DESY 22-039

# Duality Symmetric Electrodynamics in Curved Spacetime

Anton V. Sokolov<EMAIL_ADDRESS>Rudolf Peierls Centre for
Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United
Kingdom  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg,
Germany

###### Abstract

We derive Maxwell equations for electric and magnetic fields in curved
spacetime from first principles, relaxing an unnecessary assumption on the
structure of the four-potential inherent to the standard approach and thus
restoring the full consistency with the equivalence principle in the following
two important cases: first, if the electromagnetic field is considered as a
physical entity separate from the charged particles used to measure it, and
second, if hypothetical magnetically charged particles are allowed to exist.
We find that in a generic curved spacetime, the electromagnetic field has to
be described by two pairs of electric and magnetic fields, instead of the only
one pair which is enough in the flat spacetime case.

Introduction.—These days, we routinely detect kHz frequency Gravitational
Waves (GWs) with laser interferometers such as Advanced LIGO [1], Advanced
VIRGO [2] and KAGRA [3]. One of the challenges ahead is to detect GWs in other
types of experiments which would probe different wavelengths. A lot of work
has been done in this direction. In particular, it has been recently argued
that the existing axion detection experiments, or versions of them, can allow
for GW detection in the Ultra-High Frequency (UHF), i.e. above a kHz, range
[4, 5, 6, 7]. To interpret the results of such experiments, one has to model
the influence of the passing GWs on the magnetic and electric fields of the
detector, which can be done by considering Maxwell equations in the GW
background [8, 9]. These equations are obtained from the well-known
formulation of classical electrodynamics in curved spacetime [10] by A.
Einstein. As we will show, however, contrary to the flat spacetime Maxwell
equations, these equations are not symmetric with respect to the following
interchange of the electric and magnetic fields
$\mathbf{E}\rightarrow\mathbf{B}$, $\mathbf{B}\rightarrow-\mathbf{E}$, or more
generally, with respect to the $SO(2)$ electric-magnetic duality symmetry
transformations, although this symmetry has a physically important conserved
Noether charge associated to it. This issue leads to a number of tensions
within the standard approach. In particular, as it will be discussed in the
next section, whenever one aims to fully quantize the electromagnetic field
with the known methods, or considers a thought experiment involving a
hypothetical magnetic charge, the conventional coupling of electromagnetism to
gravity violates the equivalence principle (EP). Since no experiment to date
has detected violations of the EP, we argue that the coupling of
electromagnetic (EM) field to gravity fully respecting the EP is preferred
over the conventionally used coupling. Therefore, we find the corresponding
preferred general-covariant formulation of electrodynamics which is fully
consistent with the electric-magnetic duality symmetry. Intriguingly, we find
that the known set of curved spacetime Maxwell equations has to be
complemented by another set of equations featuring a separate pair of electric
and magnetic fields acting on magnetically charged particles.

Electric-magnetic duality symmetry in presence of gravity.—Maxwell equations
for free EM fields are:

$\displaystyle\partial_{\mu}F^{\mu\nu}=0\,,$ (1)
$\displaystyle\epsilon^{\mu\nu\lambda\rho}\,\partial_{\nu}F_{\lambda\rho}=0\,,$
(2)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the EM field
strength tensor, $A_{\mu}$ is the four-potential of the EM field,
$\epsilon^{\mu\nu\lambda\rho}$ is the Levi-Civita symbol, $\epsilon^{0123}=1$.
In his seminal work on General Relativity (GR) [10], using the EP as a guide,
A. Einstein states that the effects gravity exhibits on matter are equivalent
to the effects of curved spacetime. Adapting the Eqs. (1) and (2) to the case
of curved spacetime, he obtains:

$\displaystyle D_{\mu}F^{\mu\nu}=0\,,$ (4)
$\displaystyle\varepsilon^{\mu\nu\lambda\rho}D_{\nu}F_{\lambda\rho}=0\,,$

where $D_{\mu}$ is a covariant derivative,
$\varepsilon^{\mu\nu\lambda\rho}=\epsilon^{\mu\nu\lambda\rho}/\sqrt{-g}$ is
the Levi-Civita pseudotensor, $g\equiv\textrm{det}\,g_{\mu\nu}$ with
$g_{\mu\nu}$ being the spacetime metric tensor, for which we adopt the
$(-,+,+,+)$ signature. Note that
$F_{\mu\nu}=D_{\mu}A_{\nu}-D_{\nu}A_{\mu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$.

Defining electric and magnetic fields in terms of the covariant components of
$F_{\mu\nu}$ 111Note that one defines physical fields in terms of the
covariant components of the field strength tensor since the latter tensor is
proportional to the commutator of $U(1)$ covariant derivatives, while in GR,
derivatives are essentially defined with lower indices.:

$\displaystyle E_{i}=-F_{0i}\,,\;\;B_{i}=\frac{1}{2}\,\epsilon_{ijk}F_{jk}\,,$
(5)

one finds the following vector form for the Maxwell equations:

$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\mathbf{B}=0\,,$ (6)
$\displaystyle\boldsymbol{\nabla}\\!\times\\!\mathbf{E}+\frac{\partial\mathbf{B}}{\partial
t}=0\,,$ (7)
$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\left\\{\sqrt{-g}\left[g^{00}\mathcal{G}\\!\cdot\\!\mathbf{E}-\mathbf{g}\left(\mathbf{g}\\!\cdot\\!\mathbf{E}\right)+\mathcal{G}\\!\cdot\\!\left(\mathbf{g}\\!\times\\!\mathbf{B}\right)\right]\right\\}=0\,,\;\;$
(8)
$\displaystyle\boldsymbol{\nabla}\\!\times\\!\left\\{\sqrt{-g}\left[\mathbf{g}\\!\times\\!\left(\mathcal{G}\\!\cdot\\!\mathbf{E}\right)+\frac{1}{2}\,\mathrm{tr}\left(\mathcal{G}\mathbf{S}\mathcal{G}\mathbf{S}\\!\cdot\\!\mathbf{B}\right)\right]\right\\}-$
$\displaystyle\frac{\partial}{\partial
t}\left\\{\sqrt{-g}\left[g^{00}\mathcal{G}\\!\cdot\\!\mathbf{E}-\mathbf{g}\left(\mathbf{g}\\!\cdot\\!\mathbf{E}\right)+\mathcal{G}\\!\cdot\\!\left(\mathbf{g}\\!\times\\!\mathbf{B}\right)\right]\right\\}=0\,,\;\;\;\;\;$
(9)

where $\mathcal{G}\equiv g^{ij}$, $g_{i}\equiv g^{0i}$,
$(S_{i})_{jk}\equiv\epsilon_{ijk}$, Latin letters denote spatial indices [12].
Note that the first two vector equations (6) and (7) coincide with the
ordinary expressions for the no magnetic monopoles condition and the Faraday
law, respectively.

One can easily check that the Maxwell equations (6)–(9) are generally not
invariant with respect to the $SO(2)$ electric-magnetic duality
transformations:

$\displaystyle\mathbf{E}\;\rightarrow\;\mathbf{E}\cos\theta+\mathbf{B}\sin\theta$
(10)
$\displaystyle\mathbf{B}\;\rightarrow\;\mathbf{B}\cos\theta-\mathbf{E}\sin\theta\,\,,$

which however leave the flat spacetime Maxwell equations unchanged. In flat
spacetime, rotations (LABEL:rot) are known to be induced by the following
infinitesimal transformation of the transverse part of the vector-potential
$\mathbf{A}^{\text{T}}$ [13]:

$\delta\mathbf{A}^{\text{T}}=-\theta\,\boldsymbol{\nabla}^{-2}\,\boldsymbol{\nabla}\\!\times\\!\dot{\mathbf{A}}^{\text{T}}\,,$
(11)

which leaves the EM field action invariant. The latter invariance gives rise
to the conserved $U(1)$ charge:

$\displaystyle S_{0}=\int d^{3}x\,s_{0}=$ $\displaystyle\frac{1}{2}\int
d^{3}x\left\\{\mathbf{A}^{\text{T}}\\!\cdot\boldsymbol{\nabla}\\!\times\\!\mathbf{A}^{\text{T}}-\dot{\mathbf{A}}^{\text{T}}\\!\cdot\boldsymbol{\nabla}^{-2}\,\boldsymbol{\nabla}\\!\times\\!\dot{\mathbf{A}}^{\text{T}}\right\\}=$
$\displaystyle-\frac{1}{2}\int
d^{3}x\Bigl{\\{}\mathbf{B}\cdot\boldsymbol{\nabla}^{-2}\,\boldsymbol{\nabla}\\!\times\\!\mathbf{B}+\mathbf{E}\cdot\boldsymbol{\nabla}^{-2}\,\boldsymbol{\nabla}\\!\times\\!\mathbf{E}\Bigr{\\}}\,,\qquad$
(12)

which is known as helicity of EM field [14, 13, 15, 16, 17, 18]. The two terms
in Eq. (Duality Symmetric Electrodynamics in Curved Spacetime) have similar
bilinear structure in terms of $\mathbf{E}$ and $\mathbf{B}$ and their time
derivatives cancel each other due to the similarity of the structure of the
Maxwell equations (1) and (2) for $\dot{\mathbf{E}}$ and $-\dot{\mathbf{B}}$.
Introducing the dual-symmetrized spin angular momentum density of EM field
$\mathbf{s}$, one can rewrite the conservation law in terms of the
conservation of the four-current $\partial_{\mu}s^{\mu}=0$. Adapting this law
to the case of curved spacetime, one obtains:

$D_{\mu}s^{\mu}=0\,,\vspace*{-2pt}$ (13)

i.e. the current $s^{\mu}$ corresponding to the internal $U(1)$ symmetry of
the EM field (LABEL:rot) has to be covariantly conserved. However, it is easy
to see that the latter conservation is inconsistent with Eqs. (4) and (4)
(i.e. Eqs. (6)-(9)), since they explicitly break this $U(1)$ symmetry 222It
was argued in Ref. [13] that, contrary to our conclusion, conventional
electrodynamics in curved spacetime does preserve the electric-magnetic
duality symmetry; however a careful investigation shows that their calculation
is incorrect..

