# Fracton-elasticity duality on curved manifolds

Lazaros Tsaloukidis<EMAIL_ADDRESS>Max Planck Institute for the Physics
of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany  Würzburg-
Dresden Cluster of Excellence ct.qmat, 01187 Dresden, Germany  José J.
Fernández-Melgarejo<EMAIL_ADDRESS>Departamento de Electromagnetismo y
Electrónica, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain
Javier Molina-Vilaplana<EMAIL_ADDRESS>Departamento de Automática.
Universidad Politécnica de Cartagena, Calle Dr. Fleming, S/N, 30202 Cartagena,
Spain  Piotr Surówka<EMAIL_ADDRESS>Institute of Theoretical
Physics, Wrocław University of Science and Technology, 50-370, Wrocław, Poland

###### Abstract

The mechanical properties of crystals on curved substrates mix elastic,
geometric and topological degrees of freedom. In order to elucidate the
properties of such crystals we formulate the low-energy effective action that
combines metric degrees of freedom with displacement fields and defects. We
propose dualities for elasticity coupled to curved geometry formulated in
terms of tensor gauge theories. We show that the metric degrees of freedom,
evolving akin to linearized gravity are mapped to tensors with three indices.
When coupled to crystals these degrees of freedom become gapped and, in the
presence of dislocations and disclinations, multivalued. The elastic degrees
of freedom remain gapless and mapped to symmetric gauge fields with two
indices. In analogy with elasticity on flat space formulation we assume that
the trace of the total quadrupole moment is conserved. In the dual
formulation, topological defects, which act as sources for the gauge fields,
are fractons or excitations with restricted mobility. This leads to
generalized glide constraints that restrict both displacement and
gravitational defects.

##  I Introduction

Two-dimensional crystals on curved substrates emerge in a variety of systems
including colloids [1], metamaterials [2] and biomatter [3]. In order to
elucidate mechanical properties of such systems one requires a consistent
framework of crystallography on curved manifolds. The crystallization of atoms
on non-flat topographies naturally leads to the emergence of topological
defects. This follows from the fact that the equilibrium structure of crystals
is encoded in the arrangement of disclinations and dislocations. The goal of
this paper is to formulate the low-energy effective description of curved
crystalline fields and investigate the dynamics of defects. In order to
achieve this goal we propose a duality between elasticity and gauge theories
that incorporates both the metric degrees of freedom and elastic
displacements.

The flat-space theory of elasticity serves as a description of deformed solid
bodies. In the traditional formulation the deformation is parameterized by a
displacement field that captures the departure from the equilibrium
configuration. The microscopic intuition behind the displacement field is
given in terms of fluctuations of atoms around the equilibrium arrangement of
their position on the lattice. The crystalline lattice results from a
spontaneous symmetry breaking of translations in space. During crystal growth
atoms can migrate from their lattice positions and a mismatch in crystal
planes can occur. This leads to the formation of defects. Vacancies in the
lattice structure are point defects that do not cause additional stresses. On
the other hand line or plane irregularities within a crystal create non-local
stresses. We call such quantities topological. These stresses cannot be
removed without cutting the crystal and global rearrangements of the lattice
structure. In two dimensions topological defects are represented as points or
singularities of the elastic fields. In most materials the defects are created
because of impurities contained in the sample. However, in two dimensions,
crystallization on a curved substrate provides another mechanism, that
necessarily leads to the formation of defects (See Fig. (1)). This conforms to
the usual lore that topological defects are sources of curvature and torsion.
Therefore a consistent description of a crystal on a curved substrate should
account for geometry, defects and fluctuations of atoms forming the crystal
[4].

Our construction extends modern developments in the theory of elasticity,
namely fracton-elasticity dualities used to reformulate elasticity as gauge
theories [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and crystal gravity
employed to marry the dynamics of metric degrees of freedom with spontaneous
symmetry breaking of crystals [17]. We introduce topological defects as
singularities that lead to multivalued fields both in elasticity and gravity.
This extends earlier developments in this direction [18]. Our main results are
the following: 1) elasticity on curved backgrounds admits the dual formulation
in terms of tensor gauge fields, 2) disclinations and dislocations in
symmetric elasticity on curved manifolds are fractons and partial fractons
respectively, exhibiting mobility restrictions when certain constraints
regarding total dipole and quadrupole moment conservation are imposed, in
analogy with flat space. As a byproduct of our analysis we show that
linearized gravity can be formulated as a gauge theory.

Our theory opens up many directions in elasticity in curved substrates. First
of all it gives a consistent low-energy field theory approach to the problem.
This is in contrast to previous developments that are rather phenomenological
and do not fall in the effective field theory paradigm [19, 20, 21]. In
addition our work avoids many difficulties with the physical interpretation,
in terms of spontaneous symmetry breaking, of the proposals in the
mathematical literature [22] or in general relativity [23, 24], which aim at a
direct covariant formulation of elasticity. We can consistently study the
dynamics of elastic fields and defects on curved backgrounds and provide a
clear physical interpretation of the resulting dynamics. The formalism also
allows for extensions to account for new degrees of freedom both in passive
and active solids. It complements developments in soft matter physics, that
focus on the geometric properties of crystal shells (see e.g. [25, 26, 27]).

Figure 1: a) As an example of a crystal on a curved manifold we start with a
sphere. b) We allow atoms to break the rotation symmetry spontaneously by
creating a crystal on its surface. c) Curvature is visible by the necessity of
including pentagons in a hexagonal lattice, which unfolded on a plane shows
angle deficiency.

##  II Dual formulation of linearized metric fluctuations

As a first step before considering crystalline structures with curvature, we
start with a membrane described by a linearized theory of metric fluctuations
in $2+1$ dimensions

$S=\frac{1}{2}\int
d^{2}xdt\left[(\partial_{t}h_{ij})^{2}-\kappa(\partial_{k}h_{ij})^{2}\right],$
(1)

where $h_{ij}$ refers to a small metric fluctuation around a fixed background
and $\kappa$ is a constant parameter. This is the Fierz-Pauli action in the
helical projection when the time components are omitted. In general $\kappa$
could be a constant tensor, however, this renders the expressions more
cumbersome. Since it does not affect the dualization procedure, we only choose
to work with the most asymmetric term comprising the theory. Notice that
similar models have been considered in the context of polymerized membranes
[28] (for a review see [29]) or linearized gravity [30] (for a review see
[31]).

