# Vacuum energy, temperature corrections and heat kernel coefficients in
$(D+1)$-dimensional spacetimes with nontrivial topology

1Herondy Mota<EMAIL_ADDRESS>1Departamento de Física, Universidade
Federal da Paraíba,
Caixa Postal 5008, João Pessoa, Paraíba, Brazil.

###### Abstract

In this work we make use of the generalized zeta function technique to
investigate the vacuum energy, temperature corrections and heat kernel
coefficients associated with a scalar field under a quasiperiodic condition in
a $(D+1)$-dimensional conical spacetime. In this scenario we find that the
renormalized vacuum energy, as well as the temperature corrections, are both
zero. The nonzero heat kernel coefficients are the ones related to the usual
Euclidean divergence, and also to the nontrivial aspects of the
quaisperiodically identified conical spacetime topology. An interesting result
that arises in this configuration is that for some values of the quasiperiodic
parameter, the heat kernel coefficient associated with the nontrivial topology
vanishes. In addition, we also consider the scalar field in a
$(D+1)$-dimensional spacetime formed by the combination of a conical and screw
dislocation topological defects. In this case, we obtain a nonzero
renormalized vacuum energy density and its corresponding temperature
corrections. Again, the nonzero heat kernel coefficients found are the ones
related to the Euclidean and nontrivial topology divergences. For $D=3$ we
explicitly show, in the massless scalar field case, the limits of low and high
temperatures for the free energy. In the latter, we show that the free energy
presents a classical contribution.

##  I Introduction

Casimir effect is a phenomenon that arises in the realm of Quantum Field
Theory as a consequence of boundary conditions imposed on quantum fields,
namely, scalar, electromagnetic and spinor fields. In this sense, the quantum
modes of a given field in its vacuum state are altered in such a way that a
detectable nonzero Casimir force is produced, at least in the electromagnetic
case. In fact, an attractive force was predicted in 1948 by Casimir who
considered a configuration constituted by an electromagnetic field whose modes
are confined between two identical and large perfect parallel conducting
plates Casimir1948dh . Although not with a great precision, the first attempt
to detect this effect was in 1958 by Sparnaay Sparnaay1958  and confirmed,
after several decades, by others Lamoreaux:1996wh ; Lamoreaux1996wh ;
MohideenRoy1998iz ; Bressi:2002fr ; PhysRevA.81.052115 ; PhysRevA.78.020101 .

In the past years, Casimir like effects arising due to the geometrical and
topological aspects of curved spacetimes have been investigated. In these
scenarios, the quantum modes in the vacuum state are modified and a nonzero
vacuum energy is produced bordag2009advances ; BordagMohideenMostepanenko .

An interesting curved spacetime is the one with a conical topology as, for
instance, the cosmic string spacetime. The existence of cosmic strings can
produce gravitational, astrophysical and cosmological signatures and is
predicted in some extensions of the Standard Model of particle physics VS ;
hindmarsh , and in the context of string theory Copeland:2011dx ;
Hindmarsh:2011qj . Cosmic string is also a linear topological defect that is
supposed to be formed due to phase transitions in the early Universe. An
idealized and static one is characterized by a spacetime with conical topology
that has associated to it a planar angle deficit proportional to its linear
energy density, $\mu_{0}$, that is, $\Delta\phi\simeq 8\pi G\mu_{0}$, where
$G$ is the Newton’s gravitational constant. In condensed matter systems, the
cosmic string counterpart is a disclination Katanaev:1992kh ; Puntigam:1996vy
. In Refs. BezerradeMello2011nv ; MotaDispiration ; klecioQuasiPNanotubes ;
Braganca:2019mvj ; Braganca:2014qma ; Braganca:2020jci ; deFarias:2021qdg ;
deFarias:2022rju  quantum effects induced by linear topological defects, such
as the ones just described, on the vacuum expectation values of the energy-
momentum tensor, induced current density and induced density fluctuation of a
classical liquid were considered.

In the context of theories of solid and crystal continuum media there also
exists an interesting linear topological defects known as screw dislocation
Puntigam:1996vy . The combination of a screw dislocation and a cosmic string
(disclination) is also a topological defect that has a spacelike helical
structure, with chiral properties Letelier:1995ze ; Galtsov:1993ne ;
tod1994conical . In this case, there is not only a delta function singularity
in the scalar curvature, associated with the conical defect, but also in the
torsion. The latter is a characteristic of a screw dislocation and the
combination of it with a conical defect, also known as cosmic dispiration
DeLorenci:2002jv ; DeLorenci:2003wv , is formally constructed in the framework
of the Einstein-Cartan theory of gravity Letelier:1995ze .

In dealing with the technical calculation involved in the investigation of
Casimir effect it is necessary to adopt regularization and renormalization
methods to treat the infinity contributions to the vacuum energy arising in
this context. A very elegant and powerful method is the one constructed by
means of the generalized zeta function approach Hawking1977 ;
DowkerCritchley1975tf ; aleixo2021thermal . The zeta function in this case is
defined through the eigenvalues associated with differential operators such as
the Laplace-Beltrami one Elizalde1994book . The construction of this approach
is by making use of the path integral formalism since, by doing so, it is
possible to make a connection with thermodynamics, allowing us to calculate
the partition function and, as a consequence, also pave the way to obtain
temperature corrections to the vacuum energy Hawking1977 ;
DowkerCritchley1975tf ; aleixo2021thermal . In this framework, an expansion
for the heat kernel is generally adopted in order to study the possible
divergent structures associated with nonzero heat kernel coefficients
bordag2009advances ; Bordag:1998rf ; BordagMohideenMostepanenko ;
Elizalde1994book .

By making use of the generalized zeta function method described above, in the
present work, we consider the calculation of the scalar vacuum energy,
temperature corrections and nonzero heat kernel coefficients induced by two
types of spacetimes with nontrivial topology, namely, a quasiperiodically
identified conical spacetime, and also the spacetime describing a cosmic
dispiration. Concerning the former, a massive scalar field whose modes
propagate in a conical spacetime is subject to obey a quasiperiodic condition
in the azimuthal direction. Such a condition generalizes the widely known
periodic and antiperiodic conditions and has been considered in different
scenarios QuasiPNanotubes ; klecioQuasiPNanotubes ; deFarias:2021qdg ;
deFarias:2022rju ; Junior:2023feu ; Ferreira:2023uxs ; Ferreira:2022eno ;
Porfirio:2019gdy ; Mota:2016eoi . Hence, we generalize the results obtained in
Ref. cognola:1993qg , where the authors studied the vacuum energy, temperature
corrections and heat kernel coefficients in the periodic case (see also Ref.
Fursaev:1993qk ). In contrast with Ref. cognola:1993qg , in our investigation,
we take into consideration the renormalization scheme discussed in Refs.
bordag2009advances ; Bordag:1998rf , where the additional requirement that the
renormalized vacuum energy must vanish for large masses is necessary to be
adopted. All this is also investigated by considering a cosmic dispiration
spacetime and, to the best of our knowledge, is studied for the first time
here. Both spacetimes with nontrivial topology are considered in $(D+1)$
dimensions.

This work is organized as follows. In Sec.II we overview the necessary
elements, in $(D+1)$ dimensions, involved in the generalized zeta function
method such as the heat kernel and free energy definitions. In Sec.III, the
vacuum energy, temperature corrections and heat kernel coefficients are
obtained in a quaisperiodically identified $(D+1)$ conical spacetime. In
Sec.IV, by doing the same analysis, we consider the $(D+1)$ cosmic dispiration
spacetime. Finally, in Sec.V we present our conclusions. In this paper we use
natural units $\hbar=c=1$.

##  II Generalized zeta function, heat kernel and free energy

In this section we shall consider the general aspects involving the use of the
zeta function method for temperature corrections to the vacuum energy of a
massive scalar field. The zeta function provides a powerful and elegant tool
to calculate the vacuum energy at zero temperature and also to find the
corresponding nonzero temperature corrections by adopting the Euclidean
formalism, with a compact imaginary time $\tau$. The generalized zeta function
in this framework is defined as Hawking1977 ; Elizalde1994book ;
Kirsten:2010zp ; Nesterenko:2004gzu ; aleixo2021thermal

$\zeta_{N}(s)=\sum_{\sigma}\lambda^{-s}_{\sigma},$ (1)

where $\lambda_{\sigma}$ stands for the eigenvalues of a $N$-dimensional
Laplace-Beltrami operator $\hat{A}_{N}$ and may not be always discrete,
despite the summation symbol representation above. Moreover, the definition in
(1) converges for Re($s$) $>$ $N/2$ and can also be analytically extended for
Re($s$) $<$ $N/2$, with a pole at $s=N/2$ Kirsten:2010zp ; Nesterenko:2004gzu
. For static spacetimes, the Laplace-Beltrami operator $\hat{A}_{N}$ can be
expressed as

$\displaystyle\hat{A}_{N}$ $\displaystyle=$
$\displaystyle-\frac{\partial^{2}}{\partial\tau^{2}}-\nabla^{2}_{D}+m^{2}$ (2)
$\displaystyle=$
$\displaystyle-\frac{\partial^{2}}{\partial\tau^{2}}+\hat{A}_{D},$

where $N=D+1$, $\nabla^{2}_{D}$ is the Laplace operator in $D$ spatial
dimensions and $m$ is the mass of the scalar field, which is the one to be
considered here. Note that we are making use of the Euclidean representation
of the above operator, obtained by performing a Wick rotation,
$t\rightarrow-i\tau$, in the ordinary time parameter $t$ present in the
Minkowski spacetime Hawking1977 ; Elizalde1994book ; aleixo2021thermal .

For a scalar field, $\Phi_{\sigma}(\tau,{\bf r})$, whose modes propagate in a
$(D+1)$ Euclidean static spacetime the temperature correction is introduced by
imposing a periodicity condition in the imaginary time coordinate, i.e.,

$\begin{split}\Phi_{\sigma}(\tau,{\bf r})=\Phi_{\sigma}(\tau+\beta,{\bf
r}),\end{split}$ (3)

where $\beta=\frac{1}{k_{B}T}$, with $k_{B}$ being the Boltzmann constant and
$T$ the temperature. This means that, by acting the operator in Eq. (2) on the
scalar field obeying the periodicity condition above, we can obtain a solution
of the form

$\begin{split}\Phi_{\sigma}(\tau,{\bf r})=e^{-i\omega_{n}\tau}\varphi_{j}({\bf
r}),\qquad\quad\omega_{n}=\frac{2\pi n}{\beta},\end{split}$ (4)

where the quantum numbers are now represented by $\sigma=(n,j)$ and $n=0,\pm
1,\pm 2,...$ . This provides a set of eigenvalues as follows

$\begin{split}\lambda^{2}_{\sigma}=\left(\frac{2\pi
n}{\beta}\right)^{2}+\Omega^{2}_{j},\end{split}$ (5)

with $\Omega_{j}$ being the set of eigenvalues associated with the spatial
operator $\hat{A}_{D}$ defined in Eq. (2).