Thus, Eqs. (4), (4) and (13) cannot be satisfied simultaneously. This suggests
that the conventional coupling of electromagnetism to gravity introduced in
Ref. [10] is not minimal, i.e. its effects on the EM field are different from
what one would get by simply considering classical EM field in a curved
spacetime background. In this formalism, gravity violates the internal
symmetry of the EM field: it interacts with distinct helicity states of the EM
field differently. If the EM field is regarded as a physical entity in itself,
separate in principle from the charged particles used to probe it, the
breaking of its internal symmetry by gravity may be regarded as a violation of
the EP and so signify a conceptual problem within the standard formalism.
While all exhaustive quantum-field-theoretic approaches to EM interactions are
indeed based on the perspective that the EM field should be quantized
separately from the particles used to probe it, an alternative point of view
on the electromagnetic field is in principle possible in the classical theory,
see for example Ref. [20]. It is interesting that if one adopts this
alternative point of view on the EM field, there is no immediate inconsistency
anymore. Indeed, if an observer can use only electric charges to measure the
EM field, the measurement process itself breaks the electric-magnetic duality
symmetry 333Note that the breaking of the electric-magnetic duality symmetry
by sources of the EM field is of no relevance to the present discussion,
because the coupling of the EM field to gravity cannot depend on the kind of
charges sourcing the EM field: in fact, these charges could be causally
disconnected from a given gravitational field.: the symmetry is actually never
present. The full quantization of the theory in such approach to EM
interactions is however an open problem.

The tension with the EP becomes even more clear if one considers a thought
experiment involving a hypothetical magnetic charge. For example, consider a
pair of probe charges of the same mass initially at rest, one of which is
electric and another one is magnetic, in the background of static parallel
electric and magnetic fields of the same magnitude. The system obviously
respects the electric-magnetic duality symmetry. Suppose now one switches on a
generic gravitational field. According to the Eqs. (4) and (4) (i.e. Eqs.
(6)-(9)), the electric and magnetic fields acting on the probe charges change
differently. By registering the difference in motion of the charges, one can
clearly state that it was the effect of gravity, but not of the acceleration
of the laboratory; the EP is violated.

Two-potential formulation of electrodynamics in curved spacetime.—From the
results of the previous section, it may seem that the minimal coupling of EM
field to gravity does not exist. In fact, this is not true, since the
inconsistency with the EP found in the previous section can be traced back to
a certain assumption made in Ref. [10]. Indeed, consider a generic experiment
where one aims to probe the coupling of EM field to gravity. An experimenter
creates probe EM fields in the laboratory and measures their change in a
varying background gravitational field, such as the one of a GW. While the
probe EM fields are created by electric currents and thus can be described by
a regular four-potential, the same cannot be a priori stated for gravity-
induced variations of these fields. It is well-known that some physical
configurations of EM field, e.g. field of a magnetic monopole [22], cannot be
described by a regular four-potential. This means that one should not a priori
restrict the solution space for the gravity-induced EM four-potential by
solely regular configurations. However, this is exactly what the Bianchi
identity (4) does:

$\varepsilon^{\mu\nu\lambda\rho}D_{\nu}F_{\lambda\rho}=0\quad\Rightarrow\quad\left[\partial_{\nu},\partial_{\lambda}\right]A_{\rho}=0\,,$
(14)

where the latter expression follows from the former one regardless of the
metric. Such unphysical restriction on the solution space is the reason for
the inconsistency discussed in the previous section.

Not to miss any possible physical configurations of the gravity-induced EM
fields, one can either allow four-potential $A_{\mu}$ to be non-regular, so
that $\left[\partial_{\nu},\partial_{\lambda}\right]A_{\rho}$ need not be zero
everywhere, or introduce an additional four-potential $C_{\mu}$ and work in
the two-potential formulation of electrodynamics [23, 14]. For the sake of
simplicity, let us choose the second option, in particular we will build on
the two-potential formulation by Zwanziger [14]. We introduce the following
general-covariant Lagrangian density:

$\displaystyle\mathcal{L}_{1}=-\frac{1}{2}\,n^{\alpha^{\prime}}\\!n^{\gamma^{\prime}}\\!e^{\alpha}_{\alpha^{\prime}}e^{\gamma}_{\gamma^{\prime}}\sqrt{-g}\,\Biggl{\\{}\frac{1}{2}\,g_{\zeta\gamma}\varepsilon^{\zeta\beta\sigma\rho}\Bigl{(}E_{\alpha\beta}G_{\sigma\rho}-$
$\displaystyle\Bigr{.}\Biggr{.}$
$\displaystyle\Biggl{.}\Bigl{.}G_{\alpha\beta}E_{\sigma\rho}\Bigr{)}+g^{\rho\beta}\Bigl{(}E_{\alpha\beta}E_{\gamma\rho}+G_{\alpha\beta}G_{\gamma\rho}\Bigr{)}\Biggr{\\}}\,,$
(15)

where $e^{\alpha}_{\alpha^{\prime}}$ are vierbein vectors and
$n^{\alpha^{\prime}}\\!$ is a fixed Lorentz four-vector with unit norm
$n^{2}=1$; we denote Lorentz indices by Greek letters with a prime. The
dynamical variables of the theory are the two four-potentials $A_{\mu}$ and
$C_{\mu}$, which enter the Lagrangian density (Duality Symmetric
Electrodynamics in Curved Spacetime) via the gauge-invariant field strength
tensors:

$\displaystyle
E_{\alpha\beta}=D_{\alpha}A_{\beta}-D_{\beta}A_{\alpha}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\,,$
(16) $\displaystyle
G_{\alpha\beta}=D_{\alpha}C_{\beta}-D_{\beta}C_{\alpha}=\partial_{\alpha}C_{\beta}-\partial_{\beta}C_{\alpha}\,.$
(17)

In flat spacetime
$e^{\alpha}_{\alpha^{\prime}}=\delta^{\alpha}_{\alpha^{\prime}}$, the
Lagrangian density (Duality Symmetric Electrodynamics in Curved Spacetime)
reduces to the Lagrangian density introduced in Ref. [14]. Similarly to the
case of the latter article, a very interesting feature of the Lagrangian
density (Duality Symmetric Electrodynamics in Curved Spacetime) is that while
it contains a fixed four-vector $n^{\alpha^{\prime}}\\!$ and is therefore not
manifestly local-Lorentz-invariant, the motion of the system is fully local-
Lorentz-invariant 444It is even more remarkable that the Lorentz-invariance in
the flat spacetime case is known to hold also at the quantum level: the
partition function of the Zwanziger theory was shown to be Lorentz-invariant
in Refs. [25, 26]. We will illustrate that this property indeed holds for the
general-covariant case shortly after deriving the equations of motion (EOM).

Varying the action $S_{1}=\int d^{4}x\,\mathcal{L}_{1}$ with respect to the
four-potentials $A_{\mu}$ and $C_{\mu}$, we obtain the following EOM for the
electromagnetic field:

$\displaystyle\partial_{\zeta}\Bigl{(}\sqrt{-g}\,g^{\rho\beta}\gamma^{\zeta}\gamma^{\alpha}E_{\alpha\beta}-\sqrt{-g}\,g^{\zeta\beta}\gamma^{\rho}\gamma^{\alpha}E_{\alpha\beta}\Bigr{)}-$
(19)
$\displaystyle\qquad\qquad\qquad\qquad\;\;\epsilon^{\zeta\beta\sigma\rho}\partial_{\sigma}\Bigl{(}\gamma_{\zeta}\gamma^{\alpha}G_{\alpha\beta}\Bigr{)}=0\,,$
$\displaystyle\partial_{\zeta}\Bigl{(}\sqrt{-g}\,g^{\rho\beta}\gamma^{\zeta}\gamma^{\alpha}G_{\alpha\beta}-\sqrt{-g}\,g^{\zeta\beta}\gamma^{\rho}\gamma^{\alpha}G_{\alpha\beta}\Bigr{)}+$
$\displaystyle\qquad\qquad\qquad\qquad\;\;\epsilon^{\zeta\beta\sigma\rho}\partial_{\sigma}\Bigl{(}\gamma_{\zeta}\gamma^{\alpha}E_{\alpha\beta}\Bigr{)}=0\,,$

where we introduced $\gamma^{\alpha}\equiv
n^{\alpha^{\prime}}\\!e^{\alpha}_{\alpha^{\prime}}$, and used the following
identities for the Riemann tensor $R_{\alpha\beta\zeta\delta}$ and the
vierbein $e^{\alpha}_{\alpha^{\prime}}$:

$\displaystyle
R_{\alpha\beta\zeta\delta}+R_{\alpha\delta\beta\zeta}+R_{\alpha\zeta\delta\beta}=0\,,$
(20) $\displaystyle D_{\mu}e^{\alpha}_{\alpha^{\prime}}=0\,.$ (21)

Eq. (19) can be solved to obtain $\gamma^{\alpha}G_{\alpha\beta}$:

$\displaystyle\gamma^{\alpha}G_{\alpha\beta}=\frac{1}{2}\varepsilon^{\zeta\sigma\mu\nu}E_{\mu\nu}\gamma_{\zeta}g_{\sigma\beta}+P_{\beta}\,,$
(22)
$\displaystyle\gamma^{\alpha}P_{\alpha}=D_{\alpha}P^{\alpha}=0\,,\;\;\gamma^{\alpha}D_{\alpha}P^{\beta}=0\,.$
(23)

In particular, the solution (22) can be found by taking advantage of the
identity

$\displaystyle
K_{\alpha\beta}=\gamma_{\alpha}\gamma^{\mu}K_{\mu\beta}-\gamma_{\beta}\gamma^{\mu}K_{\mu\alpha}-\varepsilon_{\alpha\beta\gamma\sigma}\gamma^{\sigma}D^{\gamma}\,,\;\;\;$
(24) $\displaystyle\text{where}\quad
D^{\gamma}\equiv\frac{1}{2}\varepsilon^{\rho\gamma\sigma\zeta}\gamma_{\rho}K_{\sigma\zeta}\,,$
(25)

which holds for any antisymmetric tensor $K_{\alpha\beta}$. Indeed, applying
this identity to the tensor
$K_{\alpha\beta}=\varepsilon^{\zeta\sigma\mu\nu}E_{\mu\nu}g_{\alpha\zeta}g_{\sigma\beta}/2$,
and using the regularity of the four-potential $A_{\mu}$:

$\displaystyle\epsilon^{\mu\nu\lambda\rho}\,\partial_{\nu}E_{\lambda\rho}=0\,,$
(26)

one can transform Eq. (19) into:

$\displaystyle
D_{\zeta}\Bigl{(}L^{\rho}\gamma^{\zeta}-L^{\zeta}\gamma^{\rho}\Bigr{)}=0\,,$
(27) $\displaystyle\text{where}\quad
L_{\beta}\equiv\gamma^{\alpha}G_{\alpha\beta}-\frac{1}{2}\varepsilon^{\zeta\sigma\mu\nu}E_{\mu\nu}\gamma_{\zeta}g_{\sigma\beta}\,,\;\;\;$
(28)

from which the solution (22) trivially follows.