In our case, Eq. (1) can be viewed as a toy model to set the stage for
elasticity on a curved background and illustrate the steps that are necessary
to construct the duality for a theory that contains a metric. We ask the
question if this theory has a dual formulation as a gauge theory. In fact such
a question in the context of gravity has a long history with different
approaches that range from rewriting gravity in terms of gauge variables [32]
to developments based on gauging the algebraic structure of gravity [33, 34,
35, 36, 37]. In this work, having in mind applications to elasticity, we
perform a Hubbard-Stratonovich transformation on Eq. (1), which is at the core
of elasticity/gauge theory dualities. For given sets of conjugate tensor
fields $\phi$ and $\psi$ we can compactly write it as

$\exp\left[\frac{1}{2}{\psi M\psi}\right]=\frac{1}{\mathcal{N}}\int
D\phi\exp\left[{\frac{-1}{2}\phi M^{-1}\phi+\psi\phi}\right],\,$ (2)

where $\mathcal{N}$ is a normalization factor. Here we take the conjugate
fields to be $L_{ij}=\partial_{t}h_{ij}$, $K_{kij}=\kappa\partial_{k}h_{ij}$.
It is important to note that no gauge choice is imposed before the
dualization, which is standard in the previous approaches to crystallography
on curved substrates [26]. Picking a gauge may introduce higher-order terms in
the action, which obstructs the Hubbard-Stratonovich transformation. So far we
have assumed that the metric is smooth. We can also introduce singularities by
the following splitting $h_{ij}=\bar{h}_{ij}+h^{(s)}_{ij}$, with $(s)$
denoting the singular part. The dual action is

$S_{\text{HS}}=\frac{1}{2}\int
d^{2}xdt\left[\frac{1}{\kappa}(K_{kij})^{2}-(L_{ij})^{2}-\right.\\\
-2K_{kij}\partial_{k}h^{(s)}_{ij}+2L_{ij}\partial_{t}h^{(s)}_{ij}\Bigl{]}.$
(3)

In deriving Eq. (3), we have integrated out the smooth part, leading to the
conservation law

$\partial_{t}L_{ij}-\partial_{k}K_{kij}=0.$ (4)

This equation is just a dual representation of the wave equation for the
metric distortions. Thus it captures the dynamics of the substrate. The
defects of the theory are points in space where partial derivatives do not
commute. Their density form, as well as the current associated with them, are
given by:

$\displaystyle\tilde{\rho}_{ij}=$ $\displaystyle\
\epsilon_{ik}\epsilon_{jl}\epsilon_{mn}\partial_{n}\partial_{m}h^{(s)}_{kl},$
(5a) $\displaystyle\tilde{J}_{mij}=$ $\displaystyle\
\epsilon_{ik}\epsilon_{jl}(\partial_{m}\partial_{t}-\partial_{t}\partial_{m})h_{kl}^{(s)}.$
(5b)

The defect density is described by a tensor density. Eq. (4) is a continuity
equation that, as it will be shown below, can be resolved in terms of gauge
fields $\tilde{{A}}_{lij}$ and $\tilde{\phi}_{ij}$. We present the details in
a combined theory, in which the coupling of $h_{ij}$ to elasticity has been
introduced.

##  III Elasticity on a curved background

Crystallization occurs as a spontaneous symmetry breaking of the translation
symmetry. Such a process does not require the underlying manifold to be flat.
However, if a lattice forms on a curved substrate the curvature is encoded in
topological defects that cannot be avoided. It is important to note that
defects are singularities in the macroscopic fields, which in this case
correspond to displacements or lattice vibrations and the metric. As a result,
contrary to flat lattices, the equilibrium configuration contains defects. If
new defects are created in the equilibrium configuration, they affect both
metric and displacement degrees of freedom. Before we study the dynamics of
defects we focus on a region of space without defects. Such a configuration
can be understood similarly to the spontaneous symmetry breaking present in
superconductors. Here, the strain field, plays the role of the order parameter
for the metric [17]. The low-energy theory that describes both the rigidity of
the solid plate and the linearized fluctuations of the metric is given by

$S=\frac{1}{2}\int
d^{2}xdt[\rho_{d}(\partial_{t}u_{i})^{2}+(\partial_{t}h_{ij})^{2}-\\\
-(u_{ij}+h_{ij})C_{ijkl}(u_{kl}+h_{kl})-\kappa(\partial_{k}h_{ij})^{2}].$ (6)

In the above, $\rho_{d}$ is the material density, $u_{i}$ is the displacement
field, $u_{ij}=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i})$ the strain
field and $C_{ijkl}$ the usual 4-rank elasticity tensor whose exact form is
dictated by the symmetries present in the crystal. For simplicity we expand
around flat space, however, the procedure can be generalized as an expansion
around arbitrarily curved substrates. An important aspect of this theory is
that the metric degrees of freedom, or ”gravitons” become gapped, while the
fluctuations of atoms remain gapless. Spontaneous symmetry breaking of
elasticity requires massive Stückelberg fields as already anticipated in [13,
38]. Here the tensor of elastic coefficients plays the role of mass
parameters. We note that a generic displacement fluctuation does not belong to
the original manifold and has to be accompanied by a fluctuation of geometry.

We now extend the theory to include defects in the picture. In order to do
this consistently, we need to account both for singularities of the
displacement field and the metric. An attempt in this direction was already
done in [18], where a purely geometric theory was constructed. However, we
find it convenient to keep both the displacement fields and the metric. Thus
we proceed by splitting down the displacement field into its singular and
smooth phonon pieces. Analogously the metric is also divided into singular and
smooth parts. The reason to keep singularities in the two fields can be
understood as follows. Imagine that we have singular configurations of the
displacement field, then

$u_{ij}+h_{ij}\rightarrow
u_{ij}+h_{ij}+\frac{1}{2}(\partial_{i}u_{j}^{(s)}+\partial_{j}u_{i}^{(s)}).$
(7)

However now, we can also view this as a singular contribution to the metric

$h_{ij}^{(s)}=\frac{1}{2}(\partial_{i}u_{j}^{(s)}+\partial_{j}u_{i}^{(s)}).$
(8)

Therefore, without imposing any additional conditions, we can have
singularities contributing both to the strain and the metric.

We dualize the resulting effective action by means of the Hubbard-Stratonovich
transformation to obtain

$\displaystyle S_{\text{HS}}=\frac{1}{2}\int
d^{2}xdt\left[T_{ij}C_{ijkl}^{-1}T_{kl}-\frac{1}{\rho_{d}}(\pi_{i})^{2}+\frac{1}{\kappa}(K_{kij})^{2}\right.$
$\displaystyle\left.-(L_{ij})^{2}+2\pi_{i}\partial_{t}u_{i}^{(s)}+2L_{ij}\partial_{t}h_{ij}^{(s)}-2w_{ij}^{(s)}T_{ij}\right.$
(9) $\displaystyle\left.-2\partial_{k}h_{ij}^{(s)}K_{kij}\right],$

where, $T_{ij}$ is the generalized stress-tensor, $\pi_{i}$ is the momentum
density and $K_{kij}$ and $L_{ij}$ are their linearized gravity counterparts
and $w_{ij}^{(s)}=u_{ij}^{(s)}+h_{ij}^{(s)}$. The dual formulation of this
theory parallels Cosserat elasticity [13], where gapless displacements are
supplemented with gapped bond angles. By performing integration by parts on
the last four terms of Eq. (III), we obtain the two dynamical laws governing
the system

$\displaystyle\partial_{t}\pi_{i}-\partial_{j}T_{ij}=$ $\displaystyle\ 0,$
(10) $\displaystyle\partial_{t}L_{ij}-\partial_{k}K_{kij}=-$ $\displaystyle\
T_{ij}.$ (11)

Equation (10) above is the well-known Hooke’s law with the stress tensor
containing contributions from the bending of the plate besides stretching and
shearing. In symmetric elasticity it is resolved by introducing a first set of
gauge fields ($\phi$, $A_{ij}$)