In order to use Eqs. (4) and (5) to calculate the generalized zeta function in
$(D+1)$ dimensions, let us consider Eq. (1) in the form Hawking1977 ;
Elizalde1994book ; Kirsten:2010zp ; Nesterenko:2004gzu ; aleixo2021thermal

$\begin{split}\zeta_{N}(s)=\frac{1}{\Gamma(s)}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\xi^{s-1}e^{-\left(\frac{2\pi
n}{\beta}\right)^{2}\xi}\text{Tr}\left[e^{-\xi\hat{A}_{D}}\right]d\xi,\end{split}$
(6)

where the trace of the operator $\hat{A}_{D}$, as indicated above, is given by

$\begin{split}K_{D}(\xi)=\text{Tr}\left[e^{-\xi\hat{A}_{D}}\right]=\sum_{j}e^{-\Omega_{j}\xi}.\end{split}$
(7)

This expression also defines what is commonly known as heat kernel. Moreover,
associated with the above expression there is also a very useful expansion for
small $\xi$ which gives information about the ultraviolet divergencies that
usually appear in the calculation of the vacuum energy. Such heat kernel
expansion may be written as Elizalde1994book ; Kirsten:2010zp ;
Nesterenko:2004gzu ; bordag2009advances

$K_{D}(\xi)=\frac{e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}\sum_{p=0}^{\infty}C_{\frac{p}{2}}\xi^{\frac{p}{2}}+\text{ES},$
(8)

where $C_{\frac{p}{2}}$ are the heat kernel coefficients and ‘ES’ stands for
exponentially suppressed terms in the calculation of the integral (6). The
expressions in Eqs. (7) and (8) will play a very important role in our
analysis below.

The generalized zeta function in (6) leads to the following expression:

$\begin{split}\zeta_{N}(s)=\frac{\beta}{\sqrt{4\pi}\Gamma(s)}\left\\{\Gamma(s-1/2)\zeta_{D}(s-1/2)+2\sum_{n=1}^{\infty}\int_{0}^{\infty}\xi^{s-\frac{3}{2}}e^{-\frac{(n\beta)^{2}}{4\xi}}\text{Tr}\left[e^{-\xi\hat{A}_{D}}\right]d\xi\right\\},\end{split}$
(9)

where $\zeta_{D}(s-1/2)$ is obtained from the term $n=0$ in Eq. (6) . As a
consequence, the connection with the partition function $Z$ is written as

$\begin{split}\ln
Z=\frac{1}{2}\zeta^{\prime}_{N}(0)+\frac{1}{2}\ln\left(\frac{\pi\mu^{2}}{4}\right)\zeta_{N}(0),\end{split}$
(10)

where prime indicates derivation with respect to the argument $s$ of the zeta
function (9). The parameter $\mu$ has dimension of mass and arises as a mesure
on the space of field functions in the path integral formulation adopted
Elizalde1994book ; aleixo2021thermal .

Furthermore, the free energy follows from the partition function above and is
given by Elizalde1994book ; Kirsten:2010zp ; Nesterenko:2004gzu

$\begin{split}F&=-\frac{1}{\beta}\ln Z\\\
&=\frac{1}{2}\zeta_{D}(-1/2)-\frac{\bar{C}_{\frac{N}{2}}}{2(4\pi)^{\frac{N}{2}}}\ln(M^{2})-\frac{1}{\sqrt{4\pi}}\sum_{n=1}^{\infty}\int_{0}^{\infty}\xi^{-\frac{3}{2}}e^{-\frac{(n\beta)^{2}}{4\xi}}\text{Tr}\left[e^{-\xi\hat{A}_{D}}\right]d\xi,\end{split}$
(11)

where $M=\frac{\sqrt{\pi}e^{2}\mu}{8}$ and the coefficients
$\bar{C}_{\frac{N}{2}}$ are given by Eq. (18), also related to the heat kernel
coefficients defined in Eq. (8). While the third term in the r.h.s. of Eq.
(11) makes possible to obtain temperature corrections to the free energy, the
first two terms, by cautiously adopting an adequate renormalization procedure,
provide the vacuum energy at zero temperature Elizalde1994book ;
Kirsten:2010zp . In order to better see that, let us write

$\displaystyle
E_{0}(D)=E_{0}^{\text{fin}}(D)-\frac{\bar{C}_{\frac{N}{2}}}{2(4\pi)^{\frac{N}{2}}}\ln(M^{2}),$
(12)

where

$E_{0}^{\text{fin}}(D)=\lim_{s\rightarrow
0}\left[E_{0}(D,s)-E_{0}^{\text{div}}(D,s)\right]$ (13)

is the finite part of the first term in the r.h.s. of Eq. (11). This term can
be regularized by defining

$E_{0}(D,s)=\frac{1}{2}\zeta_{D}(s-1/2).$ (14)

Furthermore, the divergent contribution indicated in Eq. (13) can be obtained
by using the heat kernel expansion (8) in the calculation of the regularized
expression111See Refs. bordag2009advances ; Bordag:1998rf  for the case $D=3$.
in Eq. (14). This is achieved by considering the term $n=0$ in Eq. (6), which
gives

$E_{0}(D,s)=\frac{1}{2(4\pi)^{\frac{D}{2}}\Gamma\left(s-\frac{1}{2}\right)}\sum_{p=0}^{\infty}\left[\frac{\bar{C}_{p}}{s-\frac{N}{2}+p}+\frac{\bar{C}_{\frac{2p+1}{2}}}{s-\frac{N}{2}+\frac{2p+1}{2}}\right],$
(15)

with

$\bar{C}_{p}=\sum_{d=0}^{p}\frac{(-1)^{d}}{d!}C_{p-d}m^{2d},\qquad\qquad\bar{C}_{\frac{2p+1}{2}}=\sum_{d=0}^{p}\frac{(-1)^{d}}{d!}C_{\frac{2p+1-2d}{2}}m^{2d},$
(16)

where we have also used a Taylor expansion, in powers of mass, for
$e^{-m^{2}\xi}$. A carefully analysis of Eq. (15) shows that the divergent
contributions when $s\rightarrow 0$ come from the terms $p=\frac{D+1}{2}$ for
odd $D$, and $p=\frac{D}{2}$ for even $D$. Thus,

$E^{\text{div}}_{0}(D,s)=-\frac{1}{2(4\pi)^{\frac{N}{2}}}\frac{\bar{C}_{\frac{N}{2}}}{s},$
(17)

where

$\bar{C}_{\frac{N}{2}}=\sum_{d=0}^{p}\frac{(-1)^{d}}{d!}C_{\frac{N-2d}{2}}m^{2d}.$
(18)

In the massive case there can be contributions in Eq. (12), coming from Eq.
(18), that are proportional to the mass with positive powers. As a
consequence, these contributions must also be subtracted from Eq. (12) in
order to obtain a renormaized expression for the vacuum energy. In this case,
the latter should satisfy the normalization condition bordag2009advances ;
Bordag:1998rf

$\lim_{m\rightarrow\infty}E_{0}^{\text{ren}}(D)=0,$ (19)

which is plausible since in the limit for large masses we should not expect to
have a nonzero vacuum energy, which is a quantum effect.

As to the operator $\hat{A}_{D}$ in Eq. (11), it satisfies the eigenvalue
equation

$\begin{split}\hat{A}_{D}\varphi_{j}=\Omega_{j}\varphi_{j},\end{split}$ (20)

where $\Omega_{j}$ is the set of eigenvalues associated with the spatial
momenta, as pointed out in Eq. (5).

For spacetimes where the operator $\hat{A}_{D}$ presents explicitly and well
defined eigenvalues the generalized zeta function (6) is in geral not
difficult to calculate. However, in cases where it is considered curved
spacetimes or spacetimes with nontrivial topology, we should use a local
approach to calculate the heat kernel Hawking1977 . This is achieved by
considering the local heat kernel definition, that is,

$K_{D}(w,w^{\prime},\xi)=\sum_{j}e^{-\Omega_{j}\xi}\varphi_{j}(w)\varphi^{*}_{j}(w^{\prime}),$
(21)

where $w=(x^{1},x^{2},...,x^{D})$ stands for $D$ spatial coordinates. The
local heat kernel above satisfies the heat kernel equation

$\frac{\partial}{\partial\xi}K_{D}(w,w^{\prime},\xi)+A_{D}K_{D}(w,w^{\prime},\xi)=0$
(22)

with initial condition

$K_{D}(w,w^{\prime},0)=\delta^{D}(w-w^{\prime}).$ (23)

The heat kernel in $D$ spatial dimentions is, thus, obtained in terms of the
local heat kernel as follows

$K_{D}(\xi)=\int\sqrt{|g^{(D)}|}K_{D}(w,w,\xi)d^{D}w,$ (24)

where we have considered the normalization condition

$\int\sqrt{|g^{(D)}|}\varphi_{j}(w)\varphi^{*}_{j^{\prime}}(w)d^{D}w=\delta_{j,j^{\prime}}.$
(25)

The delta symbol in the r.h.s. of the above expression is understood as
Kronecker delta or Dirac delta function, depending on whether $j$ is discrete
or continuous and $|g^{(D)}|$ is the determinant of the spatial part of the
metric $g_{\mu\nu}$. Therefore, once obtained the heat kernel by performing
the integration in Eq. (24), we can in principle calculate the corresponding
generalized zeta function.

##  III Vacuum energy, temperature corrections and heat kernel coefficients in
a quasiperiodically identified conical spacetime

A conical spacetime codifies the geometrical structure associated with linear
topological defects, such as cosmic strings predicted to exist in the Universe
by some extensions of the Standard Model of Particle Physics hindmarsh ; VS
and in String Theory Copeland:2011dx ; Hindmarsh:2011qj . This defect also
exists in condensed matter systems and it is known as disclination
Katanaev:1992kh .

The line element describing a conical defect in a $(D+1)$-dimensional
Euclidean spacetime in cylindrical coordinates is given by

$ds^{2}=-d\tau^{2}-dr^{2}-r^{2}d\phi^{2}-dz^{2}-\sum_{j=4}^{D}\left(dx^{j}\right)^{2},$
(26)

where $\tau=it$ is the imaginary time, $D\geq 4$, $r\geq 0$, $0\leq\phi\leq
2\pi/q$ and $-\infty\leq(t,z,x^{j})\leq\infty$, for $j=4,...,D$. The quantity
$\Delta\phi=2\pi-2\pi/q$ gives us the planar angle deficit (or excess) caused
by the presence of the conical space time. Note that the conical defect is
assumed to be on the ($D-2)$-dimensional hypersurface $r=0$. In the usual four
dimensions, the parameter $q\geq 1$ is proportional to the linear mass density
of the cosmic string $\mu_{0}$ by $q^{-1}=1-4G\mu_{0}$, where $G$ is the
Newton’s gravitational constant MotaDispiration ; klecioQuasiPNanotubes . In
condensed matter, on the other hand, the parameter $q$ characterizes a
disclination in solids possessing a spin structure and can assume values such
that $q\geq 0$.