Now, we can substitute the solution (22) into the Eq. (19) to obtain:

$\displaystyle\partial_{\mu}\left(\sqrt{-g}\,g^{\mu\lambda}g^{\nu\rho}\bar{E}_{\lambda\rho}\right)=0\,,$
(29)

where we used the identity (24) once again, and denoted
$\bar{E}_{\lambda\rho}=E_{\lambda\rho}-\varepsilon_{\lambda\rho\mu\nu}\gamma^{\mu}P^{\nu}$.
Due to Eqs. (23), $P_{\alpha}$ is a vector field defined on the 3-surfaces
spanned locally by the vectors orthogonal to $\gamma^{\alpha}$, moreover this
vector field is covariantly constant along the integral curves of
$\gamma^{\alpha}$. Given Eq. (22), this means that $P_{\alpha}$ is fully
determined by the Cauchy data for the EM field $E_{\alpha\beta}$ and
$G_{\alpha\beta}$. It is therefore always possible to choose $P_{\beta}=0$,
that is $\bar{E}_{\lambda\rho}=E_{\lambda\rho}$. We are left with the
following local-Lorentz-covariant equation for the four-potential $A_{\mu}$:

$\displaystyle\partial_{\mu}\left(\sqrt{-g}\,g^{\mu\lambda}g^{\nu\rho}{E}_{\lambda\rho}\right)=0\,.$
(30)

Analogously to what was described in the previous paragraphs, one can first
solve the Eq. (19) to find $\gamma^{\alpha}E_{\alpha\beta}$, and then
substitute the solution into the Eq. (19). In this case, one obtains the
equation for the $C_{\mu}$ four-potential only:

$\displaystyle\partial_{\mu}\left(\sqrt{-g}\,g^{\mu\lambda}g^{\nu\rho}G_{\lambda\rho}\right)=0\,.$
(31)

We see that in the case of free electromagnetic field in curved spacetime, the
equations for the two four-potentials $A_{\mu}$ and $C_{\mu}$ are identical.
This is a manifestation of the symmetry of the Lagrangian (Duality Symmetric
Electrodynamics in Curved Spacetime), which is invariant with respect to the
$SO(2)$ rotation of the four-potentials:

$\displaystyle{A_{\mu}}\;\rightarrow\;{A_{\mu}}\cos\theta-{C_{\mu}}\sin\theta$
(32)
$\displaystyle{C_{\mu}}\;\rightarrow\;{C_{\mu}}\cos\theta+{A_{\mu}}\sin\theta\,\,.$

The latter invariance gives rise to the following conserved Noether current:

$s^{\nu}=\sqrt{-g}\,g^{\mu\lambda}g^{\nu\rho}\left(A_{\lambda}G_{\mu\rho}-C_{\lambda}E_{\mu\rho}\right)\,.$
(33)

As it will become evident after we introduce charged particles, this is
exactly the curved spacetime generalization of the helicity four-current
discussed in the previous section. One can now see that within the theory
(Duality Symmetric Electrodynamics in Curved Spacetime), EOM for the EM field
in the presence of gravity are fully consistent with the conservation of
helicity:

$D_{\mu}E^{\mu\nu}=0,\quad D_{\mu}G^{\mu\nu}=0\quad\Longleftrightarrow\quad
D_{\mu}s^{\mu}=0\,.$ (34)

Finally, let us note that the fixed four-vector $n^{\alpha^{\prime}}$
introduced in Eq. (Duality Symmetric Electrodynamics in Curved Spacetime) does
not enter the EOM (30) and (31) describing the propagation of the EM field:
the dynamics of the EM field is fully local-Lorentz-invariant. It is also
important that although we introduced the two four-potentials for the
description of the EM field instead of the only one normally used, the number
of degrees of freedom of the EM field remained the same. Similarly to the case
of the flat spacetime Maxwell equations, the curved spacetime Eqs. (19) and
(19) contain $6$ independent unknown functions, which are
$\gamma^{\alpha}E_{\alpha\beta}$ and $\gamma^{\alpha}G_{\alpha\beta}$ (note
the antisymmetricity of $E_{\alpha\beta}$ and $G_{\alpha\beta}$), and $6$
independent equations (note the divergencelessness of the left-hand-sides of
Eqs. (19) and (19)).

Motion of charged particles.—Let us now introduce charges into the theory. In
particular, we consider the following Lagrangian density for the interactions
between charged particles and the EM field:

$\displaystyle\mathcal{L}_{2}=\sqrt{-g}\,j_{\text{e}}^{\alpha}A_{\alpha}+\sqrt{-g}\,j_{\text{m}}^{\alpha}C_{\alpha}\,,$
(35)

where $j_{\text{e}}^{\alpha}$ and $j_{\text{m}}^{\alpha}$ are the electric and
magnetic four-currents, respectively. In particular, the currents are given by
the following expressions:

$\displaystyle
j_{\text{e}}^{\nu}(x)=\sum_{i}q_{i}\int\frac{\delta^{4}(x-x_{i}(\tau_{i}))}{\sqrt{-g}}\,dx_{i}^{\nu}\,,$
(36) $\displaystyle
j_{\text{m}}^{\nu}(x)=\sum_{i}g_{i}\int\frac{\delta^{4}(x-x_{i}(\tau_{i}))}{\sqrt{-g}}\,dx_{i}^{\nu}\,,$
(37)

where $x^{\nu}_{i}(\tau_{i})$ is the worldline of the $i$-th particle, $q_{i}$
($g_{i}$) is the electric (magnetic) charge of the $i$-th particle,
$\delta^{4}(x)$ is the Dirac delta distribution density. Note that from Eq.
(35), it follows that the $SO(2)$ rotation of the four-potentials (LABEL:rot2)
corresponds to the rotation in the electric-magnetic plane, so that the four-
current Eq. (33) is indeed the helicity of the EM field. The EOM for the four-
potentials $A_{\mu}$ and $C_{\mu}$ derived by varying the action $S=\int
d^{4}x\left(\mathcal{L}_{1}+\mathcal{L}_{2}\right)$ are:

$\displaystyle\partial_{\zeta}\Bigl{(}\sqrt{-g}\,g^{\rho\beta}\gamma^{\zeta}\gamma^{\alpha}E_{\alpha\beta}-\sqrt{-g}\,g^{\zeta\beta}\gamma^{\rho}\gamma^{\alpha}E_{\alpha\beta}\Bigr{)}-$
$\displaystyle\qquad\qquad\;\;\;\epsilon^{\zeta\beta\sigma\rho}\partial_{\sigma}\Bigl{(}\gamma_{\zeta}\gamma^{\alpha}G_{\alpha\beta}\Bigr{)}=-\sqrt{-g}\,j_{\text{e}}^{\rho}\,,$
(38)
$\displaystyle\partial_{\zeta}\Bigl{(}\sqrt{-g}\,g^{\rho\beta}\gamma^{\zeta}\gamma^{\alpha}G_{\alpha\beta}-\sqrt{-g}\,g^{\zeta\beta}\gamma^{\rho}\gamma^{\alpha}G_{\alpha\beta}\Bigr{)}+$
$\displaystyle\qquad\qquad\;\;\;\epsilon^{\zeta\beta\sigma\rho}\partial_{\sigma}\Bigl{(}\gamma_{\zeta}\gamma^{\alpha}E_{\alpha\beta}\Bigr{)}=-\sqrt{-g}\,j_{\text{m}}^{\rho}\,.$
(39)

As in the previous section, we can transform Eq. (Duality Symmetric
Electrodynamics in Curved Spacetime) as follows:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!D_{\zeta}\Bigl{(}L^{\rho}\gamma^{\zeta}-L^{\zeta}\gamma^{\rho}\Bigr{)}=-j^{\rho}_{\text{m}}\,,$
(40) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\text{where}\quad
L_{\beta}\equiv\gamma^{\alpha}G_{\alpha\beta}-\frac{1}{2}\varepsilon^{\zeta\sigma\mu\nu}E_{\mu\nu}\gamma_{\zeta}g_{\sigma\beta}\,,$
(41)

The solution for $\gamma^{\alpha}G_{\alpha\beta}$ is:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\gamma^{\alpha}G_{\alpha\beta}=\frac{1}{2}\varepsilon^{\zeta\sigma\mu\nu}E_{\mu\nu}\gamma_{\zeta}g_{\sigma\beta}-g_{\alpha\beta}Rj^{\alpha}_{\text{m}}+P_{\beta}\,,$
(42)

where $P_{\beta}$ has the properties detailed in Eqs. (23), so one can always
impose $P_{\beta}=0$ as discussed in the previous section;
$R\equiv\left(\gamma^{\mu}D_{\mu}\right)^{-1}$ is an integral operator with
the following kernel:

$\displaystyle R_{K}(x-y)=\int_{0}^{\infty}\\!\delta^{4}(x-y-\gamma s)\,ds\,.$
(43)

We then substitute the solution (42) into Eq. (Duality Symmetric
Electrodynamics in Curved Spacetime) to obtain:

$\displaystyle\partial_{\mu}\left(\sqrt{-g}\,g^{\mu\lambda}g^{\nu\rho}E^{\prime}_{\lambda\rho}\right)=-\sqrt{-g}\,j_{\text{e}}^{\nu}\,,$
(44)

where

$\displaystyle
E^{\prime}_{\lambda\rho}=E_{\lambda\rho}+\varepsilon_{\lambda\rho\mu\nu}\gamma^{\mu}Rj^{\nu}_{\text{m}}\,.$
(45)

Similarly, we find that Eq. (Duality Symmetric Electrodynamics in Curved
Spacetime) admits the following solution:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\gamma^{\alpha}E_{\alpha\beta}=-\frac{1}{2}\varepsilon^{\zeta\sigma\mu\nu}G_{\mu\nu}\gamma_{\zeta}g_{\sigma\beta}-g_{\alpha\beta}Rj^{\alpha}_{\text{e}}\,.$
(46)

Substituting it into Eq. (Duality Symmetric Electrodynamics in Curved
Spacetime), we obtain the following equation for the $C_{\mu}$ four-potential:

$\displaystyle\partial_{\mu}\left(\sqrt{-g}\,g^{\mu\lambda}g^{\nu\rho}G^{\prime}_{\lambda\rho}\right)=-\sqrt{-g}\,j_{\text{m}}^{\nu}\,,$
(47)

where

$\displaystyle
G^{\prime}_{\lambda\rho}=G_{\lambda\rho}-\varepsilon_{\lambda\rho\mu\nu}\gamma^{\mu}Rj^{\nu}_{\text{e}}\,.$
(48)

Eqs. (45) and (48) can be used to reformulate the conditions of the regularity
of the four-potentials (i.e. Eq. (26) and its analogue
$\epsilon^{\mu\nu\lambda\rho}\partial_{\nu}G_{\lambda\rho}=0$), in terms of
the tensors $E^{\prime}_{\mu\nu}$ and $G^{\prime}_{\mu\nu}$:

$\displaystyle\frac{1}{2}\,\epsilon^{\mu\nu\lambda\rho}\partial_{\nu}E^{\prime}_{\lambda\rho}=\sqrt{-g}\,j_{\text{m}}^{\mu}\,,$
(49)
$\displaystyle\frac{1}{2}\,\epsilon^{\mu\nu\lambda\rho}\partial_{\nu}G^{\prime}_{\lambda\rho}=-\sqrt{-g}\,j_{\text{e}}^{\mu}\,,$
(50)

where we used the conservation of the charged currents
$D_{\mu}j_{\text{e}}^{\mu}=D_{\mu}j_{\text{m}}^{\mu}=0$. Comparing these
equations with Eqs. (44) and (47), one sees that it is always possible to
impose the following condition:

$\displaystyle\frac{1}{2}\,\epsilon^{\mu\nu\lambda\rho}E^{\prime}_{\lambda\rho}=\sqrt{-g}\,g^{\nu\lambda}g^{\mu\rho}G^{\prime}_{\lambda\rho}\,.$
(51)

From Eqs. (44) and (47), we see that charged particles source the field
described by the tensors $E^{\prime}_{\mu\nu}$ and $G^{\prime}_{\mu\nu}$ in a
fully local-Lorentz-invariant way. However, Eqs. (45) and (48) show that the
latter field is related to the tensors $E_{\mu\nu}$ and $G_{\mu\nu}$ in a way
that breaks local Lorentz invariance: the difference depends on a fixed
Lorentz four-vector $n^{\alpha^{\prime}}$. It is a very interesting
observation made by Zwanziger in Ref. [14], that the flat spacetime analogues
of the would-be Lorentz-violating terms in Eqs. (45) and (48) have no
influence on physical observables: while the theory is not manifestly Lorentz-
invariant, all the possible observable processes described by it are Lorentz-
invariant. Let us now show that the curved spacetime case is similar in this
regard: all the physics described by the Lagrangian densities (Duality
Symmetric Electrodynamics in Curved Spacetime) and (35) is local-Lorentz-
invariant.

For this, we have to consider the EOM of charged particles in an EM field.
While the left-hand side of these EOM is given by the standard expression, the
right-hand side can be easily derived by extremizing the action with the
Lagrangian density (35). The result is:

$\displaystyle
g_{\lambda\rho}m_{i}u_{i}^{\nu}D_{\nu}u_{i}^{\rho}=\Bigl{(}q_{i}E_{\lambda\mu}(x_{i})+g_{i}G_{\lambda\mu}(x_{i})\Bigr{)}u_{i}^{\mu}\,,$
(52)

where $m_{i}$ is the mass of the $i$-th particle and
$u_{i}^{\nu}=dx_{i}^{\nu}/d\tau_{i}$. Let us now rewrite these EOM in terms of
the local-Lorentz-covariant tensors $E^{\prime}_{\lambda\rho}$ and
$G^{\prime}_{\lambda\rho}$ entering Eqs. (44) and (47):

$\displaystyle
g_{\lambda\rho}m_{i}u_{i}^{\nu}D_{\nu}u_{i}^{\rho}=\Bigl{(}q_{i}E^{\prime}_{\lambda\mu}(x_{i})+g_{i}G^{\prime}_{\lambda\mu}(x_{i})\Bigr{)}u_{i}^{\mu}+$
$\displaystyle\sum_{j}(q_{i}g_{j}-g_{i}q_{j})\\!\\!\int\\!\\!R_{K}(x_{i}-x_{j})\,\epsilon_{\mu\lambda\nu\rho}\gamma^{\mu}u^{\nu}_{i}u^{\rho}_{j}\,d\tau_{j}\,.\quad\;\;$
(53)

Due to the form of the kernel $R_{K}$ (Eq. (43)), the local-Lorentz-violating
term in Eq. (Duality Symmetric Electrodynamics in Curved Spacetime) is non-
zero only if the worldline of the $i$-th particle satisfies:

$\displaystyle x^{\nu}_{i}(\tau_{i})=x^{\nu}_{j}(\tau_{j})+\gamma^{\nu}s\,,$
(54)

for any $\tau_{i}\,,\tau_{j}\,,s\in(-\infty,\infty)$. Since there are more
equations than free parameters in this condition, it is satisfied only for
exceptional worldlines, i.e. the set of all the worldlines $x_{i}(\tau_{i})$
satisfying Eq. (54) is nowhere dense in spacetime. An interesting feature of
nowhere dense sets of worldlines is that they can never be measured, as every
measurement has a finite accuracy. Thus, the last term on the right-hand side
of Eq. (Duality Symmetric Electrodynamics in Curved Spacetime) can be safely
omitted, and we are left with the same form for the EOM as in Eq. (52), but
now with the EM field represented by the local-Lorentz-covariant tensors
$E^{\prime}_{\lambda\mu}$ and $G^{\prime}_{\lambda\mu}$. We therefore see that
the classical dynamics described by the theory with the Lagrangian density
$\mathcal{L}_{1}+\mathcal{L}_{2}$ is fully local-Lorentz-invariant.

One can wonder whether the local Lorentz invariance survives also at the
quantum level. The answer is yes, if the Dirac-Schwinger-Zwanziger (DSZ)
quantization condition is satisfied, in the full analogy to the flat spacetime
case. Indeed, it is enough to show that for closed particle paths, the path
integral does not depend on $n^{\alpha^{\prime}}$ [25, 26]. This means that
the following action

$\displaystyle\int
d^{4}x\,\mathcal{L}_{2}=\int_{\Sigma_{i}}\\!\\!d\Sigma^{\alpha\beta}\Bigl{(}q_{i}E_{\alpha\beta}+g_{i}G_{\alpha\beta}\Bigr{)}\,,$
(55)

where $\Sigma_{i}$ is a 2-surface with the boundary $x_{i}(\tau_{i})$, has to
be $n^{\alpha^{\prime}}$-independent modulo $2\pi n$ where $n\in\mathbb{Z}$.
Using Eqs. (45) and (48), we find the $n^{\alpha^{\prime}}$-dependent part of
this action:

$\displaystyle\sum_{j}(q_{i}g_{j}-g_{i}q_{j})\\!\\!\int_{\Sigma_{i}}\\!\\!d\Sigma^{\alpha\beta}\gamma^{\mu}\epsilon_{\alpha\beta\mu\nu}\times$
$\displaystyle\oint
dx_{j}^{\nu}\int_{0}^{\infty}\\!\\!\\!ds\;\delta^{4}(x-x_{j}-\gamma
s)=(q_{i}g_{j}-g_{i}q_{j})L_{ij}\,,\quad\;\;\;\;$ (56)

where $L_{ij}\in\mathbb{Z}$ is the Gauss linking number counting the number of
times the loop $x_{j}(\tau)$ intersects the oriented 3-surface
$\Sigma_{i}\\!\times\\!\gamma s$, $s\in[0,\infty)$. One sees that the path
integral of the theory is $n^{\alpha^{\prime}}$-independent as long as the DSZ
quantization condition is satisfied: $q_{i}g_{j}-g_{i}q_{j}=2\pi n$,
$n\in\mathbb{Z}$.

Electric and magnetic fields.—In the previous sections, we introduced the
theory of the EM field in curved spacetime using a general-covariant
formalism. We showed that the latter theory is electric-magnetic duality
invariant, see Eq. (34). Still, we started our discussion of the electric-
magnetic duality symmetry by writing down the curved spacetime Maxwell
equations (6)–(9) in terms of electric and magnetic fields and noticing the
asymmetry between $\mathbf{E}$ and $\mathbf{B}$. Let us see how the EM field
equations (44) and (47) written in terms of electric and magnetic fields are
reconciled with the electric-magnetic duality symmetry.

The key observation is that the EM field need not be defined in terms of the
only one pair ($\mathbf{E}$, $\mathbf{B}$) of electric and magnetic fields.
Indeed, the relevant dynamical variables for the EM field are four-potentials,
while the description in terms of electric and magnetic fields is a matter of
convenience. For example, whenever our system admits the presence of
hypothetical magnetic monopoles, there are two different ways to define the
magnetic field: it can either be defined as a force acting on a probe magnetic
charge at rest $\mathbf{B}_{\text{m}}$, or it can be defined in a standard way
via a force acting on an electric current $\mathbf{B}_{\text{e}}$. It turns
out that in a general curved spacetime background, these two definitions give
different results $\mathbf{B}_{\text{e}}\neq\mathbf{B}_{\text{m}}$. In
particular, guided by the EOM for the charges Eq. (Duality Symmetric
Electrodynamics in Curved Spacetime), we define the electric and magnetic
fields as follows:

$\displaystyle
E_{\text{e}}{}_{i}=-E^{\prime}_{0i}\,,\;\;B_{\text{m}}{}_{i}=-G^{\prime}_{0i}\,,$
(57) $\displaystyle
B_{\text{e}}{}_{i}=\frac{1}{2}\,\epsilon_{ijk}E^{\prime}_{jk}\,,\;\;E_{\text{m}}{}_{i}=-\frac{1}{2}\,\epsilon_{ijk}G^{\prime}_{jk}\,,$
(58)

so that $\mathbf{E}_{\text{e}}$ and $\mathbf{B}_{\text{e}}$ determine the
Lorentz force acting on the electric charges, while $\mathbf{B}_{\text{m}}$
and $\mathbf{E}_{\text{m}}$ determine the dual Lorentz force acting on the
magnetic charges. Then, the EM field equations in curved spacetime (44) and
(47) become:

$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\left\\{\sqrt{-g}\left[g^{00}\mathcal{G}\\!\cdot\\!\mathbf{E}_{\text{e}}-\mathbf{g}\left(\mathbf{g}\\!\cdot\\!\mathbf{E}_{\text{e}}\right)+\mathcal{G}\\!\cdot\\!\left(\mathbf{g}\\!\times\\!\mathbf{B}_{\text{e}}\right)\right]\right\\}=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;-\sqrt{-g}\rho_{\text{e}}\,,$
(59)
$\displaystyle\boldsymbol{\nabla}\\!\times\\!\left\\{\sqrt{-g}\left[\mathbf{g}\\!\times\\!\left(\mathcal{G}\\!\cdot\\!\mathbf{E}_{\text{e}}\right)+\frac{1}{2}\,\mathrm{tr}\left(\mathcal{G}\mathbf{S}\mathcal{G}\mathbf{S}\\!\cdot\\!\mathbf{B}_{\text{e}}\right)\right]\right\\}-$
$\displaystyle\;\frac{\partial}{\partial
t}\left\\{\sqrt{-g}\left[g^{00}\mathcal{G}\\!\cdot\\!\mathbf{E}_{\text{e}}-\mathbf{g}\left(\mathbf{g}\\!\cdot\\!\mathbf{E}_{\text{e}}\right)+\mathcal{G}\\!\cdot\\!\left(\mathbf{g}\\!\times\\!\mathbf{B}_{\text{e}}\right)\right]\right\\}=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;-\sqrt{-g}\,\mathbf{j}_{\text{e}}\,,$
(60)
$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\left\\{\sqrt{-g}\left[g^{00}\mathcal{G}\\!\cdot\\!\mathbf{B}_{\text{m}}-\mathbf{g}\left(\mathbf{g}\\!\cdot\\!\mathbf{B}_{\text{m}}\right)-\mathcal{G}\\!\cdot\\!\left(\mathbf{g}\\!\times\\!\mathbf{E}_{\text{m}}\right)\right]\right\\}=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\sqrt{-g}\rho_{\text{m}}\,,$
(61)
$\displaystyle\boldsymbol{\nabla}\\!\times\\!\left\\{\sqrt{-g}\left[\mathbf{g}\\!\times\\!\left(\mathcal{G}\\!\cdot\\!\mathbf{B}_{\text{m}}\right)-\frac{1}{2}\,\mathrm{tr}\left(\mathcal{G}\mathbf{S}\mathcal{G}\mathbf{S}\\!\cdot\\!\mathbf{E}_{\text{m}}\right)\right]\right\\}-$
$\displaystyle\;\frac{\partial}{\partial
t}\left\\{\sqrt{-g}\left[g^{00}\mathcal{G}\\!\cdot\\!\mathbf{B}_{\text{m}}-\mathbf{g}\left(\mathbf{g}\\!\cdot\\!\mathbf{B}_{\text{m}}\right)-\mathcal{G}\\!\cdot\\!\left(\mathbf{g}\\!\times\\!\mathbf{E}_{\text{m}}\right)\right]\right\\}=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\sqrt{-g}\,\mathbf{j}_{\text{m}}\,,$
(62)