$\pi_{i}=-\epsilon_{ij}\epsilon_{kl}\partial_{k}A_{lj},\quad
T_{ij}=-\epsilon_{ik}\epsilon_{jl}(\partial_{t}A_{kl}+\partial_{k}\partial_{l}\phi).$
(12)

Now, as it is commonly done in the context of elasticity/gauge duality, it is
convenient to define the generalized electric and magnetic fields as

$B_{i}=\epsilon_{ij}\pi_{j},\qquad E_{ij}=\epsilon_{ik}\epsilon_{jl}T_{kl}.$
(13)

In this way Eq. (10) takes the Faraday equation form. The second equation at
hand, Eq. (11), captures the dynamics of defects from the metric fluctuations
with the stress-tensor acting as a source. Here we introduce a second set of
gauge fields $(\tilde{\phi}_{ij},\ \tilde{A}_{ijk})$, first by plugging the
$T_{ij}$ from Eq. (12), to get:

$\displaystyle\
\tilde{B}_{ij}=L_{ij}-\epsilon_{ik}\epsilon_{jl}A_{kl}=\epsilon_{ik}\epsilon_{jl}\partial_{m}\tilde{A}_{mkl},$
(14) $\displaystyle\tilde{E}_{mij}=$ $\displaystyle\
K_{mij}+\epsilon_{im}\epsilon_{jl}\partial_{l}\phi=\epsilon_{ik}\epsilon_{jl}(\partial_{t}\tilde{A}_{mkl}+\epsilon_{mn}\partial_{n}\tilde{\phi}_{kl}).$
(15)

Note that while the rank-2 magnetic field tensor $\tilde{B}_{ij}$ is symmetric
under the permutation ($i\leftrightarrow j$), the rank-3 electric field tensor
$\tilde{E}_{kij}$ is not. In total the theory includes four gauge fields,
closely similar to the Cosserat theory of elasticity.

Eq. (11) means that $L_{ij}$ is not a conserved quantity, however, both
$L_{ij}$ and $K_{kij}$ are observables of the theory and they should be
invariant under a specific class of gauge transformations that leaves both of
these quantities unchanged, i.e. $\delta L_{ij}=0$ and $\delta K_{kij}=0$.
These gauge transformations can be constructed explicitly

$\displaystyle\hskip 14.22636pt\delta\phi=\partial_{t}f,\hskip
42.67912pt\delta A_{ij}=-\partial_{i}\partial_{j}f,$ (16)
$\displaystyle\delta\tilde{\phi}_{ij}$
$\displaystyle=-\partial_{t}\tilde{f}_{ij},\,\qquad\delta\tilde{A}_{mij}=\delta_{m\left(i\right.}\partial_{\left.j\right)}f+\epsilon_{mn}\partial_{n}\tilde{f}_{ij}\,,$
(17)

where $f(\vec{x},t)$ is any space-time scalar function and
$\tilde{f}_{ij}(\vec{x},t)$ is another space-time dependent tensor that is
symmetric under exchange $i\leftrightarrow j$. By using equations (14) and
(15) now, we can formulate the dual Lagrangian of the theory as function of
the gauge fields

$\displaystyle\mathcal{L}_{\text{dual}}=\frac{1}{2}\left[\tilde{C}^{-1}_{mnab}E_{mn}E_{ab}-\frac{(B_{i})^{2}}{\rho_{d}}+\frac{1}{\kappa}(\tilde{E}_{kij}-\epsilon_{ik}\epsilon_{jl}\partial_{l}\phi)^{2}\right.$
$\displaystyle\left.-(\tilde{B}_{ij}+\epsilon_{ik}\epsilon_{jl}A_{kl})^{2}\right]+\rho\phi-
A_{ij}J_{ij}+\tilde{\rho}_{ij}\tilde{\phi}_{ij}-\tilde{A}_{kij}\tilde{J}_{kij},$
(18)

where we have defined
$\tilde{C}^{-1}_{mnab}=\epsilon_{mi}\epsilon_{nj}\epsilon_{ak}\epsilon_{bl}C^{-1}_{ijkl}$
for the rotated inverse elasticity tensor and we have also introduced the
sources of defects, $\rho$, $\tilde{\rho}_{ij}$ and currents $J_{ij}$,
$\tilde{J}_{kij}$ in our theory. The first two terms and their respective
sources in the equation above, are the familiar theory of fracton-elasticity
duality [9]. From Eq. (III) the continuity equations read

$\displaystyle\partial_{t}\rho-\partial_{i}\partial_{j}J_{ij}-\partial_{j}\tilde{J}_{iij}=0\,,$
(19)
$\displaystyle\qquad\partial_{t}\tilde{\rho}_{ij}-\epsilon_{nm}\partial_{n}\tilde{J}_{mij}=0,$
(20)

with the sources and currents

$\displaystyle\rho=$ $\displaystyle\
\epsilon_{ik}\epsilon_{jl}\partial_{i}\partial_{j}u_{kl}^{(s)},$ (21a)
$\displaystyle J_{ij}=$ $\displaystyle\
\epsilon_{ik}\epsilon_{jl}(\partial_{t}\partial_{k}-\partial_{k}\partial_{t})u_{l}^{(s)},$
(21b)

and $\tilde{\rho}_{ij}$ and $\tilde{J}_{kij}$ given by Eqns. (5a) and (5b)
respectively. In order now to study the fractonic behavior of the system, we
begin by calculating the canonical momenta conjugate to the tensor fields
$\Pi_{ij}=\left(\frac{\partial\mathcal{L}}{\partial\dot{A}_{ij}}\right)$ and
$\tilde{\Pi}_{ijk}=\left(\frac{\partial\mathcal{L}}{\partial\dot{\tilde{A}}_{ijk}}\right)$.
The Hamiltonian density then reads

$\mathcal{H}=\Pi_{ij}\dot{A}_{ij}+\tilde{\Pi}_{ijk}\dot{\tilde{A}}_{ijk}-\mathcal{L}.$
(22)

Requiring gauge invariance of Eq. (22), we arrive at

$\partial_{i}\partial_{j}\Pi_{ij}+\partial_{j}\tilde{\Pi}_{iij}=-\rho,$ (23)
$\epsilon_{km}\partial_{m}\tilde{\Pi}_{kij}=-\tilde{\rho}_{ij}.$ (24)

To demonstrate that disclinations behave similarly to fractons, we evaluate
the contributions from the two types of charges present to the system’s
overall dipole moment. We then proceed to derive the total dipole moment

$\displaystyle P_{a}+\tilde{P}_{a}=\int\left(\rho
x_{a}+\epsilon_{ia}\tilde{\rho}_{ij}x_{j}d^{2}x\right)d^{2}x=\hskip
14.22636pt$ $\displaystyle\hskip 99.58464pt=\frac{1}{\kappa}\int
K_{aii}d^{2}x+\text{const}.$ (25)