We wish to consider the scalar field’s quantum modes propagating in the
$(D+1)$-dimensional Euclidean spacetime described by the line element in Eq.
(26). The scalar field is also required to obey a quasiperiodic condition
given by

$\varphi\left(r,\phi,z,x^{j}\right)=e^{-2\pi
i\alpha}\>\varphi\left(r,\phi+2\pi/q,z,x^{j}\right),$ (27)

where $0\leq\alpha<1$ is a constant that regulates the phase angle in Eq.
(27). The special case where $\alpha=0$ corresponds to the periodic condition,
while the special case with $\alpha=1/2$ corresponds to the antiperiodic
condition.

From the eigenvalue equation (20), and taking into consideration the line
element (26), the spatial part of the scalar field (4) can be obtained by
solving the equation

$\left[-\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial
r}\right)-\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\phi^{2}}-\frac{\partial^{2}}{\partial
z^{2}}+\sum_{j=4}^{D}\frac{\partial^{2}}{\partial
x^{j2}}+m^{2}\right]\varphi_{j}(w)=\lambda_{j}\varphi_{j}(w),$ (28)

where $j$ stands for the spatial quantum modes.

The complete set of normalized solutions for the eigenvalue equation (28)
under the quasiperiodic condition (27) on a conical spacetime is given by
MotaDispiration ; klecioQuasiPNanotubes ; deFarias:2021qdg

$\varphi_{j}\left(\mathbf{r}\right)=\left[\frac{q\eta}{(2\pi)^{D-1}}\right]^{\frac{1}{2}}e^{i{\bf
p}\cdot{\bf r}_{\parallel}+i\nu z+iq(\ell+\alpha)\phi}J_{q|\ell+\alpha|}(\eta
r),$ (29)

where ${\bf r}_{\parallel}$ and ${\bf p}$ stand, respectively, for the
coordinates of the extra dimensions and their corresponding momenta,
$J_{\mu}(x)$ is the Bessel function of the first kind and
$j=\left(p,\nu,\eta,\ell\right)$ is the set of quantum numbers. In this case,
the eigenvalues are found to be

$\Omega_{j}=p^{2}+\nu^{2}+\eta^{2}+m^{2}.$ (30)

Now we can use Eqs. (29) and (30) with Eq. (21). The sum for the set of
quantum numbers of the problem takes the form

$\sum_{j}=\int
d^{D-3}p\int_{-\infty}^{\infty}d\nu\int_{0}^{\infty}d\eta\sum_{\ell=-\infty}^{\infty}.$
(31)

Consequently, the local heat kernel defined in Eq. (21) is written as

$\begin{split}K_{D}(w,w^{\prime},\xi)=\frac{q}{(2\pi)^{D-1}}\sum_{j}e^{-\Omega_{j}\xi}e^{i\ell\Delta\phi+i\nu\Delta
z+i{\bf p}\cdot\Delta{\bf r}_{\parallel}}\eta J_{q|\ell+\alpha|}(\eta
r)J_{q|\ell+\alpha|}(\eta^{\prime}r),\end{split}$ (32)

where $\Delta\phi=\phi-\phi^{\prime}$, $\Delta z=z-z^{\prime}$ and $\Delta{\bf
r}_{\parallel}={\bf r}_{\parallel}-{\bf r}_{\parallel}^{\prime}$. The type of
calculation involved in the above expression has been developed in several
works, as for instance in Refs. MotaDispiration ; klecioQuasiPNanotubes ;
deFarias:2021qdg ; BezerradeMello:2014phm . In this sense, by using the method
adopted in these works we found

$\begin{split}K_{D}(w,w^{\prime},\xi)=&\frac{e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}\Biggl{\\{}\sum_{\ell}\frac{e^{i\alpha(2\pi\ell-q\Delta\phi)}}{e^{\frac{\Delta
R_{\ell}^{2}}{4\xi}}}-\frac{q}{2\pi i}\sum_{b=\pm
1}be^{ibq\alpha\pi}\int_{0}^{\infty}dy\>\frac{\cosh[qy(1-\alpha)]-\cosh(qy\alpha)e^{-iq(\Delta\phi+b\pi)}}{e^{\frac{\Delta
R_{y}^{2}}{4\xi}}\left[\cosh(qy)-\cos(q\Delta\phi+bq\pi)\right]}\Biggl{\\}},\end{split}$
(33)

where the index $\ell$ must obey the restriction BezerradeMello:2014phm

$-\frac{q}{2}+\frac{q\Delta\phi}{2\pi}\leq\ell\leq\frac{q}{2}+\frac{q\Delta\phi}{2\pi}.$
(34)

Moreover, we have also introduced the definition

$\displaystyle\Delta R_{\ell}^{2}$ $\displaystyle=$ $\displaystyle\Delta
z^{2}+\Delta r_{\parallel}^{2}+r^{2}+r^{\prime
2}-2rr^{\prime}\cos\left(\frac{2\pi\ell}{q}-\Delta\phi\right)$
$\displaystyle\Delta R_{y}^{2}$ $\displaystyle=$ $\displaystyle\Delta
z^{2}+\Delta r_{\parallel}^{2}+r^{2}+r^{\prime 2}+2rr^{\prime}\cosh(y).$ (35)

Upon taking the coincidence limit $w^{\prime}\rightarrow w$ in Eq. (33) we
have

$K_{D}(w,w,\xi)=K_{D}^{\text{E}}(w,w,\xi)+\frac{e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}\Biggl{\\{}2\sum_{\ell=1}^{[q/2]}\cos(2\pi\ell\alpha)e^{-\frac{(2rs_{\ell})^{2}}{4\xi}}-\frac{q}{\pi}\int_{0}^{\infty}dy\>M(y,\alpha,q)\>e^{-\frac{(2rs_{y})^{2}}{4\xi}}\Biggl{\\}},$
(36)

where $s_{\ell}=\sin(\pi\ell/2)$, $s_{y}=\cosh(y/2)$ and $[q/2]$ stands for
the integer part of $q/2$ and in the case it is an integer the corresponding
term in the sum should be taken with the coefficient 1/2. Note also that

$M(y,\alpha,q)=\frac{\cosh[qy(1-\alpha)]\sin(q\pi\alpha)+\cosh(qy\alpha)\sin[q\pi(1-\alpha)]}{\cosh(qy)-\cos(q\pi)}.$
(37)

The first term in the r.h.s. of Eq. (36) is the Euclidean local heat kernel
contribution that comes from the term $\ell=0$ in the sum present in Eq. (33).
This contribution is given by

$K_{D}^{\text{E}}(w,w,\xi)=\frac{1}{(4\pi\xi)^{\frac{D}{2}}}e^{-m^{2}\xi}.$
(38)

By using (24) and (36) the heat kernel is found to be

$K_{D}(\xi)=\frac{e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}C_{0}+\frac{\xi
e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}C_{1},$ (39)

where $C_{0}$ and $C_{1}$ are the heat kernel coefficients obtained by
comparison of the above expression with the heat kernel expansion in Eq. (8).
These nonzero coefficients are given by

$C_{0}=V_{D}$ (40)

and

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle h(q,\alpha)V_{D-2}$ (41)
$\displaystyle=$
$\displaystyle\Biggl{\\{}\sum_{\ell=1}^{[q/2]}\frac{\cos(2\pi\ell\alpha)}{s_{\ell}^{2}}-\frac{q}{2\pi}\int_{0}^{\infty}dy\>\frac{M(y,\alpha,q)}{s_{y}^{2}}\Biggl{\\}}\frac{2\pi}{q}V_{D-2},$

where $V_{D}$ and $V_{D-2}$ are infinite volumes. The $C_{0}$ coefficient is
associated with the Euclidean heat kernel in the first term in the r.h.s. of
Eq. (66). The latter provides a divergent contribution to the zeta function
$\zeta_{D}(s-1/2)$ that can be obtained from the term $n=0$ in (6). The
coefficient $C_{1}$ shows a dependence on the parameter $q$ and also on
$\alpha$. Thus, it exists as a consequence of the topology of the conical
spacetime as well as of the quasiperiodic condition. It is associated with
additional ultraviolet divergencies that should be subtracted as we shall see
below. We have checked that for $\alpha=0$ the coefficient $C_{1}$ above
recovers known results found in Refs. cognola:1993qg ; Fursaev:1993qk , as it
is shown in Eq. (49). In contrast, for $q=1$, we have only a contribution due
to the quasiperiodic condition that comes from the second term in the r.h.s.
of Eq. (41). Thus, here, we have generalized the results of Refs.
cognola:1993qg ; Fursaev:1993qk  by including the quasiperiodicity.

In Fig.1 we have plotted, in terms of $\alpha$, the function $h(q,\alpha)$ in
Eq. (41) which carries the nontrivial topology information of the heat kernel
coefficient $C_{1}$. From the plots, we can see that the minimum value for
$C_{1}$ is always when $\alpha=1/2$, the antiperiodic case. The most
interesting information in the plots, though, is that for $q>1$ there are some
values of $\alpha$ such that the coefficient $C_{1}$ vanishes. This eliminates
the divergent contribution in Eq. (43) due to the nontrivial topology of the
spacetime, leaving only the usual Euclidean (Minkowski) divergence related to
the coefficient $C_{0}$.

Figure 1: Function $h(q,\alpha)$ plotted in terms of $\alpha$, for several
values of $q$.

From (66) and (6) we have the generalized zeta function as given by

$\displaystyle\Gamma(s-1/2)\zeta_{D}(s-1/2)$ $\displaystyle=$
$\displaystyle\frac{m^{-2\left(s-\frac{D+1}{2}\right)}}{(4\pi)^{\frac{D}{2}}}\Gamma\left(s-\frac{D+1}{2}\right)C_{0}$
(42) $\displaystyle+$
$\displaystyle\frac{m^{-2\left(s-\frac{D-1}{2}\right)}}{(4\pi)^{\frac{D}{2}}}\Gamma\left(s-\frac{D-1}{2}\right)C_{1},$

from which the regularized vacuum energy can be obtained by using Eq. (14).
Note that the expression above depends on the spatial dimension, that is, if
$D$ is even we have a straightforward finite result, after taking the limit
$s\rightarrow 0$, while if $D$ is odd there will be divergent contributions,
along with finite ones. In this case, the divergent contributions to the
vacuum energy is written as

$2(4\pi)^{\frac{D+1}{2}}E_{0}^{\text{div}}(D,s)=-\frac{1}{s}\left[\frac{(-1)^{\frac{D+1}{2}}m^{D+1}}{\Gamma\left(\frac{D+3}{2}\right)}C_{0}+\frac{(-1)^{\frac{D-1}{2}}m^{D-1}}{\Gamma\left(\frac{D+1}{2}\right)}C_{1}\right].$
(43)

By comparing the expression above with Eq. (17), it is clear that

$\bar{C}_{\frac{N}{2}}=\left[\frac{(-1)^{\frac{D+1}{2}}m^{D+1}}{\Gamma\left(\frac{D+3}{2}\right)}C_{0}+\frac{(-1)^{\frac{D-1}{2}}m^{D-1}}{\Gamma\left(\frac{D+1}{2}\right)}C_{1}\right]$
(44)

and, as a consequence, the second term in the r.h.s. of Eq. (12) is completely
determined. Furthermore, still for odd $D$, upon taking the limit
$s\rightarrow 0$ of Eq. (42) there will be finite contributions that
logarithmically depend on the mass $m$, as well as on the mass with positive
powers. Thus, after subtracting the divergent contribution (43) according to
the prescription (13), from Eq. (12), we obtain for odd $D$ that

$\displaystyle 2(4\pi)^{\frac{D+1}{2}}E_{0}(D)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\frac{D+3}{2}}m^{D+1}}{\Gamma\left(\frac{D+3}{2}\right)}C_{0}\left[\frac{1}{\Gamma\left(\frac{D+3}{2}\right)}-\ln\left(\frac{4M^{2}}{m^{2}}\right)\right]$
(45) $\displaystyle+$
$\displaystyle\frac{(-1)^{\frac{D+1}{2}}m^{D-1}}{\Gamma\left(\frac{D+1}{2}\right)}C_{1}\left[\frac{1}{\Gamma\left(\frac{D+1}{2}\right)}-\ln\left(\frac{4M^{2}}{m^{2}}\right)\right],$

where the logarithmic mass terms from the finite contribution and the second
term in the r.h.s. of Eq. (12) have been put together. Note that the massless
limit of the above expression vanish.

