Ministry of Education, Science, Culture and Sport RA.

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