The relation above extends the one originally formulated by Pretko [9, 10] for
a flat background. As in that case, the second constant term on the right-hand
side arises from the integrals of the boundary terms, which are unaffected by
local operations within the bulk of the region. In this context, we observe
that the total dipole moment for disclination defects is not universally
conserved. Intriguingly, it is connected to the behavior of specific
components of the metric and only becomes a conserved quantity when
$K_{aii}=0$. This is comparable to the situation in Cosserat elasticity, where
two separate dipole moments are linked to two different kinds of defects, as
indicated in [13]. This result indicates that disclinations can manifest
fractonic properties. To this end, we follow [9] and express the integrand in
Eq. (25) in terms of elastic variables as,

$\frac{1}{\kappa}K_{aii}=\partial_{a}h_{ii}=\partial_{a}(\bar{h}_{ii}+h^{(s)}_{ii})=0\,.$
(26)

The smooth part represents volume changes attributed to the regular
deformations in the metric. These, as noted in [9] regarding creation of
vacancies in the crystal can be set to zero i.e. $\partial_{a}\bar{h}_{ii}=0$.
Consequently, the singular component now signifies volume alterations that are
contingent on the disclination defects present in the metric. Should that term
be zero, as expressed by

$\partial_{a}h_{ii}^{(s)}=0\,.$ (27)

then disclinations will indeed behave like fractons, as the total dipole
moment of the system becomes conserved. An immediate result of all the above
is that the trace of the metric defects density $\tilde{\rho}_{ii}$ is now
equal to zero (by use of Eq. (20) we also get
$\epsilon_{nm}\partial_{n}J_{mii}=0$).

##  IV Glide constraint on curved manifold

A comparison with the theory on flat background gives us a physical intuition
about the underpinning mobility restrictions mechanism. There, a physical
requirement that the solid does not contain vacancies and interstitials leads
to the so-called glide constraint, which states that the trace $J_{ii}$
vanishes [39]. In our work this is no longer the only requirement. Calculating
the quadrupole moment from both contributions results in

$\displaystyle Q_{ii}+\tilde{Q}_{ii}$ $\displaystyle=\int\left(\rho
x_{i}^{2}+\epsilon_{im}\tilde{\rho}_{ij}x_{j}x_{m}\right)d^{2}x=$
$\displaystyle=\int\left[2\tilde{C}_{iikl}^{-1}T_{kl}+\frac{1}{\kappa}K_{aii}x_{a}\right]d^{2}x+\text{const}.$
(28)

In the relation above, the first term on the right hand side is the familiar
result of Pretko [10] regarding volume changes connected to the presence of
vacancies or interstitials in the crystal (expressed in elastic variables
instead of their electromagnetic counterparts). Since the stress tensor
$T_{ij}$ is dual to the generalized strain tensor $w_{ij}$, this term now
captures contributions to the volume change coming from the trace of the
metric too. Should that term vanish along with Eq. (27), imposing dipole
moment conservation, then the trace of the total quadrupole moment of the
system is also a conserved quantity. Dislocations then become lineons, i.e.
only able to move along their respective Burgers vector. The new modified
constraint on a curved background can be formulated as two distinct sub-cases

$\text{1st case}:\quad u_{ii}^{(s)}=h_{ii}^{(s)}=0,\qquad J_{ii}=0,\,\hskip
14.22636pt\tilde{J}_{mii}=0,$ (29) $\text{2nd case}:\quad
u_{ii}^{(s)}=-h_{ii}^{(s)},\quad\tilde{J}_{mii}=-(\partial_{m}J_{ii}+\partial_{m}\partial_{i}\pi_{i}^{(s)}),$
(30)

where $\pi_{i}^{(s)}$ acts as the current for vacancies-interstitials of the
displacement field. The first case is simply the generalized glide constraint
in the curved background. The traces of both kinds of currents represent the
defects present in the theory. In the second case the presence of any
vacancies or interstitials originating from the displacement field are
canceled out by their counterparts coming from the metric tensor. This
cancellation leads to a relation between the two kinds of currents in the
theory.

We conclude that disclocations can retain their fractonic behavior on curved
manifolds provided that their motion perpendicular to the Burgers vector is
not induced either by the presence of vacancies or interstitials or certain
configurations of the strain tensor and the metric.

##  V Discussion

In this work we have shown that elasticity on curved manifolds can be
formulated in terms of tensor gauge variables. Elastic defects remain
fractonic and exhibit mobility constraints, provided that one of the two
generalized glide-constraints is satisfied. In accord with previous
expectations we show that the duality does not map elasticity to tensor gauge
theories coupled to curvature, which would break the gauge invariance [40, 41,
42]. A resolution of this puzzle is that in the dual language, metric
fluctuations are mapped to additional tensor gauge variables.

## Acknowledgments

J.J.F.-M. and J.M.-V. thank the financial support of Spanish Ministerio de
Ciencia e Innovación PID2021-125700NAC22. J.M.-V. also thanks Programa
Recualificación del Sistema Universitario Español 2021–2023. P.S. acknowledges
support form the Polish National Science Centre (NCN) Sonata Bis grant
2019/34/E/ST3/00405. P.S. thanks University of Murcia for hospitality and
support from the Talento program. L.T. has been supported in part by the
Deutsche Forschungsgemeinschaft through the cluster of excellence ct.qmat
(Exzellenzcluster 2147, Project 390858490).

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##  Appendix A Hubbard Stratonovich Transformation

The theory under consideration is given by

$\displaystyle\mathcal{Z}=\int D[\Phi]\,\exp\left(-S\right),\,$ (31)

with $D[\Phi]\equiv
D[w_{ij},\partial_{t}u_{i},\partial_{t}h_{ij},\partial_{k}h_{ij}]$, the
measure of the fields entering the action of the theory

$S=\frac{1}{2}\int
d^{2}xdt\left[\rho_{d}(\partial_{t}u_{i})^{2}+(\partial_{t}h_{ij})^{2}-(u_{ij}+h_{ij})C_{ijkl}(u_{kl}+h_{kl})-\kappa(\partial_{k}h_{ij})^{2}\right],$
(32)

where $w_{ij}=u_{ij}+h_{ij}$ is the generalized symmetric strain tensor and
$u_{ij}=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i})$ the usual symmetric
elasticity strain tensor. The constants $\rho_{d}$ represent the material 2d
density and $b$ acts as an elastic modulus for the gravity part respectively.

Through a Hubbard-Stratonovich transformation we get:

$\begin{split}\mathcal{Z}_{\text{HS}}=\frac{1}{N}\int\,D[\widetilde{\Phi}\,,\Phi]\,\exp(-S_{\text{HS}}),\end{split}$

with $D[\widetilde{\Phi}\,,\Phi]\equiv
D[\pi_{i},L_{ij},T_{ij},K_{kij},w_{ij},\partial_{t}u_{i},\partial_{t}h_{ij},\partial_{k}h_{ij}]$
being the measure in terms of the fields and the HS-conjugate fields and

$S_{\text{HS}}=\frac{1}{2}\int
d^{2}xdt\left[T_{ij}C^{-1}_{ijkl}T_{kl}-\frac{1}{\rho_{d}}(\pi_{i})^{2}+\frac{1}{\kappa}(K_{kij})^{2}-(L_{ij})^{2}+2\pi_{i}\partial_{t}u_{i}+2L_{ij}\partial_{t}h_{ij}-2w_{ij}T_{ij}-2\partial_{k}h_{ij}K_{kij}\right],$
(33)

where we took $T_{0j}=\pi_{j}$ and $K_{0ij}=L_{ij}$.