Although now the vacuum energy in Eq. (45) is finite, it still depends on the
mass scale $M$ due to the ambiguity in the r.h.s. of Eq. (12). Moreover, the
expression above increases without bounds with the mass $m$, in contrast with
the fact that the vacuum energy is purely a quantum phenomenon and, hence,
should vanish in the limit of very large masses. In this sense, the additional
requirement encoded in Eq. (19) should be satisfied in order to obtain a
renormalized vacuum energy bordag2009advances ; Bordag:1998rf . This means
that, we must also subtract all the terms in the vacuum energy that increase
with the mass $m$. In fact, a brief inspection of Eq. (45) for odd $D$ and of
Eq. (42) for even $D$ shows that all terms present in theses expressions
increase with the mass and are to be subtracted. Of course, this leads to a
zero renormalized vacuum energy in the quasiperiodically identified conical
spacetime considered, that is,

$E^{\text{ren}}_{0}(D)=0,$ (46)

which trivially satisfies the requirement in Eq. (19). The result above for
the renormalized vacuum energy has not been pointed out in the literature so
far for the system analyzed in this section, including the periodic case
$\alpha=0$. Moreover, a similar result was obtained in Refs.
Khusnutdinov:1998tf , where the authors considered the scalar vacuum energy in
the cosmic string spacetime under Dirichlet boundary condition.

For $\alpha=0$ and integer values of $q$, we can make the following
replacement in the coefficient $C_{1}$ in Eq. (41):

$\sum_{\ell=1}^{[q/2]}\rightarrow\frac{1}{2}\sum_{\ell=1}^{q-1}$ (47)

with

$M(y,0,q)=\frac{\sin(q\pi)}{\cosh(qy)-\cos(q\pi)}.$ (48)

If $q$ is an integer, it is obvious that, by using (47), the heat kernel
coefficient $C_{1}$ for $\alpha=0$ reduces to

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{q}V_{D-2}\frac{1}{2}\sum_{\ell=1}^{q-1}\frac{1}{s_{\ell}^{2}}$
(49) $\displaystyle=$ $\displaystyle
V_{D-2}\frac{\pi}{3}\left(q-\frac{1}{q}\right),$

which is in essence the result found in Refs. cognola:1993qg ; Fursaev:1993qk
. The expression above is an analytic function of $q$ and, by analytic
continuation, should be valid for any $q$.

On the other hand, in the case $\alpha=1/2$, we can still make use of Eq. (47)
but only if $q$ is an even integer. Hence, by noting that $M(y,1/2,q)=0$ from
Eq. (37), we have

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{q}V_{D-2}\frac{1}{2}\sum_{\ell=1}^{q-1}\frac{(-1)^{\ell}}{s_{\ell}^{2}}$
(50) $\displaystyle=$ $\displaystyle-
V_{D-2}\frac{\pi}{6}\left(q+\frac{2}{q}\right).$

This result although has been obtained by considering even integer values of
$q$, it is by analytic continuation in fact valid for any value of $q$. In
particular, for $q=1$, we have $C_{1}=-\frac{\pi}{2}V_{D-2}$. Note that both
expressions in Eqs. (49) and (50) are in agreement with the plots shown in
Fig.1.

###  III.1 Temperature corrections

Let us now turn to the calculation of possible temperature corrections to the
renormalized vacuum energy. This is achieved by making use of the third term
in the r.h.s. of Eq. (11), i.e.,

$\begin{split}F_{T}=-\frac{1}{\sqrt{4\pi}}\sum_{n=1}^{\infty}\int_{0}^{\infty}\xi^{-\frac{3}{2}}e^{-\frac{(n\beta)^{2}}{4\xi}}\text{Tr}\left[e^{-\xi\hat{A}_{D}}\right]d\xi.\end{split}$
(51)

Thus, by taking into consideration Eq. (39) the above expression provides

$\begin{split}F_{T}=-\frac{2^{\frac{D+3}{2}}m^{D+1}C_{0}}{(4\pi)^{\frac{D+1}{2}}}\sum_{n=1}^{\infty}f_{\frac{D+1}{2}}(mn\beta)-\frac{2^{\frac{D+1}{2}}m^{D-1}C_{1}}{(4\pi)^{\frac{D+1}{2}}}\sum_{n=1}^{\infty}f_{\frac{D-1}{2}}(mn\beta),\end{split}$
(52)

for the massive scalar field case. The function $f_{\mu}(z)$ is defined in
terms of the Macdonald function $K_{\mu}(z)$ as

$\begin{split}f_{\mu}(z)=\frac{K_{\mu}(z)}{z^{\mu}}.\end{split}$ (53)

The massless limit is obtained by using the asymptotic expression of the
Macdonald function for small arguments, that is,
$K_{\mu}(z)\simeq\frac{\Gamma(\mu)}{2}\left(\frac{2}{z}\right)^{\mu}$
abramowitz ; gradshteyn2000table . This gives

$\begin{split}F_{T}=-\frac{2^{D+1}C_{0}}{(4\pi)^{\frac{D+1}{2}}\beta^{D+1}}\Gamma\left(\frac{D+1}{2}\right)\zeta_{\text{R}}(D+1)-\frac{2^{D-1}C_{1}}{(4\pi)^{\frac{D+1}{2}}\beta^{D-1}}\Gamma\left(\frac{D-1}{2}\right)\zeta_{\text{R}}(D-1),\end{split}$
(54)

where $\zeta_{\text{R}}(s)$ is the Riemann zeta function.

In particular, for $D=3$, we find

$\begin{split}F_{T}=-\frac{\pi^{2}}{90}C_{0}\frac{(k_{B}T)^{4}}{(\hbar
c)^{4}}-\frac{1}{24}C_{1}\frac{(k_{B}T)^{2}}{(\hbar c)},\end{split}$ (55)

for the massless scalar field case. Note that the first term in the r.h.s. is
the scalar black body radiation contribution and it is related to the heat
kernel coefficient $C_{0}$, in general associated with the Minkowski spacetime
Bezerra:2011nc ; PhysRevD.83.104042 . On the other hand, the second term is a
contribution due to the spacetime conical topology and quasiperiodicity. This
term, as we can see, it is related to the heat kernel coefficient $C_{1}$. As
we have learned, nonzero heat kernel coefficients indicate the existence of
divergences and, as such, both terms in Eqs. (52), (54) and (55) should be
subtracted. With this, we obtain a zero renormalized temperature correction
contribution, i.e.,

$F_{T}^{\text{ren}}=0.$ (56)

In the massless case, and for $D=3$, it has also been argued that terms of the
type

$\displaystyle\alpha_{0}\frac{(k_{B}T)^{4}}{(\hbar
c)^{3}},\qquad\qquad\alpha_{1}\frac{(k_{B}T)^{3}}{(\hbar
c)^{2}},\qquad\qquad\alpha_{2}\frac{(k_{B}T)^{2}}{\hbar c},$ (57)

must be subtracted in the renormalization process in order to obtain a correct
classical contribution in the high-temperature limit Bezerra:2011nc ;
PhysRevD.83.104042 ; Mota:2022qpf . In other words, the expressions above are
all of quantum nature and should not dominate in the high-temperature limit.
In Refs. Bezerra:2011nc ; PhysRevD.83.104042 ; Mota:2022qpf  it has also been
pointed out that the parameters $\alpha_{0}$, $\alpha_{1}$ and $\alpha_{2}$
are related to the heat kernel coefficients and we have indeed shown here that
this is in fact the case, at least for $\alpha_{0}$ and $\alpha_{2}$. Upon
comparing Eqs. (57) and (55) we conclude that

$\displaystyle\alpha_{0}=-\frac{\pi^{2}}{90}C_{0},\qquad\qquad\alpha_{2}=-\frac{1}{24}C_{1}.$
(58)

Of course, the discussion above is to say, in other words, that terms
containing any heat kernel coefficient must be subtracted.

In the next section we shall analyze, by using the same technique, the vacuum
energy and temperature corrections arising as a consequence of the nontrivial
topology of a cosmic dispiration, a combination of a conical defect, like a
cosmic string or disclination, with another defect known as screw dislocation.

##  IV VACUUM ENERGY AND TEMPERATURE CORRECTIONS IN A COSMIC DISPIRATION
SPACETIME

We want now to consider a spacetime geometry that is a combination of two
topological defects, that is, a conical and a screw dislocation, forming what
is known as cosmic dispiration spacetime Galtsov:1993ne ; DeLorenci:2003wv ;
DeLorenci:2002jv . The line element that describes an idealized
($D+1$)-dimensional spacetime that characterizes this combination, in
cylindrical coordinates, is given by MotaDispiration

$ds^{2}=dt^{2}-dr^{2}-r^{2}d\phi^{2}-(dz+\kappa
d\phi)^{2}-\sum_{i=4}^{D}(dx^{i})^{2},$ (59)

where $\kappa$ is a constant parameter associated with the screw dislocation
and that has dimension of length, $D>3$ and $(r,\phi,z,x^{4},...,x^{D})$ are
the generalized cylindrical coordinates taking values in the ranges $r\geq 0$,
$0\leq\phi\leq\phi_{0}=2\pi/q$ and $-\infty<(t,z,x^{i})<+\infty$, for
$i=4,...,D$. The conical defect carries all the properties described below Eq.
(26).

From the eigenvalue equation (20), and taking into consideration the line
element (59), the spatial part of the scalar field (4) can be obtained by
solving the equation

$\displaystyle\left[-\frac{1}{r}\frac{\partial}{\partial
r}\left(r\frac{\partial}{\partial
r}\right)-\frac{1}{r^{2}}\left(\frac{\partial}{\partial\phi}-\kappa\frac{\partial}{\partial
z}\right)^{2}-\frac{\partial^{2}}{\partial
z^{2}}-\sum_{i=4}^{D}\frac{\partial^{2}}{\partial
x^{i2}}+m^{2}\right]\varphi_{j}(w)=\Omega_{j}\varphi_{j}(w),$ (60)

where $j$ stands for the spatial quantum modes.