where the notation from Eqs. (6)–(9) is used, and the components of the four-
currents are introduced $j^{\mu}_{i}=(\rho_{i},\mathbf{j}_{i})$. These
equations can be complemented by the conditions of the regularity of the four-
potentials Eqs. (49) and (50):

$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\mathbf{B}_{\text{e}}=\sqrt{-g}\,\rho_{\text{m}}\,,$
(63)
$\displaystyle\boldsymbol{\nabla}\\!\times\\!\mathbf{E}_{\text{e}}+\frac{\partial\mathbf{B}_{\text{e}}}{\partial
t}=-\sqrt{-g}\,\mathbf{j}_{\text{m}}\,,$ (64)
$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\mathbf{E}_{\text{m}}=\sqrt{-g}\,\rho_{\text{e}}\,,$
(65)
$\displaystyle\boldsymbol{\nabla}\\!\times\\!\mathbf{B}_{\text{m}}-\frac{\partial\mathbf{E}_{\text{m}}}{\partial
t}=\sqrt{-g}\,\mathbf{j}_{\text{e}}\,.$ (66)

The relations between different electric and magnetic fields Eqs. (51) in the
3-vector form are:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{E}_{\text{e}}=\sqrt{-g}\left[\mathbf{g}\\!\times\\!\left(\mathcal{G}\\!\cdot\\!\mathbf{B}_{\text{m}}\right)-\frac{1}{2}\,\mathrm{tr}\left(\mathcal{G}\mathbf{S}\mathcal{G}\mathbf{S}\\!\cdot\\!\mathbf{E}_{\text{m}}\right)\right]\,,$
(67)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{B}_{\text{e}}=-\sqrt{-g}\left[g^{00}\mathcal{G}\\!\cdot\\!\mathbf{B}_{\text{m}}-\mathbf{g}\left(\mathbf{g}\\!\cdot\\!\mathbf{B}_{\text{m}}\right)-\mathcal{G}\\!\cdot\\!\left(\mathbf{g}\\!\times\\!\mathbf{E}_{\text{m}}\right)\right]\,,\qquad\vspace*{-2pt}$
(68)

or equivalently:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{B}_{\text{m}}=-\sqrt{-g}\left[\mathbf{g}\\!\times\\!\left(\mathcal{G}\\!\cdot\\!\mathbf{E}_{\text{e}}\right)+\frac{1}{2}\,\mathrm{tr}\left(\mathcal{G}\mathbf{S}\mathcal{G}\mathbf{S}\\!\cdot\\!\mathbf{B}_{\text{e}}\right)\right]\,,$
(69)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{E}_{\text{m}}=-\sqrt{-g}\left[g^{00}\mathcal{G}\\!\cdot\\!\mathbf{E}_{\text{e}}-\mathbf{g}\left(\mathbf{g}\\!\cdot\\!\mathbf{E}_{\text{e}}\right)+\mathcal{G}\\!\cdot\\!\left(\mathbf{g}\\!\times\\!\mathbf{B}_{\text{e}}\right)\right]\,.\qquad\vspace*{-2pt}$
(70)

The system of Eqs. (Duality Symmetric Electrodynamics in Curved
Spacetime)–(66) is symmetric with respect to the $SO(2)$ electric-magnetic
duality transformations of the electric and magnetic fields supplemented by
the interchange $\sigma$ of the indices e and m:

$\displaystyle\mathbf{E}_{\alpha}\;\rightarrow\;\mathbf{E}_{\sigma(\alpha)}\cos\theta+\mathbf{B}_{\sigma(\alpha)}\sin\theta$
(71)
$\displaystyle\mathbf{B}_{\alpha}\;\rightarrow\;\mathbf{B}_{\sigma(\alpha)}\cos\theta-\mathbf{E}_{\sigma(\alpha)}\sin\theta\,\,,\vspace*{-3pt}$

as well as by the similar $SO(2)$ rotation of the charges and currents.

Since there are only electrically charged particles in the laboratory, the
part of the EM field EOM (Duality Symmetric Electrodynamics in Curved
Spacetime)–(66) relevant for the GW detection features only the
$\mathbf{E}_{e}$ and $\mathbf{B}_{e}$ fields and therefore has the same
structure as Eqs. (6)–(9). This means that the duality-symmetric theory of the
EM field in curved spacetime presented here yields no novel predictions for
the GW detection experiments.

Conclusions.—Ever since the start of the XXth century, there has been a
prevailing point of view in physics community that the electromagnetic field
represents a physical entity separate from charged particles used to measure
it. In this article, we showed that this perspective, which is currently
adopted in any exhaustive quantization scheme, leads to the tension within the
standard description of the electromagnetic field in General Relativity, since
the core principle of General Relativity – the equivalence principle – as
applied to the electromagnetic field itself, is violated. In particular, we
showed that in the standard description, the otherwise conserved Noether
charge corresponding to the electric-magnetic duality symmetry of the
electromagnetic field is no longer conserved in the presence of gravity. We
also showed that even if one rejects the above-mentioned notion of the
electromagnetic field as an entity independent of the charges used to probe
it, the standard description still conflicts with the equivalence principle
whenever hypothetical magnetic charges are allowed to exist. Meanwhile, the
possibility that magnetic charges exist is considered highly likely within the
theoretical physics community due to the phenomenon of the quantization of
charge observed in nature [22], due to the predictions of Grand Unified
Theories [27, 28], as well as due to the implications of the general
requirement that quantum theory and gravity coexist [29].

Motivated by these tensions within the standard description, we constructed an
electric-magnetic duality symmetric theory of the interactions between
electromagnetic field and gravity. In particular, we proposed a general-
covariant generalization of the flat spacetime two-potential formalism
developed earlier in Ref. [14]. We showed that although the Lagrangian of the
theory is not manifestly local-Lorentz-invariant, all the physics described by
it respects local Lorentz symmetry. We also found an interesting difference
with respect to the flat spacetime case: in general, the electromagnetic field
in the presence of gravity has to be described in terms of two pairs of
electric and magnetic fields. While the one pair of fields can be defined by
their action on a probe electric charge or current, the other pair can be
defined by their action on a probe magnetic charge or current. The duality
symmetry transformation then necessarily involves an interchange of these two
pairs of fields.

###### Acknowledgements.

Acknowledgments.— The author thanks an anonymous referee and Andreas Ringwald
for valuable comments on the manuscript, as well as Kevin Zhou, Sebastian A.
R. Ellis and Jai-chan Hwang for fruitful discussions. The author is funded by
the UK Research and Innovation grant MR/V024566/1. During his time at DESY,
the author was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC 2121 Quantum
Universe – 390833306.

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# Transformer-based Entity Legal Form Classification

Alexander Arimond  Sociovestix LabsEdinburgh, Scotland, UK
<EMAIL_ADDRESS>,  Mauro Molteni  Sociovestix LabsEdinburgh,
Scotland, UK<EMAIL_ADDRESS>,  Dominik Jany  GLEIFBasel,
Switzerland<EMAIL_ADDRESS>,  Zornitsa Manolova  GLEIFBasel,
Switzerland<EMAIL_ADDRESS>,  Damian Borth  University of St.
Gallen (HSG)St. Gallen, Switzerland<EMAIL_ADDRESS>and  Andreas G.F.
Hoepner  University College DublinDublin, Ireland<EMAIL_ADDRESS>
###### Abstract.

We propose the application of Transformer-based language models for
classifying entity legal forms from raw legal entity names. Specifically, we
employ various BERT variants and compare their performance against multiple
traditional baselines. Our evaluation encompasses a substantial subset of
freely available Legal Entity Identifier (LEI) data, comprising over 1.1
million legal entities from 30 different legal jurisdictions. The ground truth
labels for classification per jurisdiction are taken from the Entity Legal
Form (ELF) code standard (ISO 20275).

Our findings demonstrate that pre-trained BERT variants outperform traditional
text classification approaches in terms of F1 score, while also performing
comparably well in the Macro F1 Score. Moreover, the validity of our proposal
is supported by the outcome of third-party expert reviews conducted in ten
selected jurisdictions.

This study highlights the significant potential of Transformer-based models in
advancing data standardization and data integration. The presented approaches
can greatly benefit financial institutions, corporations, governments and
other organizations in assessing business relationships, understanding risk
exposure, and promoting effective governance.

transformer, language models, entity legal forms, standardization

††doi: XXXXXXX.XXXXXXX††copyright: none

##  1\. Introduction

Table 1. Examples of inconsistent legal form representations | Legal name | Jurisdiction | Legal form name | ELF code
---|---|---|---|---
(a) | Dean Quarry Apartments LLC | US-NY | Limited Liability Company | SDX0
(b) | RUBICON TECHNOLOGY MANAGEMENT L.L.C. | US-DE | Limited Liability Company | HZEH
(c) | LOCKWOOD RIVERFRONT HOTEL, LLC | US-DE | Limited Liability Company | HZEH
(d) | GIANT Weilerswist g21 GmbH | DE | Gesellschaft mit beschränkter Haftung | 2HBR
(e) | Selbstfahrer Union G.m.b.H. | DE | Gesellschaft mit beschränkter Haftung | 2HBR
(f) | Interproximal AB | SE | Aktiebolag | XJHM
(g) | Konstlist i Heby Aktiebolag | SE | Aktiebolag | XJHM
(h) | Aktiebolaget Clas Grönwalls Lantbrukstjänst ilkividation | SE | Aktiebolag | XJHM
(i) | Infrastrukturentwicklungsgesellschaft Hilden mbH | DE | Gesellschaft mit beschränkter Haftung | 2HBR
(j) | Katholische Kirchengemeinde Maria Königin Lingen | DE | Körperschaft des öffentlichen Rechts | SQKS
(k) | むつ小川原風力合同会社 | JP | 合同会社 | 7QQ0
(l) | 合同会社まつお | JP | 合同会社 | 7QQ0

Identifying and understanding an entity’s legal form is a crucial component in
many financial and business related processes. The legal form and structure of
corporations can inform how to conduct transactions effectively and serves as
a risk indicator. Legal forms indicate the owner’s responsibility in case of
financial problems. Depending on the legal form, an entity’s owner might only
be liable to a limited amount of money. This may determine whether an entity
tends to make risky decisions which in turn will affect investors and
creditors (Jensen and Meckling, 1976; Campos and Requejo, 2003). Automating
the process of identifying an entity’s legal form can therefore lower risk,
create transparency and increase operational efficiency by enabling straight-
through processing (STP) capabilities.