##  Appendix B Dual Gauge Theory

To introduce defects in the theory, one splits the fields
$u_{i}=\overline{u}_{i}+u^{(s)}_{i}$ and
$h_{ij}=\overline{h}_{ij}+h^{(s)}_{ij}$, into a smooth part represented by the
bar, and $(s)$ a singular part. Plugging this into the action and carrying out
integration by parts on the smooth fields $\bar{u}_{i}$ and $\bar{h}_{ij}$ one
gets

$\displaystyle S^{*}_{\text{HS}}$ $\displaystyle=\frac{1}{2}\int
d^{2}xdt\left[T_{ij}C^{-1}_{ijkl}T_{kl}-\frac{1}{\rho_{d}}(\pi_{i})^{2}+\frac{1}{\kappa}(K_{kij})^{2}-(L_{ij})^{2}\right]+S_{\rm
source},$ (34) $\displaystyle S_{\rm source}$ $\displaystyle=\int
d^{2}xdt\left[\pi_{i}\partial_{t}u^{(s)}_{i}+L_{ij}\partial_{t}h^{(s)}_{ij}-(u^{(s)}_{ij}+h^{(s)}_{ij})T_{ij}-\partial_{k}h^{(s)}_{ij}K_{kij}\right]$
(35)

from which two conservation laws can be derived. The first one is the usual
Hooke’s law

$\partial_{t}\pi_{i}-\partial_{j}T_{ij}=0\,.$ (36)

This equation can be resolved in terms of set of gauge potentials
$(\phi,A_{ij})$ as:

$\displaystyle\pi_{i}$
$\displaystyle=-\epsilon_{ij}\epsilon_{kl}\partial_{k}A_{lj}=-\epsilon_{ij}B_{j}\,,$
(37) $\displaystyle T_{ij}$
$\displaystyle=-\epsilon_{ik}\epsilon_{jl}(\partial_{t}A_{kl}+\partial_{k}\partial_{l}\phi)=\epsilon_{ik}\epsilon_{jl}E_{kl}\,,$

where

$B_{i}=\epsilon_{jk}\partial_{j}A_{ki}\,,\qquad
E_{ij}=-\partial_{t}A_{ij}-\partial_{i}\partial_{j}\phi\,,$ (38)

with $E_{ij}$ being the dual electric tensor field and $B_{i}$ the dual
magnetic vector field.

The second conservation law is given by

$\partial_{t}L_{ij}-\partial_{k}K_{kij}+T_{ij}=0\,,$ (39)

which, written in terms of the gauge fields introduced above, can be written
as

$\partial_{t}(L_{ij}-\epsilon_{ik}\epsilon_{jl}A_{kl})-\partial_{k}(K_{kij}+\epsilon_{ik}\epsilon_{jl}\partial_{l}\phi)=0\,.$
(40)

This second equation can be resolved by a second set of gauge fields
$(\tilde{\phi}_{kl},\tilde{A}_{klm})$

$\displaystyle\tilde{B}_{ij}$
$\displaystyle=L_{ij}-\epsilon_{ik}\epsilon_{jl}A_{kl}=\epsilon_{ik}\epsilon_{jl}\partial_{m}\tilde{A}_{mkl}\,,$
(41) $\displaystyle\tilde{E}_{mij}$
$\displaystyle=K_{mij}+\epsilon_{im}\epsilon_{jl}\partial_{l}\phi=\epsilon_{ik}\epsilon_{jl}(\partial_{t}\tilde{A}_{mkl}+\epsilon_{mn}\partial_{n}\tilde{\phi}_{kl})\,,$

which allows one to write it as a Faraday type equation as

$\partial_{t}\,\tilde{B}_{ij}-\partial_{k}\,\tilde{E}_{kij}=0,$ (42)

where now $\tilde{E}_{mij}$ is the dual rank-3 tensor electric field
(symmetric under $i\longleftrightarrow j$) and $\tilde{B}_{ij}$ is the rank-2
tensor magnetic field.

In terms of these dual gauge fields, the Lagrangian of the theory can be
written only in terms of the electric and magnetic fields, including sources,
as:

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}\tilde{C}^{-1}_{ijkl}E_{ij}E_{kl}-\frac{1}{2\rho_{d}}(B_{i})^{2}+\frac{1}{2\kappa}(\tilde{E}_{kij}-\epsilon_{ik}\epsilon_{jl}\partial_{l}\phi)^{2}-\frac{1}{2}(\tilde{B}_{ij}+\epsilon_{ik}\epsilon_{jl}A_{kl})^{2}+\mathcal{L}_{\rm
source}\,,$ (43) $\displaystyle\mathcal{L}_{\rm source}$
$\displaystyle=\phi\rho-
A_{kl}J_{kl}+\tilde{\phi}_{ij}\tilde{\rho}_{ij}-\tilde{A}_{mkl}\tilde{J}_{mkl}\,,$

with $\mathcal{L}_{\rm srcs}$ being discussed below. We note that the above
Lagrangian is gauge invariant. That is, although the fields $\tilde{E}_{kij}$
and $\tilde{B}_{ij}$ are not gauge invariant, the combinations
$(\tilde{E}_{kij}-\epsilon_{ik}\epsilon_{jl}\partial_{l}\phi)$ and
$(\tilde{B}_{ij}+\epsilon_{ik}\epsilon_{jl}A_{kl})$ are gauge invariant as
required for a dual gauge theory of the elasticity theory considered here.

##  Appendix C Sources

As implicitly shown above, we can write the action $S_{\rm srcs}$ in terms of
the gauge fields as:

$\displaystyle S_{\rm source}$ $\displaystyle=\int
d^{2}xdt\left[\pi_{i}\partial_{t}u_{i}^{(s)}+L_{ij}\partial_{t}h_{ij}^{(s)}-(u^{(s)}_{ij}+h^{(s)}_{ij})T_{ij}-\partial_{k}h_{ij}K_{kij}\right]$
(44) $\displaystyle=\int d^{2}xdt\left[\phi\rho-
A_{kl}J_{kl}+\tilde{\phi}_{ij}\tilde{\rho}_{ij}-\tilde{A}_{mkl}\tilde{J}_{mkl}\right]\,,$

by defining the sources in the dual gauge theory as

$\displaystyle\rho$
$\displaystyle=\epsilon_{ik}\epsilon_{jl}\partial_{i}\partial_{j}u_{kl}^{(s)}\,,\qquad
J_{ij}=\epsilon_{ik}\epsilon_{jl}(\partial_{t}\partial_{k}-\partial_{k}\partial_{t})u_{l}^{(s)},\,$
(45) $\displaystyle\tilde{\rho}_{ij}$
$\displaystyle=\epsilon_{ik}\epsilon_{jl}\epsilon_{mn}\partial_{n}\partial_{m}h^{(s)}_{kl},\,\qquad\tilde{J}_{mij}=\epsilon_{ik}\epsilon_{jl}(\partial_{m}\partial_{t}-\partial_{t}\partial_{m})h_{kl}^{(s)}\,.$
(46)