The complete set of normalized solutions for the eigenvalue equation (60) is
given by MotaDispiration

$\displaystyle\varphi_{j}\left(\mathbf{r}\right)=\left[\frac{q\eta}{(2\pi)^{D-1}}\right]^{\frac{1}{2}}J_{|q\ell-\kappa\nu|}(\eta
r)e^{i\ell q\phi+i\nu z+i{\bf p}\cdot{\bf r_{\parallel}}},$ (61)

where ${\bf r}_{\parallel}$ and ${\bf p}$ stand, respectively, for the
coordinates of the extra dimensions and their corresponding momenta,
$J_{\mu}(x)$ is the Bessel function of the first kind and
$j=\left(p,\nu,\eta,\ell\right)$ is the set of quantum numbers. Note that the
eigenvalues are still given by Eq. (30).

The local heat kernel can be obtained by using Eq. (21) and the complete set
of normalized solutions (61), i.e.,

$\begin{split}K_{D}(w,w^{\prime},\xi)=\frac{q}{(2\pi)^{D-1}}\sum_{j}e^{-\Omega_{j}\xi}e^{i(q\ell-\kappa\nu)\Delta\phi+i\nu\Delta
Z+i{\bf p}\cdot\Delta{\bf r}_{\parallel}}\eta J_{|q\ell-\kappa\nu|}(\eta
r)J_{|q\ell-\kappa\nu|}(\eta r^{\prime}),\end{split}$ (62)

where the sum in $j$ above is defined in Eq. (31) and we have used the
coordinate transformation $Z=z+\kappa\phi$ in Eq. (61) (see Ref.
MotaDispiration  for more details). The type of calculation involved in the
above expression has been developed in Ref. MotaDispiration  and, in this
sense, by following the same steps we found

$\begin{split}K_{D}(w,w^{\prime},\xi)=&\frac{e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}\Biggl{\\{}\sum_{\ell}e^{-\frac{\Delta\rho_{\ell}^{2}+\Delta
Z_{n}^{2}}{4\xi}}-\frac{q}{\pi^{2}}\sum_{k=-\infty}^{\infty}\int_{0}^{\infty}dy\;e^{-\frac{\Delta\rho_{y}^{2}+\Delta
Z_{k}^{2}}{4\xi}}M_{k,q}(\Delta\phi,y)\Biggl{\\}},\end{split}$ (63)

where the form of the function $M_{k,q}(\Delta\phi,y)$ has been obtained in
Ref. MotaDispiration  and, in the coincidence limit, that is, for
$\Delta\phi=0$, it is given by Eq. (67). The sum in $\ell$ must also obey the
restriction in Eq. (34) and

$\displaystyle\Delta\rho_{n}^{2}$ $\displaystyle=$ $\displaystyle\Delta
r_{\parallel}^{2}+r^{2}+r^{\prime
2}-2rr^{\prime}\cos\left(\frac{2\pi\ell}{q}-\Delta\phi\right),$
$\displaystyle\Delta\rho_{y}^{2}$ $\displaystyle=$ $\displaystyle\Delta
r_{\parallel}^{2}+r^{2}+r^{\prime 2}+2rr^{\prime}\cosh(y).$ (64)

Note that we have also defined $\bar{\kappa}=\frac{2\pi\kappa}{q}$ and $\Delta
Z_{\ell}=\Delta Z-\bar{\kappa}b$, where $b=\ell$ and $b=k$ in the first and
second terms in the r.h.s. of Eq. (63), respectively.

The quantity of interest here is the one in Eq. (63) taken in the coincidence
limit $w^{\prime}\rightarrow w$. This gives

$\displaystyle K_{D}(w,w,\xi)$ $\displaystyle=$ $\displaystyle
K_{D}^{\text{E}}(w,w,\xi)$ (65) $\displaystyle+$
$\displaystyle\frac{e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}\Biggl{\\{}2\sum_{\ell=1}^{[q/2]}e^{-\frac{\bar{\kappa}^{2}\ell^{2}}{4\xi}}e^{-\frac{r^{2}s_{\ell}^{2}}{\xi}}-\frac{q}{\pi^{2}}\sum_{k=-\infty}^{\infty}\int_{0}^{\infty}dy\;e^{-\frac{\bar{\kappa}^{2}k^{2}}{4\xi}}e^{-\frac{r^{2}s_{y}^{2}}{\xi}}M_{k,q}(0,y)\Biggl{\\}},$

with the Euclidean local heat kernel given by Eq. (38),
$s_{\ell}=\sin(\pi/2)$, $s_{y}=\cosh(y/2)$ and $[q/2]$ stands for the integer
part of $q/2$ and in the case it is an integer the corresponding term in the
sum should be taken with the coefficient 1/2 MotaDispiration .

Now, the heat kernel is obtained by integrating Eq. (65) in whole space,
providing the following expression:

$\displaystyle K_{D}(\xi)$ $\displaystyle=$
$\displaystyle\frac{e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}V_{D}$ (66)
$\displaystyle+$ $\displaystyle\frac{\xi
e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}\Biggl{\\{}\sum_{\ell=1}^{[q/2]}\frac{e^{-\frac{\bar{\kappa}^{2}\ell^{2}}{4\xi}}}{s_{\ell}^{2}}-\frac{q}{2\pi^{2}}\sum_{k=-\infty}^{\infty}\int_{0}^{\infty}dy\;\frac{e^{-\frac{\bar{\kappa}^{2}k^{2}}{4\xi}}}{s_{y}^{2}}M_{k,q}(0,y)\Biggl{\\}}\frac{2\pi}{q}V_{D-2},$

where

$\displaystyle
M_{k,q}(0,y)=\frac{\left(\frac{q}{2}+k\right)}{\left(\frac{q}{2}+k\right)^{2}+\left(\frac{y}{\phi_{0}}\right)^{2}}.$
(67)

By comparing Eq. (66) with the heat kernel expansion in Eq. (8) we can infer
that the only nonzero heat kernel coefficients are $C_{0}=V_{D}$ and $C_{1}$
given by Eq. (69). Then, we have

$\displaystyle K_{D}(\xi)$ $\displaystyle=$
$\displaystyle\frac{e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}C_{0}+\frac{\xi
e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}C_{1}$ (68) $\displaystyle+$
$\displaystyle\frac{2\pi}{q}V_{D-2}\frac{\xi
e^{-m^{2}\xi}}{(4\pi\xi)^{\frac{D}{2}}}\Biggl{\\{}\sum_{n=1}^{[q/2]}\frac{e^{-\frac{\bar{\kappa}^{2}n^{2}}{4\xi}}}{s_{n}^{2}}-\frac{q}{2\pi^{2}}\sideset{}{{}^{\prime}}{\sum}_{k=-\infty}^{\infty}\int_{0}^{\infty}dy\;\frac{e^{-\frac{\bar{\kappa}^{2}k^{2}}{4\xi}}}{s_{y}^{2}}M_{k,q}(0,y)\Biggl{\\}},$

where prime in the sum in $k$ indicates that $k=0$ is excluded. The latter
provides the term with the coefficient $C_{1}$ in the above expression, with

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle-\frac{2\pi
V_{D-2}}{q}\int_{0}^{\infty}\frac{dy}{\cosh^{2}(y/2)}\frac{1}{(\pi^{2}+y^{2})}$
(69) $\displaystyle\simeq$ $\displaystyle-\frac{\pi}{3q}V_{D-2}.$

Note that, numerically, the integral above is approximately given by $1/6$.
One interesting aspect about Eq. (66) is that in the case $\kappa=0$, the sum
in $k$ of the function $M_{k,q}(0,y)$ converges to MotaDispiration

$\displaystyle M_{q}(0,y)$ $\displaystyle=$
$\displaystyle\sum_{k=-\infty}^{\infty}M_{k,q}(0,y)$ (70) $\displaystyle=$
$\displaystyle\frac{\pi\sin(q\pi)}{\cosh(qy)-\cos(q\pi)}.$

This allows us to recover as a limiting case the heat kernel (36) for
$\alpha=0$, associated with a conical spacetime. Consequently, the heat kernel
coefficient $C_{1}$ becomes (49). However, as we can see from Eq. (69), if we
assume $\kappa\neq 0$ the structure of $C_{1}$ changes as a consequence of the
different topology of the spacetime we are considering in this section.
Surprisingly, the coefficient above does not depend on $\kappa$, although its
form is due to $\kappa\neq 0$.

The next step is to compute the vacuum energy by making use of the zeta
function (6) for $n=0$ and Eq. (14). In this case, the zeta function is found
to be

$\displaystyle\Gamma(s-1/2)\zeta_{D}(s-1/2)$ $\displaystyle=$
$\displaystyle\frac{m^{-2\left(s-\frac{D+1}{2}\right)}}{(4\pi)^{\frac{D}{2}}}\Gamma\left(s-\frac{D+1}{2}\right)C_{0}+\frac{m^{-2\left(s-\frac{D-1}{2}\right)}}{(4\pi)^{\frac{D}{2}}}\Gamma\left(s-\frac{D-1}{2}\right)C_{1}$
(71) $\displaystyle+$
$\displaystyle\frac{2\pi}{q}V_{D-2}\frac{2^{\frac{D+1-2s}{2}}m^{D-1-2s}}{(4\pi)^{\frac{D}{2}}}$
$\displaystyle\times$
$\displaystyle\Biggl{\\{}\sum_{\ell=1}^{[q/2]}\frac{f_{\frac{D-1-2s}{2}}(m\bar{\kappa}\ell)}{s_{\ell}^{2}}-\frac{q}{2\pi^{2}}\int_{0}^{\infty}\;\frac{dy}{s_{y}^{2}}\sideset{}{{}^{\prime}}{\sum}_{k=-\infty}^{\infty}f_{\frac{D-1-2s}{2}}(|m\bar{\kappa}k|)M_{k,q}(0,y)\Biggl{\\}},$

where the function $f_{\mu}(z)$ has been defined in Eq. (53) and the first two
terms in the r.h.s. resemble the form of the zeta function in Eq. (42), but in
this case with $C_{1}$ given by (69). These terms, as we have argued before,
have to be subtracted in order for the renomalized vacuum energy to satisfy
the requirement in Eq. (19). This also includes, in the massive case, the
subtraction of the term that contains the coefficient $\bar{C}_{\frac{N}{2}}$
in Eq. (12). In our case, this coefficient has also the same structure of Eq.
(44), but again with $C_{1}$ given by (69).