Knowing the entity legal form also has high value in data integration and
deduplication tasks (Eurostat, 2010). Depending on the task at hand, entity
data needs to be linked across several sources, while often times there is no
common unique identifier available. As a consequence, entity linkage methods
typically aim to identify similarities between legal names, addresses and
other features of the entities. Due to differing representations of such
features across sources, this methodology usually results in many ambiguous
entity mapping pairs and thus requires additional manual effort. Introducing
the entity’s legal form via standardized data items adds an additional layer
of confidence into such entity linkage tasks, enabling robust mapping pairs
across multiple datasets (Kruse et al., 2021), as each entity can only have
one single legal form.

The wide range of entity legal forms existing within and between different
jurisdictions has made it challenging for organizations to categorize and
structure this information effectively. This task becomes even more difficult
due to the similarities in types and textual representation of these legal
forms across jurisdictions. For instance, as one can see in the examples (a)
and (b) in Table 1, different jurisdictions can have their distinct versions
of a ”Limited Liability Company” (LLC), where each one operates under its
specific legal framework. It is essential to note that even if multiple
jurisdictions, like US-Delaware and US-New York, have LLCs, they are distinct
entity forms. The complexity therefore lies in handling the diversity of
entity legal forms across jurisdictions and the need to distinguish between
seemingly similar legal structures.

Furthermore, the representation of legal forms within entity names is
influenced by the cultural and linguistic context of the jurisdiction and the
specific legislation used to manage registrant information. Expert interviews
have highlighted that, for example, many entities in France are not legally
obligated to include their legal form in their names.

Additionally, entity legal forms of the same type may be inconsistently
represented, especially across different data sources. These discrepancies
often manifest in different punctuation styles, as shown in the examples (d)
and (e) in Table 1, or in the use of various forms of abbreviations as
demonstrated by the entities (f), (g) and (h): the legal form ”Aktiebolag” can
be written out in full at the end of the name or represented by the
abbreviation ”AB”. Moreover, it can also appear at the beginning of the
entity’s name as ”Aktiebolaget”. In Germany, according to its official
business register of the German Federal Statistical Office 2018, of the names
of the more than 9,500 entities that are stock corporations, 13% include the
term ”Aktiengesellschaft” while 87% include the abbreviation ”AG”. Similar
figures result for the almost 700,000 private limited companies (Gesellschaft
mit beschränkter Haftung, GmbH) (Laurent, 2021).

Inconsistencies are further introduced by variations in the capitalization of
individual characters or the entire name. Many business registers default to
representing all information in uppercase letters, which adds to the
complexity, as shown with examples (b) and (c) in Table 1. Also, the
intermingling of legal form and name elements as in example (i), the absence
of any mention of the legal form like in example (j), or the usage of non-
latin characters in examples (k) and (l), exacerbates the challenge of
accurately detecting legal forms, in particular if data users lack local
domain expertise. In summary, the significant variability in representation
presents considerable obstacles for any automated identification approach.

In this paper, we explore the application of machine learning and deep
learning techniques to accurately classify entity legal forms. Our focus lies
specifically on utilizing the Entity Legal Form (ELF) code (ISO
20275)111https://www.iso.org/standard/67462.html as a fundamental basis for
classification. The ELF code standard serves as a comprehensive solution for
standardized legal form representation and is incorporated in the open LEI
data, which encompasses over 2.4 million entities worldwide as of July 2023.
Consequently, the LEI data and the ELF code standard provide an optimal data
source for training classifiers dedicated to legal form detection.

We address the challenge of legal form classification by employing a novel
approach using Transformer language models based on the BERT architecture in
combination with standardized ELF codes. The results of the Transformer models
are compared to a traditional Bag-of-Words setting. To evaluate the
performance of both Transformer and traditional approaches, a substantial
subset of LEI data comprising over 1 million legal entities from 30 different
legal jurisdictions is employed as the evaluation dataset. Lastly, we
conducted an expert review using 7,256 entities, further corroborating the
plausibility of our findings.

###  1.1. LEI Data

As a basis for our work we use the open and freely available Global LEI
Repository. As of July 2023, there are over 2.4 million entities listed in the
LEI system. This global directory is centered on the Legal Entity Identifier
(LEI). The LEI is a 20-digit alpha-numeric code based on the ISO 17442
standard222https://www.iso.org/standard/78829.html. It connects to the key
reference information that enables clear and unique identification of legal
entities, and greatly enhances transparency by making available dependable and
trusted data free of charge. The reference data is maintained by so-called LEI
issuers that are distributed across the globe. These LEI issuing organizations
apply their local knowledge to verify the information of each LEI record. ELF
code, legal name and jurisdiction are part of each LEI
record333https://www.gleif.org/en/lei-data/access-and-use-lei-data. The data
is easily accessible as csv, json, xml444https://www.gleif.org/en/lei-
data/gleif-golden-copy/download-the-golden-copy/ and via a freely accessible
API555https://www.gleif.org/en/lei-data/gleif-api. The Global Legal Entity
Identifier Foundation (GLEIF) ensures high data quality and offers transparent
data quality reporting666https://www.gleif.org/en/lei-data/gleif-data-quality-
management/quality-reports 777https://dq-dashboard.gleif.org/dashboard and can
therefore be considered a suitable source for our work.

###  1.2. Entity Legal Forms (ELF) Code List

The ELF code, established by the International Organization for
Standardization (ISO)11footnotemark: 1, is a unique 4-digit alpha-numeric code
and serves as a comprehensive solution for standardized legal form
representation. As of July 2023 there are 3,250 legal forms in 175
jurisdictions worldwide available888https://www.gleif.org/en/about-lei/code-
lists/iso-20275-entity-legal-forms-code-list in version 1.4.1 of the openly
accessible ELF code list. GLEIF has been acting as the maintenance agency
secretariat of the ELF code list since 2017, regularly introducing new legal
forms and jurisdictions. By way of example, Table 3 shows the ELF codes for
US-Delaware for entities that have been revalidated
recently999Entity.EntityStatus == ”ACTIVE” and Registration.RegistrationStatus
== ”ISSUED”. Each jurisdiction has its unique list of legal forms. Legal forms
that appear to be similar across jurisdictions must be treated individually
due to differing legislature within each jurisdiction. ELF code 8888 is a so-
called reserved code that is used in case no specific legal form can be
assigned. Notably, the legal forms are not equally distributed within US-
Delaware. Similarly imbalanced legal form data can be observed in all
jurisdictions.

Table 2. Legal forms and their ELF codes in US-Delaware for entities in scope99footnotemark: 9 Legal form name | ELF code | #Entities
---|---|---
Commercial Bank | 9ASJ | 2
Corporation | XTIQ | 5,379
Limited Liability Company | HZEH | 30,553
Limited Liability Limited Partnership | TGMR | 44
Limited Liability Partnership | 1HXP | 64
Limited Partnership | T91T | 9,707
Non-deposit Trust Company | MIPY | 2
Partnership | QF4W | 16
Savings Bank | JU79 | 0
Statutory Trust | 4FSX | 1,266
Unincorporated Nonprofit Association | 12N6 | 1
Legal form not yet in code list | 8888 | 7,118

Using this standardized and up-to-date list of entity legal forms, data users
are enabled to uniquely identify the legal forms of entities around the globe
without having any local knowledge about language or legislature. This
benefits data users as they do not need to carry out costly legal name
analysis. For instance, there are more than 500 legal form checks implemented
in Deutsche Bundesbank’s Financial Statement Data Pool on approximately
125,000 entities each year. These checks would not be necessary, if all
entities were assigned with an ELF code (Laurent, 2021).

The remainder of this work is structured as follows: In Section 2, we provide
an overview of related work. Section 3 follows with an explanation of our
baseline setup with traditional Bag-of-Words, as well as a description of the
methodology of using Transformers in the context of legal form classification.
The experimental setup and results are outlined in Section 4, while Section 5
showcases the results of an expert review. In Section 6, the paper concludes
with a summary of the work.

##  2\. Related work

In this section, we present our literature study and existing approaches to
entity legal form classification, while highlighting similarities and
differences to our work.

Schild et al. (Schild et al., 2017) presented a technical report that
demonstrated record linkage on company entities using machine learning. They
applied and evaluated their approach by matching seven distinct German
datasets. Their matching approach involved comprehensive preprocessing,
including legal form pattern extraction and standardization, achieved through
regular expression matching. Despite impressive results (90% coverage),
creating meaningful regular expressions required local expertise in German
legal forms and is difficult to scale to non-german jurisdictions. In
contrast, our proposed approach is more automated, with the underlying model
autonomously identifying relevant tokens representing legal forms.

A similar rule-based approach is taken by the open-source Python library
cleanco101010https://github.com/psolin/cleanco. This library aims to detect
and eliminate the legal form from a company’s name by incorporating an
extensive list of legal form abbreviations and commonly encountered suffixes.
According to the main author111111https://paulsolin.com/2014/03/16/cleanco-a-
python-module-for-checking-business-names/, the rules were mainly distilled
from Wikipedia
articles121212https://en.wikipedia.org/wiki/List_of_legal_entity_types_by_country.
The library’s coverage encompasses 16 distinct entity types mapped to 66
countries. Notably, the library does not apply the concept of jurisdictions
within countries. For instance, it does not differentiate individual states
within the United States of America. Assigning the legal forms to their
respective states is necessary as each state represents a unique jurisdiction
in which similar sounding legal forms can have different legal implications.
Also, the legal forms that are returned to the user do not represent the
actual local name of the legal form but are instead translated into an English
legal form that is assumed similar to the actual legal form. This can cause
ambiguity as legal forms are unique within their respective jurisdictions and
should therefore be represented in their local naming convention to avoid any
misjudgement. In our work we avoid any ambiguity of legal forms by utilizing
ELF codes that provide a unique link to the local name of the legal form.