##  Appendix D Gauge Transformations

In our theory, since by dropping the terms containing metric fluctuations one
should recover the original fracton elasticity duality gauge transformations
associated to the first set of gauge fields (given by an arbitrary function
$f$), we choose to retain those as fixed and work on finding those for the
second set of gauge fields (given by an arbitrary function $\tilde{f}_{kl}$).
In this sense we have:

$\delta
L_{ij}=\epsilon_{ik}\epsilon_{jl}[\partial_{m}(\delta\tilde{A}_{mkl})+\delta
A_{kl}]=\partial_{m}(\delta\tilde{A}_{mkl})-\partial_{k}\partial_{l}f=0\Rightarrow\delta\tilde{A}_{mkl}=\delta_{m\left(k\right.}\partial_{\left.l\right)}f+\tilde{F}_{mkl}.$

$\tilde{F}_{mkl}$ is a function that is symmetric in the last two indices and
also has the symmetry $\partial_{m}\tilde{F}_{mkl}=0$. For reasons that will
become obvious in a bit we choose to set it equal to

$\tilde{F}_{mkl}=\epsilon_{mn}\partial_{n}\tilde{f}_{kl}.$

In addition

$\begin{split}\delta
K_{mij}=\epsilon_{ik}\epsilon_{jl}[\partial_{t}(\delta\tilde{A}_{mkl})-\delta_{m\left(k\right.}\partial_{\left.l\right)}(\delta\phi)+\epsilon_{mn}\partial_{n}(\delta\tilde{\phi}_{kl})]=\hskip
56.9055pt\\\
=\partial_{t}\left[\delta_{m\left(k\right.}\partial_{\left.l\right)}f+\epsilon_{mn}\partial_{n}\tilde{f}_{kl}\right]-\delta_{m\left(k\right.}\partial_{\left.l\right)}\partial_{t}f+\epsilon_{mn}\partial_{n}(\delta\tilde{\phi}_{kl})=\partial_{t}\partial_{n}\tilde{f}_{kl}+\partial_{n}(\delta\tilde{\phi}_{kl})=0\end{split}$

where the last equality is fulfilled requiring that
$\delta\tilde{\phi}_{kl}=-\partial_{t}\tilde{f}_{kl}$. Finally we have,

$\displaystyle\delta\phi$ $\displaystyle=\partial_{t}f\,,\qquad\delta
A_{ij}=-\partial_{i}\partial_{j}f,$ (47)
$\displaystyle\delta\tilde{\phi}_{ij}$
$\displaystyle=-\partial_{t}\tilde{f}_{ij},\,\qquad\delta\tilde{A}_{mij}=\delta_{m\left(i\right.}\partial_{\left.j\right)}f+\epsilon_{mn}\partial_{n}\tilde{f}_{ij}\,.$
(48)

where again we note that $\tilde{f}_{ij}=\tilde{f}_{ji}$.

##  Appendix E Continuity equations

The continuity equations for our theory can be derived by

$\begin{split}\delta S_{\rm source}=\int
d^{2}xdt\left[\rho(\delta\phi)-J_{ij}(\delta
A_{ij})+\tilde{\rho}_{ij}(\delta\tilde{\phi}_{ij})-\tilde{J}_{mij}(\delta\tilde{A}_{mij})\right]=\hskip
22.76228pt\\\ =\int
d^{2}xdt\left[\rho(\partial_{t}f)+J_{ij}(\partial_{i}\partial_{j}f)-\tilde{\rho}_{ij}(\partial_{t}\tilde{f}_{ij})-\tilde{J}_{mij}[\delta_{m\left(i\right.}\partial_{\left.j\right)}f+\epsilon_{mn}\partial_{n}\tilde{f}_{ij}]\right]=\\\
=\int
d^{2}xdt\left[\left(-\partial_{t}\rho+\partial_{i}\partial_{j}J_{ij}+\partial_{j}\tilde{J}_{iij}\right)f+\left(\partial_{t}\tilde{\rho}_{ij}-\epsilon_{nm}\partial_{n}\tilde{J}_{mij}\right)\tilde{f}_{ij}\right]=0\,,\end{split}$

from which we obtain

$\partial_{t}\rho-\partial_{i}\partial_{j}J_{ij}-\partial_{j}\tilde{J}_{iij}=0\,,\qquad\partial_{t}\tilde{\rho}_{ij}-\epsilon_{nm}\partial_{n}\tilde{J}_{mij}=0\,,$
(49)

where the signs are conventional. They are a consequence of how the source
terms are introduced in the action.

##  Appendix F Equations of motion - Maxwell equations for the symmetric
tensor gauge fields

We can now use the Lagrangian in Eq. (43) to derive the Maxwell equations of
our theory. The two sets of Maxwell equations are as follows:

1st set

$\tilde{C}_{ijkl}^{-1}\partial_{i}\partial_{j}E_{kl}-\left(\frac{1}{\kappa}\epsilon_{ik}\epsilon_{jl}\partial_{l}\tilde{E}_{kij}-\frac{2}{\kappa}\partial_{m}^{2}\phi\right)=-\rho\,,\quad{\rm(Gauss\,\,Law).}$
$\partial_{t}B_{i}+\epsilon_{jk}\partial_{j}E_{ki}=0\,,\qquad{\rm(Faraday\,\,Law)}.$
(50)
$\tilde{C}_{ijkl}^{-1}\partial_{t}E_{kl}+\epsilon_{\left(ik\right.}\partial_{k}B_{\left.j\right)}+\left(\epsilon_{ik}\epsilon_{jl}\tilde{B}_{kl}+A_{ij}\right)=-J_{ij}\,,\quad\text{\rm(Amp\\`{e}re\,\,Law),}$

2nd set

$\frac{1}{\kappa}\left(\epsilon_{ik}\epsilon_{jl}\epsilon_{mn}\partial_{n}\tilde{E}_{mkl}+\epsilon_{ik}\partial_{k}\partial_{j}\phi\right)=-\tilde{\rho}_{ij},\qquad{\rm(Gauss\,\,Law),}$
$\partial_{t}\tilde{B}_{ij}-\partial_{m}\tilde{E}_{mij}=0,\qquad{\rm(Faraday\,\,Law),}$
(51)
$\frac{1}{\kappa}\left(\epsilon_{ik}\epsilon_{jl}\partial_{t}\tilde{E}_{mkl}-\delta_{mi}\partial_{t}\partial_{j}\phi\right)-\epsilon_{ik}\epsilon_{jl}\partial_{m}\tilde{B}_{kl}-\partial_{m}A_{ij}=-\tilde{J}_{mij},\qquad\text{\rm(Amp\\`{e}re\,\,
Law).}$

In the first set above, setting the terms in the parentheses equal to zero,
one may recover the laws governing the fracton-elasticity duality derived in
[9].