The renormalized vacuum energy per unit ‘volume’ $V_{D-2}$, from Eqs. (71),
(12), (13) and (14), is given by

$\displaystyle\mathcal{E}^{\text{ren}}_{0}(D)=-\frac{2\pi}{q}\frac{2^{\frac{D-1}{2}}m^{D-1}}{(4\pi)^{\frac{D+1}{2}}}\Biggl{\\{}\sum_{\ell=1}^{[q/2]}\frac{f_{\frac{D-1}{2}}(m\bar{\kappa}\ell)}{s_{\ell}^{2}}-\frac{q}{2\pi^{2}}\int_{0}^{\infty}\;\frac{dy}{s_{y}^{2}}\sideset{}{{}^{\prime}}{\sum}_{k=-\infty}^{\infty}f_{\frac{D-1}{2}}(|m\bar{\kappa}k|)M_{k,q}(0,y)\Biggl{\\}}.$
(72)

It is clear that the above expression satisfies the requirement (19) as a
consequence of the asymptotic limit of the Macdonald function for large
arguments, i.e., $K_{\mu}(z)\simeq\sqrt{\frac{\pi}{2z}}e^{-z}$ abramowitz ;
gradshteyn2000table . Hence, in the limit $m\bar{\kappa}\rightarrow\infty$,
the vacuum energy density above is exponentially suppressed. In Fig.2 we have
plotted, in terms of $m\kappa$, the vacuum energy density above for $D=3$. We
can see that it is always negative and goes to zero as $m\kappa$ increases, in
agreement with the condition (19). In contrast, when $m\kappa\rightarrow 0$,
the vacuum energy density (72) converges to values associated with the
massless case in Eq. (73) also for $D=3$. Note that for $q<1$ the vacuum
energy density increases and for $q>1$ it decreases. The case $q=1$ is the
vacuum energy density induced solely by the screw dislocation.

Figure 2: Plot of the vacuum energy density (72) for $D=3$, considering
different values of $q$.

The massless case of the vacuum energy density can be obtained by taking the
limit of Eq. (72) for small arguments of the Macdonald function
$K_{\mu}(z)\simeq\frac{\Gamma(\mu)}{2}\left(\frac{2}{z}\right)^{\mu}$
abramowitz ; gradshteyn2000table . This provides

$\displaystyle\mathcal{E}^{\text{ren}}_{0}(D)=-\left(\frac{q}{\pi}\right)^{D-2}\frac{\Gamma\left(\frac{D-1}{2}\right)}{(4\pi)^{\frac{D+1}{2}}}\frac{1}{\kappa^{D-1}}\Biggl{\\{}\sum_{\ell=1}^{[q/2]}\frac{1}{\ell^{D-1}s_{\ell}^{2}}-\frac{q}{2\pi^{2}}\int_{0}^{\infty}\;\frac{dy}{s_{y}^{2}}g(D,y,q)\Biggl{\\}},$
(73)

where

$\displaystyle
g(D,y,q)=\sideset{}{{}^{\prime}}{\sum}_{k=-\infty}^{\infty}\frac{1}{k^{D-1}}M_{k,q}(0,y).$
(74)

Note that the vacuum energy densities above are defined for $\kappa\neq 0$.
The case $\kappa=0$ needs to be consider from Eq. (66) and reproduces the
conical spacetime discussion considered in the previous section. Also, in the
massless scalar field case, $\bar{C}_{\frac{N}{2}}=0$ and we do not have the
ambiguity present in the free energy defined in Eq. (11), neither it is needed
a condition of the type (19). Moreover, it is evident that the vacuum energy
density in Eq. (73) vanishes in the limit $\kappa\rightarrow\infty$. In
contrast, for $q=1$, we obtain only the screw dislocation contribution given
by the second term in the r.h.s. of Eqs. (72) and (73).

###  IV.1 Temperature corrections

We turn our analysis now to the temperature corrections to the vacuum energy
densities (72) and (73). This is achieved by making use of Eqs. (51) and (68).
In the latter, the first two terms in the r.h.s. give contributions that
depend on the heat kernel coefficients $C_{0}$ and $C_{1}$ and must be
subtracted. For the massless scalar field case, these same two terms provide
contributions of the type presented in Eq. (54) and, in particular, for $D=3$,
of the type presented in Eq. (55) in which the term containing $C_{0}$ is the
scalar black body radiation contribution Bezerra:2011nc ; PhysRevD.83.104042 ;
Mota:2022qpf .

Figure 3: For $D=3$, plot of the free energy (75) per unit of $V_{1}$, in
terms of $\kappa k_{B}T$, for different values of $m\kappa$ and fixing four
values of $q$.

By subtracting the terms discussed above, we have the renormalized free energy
in the massive case as being

$\displaystyle F^{\text{ren}}_{T}=-\frac{2^{\frac{D+3}{2}}\pi
m^{D-1}}{q(4\pi)^{\frac{D+1}{2}}}V_{D-2}\sum_{n=1}^{\infty}\Biggl{\\{}\sum_{\ell=1}^{[q/2]}\frac{f_{\frac{D-1}{2}}\left(mR_{n,\ell}\right)}{s_{\ell}^{2}}-\frac{q}{2\pi^{2}}\int_{0}^{\infty}\;\frac{dy}{s_{y}^{2}}\sideset{}{{}^{\prime}}{\sum}_{k=-\infty}^{\infty}f_{\frac{D-1}{2}}\left(mR_{n,k}\right)M_{k,q}(0,y)\Biggl{\\}},$
(75)

where

$\displaystyle R_{n,b}=\sqrt{(n\beta)^{2}+(b\bar{\kappa})^{2}},$ (76)

with $b$ standing for $\ell$ in the first tem in the r.h.s. of (75) and for
$k$ in the second term.

Upon taking the limit of small arguments for the Macdonald function we can
obtain, from Eq. (75), the massless case. This is given by

$\displaystyle
F^{\text{ren}}_{T}=-\frac{2^{D}\pi\Gamma\left(\frac{D-1}{2}\right)}{q(4\pi)^{\frac{D+1}{2}}}V_{D-2}\sum_{n=1}^{\infty}\Biggl{\\{}\sum_{\ell=1}^{[q/2]}\frac{1}{s_{\ell}^{2}R^{D-1}_{n,\ell}}-\frac{q}{2\pi^{2}}\int_{0}^{\infty}\;\frac{dy}{s_{y}^{2}}\sideset{}{{}^{\prime}}{\sum}_{k=-\infty}^{\infty}\frac{M_{k,q}(0,y)}{R^{D-1}_{n,k}}\Biggl{\\}}.$
(77)

In particular, for $D=3$, we have

$\displaystyle F^{\text{ren}}_{T}=-\frac{1}{2\pi
q}V_{1}\sum_{n=1}^{\infty}\Biggl{\\{}\sum_{\ell=1}^{[q/2]}\frac{1}{s_{\ell}^{2}R^{2}_{n,\ell}}-\frac{q}{2\pi^{2}}\int_{0}^{\infty}\;\frac{dy}{s_{y}^{2}}\sideset{}{{}^{\prime}}{\sum}_{k=-\infty}^{\infty}\frac{M_{k,q}(0,y)}{R^{2}_{n,k}}\Biggl{\\}},$
(78)

where $V_{1}$ is in fact an infinite length in the $z$-direction. The
expressions above for the free energy is completely convergent and depends on
the conical parameter $q$, the screw dislocation parameter $\kappa$, the
temperature $T$ and of the spatial dimension $D$. In Fig.3 we have plotted,
for $D=3$, the free energy (75) per unit of $V_{1}$ in terms of $\kappa
k_{B}T$, assuming different values of $m\kappa$ and fixing four values of $q$.
In each case, the plots show that the free energy goes to zero as
$T\rightarrow 0$, as we should expect. In contrast, in the high-temperature
limit, the free energy reaches the classical limit, that is,
$F^{\text{ren}}_{T}\varpropto k_{B}T$, which is shown by the straight lines in
the plot for $q=2.5$. The plots also show that the free energy increases as
$m\kappa$ increases.

Below we shall analyze the interesting cases of the high- and low-temperature
limits of the massless free energy for $D=3$, given by Eq. (78).

####  IV.1.1 High-temperature limit

Let us first consider the high-temperature limit, $\pi\bar{\kappa}k_{B}T\gg
1$, of the expression in Eq. (78). For this, we shall write the sum in $n$ as

$\displaystyle\frac{1}{\beta^{2}}\sum_{n=1}^{\infty}\frac{1}{\left[n^{2}+\left(\frac{b\bar{\kappa}}{\beta}\right)^{2}\right]}=-\frac{1}{2b^{2}\bar{\kappa}^{2}}+\frac{\pi
k_{B}T}{2b\bar{\kappa}}\coth(b\bar{\kappa}\pi k_{B}T),$ (79)

where again $b=\ell$ for the first term in the r.h.s. of the free energy (78)
while $b=k$ for the second term. Then, after substituting (79) in (78), for
$\pi\bar{\kappa}k_{B}T\gg 1$, we have

$\displaystyle
F^{\text{ren}}_{T}\simeq-\frac{k_{B}T}{8\pi\kappa}V_{1}\Biggl{\\{}\sum_{\ell=1}^{[q/2]}\frac{1}{\ell
s_{\ell}^{2}}-\frac{q}{2\pi^{2}}\int_{0}^{\infty}\;\frac{dy}{s_{y}^{2}}\sideset{}{{}^{\prime}}{\sum}_{k=-\infty}^{\infty}\frac{M_{k,q}(0,y)}{k}\Biggl{\\}}.$
(80)

A strightfoward dimensional analysis shows that the expression above is the
classical limit of Eq. (78), as it should be. This corroborate with the fact
that terms of quantum nature of the type (55) must be subtracted in order to
obtain a renormalized free energy that has a dominating classical contribution
at high temperatures. This asymptotic limit is confirmed by the straight lines
in the plot of Fig.3 for $q=2.5$. In fact, this happens for any value of $q$.

####  IV.1.2 Low-temperature limit

We want now to obtain an expression for low temperatures by considering the
limit $\pi\bar{\kappa}k_{B}T\ll 1$ of Eq. (78). For this, it is convenient to
re-write the sum in $n$ for the fist term in the r.h.s. of (78) as follows

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{2}\beta^{2}}\frac{1}{\left[1+\left(\frac{\ell\bar{\kappa}}{n\beta}\right)^{2}\right]}=\frac{1}{\beta^{2}}\sum_{p=0}^{\infty}(-1)^{p}\left(\frac{\ell\bar{\kappa}}{\beta}\right)^{2p}\zeta_{\text{R}}(2p+2),$
(81)

where a binomial expansion has been adopted for
$\frac{\ell\bar{\kappa}}{n\beta}\ll 1$.

On the other hand, for the second term in the r.h.s. of Eq. (78) we can
consider

$\displaystyle\frac{1}{\bar{\kappa}^{2}}\sum_{n=1}^{\infty}\sideset{}{{}^{\prime}}{\sum}_{k=-\infty}^{\infty}\frac{M_{k,q}(0,y)}{\left[k^{2}+\left(\frac{n\beta}{\bar{\kappa}}\right)^{2}\right]}\simeq\left[-\frac{2\pi^{2}}{q(\pi^{2}+y^{2})}-\frac{\pi\sin(\pi
q)}{\cos(\pi
q)-\cosh(qy)}\right](k_{B}T)^{2}\zeta_{\text{R}}(2)+\pi^{2}\kappa(k_{B}T)^{3}\zeta_{\text{R}}(3),$
(82)

where we have first performed the sum in $k$. In the resulting expression we
have taken a series expansion for $\frac{n\beta}{\bar{\kappa}}\gg 1$ and in
the end performed the sum in $n$ for the first two leading terms of the
expansion, which gave rise to the Riemann zeta functions in (82).