Loster et al. (Loster et al., 2018) have worked on dissecting company names
into various elements including the core name, legal form, personal names and
location names. To address this task, they formalized the problem as a
supervised sequence labelling task and evaluated a linear chain conditional
random field classifier. Furthermore, they explored an extensive set of
potential features on a manually annotated dataset consisting of 1,500
randomly selected company names, primarily of German origin. For 1,484 out of
the 1,500 annotated legal names they were able to annotate one part of the
legal name as a representation of the entity’s legal form. For recognition of
the legal form tag, their approach achieved a 98% F1-Score. We showcase in our
work how Transformer based ELF code classification models can be utilized to
attribute legal form relevance on token level.

Kruse et al. (Kruse et al., 2021) explicitly tackle the problem of legal form
classification in the context of data integration in the absence of identifier
keys. In particular, they claim that extracting the legal form into a separate
attribute (besides name, address and others) can enhance the performance of
record linkage. Their hybrid approach to legal form extraction - which they
evaluated on a German-only subset - involves a rule-based component for
identifying the legal form. We not only claim that this can be achieved
through machine learning as presented in this work, but that its beneficial to
learn statistical relationships in cases where names lack explicit
representation of the corresponding legal form.

##  3\. Methodology

In this section, we present how to identify entity legal forms by applying
text classification techniques. We distinguish between a traditional approach,
and a modern approach based on Transformer neural networks.

###  3.1. Traditional Approach

Text classification has been widely studied in the field of Natural Language
Processing (NLP) and has important applications like sentiment analysis, news
categorization, email classification and spam filtering (Sebastiani, 2002;
Aggarwal and Zhai, 2012). In this regard, a wide variety of traditional
machine learning techniques have been successfully applied for decades, such
as Naive Bayes classifiers, Decision Tree classifiers, Random Forests and
Support Vector Machines.

Our baseline approach to legal form classification follows a traditional text
classification pipeline setup, consisting of pre-processing of input text, a
feature selection step and the training of a classifier.

The pre–processing ”cleans up” the input text to achieve a minimum degree of
harmonization, which in turn eases subsequent processing and enhances
classification performance. For this, we compared two pre-processing
approaches:

  * (a)

we apply a simple approach by transforming each input name string to lower
case letters. This ensures a minimal degree of harmonization of the input
names. This is the default setup.

  * (b)

we adopt a more extensive set of harmonization rules by following ideas for
record linkage as presented in (Magerman et al., 2006) and (Magnani and
Montesi, 2007). This includes (1) converting the string to lower case, (2)
replacing diacritics, (3) replacing multi-spaces, (4) removing double
quotation marks, (5) replacing trailing non-alphanumeric characters, (6)
correcting commas and periods, (7) applying purge rules and (8) replacing
multi-spaces. Our purge step removes special characters like ”-”, ”(”, ”)”,
”;”, ”/”, ”,” with simple white spaces and converts ” & ” and ” + ” to ” and
”. We denote this setup with ” + prep”.

In the feature selection step, we chose to transform each input legal entity
name into their Bag-of-Words representation, due to its simplicity for
classification purposes (Aggarwal and Zhai, 2012). For this, we split the
(pre-processed) legal name strings at their white spaces into their set of
words. In typical document classification scenarios, each word gets assigned
its frequency in the document. In contrast, since legal names are comparably
short and therefore rare to have duplicate words, we decided to only count
each word once by default, resulting in a binary representation vector:

(1) $X_{j}=\begin{cases}1&\text{if legal name contains word}w_{j}\\\
0&\text{otherwise}\end{cases}$

The words in the training corpus constitute the so called vocabulary. Its size
is comparably large, as legal names by nature consist of very distinctive
words. This also means that the vector dimensionality is very high. Also, it
should be noted that this kind of representation disregards information about
the position of words in the input name.

For classification we experimented with several methods, the first to mention
is the Complement Naive Bayes (CNB) algorithm, which is an adaptation of the
standard Multinomial Naive Bayes algorithm, which is particularly suited for
imbalanced data sets (Rennie et al., 2003). Furthermore, we applied a Decision
Tree classifier (DT), which divides the underlying data space with the use of
different text features (Aggarwal and Zhai, 2012). For a given text instance,
the most likely partition can then be assigned by applying a set of learned
decision rules, which is then used for the purposes of classification. We also
applied Random Forest (RF) (Breiman, 2001), which - as an ensemble method -
fits a number of decision tree classifiers on sub-samples of the data and uses
averaging to improve predictive accuracy and avoid over-fitting. As a last
method we have used Support Vector Machines (SVM), which attempt to partition
the data with the use of non-linear delineations between the different classes
(Aggarwal and Zhai, 2012). For all presented traditional classifiers, we make
use of the implementations of the python machine learning library scikit-
learn131313https://scikit-learn.org in its version 1.3.0. We do not apply any
modifications to the default parameter settings as defined by the library.

###  3.2. Transformers

In recent years, with the rise of deep learning, and in particular the
Transformer model (Vaswani et al., 2017), a new class of neural network
language models have surpassed traditional machine learning–based approaches
in many text classification benchmarks (Minaee et al., 2021).

As our second approach to entity legal form classification, we chose to
explore the usage of Transformer language models. One of the most popular of
these models are Bidirectional Encoder Representations from Transformers
(BERT) (Devlin et al., 2018) by Google, which can be pre-trained on large
corpuses of unlabelled text. A pre-trained BERT model can be easily fine-tuned
to various NLP tasks - including text classification - and as such enables
low-resource tasks to benefit from deep architectures.
Google141414https://github.com/google-research/bert itself, as well as
HuggingFace (Wolf et al., 2019) provide many variants of BERT, including the
original “base” English and multilingual versions, to be further used and
fine-tuned by the community. In this study, we apply BERT to the entity legal
form classification problem, evaluating different variants of pre-trained BERT
models.

Language models come along with pre-trained tokenizers specifically tied to
the model. BERT uses WordPiece, which tokenizes the input string into sub-word
units, resulting in a vocabulary size of 30,000 tokens. By using sub-word
units, it can effectively deal with rare words (as, for instance, given in
legal names), and allows for a good balance between the flexibility of single
characters and the efficiency of full words (Wu et al., 2016). We omitted any
custom pre-processing or tokenization in favor of following the end-to-end
processing as defined by the BERT variants151515Some models may incorporate
specific pre–processing steps, such as uncased versions of BERT that strip our
accent markers..

In contrast to the Bag-of-Words model, which loses information about the
position of words, BERT is a sequence model. The output of the model is an
embedding vector that is supposed to capture the whole sequence of tokens,
including positional context information. When fine-tuning the model for
classification, the sequence embedding serves as input to a classification
head (usually a single layer of randomly initialized weights). During
training, not only the classification head, but also the pre-trained weights
within the model are tuned to the task. We will outline in section 4 how these
characteristics can be helpful to identify those tokens of a legal name that
are relevant for the legal form attribution.

Due to its availability and strong performance, a zoo of pre-trained and fine-
tuned variations of BERT has emerged, in particular with the goal to account
for the usage in different languages. This includes models such as German BERT
and Italian BERT by DMBMDZ161616https://github.com/dbmdz/berts and
deepset171717https://www.deepset.ai/german-bert, BETO Spanish BERT (Cañete et
al., 2020), BERTje as a Dutch BERT (de Vries et al., 2019), Danish
BERT181818https://github.com/certainly.io/nordic_bert by
Certainly191919https://certainly.io/blog/danish-bert-model/, Swedish
BERT202020https://huggingface.co/KB/bert-base-swedish-cased by the National
Library of Sweden / KLab, BERT for Finnish (Virtanen et al., 2019), Polbert
for Polish212121https://huggingface.co/dkleczek/bert-base-polish-cased-v1 by
Darek Kłeczek, BERTimbau for Portuguese (Souza et al., 2020), and Japanese
BERT222222https://github.com/cl-tohoku/bert-japanese/tree/v1.0 by Tohoku
University.

For each jurisdiction, we evaluated a different set of BERT variants, mainly
driven by the official language(s) within the respective jurisdiction. For
instance, in US jurisdictions we tested the standard BERT base variants
trained on an English corpus, whereas in non-English jurisdictions we tested
language-specific variants. As an exception to this we have tested on all
jurisdictions - except on Japan - the multilingual version of
BERT232323https://huggingface.co/bert-base-multilingual-uncased, which is pre-
trained on 102 different languages. By doing so we aim to better catch
language-specific intricacies in legal names.

Another important aspect is the ”casing” of the models. Some BERT variants are
trained as ”cased” models, whereas others are trained as ”uncased”. The
difference is in the way the text is handed to the model’s tokenizer. In the
first case, the text is tokenized as is, including any capitalized letters. In
the latter case, the text is lower–cased before training. We decided to test
both, the cased and the uncased versions, where available.

The commonly available BERT models are usually trained on large corpuses of
text sentences with the goal of generalizing to many domains. Since the
majority of these sentences should not be related to any legal or financial
context, we came to question if a more domain specific model might lead to
better performance. We therefore evaluated FinBERT (Huang et al., 2023; Yang
et al., 2020), which has been specifically pre-trained for the financial
domain on corporate 10-K & 10-Q ports, as well as earning call transcripts and
analyst reports.

Regardless of the specific model variation or underlying jurisdiction data, we
optimized each model for 5 epochs with an AdamW optimizer as described in
(Loshchilov and Hutter, 2017), with a learning rate $\gamma=0.00002$ and a
weight decay of $\lambda=0.01$.

##  4\. Evaluation

In this section, we explain our experimental setup, evaluate classifcation
performance of Traditional and Transformer approaches, and present various
findings related to entity legal form classification.