##  Appendix G System Hamiltonian

The Lagrangian (43) in terms of the gauge fields reads

$\begin{split}&\mathcal{L}=\frac{1}{2}\tilde{C}_{ijkl}^{-1}(\partial_{t}A_{ij}+\partial_{i}\partial_{j}\phi)(\partial_{t}A_{kl}+\partial_{k}\partial_{l}\phi)-\frac{1}{2\rho_{d}}\partial_{m}A_{ni}(\partial_{m}A_{ni}-\partial_{n}A_{mi})+\\\
&+\frac{1}{2\kappa}\left[\epsilon_{ik}\epsilon_{jl}(\partial_{t}\tilde{A}_{mkl}+\epsilon_{mn}\partial_{n}\tilde{\phi}_{kl})-\epsilon_{im}\epsilon_{jl}\partial_{l}\phi\right]^{2}-\frac{1}{2}\left(\epsilon_{ik}\epsilon_{jl}\partial_{m}\tilde{A}_{mkl}+\epsilon_{ik}\epsilon_{jl}A_{kl}\right)^{2}+\\\
&+\phi\rho-
A_{ij}J_{ij}+\tilde{\phi}_{ij}\tilde{\rho}_{ij}-\tilde{A}_{mkl}\tilde{J}_{mkl}.\end{split}$
(52)

In order to write the Hamiltonian density we obtain the canonical momenta
which are given by:

$\Pi_{ij}=\left(\frac{\partial\mathcal{L}}{\partial\dot{A}_{ij}}\right)=\tilde{C}_{ijkl}^{-1}(\partial_{t}A_{ij}+\partial_{i}\partial_{j}\phi)=-\tilde{C}_{ijkl}^{-1}E_{kl},$
(53)
$\tilde{\Pi}_{ijk}=\left(\frac{\partial\mathcal{L}}{\partial\dot{\tilde{A}}_{ijk}}\right)=\frac{1}{\kappa}\left(\partial_{t}\tilde{A}_{ijk}+\epsilon_{im}\partial_{m}\tilde{\phi}_{jk}-\delta_{ij}\partial_{k}\phi\right).$
(54)

Conjugated momenta for the fields $\phi$ and $\tilde{\phi}_{ij}$ cannot be
defined as the Lagrangian does not depend on their time derivatives. They
instead behave as Lagrange multipliers. With this, the Hamiltonian density can
be written as,

$\mathcal{H}=\Pi_{ij}\dot{A}_{ij}+\tilde{\Pi}_{ijk}\dot{\tilde{A}}_{ijk}-\mathcal{L}.$
(55)

Varying $\mathcal{H}$ above and using the conservation equations and the form
of the gauge transformations stated above, one obtains the same Gauss law’s
derived before as

$\partial_{i}\partial_{j}\Pi_{ij}+\partial_{j}\tilde{\Pi}_{iij}=-\rho\,,\qquad\epsilon_{km}\partial_{m}\tilde{\Pi}_{kij}=-\tilde{\rho}_{ij}\,.$
(56)

##  Appendix H Fractonic Behavior

Following an argument similar to [10] we first compute the dipole moment
associated to the charges in our theory. For the charge $\rho$ associated with
the first set of gauge fields we have

$\displaystyle\begin{split}P_{a}&=\int\rho\,x_{a}d^{2}x=-\epsilon_{im}\epsilon_{jn}\int\left[\tilde{C}_{ijkl}^{-1}\partial_{m}\partial_{n}T_{kl}+\frac{1}{\kappa}\partial_{m}K_{nij}\right]x_{a}d^{2}x\\\
&=-\epsilon_{im}\epsilon_{jn}\int\left[\partial_{m}\left(\tilde{C}_{ijkl}^{-1}x_{a}\partial_{n}T_{kl}\right)-\delta_{ma}\left(\tilde{C}_{ijkl}^{-1}\partial_{n}T_{kl}\right)+\frac{1}{\kappa}\partial_{m}\left(x_{a}K_{mij}\right)-\frac{1}{\kappa}\delta_{ma}K_{nij}\right]d^{2}x\,.\end{split}$

Noting that the total divergences were switched to surface integrals through
the Gauss-Ostrogradsky theorem, one may write

$P_{a}-\frac{1}{\kappa}\int\left(K_{aii}-K_{iia}\right)d^{2}x=\text{const}\,.$
(57)

For the tensor charge $\tilde{\rho}_{kl}$ associated to the second set of
gauge fields, we define the dipole moment as,

$\begin{split}\tilde{P}_{l}&=\int\epsilon_{il}\tilde{\rho}_{ij}x_{j}d^{2}x=\frac{1}{\kappa}\int\epsilon_{il}\epsilon_{im}\epsilon_{jn}\epsilon_{kp}\partial_{p}K_{kmn}x_{j}d^{2}x=\frac{1}{\kappa}\int\epsilon_{jn}\epsilon_{kp}\partial_{p}K_{kln}x_{j}d^{2}x=\\\
&=\frac{\epsilon_{jn}\epsilon_{kp}}{\kappa}\int\left[\partial_{p}\left(K_{kln}x_{j}\right)-\delta_{jp}K_{kln}\right]d^{2}x\Rightarrow\tilde{P}_{l}-\frac{1}{\kappa}\int
K_{iil}d^{2}x=\text{const}.\end{split}$ (58)

Therefore, for the total dipole moment of our theory we have

$P_{a}+\tilde{P}_{a}=\frac{1}{\kappa}\int K_{aii}d^{2}x+\text{const}\,.$ (59)

Interestingly, when $K_{aii}=0$, the total dipole moment is conserved. This is
analogous to the case of Cosserat elasticity in which one also gets two
independent dipole moments associated with two different kind of defects [13].
This shows that disclinations can have fractonic behavior. To this end, we
follow [9] and we express the right-hand side of (59) in terms of elastic
variables as,

$\frac{1}{\kappa}K_{aii}=\partial_{a}h_{ii}=\partial_{a}(\bar{h}_{ii}+h^{(s)}_{ii})=0\,.$
(60)

The smooth part denoted by the bar represents regular volume changes
attributed to the regular changes in the metric. These, as noted in [9]
regarding creation of vacancies in the crystal can sensibly be set to zero
i.e. $\partial_{a}\bar{h}_{ii}=0$. Therefore, the singular part now represents
volume changes depending on the disclinational defects present on the metric.
Should that term be zero, i.e.