Consequently, upon substituting Eqs. (81) and (82) in Eq. (78), up to third
order in $k_{B}T$, we have

$\displaystyle
F^{\text{ren}}_{T}\simeq-\left[h(q,0)+\frac{\pi}{3q}\right]\frac{\zeta_{\text{R}}(2)}{4\pi^{2}}V_{1}(k_{B}T)^{2}+\kappa\frac{\zeta_{\text{R}}(3)}{2\pi}V_{1}(k_{B}T)^{3},$
(83)

where $h(q,0)$ is a dimensionless function defined in Eq. (41). It is clear
that in the absence of the conical defect, i.e., $q=1$, there is still
contributions due to the screw dislocation, with $h(1,0)=0$. Again, a
straightforward dimensional analysis shows that the terms present in Eq. (83)
are of quantum nature and go to zero as $T\rightarrow 0$, which is clear in
the plots of Fig.3 . Hence, in this limit, only the vacuum energy density at
zero temperature (73) survives.

##  V Conclusions

We have investigated scalar quantum vacuum fluctuations effects on the vacuum
energy, temperature corrections and heat kernel coefficients arising from the
nontrivial topology of $(D+1)$-dimensional quasiperiodically identified
conical and cosmic dispiration spacetimes. In the quasiperiodically identified
conical spacetime we have found a zero renormalized vacuum energy, as well as
zero temperature corrections. The heat kernel coefficients obtained are the
ones associated with the Euclidean divergence and also with the nontrivial
topology of the spacetime. For the latter, as it is shown in Fig.1, we have
learned that for some values of the quasiperiodic parameter $\alpha$ we obtain
a zero value for the coefficient $C_{1}$, leaving only the coefficient $C_{0}$
related to the Euclidean divergence. Thus, we have generalized the results
obtained in Ref. cognola:1993qg  for the heat kernel coefficient $C_{1}$ and
recovered the author’s results when $\alpha=0$, as it has been shown in Eq.
(49). Also, the results that we have presented in Eqs. (46) and (56) are due
to the renormalization scheme discussed in Refs. bordag2009advances ;
Bordag:1998rf .

Regarding the cosmic dispiration spacetime we have obtained, also adopting the
renormalization scheme discussed in Refs. bordag2009advances ; Bordag:1998rf ,
a nonzero vacuum energy density, its corresponding temperature corrections and
the heat kernel coefficients $C_{0}$, due to the Euclidean divergence, and
$C_{1}$ due to the nontrivial topology divergence given by Eq. (69). The
induced vacuum energy density has been plotted in Fig.2, which shows that the
renormalized vacuum energy density increases for $q<1$ and decreases for
$q>1$. For $D=3$, in the massless scalar field case, we have presented
expressions for the low and high temperature regimes. In the latter, we have
shown that the free energy provides a pure classical contribution given by Eq.
(80), also confirmed by the straight lines in the plot of Fig.3 for $q=2.5$.
The plots of Fig.3 also show that the temperature corrections go to zero as
$T\rightarrow 0$, in agreement with the low-temperature expression in Eq.
(83).

###### Acknowledgements.

The author is partially supported by the National Council for Scientific and
Technological Development (CNPq) under grant No 311031/2020-0.

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###### Abstract

Let $C_{1}$ and $C_{2}$ be codes with code distance $d$. Codes $C_{1}$ and
$C_{2}$ are called weakly isometric, if there exists a mapping
$J:C_{1}\rightarrow C_{2}$, such that for any $x,y$ from $C_{1}$ the equality
$d(x,y)=d$ holds if and only if $d(J(x),J(y))=d$. Obviously two codes are
weakly isometric if and only if the minimal distance graphs of these codes are
isomorphic. In this paper we prove that Preparata codes of length $n\geq
2^{12}$ are weakly isometric if and only if these codes are equivalent. The
analogous result is obtained for punctured Preparata codes of length not less
than $2^{10}-1$.

On weak isometries of Preparata codes

Ivan Yu. Mogilnykh
Sobolev Institute of Mathematics, Novosibirsk, Russia

e-mail<EMAIL_ADDRESS>
Submitted to Problems of Information Transmission on 11th of January 2009.

##  1 Introduction

Let $E^{n}$ denote all binary vectors of length n. The Hamming distance
between two vectors from $E^{n}$ is the number of places where they differ.
The weight of vector $x\in E^{n}$ is the distance between this vector and the
all-zero vector $0^{n}$, and the support of $x$ is the set
$supp(x)=\\{i\in\\{1,\ldots,n\\}:x_{i}=1\\}$.

A set $C,$ $C\subset E^{n}$, is called a code with parameters $(n,M,d)$, if
$|C|=M$ and the minimal distance between two codewords from $C$ equals $d$. We
say that a code $C$ is reduced if it contains all-zero vector.

A collection of $k$-subsets (referred to as blocks) of a $n$-set such that any
$t$-subset occurs in $\lambda$ blocks precisely is called a
$(\lambda,n,k,t)$-design.

The minimal distance graph of a code $C$ is defined as the graph with all
codewords of $C$ as vertices, with two vertices being connected if and only if
the Hamming distance between corresponding codewords equals to the code
distance of the code $C$. Two codeword of $C$ are called $d$-adjacent if the
Hamming distance equals code distance $d$ of the code $C$.

Two codes $C_{1}$ and $C_{2}$ of length $n$ are called equivalent, if an
automorphism $F$ of $E^{n}$ exists such that $F(C_{1})=C_{2}$. A mapping
$I:C_{1}\rightarrow C_{2}$ of two codes $C_{1}$ and $C_{2}$ is called an
isometry between codes $C_{1}$ and $C_{2}$, if the equality
$d(x,y)=d(I(x),I(y))$ holds for all $x$ and $y$ from $C_{1}$. Then codes
$C_{1}$ and $C_{2}$ are called isometric. A mapping $J:C_{1}\rightarrow C_{2}$
is called a weak isometry of codes $C_{1}$ and $C_{2}$ (and codes $C_{1}$ and
$C_{2}$ weakly isometrical), if for any $x,y$ from $C_{1}$ the equality
$d(x,y)=d$ holds if and only if $d(J(x),J(y))=d$ where $d$ is the code
distance of code $C_{1}$. Obviously two codes are weakly isometric if and only
if the minimal distance graphs of these codes are isomorphic. In [2]
Avgustinovich established that any two weakly isometric 1-perfect codes are
equivalent. In [5] it was proved that this result also holds for extended
1-perfect codes.

In this paper any weak isometry of two Preparata codes (punctured Preparata
codes) is proved to be an isometry of these codes. Moreover, weakly isometric
Preparata codes (punctured Preparata codes) of length $n\geq 2^{12}$ (of
length $n\geq 2^{10}-1$ respectively) are proved to be equivalent.

This topic is closely related with problem of metrical rigidity of codes. A
code $C$ is called metrically rigid if any isometry $I:C\rightarrow E^{n}$ can
be extended to an isometry (automorphism) of the whole space $E^{n}$.
Obviously any two metrically rigid isometric codes are equivalent. In [4] it
was established that any reduced binary code of length $n$ containing
2-$(n,k,\lambda)$-design is metrically rigid for any $n\geq k^{4}$.

A maximal binary code of length $n=2^{m}$ for even $m$, $m\geq 4$ with code
distance 6 is called a Preparata code $\overline{P^{n}}$. Punctured Preparata
code is a code obtained from Preparata code by deleting one coordinate. By
$P^{n}$ we denote a punctured Preparata code of length $n$. Preparata codes
and punctured Preparata codes have some useful properties. All of them are
distance invariant [1], strongly distance invariant [3]. Also a punctured
Preparata code is contained in the unique 1-perfect code [6]. An arbitrary
punctured Preparata code is uniformly packed [1]. As a consequence of this
property, codewords of minimal weight of a Preparata code (punctured Preparata
code) form a design. The last property is crucial in proving the main result
of this paper.

##  2 Weak isomery of punctured Preparata codes

In this section we prove that any two punctured Preparata codes of length $n$
with isomorphic minimum distance graphs are isometric. Moreover, these codes
are equivalent for $n\geq 2^{10}-1$. First we give some preliminary
statements.

######  Lemma 1.

[1]. Let $P^{n}$ be an arbitrary reduced punctured Preparata code. Then
codewords of weight 5 of the code $P^{n}$ form 2-(n,5,(n-3)/3) design.

Taking into account a structure of the design from this lemma we obtain

######  Corollary 1.

Let $P^{n}$ be an arbitrary reduced punctured Preparata code and $r,s$ be
arbitrary elements of the set $\\{1,\ldots,n\\}$. Then there exists exactly
one coordinate $t$ such that all codewords of minimal weight of the code
$P^{n}$ with ones in coordinates $r$ and $s$ has zero in the coordinate $t$.

Let $C$ be a code with code distance $d$ and $x$ be an arbitrary codeword of
$C$ of weight $i$. Denote by $D_{i,j}(x)$ the set of all codewords of $C$ of
weight $j$ which are $d$-adjacent with vector $x$. In case when $C$ is a
punctured Preparata code we give some properties of the set $D_{i,j}(x)$ that
make the structure of minimal distance graph of this code more clear.

######  Lemma 2.

Let $x$ be an arbitrary codeword of a punctured Preparata code $P^{n}$. Then
any vector from $D_{i,i-1}(x)$ ($D_{i,i-3}(x)$ $D_{i,i-5}(x)$ respectively)
has exactly 3 (4 and 5 resp.) zero coordinates from $supp(x)$ and exactly 2 (1
and 0 resp.) nonzero coordinates from $\\{1,\ldots,n\\}\backslash supp(x)$.

Proof. Suppose a vector $y\in D_{i,i-k}(x)$ has $m_{k}$ zero coordinates from
$supp(x)$. Then it has exactly $m_{k}-k$ nonzero coordinates from the set
$\\{1,\ldots,n\\}\backslash supp(x)$. Since $d(x,y)=5$ we have
$m_{k}=(5+k)/2$, which implies the required property for $k=1,3,5$ .
$\blacktriangle$

Let $x$ be a codeword of weight $i$ from a $P^{n}$ ; $m,l$ be arbitrary
coordinates from $supp(x)$. We denote by $A_{m,l}(x)$ $(B_{m,l}(x)$ and
$C_{m,l}(x)$) the sets $D_{i,i-1}(x)$ ($D_{i,i-3}(x)$ and $D_{i,i-5}(x)$
respectively) with coordinates $m$ and $l$ equal to zero.

######  Lemma 3.

Let $x\in P^{n}$, $m,l\in supp(x)$ and $u,v$ be arbitrary codewords of $P^{n}$
with zeros in coordinates $m$ and $l$ that are at distance $5$ from $x$. Then
$u,v$ do not share zero coordinates in $supp(x)\setminus\\{m,l\\}$ and do not
share coordinates equal to one in the set $\\{1,\ldots,n\\}\backslash
supp(x)$.

Proof. Let us suppose the opposite. Then the vectors $x+u$ and $x+v$ of weight
five share at least three coordinates with ones in them and therefore

$d(u,v)=d(x+u,x+v)\leq 4$

holds. Since code distance of the code $P^{n}$ equals 5 we get a
contradiction. $\blacktriangle$

######  Lemma 4.

Let $x$ be an arbitrary codeword of weight $i$ from a punctured Preparata
code. Then the following inequalities hold:

$(i-3)C_{i}^{2}\leq
3|D_{i,i-1}(x)|+12|D_{i,i-3}(x)|+30|D_{i,i-5}(x)|\leq(i-2)C_{i}^{2}.$ (1)

Proof. Fix two coordinates $m$ and $l$ from $supp(x)$. By Lemma 2 an arbitrary
vector from $A_{m,l}(x)$ ($B_{m,l}$ and $C_{m,l}$) has exactly one zero
coordinate (two and three respectively) from $supp(x)\backslash\\{m,l\\}$.
Then taking into account Lemma 3 the number of coordinates from
$supp(x)\backslash\\{m,l\\}$ which are zero for vectors from $A_{m,l}$,
$B_{m,l}$ and $C_{m,l}$ equals $|A_{m,l}(x)|$, $2|B_{m,l}|$ and $3|C_{m,l}|$
respectively. Therefore the number of coordinates from the
$supp(x)\backslash\\{m,l\\}$ which are zero for vectors from $A_{m,l}\cup
B_{m,l}\cup C_{m,l}$ equals

$|A_{m,l}(x)|+2|B_{m,l}(x)|+3|C_{m,l}(x)|.$

Since $x$ is a vector of weight $i$ and $m,l\in supp(x)$, this number does not
exceed $i-2$. From the other hand by Corollary 1 there exists at most one
coordinate from $supp(x)\backslash\\{m,l\\}$ such that all vectors from
$A_{m,l}\cup B_{m,l}\cup C_{m,l}$ have one in it. Thus we have:

$i-3\leq|A_{m,l}(x)|+2|B_{m,l}(x)|+3|C_{m,l}(x)|\leq i-2.$

Summing these inequalities for all $m,l\in supp(x)$ we obtain

$(i-3)C_{i}^{2}\leq\sum_{m,l\in supp(x)}|A_{m,l}(x)|+2\sum_{m,l\in
supp(x)}|B_{m,l}(x)|+3\sum_{m,l\in supp(x)}|C_{m,l}(x)|\leq(i-2)C_{i}^{2}$ (2)

As an arbitrary vector from $D_{i,i-1}(x)$ has exactly 3 zero coordinates from
$supp(x)$, any such vector is counted $C_{3}^{2}$ times in the sum
$\sum_{m,l\in supp(x)}|A_{m,l}(x)|$. Then

$\sum_{m,l\in supp(x)}|A_{m,l}(x)|=C_{3}^{2}|D_{i,i-1}(x)|.$

Analogously we get:

$\sum_{m,l\in supp(x)}|B_{m,l}(x)|=C_{4}^{2}|D_{i,i-3}(x)|,$ $\sum_{m,l\in
supp(x)}|C_{m,l}(x)|=C_{5}^{2}|D_{i,i-5}(x)|.$

So from (2) we get (1). $\blacktriangle$

Now we prove the main result using Lemmas 2 and 4.

######  Theorem 1.

The minimal distance graphs of two punctured Preparata codes are isomorphic if
and only if these codes are isometric.

Proof. It is obvious that if two punctured Preparata codes are isometric then
they are weakly isometric.

Let $J:P^{n}_{1}\rightarrow P^{n}_{2}$ be a weak isometry of two punctured
Preparata codes $P_{1}^{n}$ and $P_{2}^{n}$ of length $n$. Without loss of
generality suppose that $0^{n}\in P^{n}_{1}$, $J(0^{n})=0^{n}$. We now show
that mapping $J$ is an isometry. For proving this it is sufficient to show
that $wt(J(x))=wt(x)$ for all $x\in P^{n}_{1}$.

Suppose $z$ is a codeword of the code $P^{n}_{1}$, such that $wt(J(z))\neq
wt(z)=i$ holds and the mapping $J$ preserves weight of all codewords of weight
smaller that $i$. The vector $z$ satisfying these conditions we call critical
Since $J(0^{n})=0^{n}$ and the mapping $J$ preserves the distance between all
codewords at distance 5, we have $i\geq 6$. We prove that there is no critical
codewords in $P^{n}_{1}$. From $0^{n}\in P^{n}_{1}$ holds that the weak
isometry $J$ preserves a parity of weight of a vector and therefore $wt(J(z))$
equals either $i+2$ or $i+4$.

Suppose $wt(J(z))=i+2$. Since $J$ is a weak isometry and $z$ is a critical
vector we have the following: $|D_{i+2,i-1}(J(z))|=|D_{i,i-1}(z)|$,
$|D_{i+2,i-3}(J(z))|=|D_{i,i-3}(z)|$, $|D_{i,i-5}(z)|=0$. Taking into account
these equalities, from the inequalities of Lemma 4 for vectors $z$ and $J(z)$
we get

$(i-3)C_{i}^{2}\leq 3|D_{i,i-1}(z)|+12|D_{i,i-3}(z)|,$ (3)

$3|D_{i+2,i+1}(J(z))|+12|D_{i,i-1}(z)|+30|D_{i,i-3}(z)|\leq iC_{i+2}^{2}.$ (4)

Multiplying both sides of inequality (3) by $-4$ we get

$-12|D_{i,i-1}(z)|-48|D_{i,i-3}(z)|\leq-4(i-3)C_{i}^{2}.$

Summing this inequality with (4) we get

$3|D_{i+2,i+1(J(z))}|-18|D_{i,i-3}(z)|\leq iC_{i+2}^{2}-4(i-3)C_{i}^{2},$

and therefore

$|D_{i,i-3}(z)|\geq\frac{4(i-3)C_{i}^{2}-iC_{i+2}^{2}}{18}.$ (5)

In particular, from the inequality (5) we have $|D_{i,i-3}(z)|\geq 1$ for
$i=6$ and $i=7$. But there is no codewords of weight 3 and 4 in the $P_{1}$
since $P_{1}$ is reduced code with code distance 5. Therefore $i\geq 8$. From
Lemma 4 we have the following

$|D_{i,i-3}(z)|\leq\frac{(i-2)C_{i}^{2}}{12}.$ (6)

But for $i\geq 10$ the inequality
$3(i-2)C_{i}^{2}<2(4(i-3)C_{i}^{2}-iC_{i+2}^{2})$ holds. This contradicts with
(5) and (6).

So it is only remains to prove that there are no codewords of weight 8 and 9,
such that their images under the mapping $J$ have weights 10 and 11
respectively. Obviously the Hamming distance between any two vectors from
$D_{i+2,i-3}(J(z))$ is not less than 6\. By Lemma 2 all ones coordinates of
each vector from $D_{i+2,i-3}(J(z))$ are in set $supp(J(z))$. So
$|D_{i+2,i-3}(J(z))|$ does not exceed the cardinality of maximal constant
weight code of length $i+2$, with all code words of weight being equal $i-3$
and being at distance not less than 6 pairwise. For $i=8$ and $i=9$ the
cardinalities of such codes equal to 6 and 11 respectively, but from (5) we
have

$|D_{10,5}(J(z))|=|D_{8,5}(z)|\geq 12,|D_{11,6}(J(z))|=|D_{9,6}(z)|\geq 21,$

a contradiction. Therefore there is no critical vectors $z$ in $P_{1}^{n}$,
$wt(z)=i$, such that $wt(J(z))=i+2$.

Suppose $wt(J(z))=i+4$. In this case we have
$|D_{i,i-3}(z)|=|D_{i,i-5}(z)|=0$, $|D_{i+4,i-1}(J(z))|=|D_{i,i-1}(z)|$. Using
these equalities we have from the inequalities of Lemma 4 for the vectors $z$
and $J(z)$ the following:

$(i-3)C_{i}^{2}\leq 3|D_{i,i-1}(z)|,$

$30|D_{i,i-1}(z)|\leq(i+2)C_{i+4}^{2}.$

From these last two inequalities we obtain

$\frac{(i-3)C_{i}^{2}}{3}\leq\frac{(i+2)C_{i+4}^{2}}{30},$

and therefore

$10i(i-1)(i-3)\leq(i+4)(i+3)(i+2)$

that implies

$10i(i-1)(i-3)\leq 2i(i+3)(i+2).$

Since last inequality does not hold for $i\geq 6$ there is no critical vectors
in $P_{1}^{n}$ and therefore the mapping $J$ is an isometry. $\blacktriangle$

In [4] the following theorem was proved

######  Theorem 2.

Any reduced code of length $n$, that contains a $2-(n,k,\lambda)$-design is
metrically rigid for $n\geq k^{4}$.

Taking into account that by Lemma 1 any punctured reduced Preparata code
contains 2-$(n,5,(n-3)/4)$-design applying Theorems 1 and 2 we get

######  Corollary 2.

Let $n\geq 2^{10}-1$. Two punctured Preparata codes of length $n$ are
equivalent if and only if the minimal distance graphs of these codes are
isomorphic.

##  3 Weak isometry of Preparata codes

Using the analogous considerations, Theorems 1,2 and Corollary 2 can easily be
extended for extended Preparata codes. We now give the analogues of Lemmas 1-4
omitting their proofs.

######  Lemma 5.

([1]) Let $\overline{P^{n}}$ be an arbitrary reduced Preparata code. Then
codewords of weight 6 of code $\overline{P^{n}}$ form 3-(n,6,(n-4)/3)-design.

######  Lemma 6.

Let $x$ be an arbitrary codeword of a Preparata code $\overline{P^{n}}$,
$wt(x)=i$. Then any vector from $D_{i,i-2}(x)$ ($D_{i,i-4}(x)$ $D_{i,i-6}(x)$
respectively) has exactly 4 (5 and 6 respectively) zero coordinates from
$supp(x)$ and exactly 2 (1 and 0 respectively) nonzero coordinates from
$\\{1,\ldots,n\\}\backslash supp(x)$.

######  Lemma 7.

Let $x\in\overline{P^{n}}$, $m,l,k\in supp(x)$, and $u,v$ be arbitrary
codewords of $\overline{P^{n}}$ at distance $6$ from the vector $x$ with zero
coordinates in positions $m$, $l$, $k$. Then there is no coordinate from
$supp(x)\setminus\\{m,l,k\\}$ such that $u,v$ have zeros in it and there is no
coordinate from $\\{1,\ldots,n\\}\backslash supp(x)$ such that $u,v$ have ones
in it.

######  Lemma 8.

Let $x$ be an arbitrary codeword of weight $i$ from a Preparata code. Then the
following inequalities hold:

$C_{i}^{3}(i-4)\leq 4|D_{i,i-2}(x)|+20|D_{i,i-4}(x)|+60|D_{i,i-6}(x)|\leq
C_{i}^{3}(i-3).$ (7)

Using Lemmas 5-8 and the same arguments as in the proof of Theorem 1 the
following theorem it is not difficult to prove

######  Theorem 3.

The minimal distance graphs of two Preparata codes are isomorphic if and only
if the codes are isometric.

From this theorem, Lemma 5 and Theorem 2 we get

######  Corollary 3.

Let $n\geq 2^{12}$. Two Preparata codes of length $n$ are equivalent if and
only if the minimal distance graphs of these codes are isomorphic.

The Author is deepfuly grateful to Faina Ivanovna Soloveva for introducing
into the topic, problem statement and all around support of this work.

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