$\partial_{a}h_{ii}^{(s)}=0\,.$ (61)

then indeed the disclinations behave like fractons as the total dipole moment
of the system is conserved. Assuming this condition is fulfilled, this
immediately implies

$\displaystyle\tilde{\rho}_{ii}$
$\displaystyle=\epsilon_{mn}\partial_{n}\partial_{m}h^{(s)}_{ii}=0,\,$ (62)
$\displaystyle\tilde{J}_{mii}$
$\displaystyle=\partial_{m}\partial_{t}h^{(s)}_{ii}\,.$

Now, in order to analyze the dislocations, we have to look into the quadrupole
moment. Regarding the scalar charge $\rho$ we have

$\begin{split}Q_{ii}&=\int\rho\,x_{i}^{2}d^{2}x=-\epsilon_{im}\epsilon_{jn}\int\left(\tilde{C}_{ijkl}^{-1}\partial_{m}\partial_{n}T_{kl}+\frac{1}{\kappa}\partial_{m}K_{nij}\right)x_{i}^{2}d^{2}x\\\
&=-\epsilon_{im}\epsilon_{jn}\int\left[\tilde{C}_{ijkl}^{-1}\partial_{m}(\partial_{n}T_{kl}x^{2})-\tilde{C}_{ijkl}^{-1}(\partial_{n}T_{kl})(\partial_{m}x^{2})+\frac{1}{\kappa}\partial_{m}(K_{nij}x^{2})-\frac{2}{\kappa}K_{nij}x_{m}\right]d^{2}x\\\
&=\epsilon_{im}\epsilon_{jn}\int\left(\tilde{C}_{ijkl}^{-1}\partial_{n}(2T_{kl}x_{m})+2\tilde{C}_{ijkl}^{-1}T_{kl}\delta_{mn}-\frac{2}{\kappa}K_{nij}x_{m}\right)d^{2}x+\text{const}\,,\end{split}$

which can be compactly written as

$Q_{ii}=2\int\left[\tilde{C}_{iikl}^{-1}T_{kl}-\frac{1}{\kappa}\left(K_{kkl}x_{l}-K_{kll}x_{k}\right)\right]d^{2}x+\text{const}.$
(63)

On the other hand, referring to the tensor charge $\tilde{\rho}_{ij}$ we have

$\begin{split}\tilde{Q}_{ml}&=\int\epsilon_{il}\tilde{\rho}_{ij}x_{j}x_{m}d^{2}x=\frac{1}{\kappa}\int\epsilon_{jn}\epsilon_{kp}\partial_{p}K_{kln}x_{j}x_{m}d^{2}x=\frac{\epsilon_{jn}\epsilon_{kl}}{\kappa}\int\partial_{p}K_{kln}x_{j}x_{m}d^{2}x=\\\
&=\frac{\epsilon_{jn}\epsilon_{kl}}{\kappa}\int\left[\partial_{p}(K_{kln}x_{j}x_{m})-K_{kln}\partial_{p}(x_{j}x_{m})\right]d^{2}x=-\frac{\epsilon_{jn}\epsilon_{kp}}{\kappa}\int
K_{kln}(\delta_{pj}x_{m}+x_{j}\delta_{pm})d^{2}x=\\\
&=\frac{1}{\kappa}\int\left(K_{kkl}x_{m}-\epsilon_{jn}\epsilon_{km}K_{kln}x_{j}\right)d^{2}x+\text{const}\,\end{split}$

which finally gives

$\tilde{Q}_{ml}=\frac{1}{\kappa}\int\left(2K_{kkl}x_{m}-K_{klm}x_{k}\right)d^{2}x+\text{const}.$
(64)

With this, by taking the trace on $\tilde{Q}_{ml}$ one may write

$Q_{ii}+\tilde{Q}_{ii}=\int\left[2\tilde{C}_{iikl}^{-1}T_{kl}+\frac{1}{\kappa}K_{aii}x_{a}\right]d^{2}x+\text{const}\,.$
(65)

With this, requiring that the dipole moment is conserved implies that the
second term in the right-hand side in (65) is zero. The first term being zero
is exactly the regular constraint imposed in [9] along with a term originating
from the metric that has a similar contribution (the stress-tensor $T_{ij}$ is
dual to $w_{ij}$).

We have for each case separately the following:

1st case: The trace of both the strain tensor and the metric tensor are equal
to zero. In this simple case which proves to be the analog of Pretko’s theory
in curved space, the densities and currents simplify to:

$u_{ii}^{(s)}=h_{ii}^{(s)}=0,\hskip 56.9055pt\tilde{\rho}_{ii}=0,\hskip
56.9055ptJ_{ii}=0,\hskip 56.9055pt\tilde{J}_{mii}=0,$ (66)
$\rho=-\partial_{i}\partial_{j}u_{ij}^{(s)},\hskip
56.9055pt\tilde{\rho}_{ij}=\epsilon_{nm}\partial_{n}\partial_{m}h_{ij}^{(s)},$
(67)
$J_{ij}=\partial_{i}\partial_{t}u_{j}^{(s)}-\partial_{k}\partial_{t}u_{k}^{(s)}\delta_{ij},\hskip
56.9055pt\tilde{J}_{mij}=(\partial_{t}\partial_{m}-\partial_{m}\partial_{t})h_{ij}^{(s)}.$
(68)

The defects coming from the displacement field and the metric both contribute
to the volume changes but without a relation between their currents.

2nd case: Here we require that the generalized strain tensor $w_{ij}$ be equal
to zero in order to cancel the contribution for the volume changes in the
first term of Eq. (65). In total we have the conditions:

$u_{ii}^{(s)}=-h_{ii}^{(s)},\hskip
56.9055pt\partial_{a}u_{ii}^{(s)}=\partial_{a}h_{ii}^{(s)}=0,\hskip
56.9055pt\tilde{\rho}_{ii}=0,\hskip 56.9055pt$ (69)
$\rho=\partial^{2}u_{kk}^{(s)}-\partial_{i}\partial_{j}u_{ij}^{(s)},\hskip
56.9055pt\tilde{\rho}_{ij}=\epsilon_{nm}\partial_{n}\partial_{m}h_{ij}^{(s)},$
(70)
$J_{ij}=(\partial_{i}\partial_{t}-\partial_{t}\partial_{i})u_{j}^{(s)}-\partial_{k}\partial_{t}u_{k}^{(s)}\delta_{ij},\hskip
56.9055pt\tilde{J}_{mij}=-\partial_{m}\partial_{t}(u_{kk}^{(s)}\delta_{ij}+h_{ij}^{(s)}),$
(71) $\tilde{J}_{mii}=-\partial_{m}\partial_{t}u_{ii}^{(s)},\hskip
56.9055ptJ_{ii}=\partial_{t}\partial_{i}u_{i}^{(s)}-\partial_{i}\partial_{t}u_{i}^{(s)}.$
(72)

By combining the two relations above we get:

$\tilde{J}_{mii}=-(\partial_{m}J_{ii}+\partial_{m}\partial_{i}\partial_{t}u_{i}^{(s)})=-(\partial_{m}J_{ii}+\partial_{m}\partial_{i}\pi_{i}^{(s)})\,.$
(73)

The Ampère Law in the first set of Maxwell equations, can be rewritten to take
the form of a continuity equation if someone reverts back to elastic
variables. We have:

$\partial_{t}n_{d}+\partial_{i}\pi_{i}^{(s)}+\partial_{t}\tilde{\pi}^{(s)}=-J_{ii}\,,$
(74)

where $\tilde{\pi}^{(s)}=L_{ii}^{(s)}=\partial_{t}h_{ii}^{(s)}$ and $n_{d}$
the vacancy-interstitial density coming from the displacement field. This
relation is again the curved background counterpart of the one Pretko derived
regarding the continuity equation for the defects. We now combine the last two
to arrive at:

$\tilde{J}_{mii}=\partial_{m}\partial_{t}\left(n_{d}+\tilde{\pi}^{(s)}\right).$
(75)

The singular part of the trace of the metric dual tensor $L_{ij}$ effectively
works as the vacancy-interstitial density along with $n_{d}$ for the metric
defect current.