Table 3. Comparison of Traditional and Transformer models. For Traditional models, the best performing model is shown for both F1 and Macro F1 (F1-M) score separately. For Transformers, the best BERT variant is selected solely by F1 score, but both its scores are shown. LEI data | Traditional | Transformer
---|---|---
Jurisdiction | #entities | #ELF | Best variant by F1 | Best variant by F1-M | Best variant by F1 | F1 | F1-M
DE | 135,079 | 31 | RF + prep | 0.9537 | DT + prep | 0.5906 | bert-base-german-uncased | 0.9616 | 0.6174
IT | 104,968 | 50 | SVC + prep | 0.8990 | DT + prep | 0.3218 | bert-base-italian-uncased | 0.9010 | 0.3121
NL | 89,748 | 20 | RF + prep | 0.9812 | RF + prep | 0.7529 | bert-base-multilingual-uncased | 0.9847 | 0.7676
IN | 87,491 | 35 | SVC + prep | 0.8845 | DT | 0.4872 | bert-base-multilingual-uncased | 0.8862 | 0.4705
ES | 84,231 | 41 | RF + prep | 0.9491 | DT + prep | 0.5219 | bert-base-multilingual-uncased | 0.9505 | 0.5191
GB | 74,847 | 29 | SVC + prep | 0.9666 | DT + prep | 0.4081 | bert-base-uncased | 0.9690 | 0.4047
FR | 59,973 | 165 | SVC + prep | 0.5769 | DT | 0.1890 | bert-base-multilingual-cased | 0.5710 | 0.1107
DK | 56,226 | 22 | RF + prep | 0.9349 | RF + prep | 0.5870 | danish-bert-botxo | 0.9444 | 0.5941
US-DE | 54,156 | 12 | SVC | 0.9871 | RF + prep | 0.6094 | finbert-pretrain | 0.9878 | 0.5719
SE | 48,083 | 18 | RF + prep | 0.9789 | RF + prep | 0.5424 | bert-base-multilingual-uncased | 0.9854 | 0.5647
FI | 35,587 | 52 | RF + prep | 0.9839 | DT + prep | 0.5618 | bert-base-finnish-uncased-v1 | 0.9858 | 0.5978
LU | 33,683 | 28 | SVC | 0.8565 | DT + prep | 0.4279 | bert-base-multilingual-uncased | 0.8761 | 0.3817
NO | 32,996 | 27 | RF + prep | 0.9888 | DT + prep | 0.6815 | bert-base-multilingual-uncased | 0.9910 | 0.5942
AT | 24,433 | 21 | RF + prep | 0.9411 | DT + prep | 0.5496 | bert-base-german-uncased | 0.9635 | 0.6001
BE | 23,969 | 41 | SVC + prep | 0.5089 | DT | 0.1444 | bert-base-multilingual-uncased | 0.5344 | 0.1391
KY | 20,541 | 13 | RF + prep | 0.7280 | DT + prep | 0.4627 | bert-base-multilingual-uncased | 0.7108 | 0.3844
PL | 20,173 | 36 | DT + prep | 0.9898 | DT + prep | 0.6252 | bert-base-polish-uncased-v1 | 0.9879 | 0.5355
AU | 15,350 | 13 | SVC + prep | 0.8887 | DT + prep | 0.3301 | bert-base-multilingual-uncased | 0.8861 | 0.3198
IE | 15,294 | 19 | RF | 0.9189 | DT | 0.5116 | bert-base-uncased | 0.9251 | 0.4569
VG | 15,086 | 9 | SVC + prep | 0.8663 | DT | 0.2696 | bert-base-multilingual-uncased | 0.8374 | 0.1622
CZ | 14,477 | 52 | RF + prep | 0.9893 | RF + prep | 0.4355 | bert-base-multilingual-uncased | 0.9908 | 0.3824
EE | 13,824 | 13 | RF + prep | 0.9954 | RF + prep | 0.6291 | bert-base-multilingual-uncased | 0.9965 | 0.6329
CH | 13,742 | 28 | RF + prep | 0.9211 | RF + prep | 0.4066 | bert-base-multilingual-uncased | 0.9367 | 0.3902
HU | 10,041 | 33 | RF + prep | 0.9326 | DT | 0.5791 | bert-base-multilingual-uncased | 0.9265 | 0.4511
JP | 9,690 | 12 | RF + prep | 0.8968 | DT + prep | 0.2598 | bert-base-japanese | 0.9828 | 0.4400
LI | 9,458 | 13 | CNB + prep | 0.9522 | CNB + prep | 0.7708 | bert-base-multilingual-uncased | 0.9525 | 0.6616
US-MA | 6,987 | 13 | RF | 0.9548 | DT | 0.5107 | bert-base-multilingual-uncased | 0.9501 | 0.4969
PT | 6,427 | 20 | RF + prep | 0.9129 | RF + prep | 0.2950 | bert-base-multilingual-uncased | 0.9088 | 0.2566
US-CA | 6,176 | 14 | RF + prep | 0.9362 | RF + prep | 0.4067 | bert-base-uncased | 0.9399 | 0.3896
US-NY | 4,836 | 10 | RF + prep | 0.9520 | DT + prep | 0.4998 | bert-base-uncased | 0.9582 | 0.5250

LEI dataset preparation: We have evaluated the presented approaches on a
subset of the publicly available LEI data242424Downloaded as 2022/09/14 8:00
GMT from https://www.gleif.org/en/lei-data/gleif-golden-copy/download-the-
golden-copy., containing over 1.1 million legal entities from the 30 largest
jurisdictions measured by their total number of entities252525We did not
include China because their class distribution is heavily skewed towards the
ELF code ECAK (企业) with 97.8% of all its entities. Also, we did not include
Canada due to difficulties in resolving legal forms on state and on sub-
division level.. We prepared a separate dataset for each
jurisdiction262626Based on the field Entity.LegalJurisdiction, including the
entity legal name as well as the ELF code. We only selected legal entities
that are active and have recently been revalidated99footnotemark: 9. This
ensures that the ELF codes assigned to the entities utilize the latest version
of the ELF code list.

The number of samples per jurisdiction varies from a minimum of 4,836 entities
(US-New York), a median of 23,969 entities (Belgium) and a maximum of 135,079
samples (Germany). The number of class labels varies, ranging from a minimum
of nine ELF codes (British Virgin Islands) to a median of 21 ELF codes
(Austria) and a maximum of 165 ELF codes (France)272727This does not
necessarily include all defined ELF codes for the respective jurisdictions,
but only those assigned to any entity within the data in scope..

Table 3 shows the results as a comparison of Traditional and Transformer-based
models. We chose to evaluate our model in terms of F1 score to account for
precision and recall simultaneously. Additionally, we look at Macro F1 Score
(F1-M), which is useful for multiclass classification problems where classes
are imbalanced. Each model has been cross-validated with 5 stratified, non-
overlapping folds. The scores have been computed on the concatenated
predictions of all folds.

General findings: As a first remark, the classification performance varies a
lot in between jurisdictions, which emphasizes the uniqueness of each
jurisdiction in terms of the challenges it poses to the classification task.
These challenges include the varying number of samples for training, and even
more the varying number of target classes and class distribution balance. Not
at last though, there are jurisdiction-specific intriciacies which have impact
on the quality of the data and therefore influence the performance. Notably,
Belgium (BE) and France (FR) exhibit significantly lower scores than other
jurisdictions. In Belgium and France we generally observe that the majority of
entities do not carry any legal form information within their legal names.
Furthermore, France comes with an exceptionally high number of 165 ELF codes
as target classes.

Traditional vs Transformer: Considering Traditional methods only, we found
Random Forest with pre–processing to perform best in 17 jurisdictions for F1
score, while for the Macro F1 score, a Decision Tree Classifier with
pre–processing turned out to be the best in 13 jurisdictions, followed by RF +
prep in 9 jurisdictions. When comparing both methodologies, we see that
Transformers outperform the Traditional pipeline in 22 out of 30 cases in the
F1 score. On the Macro F1 score, we find that the Transformers still
outperform in 9 cases. Its strongest competitor here is the Decision Tree
(with and without pre–processing), to which it loses 15 times on the F1-M
score. This can be explained by a tendency of the Transformer to being
beneficial for the majority classes, while being comparably less accurate to
classify weakly represented classes with just a few samples.

The effect of pre–processing: For the Traditional models, we evaluated the
impact of adding pre–processing to the general performance of the pipeline. We
compared for each classification method the number of instances in which the
pre–processing enabled pipeline outperforms its simpler counterpart. For the
30 jurisdictions and for both scores (in total: 60 cases), incorporating the
pre–processing was superior to omitting it in 32 (CNB + prep), 45 (DT + prep),
48 (RF + prep), and 46 (SVM + prep) cases respectively. This strongly suggests
that the presented pre-processing is indeed supportive for the task within the
traditional setup.

The effect of language: Looking at the Transformer models, it turned out that
the multilingual BERT version performs best in 18 jurisdictions. This is
mostly due the fact that in many cases we have not found suitable language-
specific models for fine-tuning. However, for jurisdictions like Switzerland
and Luxembourg, which have multiple administrative languages, or
Liechtenstein, which exhibits a large amount of multi-language names, using a
fine-tuned multilingual BERT version performed better than a fine-tuned German
BERT. Also, the multilingual model for example slightly outperformed BERTje
(F1 0.9834, F1-M 0.7582) in the Netherlands. For some jurisdictions though, we
were able to find language-specific models that outperformed the multilingual
version, including Germany, Italy, Denmark, Finland, Austria, Poland and
Japan. Our results indicate that language-specific models are often
preferrable, but it is fair and sometimes beneficial to use a multilingual
model as a default.

In the case of Japan, the Transformer clearly outperformed the Traditional
approaches. One reason being that no delimiters in the Japanese language exist
to tokenize at (Hirabayashi et al., 2020), which causes problems to the Bag-
of-Words approach. The Japanese Bert developed by Tohoku University, though,
uses a custom Japanese Tokenizer, and thus is able to achieve a high F1 score
of 0.9828.

The effect of capitalization: Regarding the capitalization of the models, we
found that throughout all jurisdictions, the uncased variants perform better
than the cased variants. This may be attributed to the fact that many entities
are reported in a fully upper-cased fashion. The portion of upper-cased legal
names varies across all jurisdictions. A cased model, which is relying on the
input text using upper and lower case characters, might be negatively affected
by fully upper-cased entity names, whereas an uncased variant might be more
robust against such deficiencies in the data.

The effect of domain specific pre-training: For the subset of mainly English
speaking jurisdictions (GB, US-DE, US-MA, US-CA, US-NY, AU, IE, VG, KY) we
evaluated FinBERT. Even though the performance was mostly close, FinBERT, in
general, was not able to outperform the standard bert-base and multilingual
models. The only exception is US-Delaware, for which it was on par with bert-
base-uncased in the F1 score, and even outperformed it at the F1-M score
(0.5719 vs. 0.5248).

Learning of non-obvious relationships: There are cases of entity types for
which there exist no explicit rules to include legal form information within a
legal name. However, presuming sufficient domain knowledge, the legal form can
sometimes be determined from non-obvious name characteristics. For instance,
the legal entity ”Langholtgaard” is registered with ELF code
”Enkeltmandsvirksomhed” (FUKI). The name indicates that the entity is a farm.
It turns out that in Denmark, the legal form of ”Enkeltmandsvirksomhed”
(english: ”sole proprietor”) is predominantly used for farms. These kinds of
relationships are hard to cover with rule-based approaches. The presented
techniques are generally able to statistically model such patterns and thus
acquire non-obvious domain expertise.

Importance of sequence modelling: In our analysis, we found that detecting
legal forms accurately sometimes requires considering the order of tokens in
the entity names. We observed several cases where the transformer model
successfully identified the legal form by taking the token order into account.
However, we did not quantify this behavior in our study.

To illustrate the point, let’s consider two German entities: Entity A has the
legal name ”Unsere Kinder, unsere Zukunft – Stiftung der Volksbank Odenwald
eG”, while entity B has the name ”Volksbank Odenwald eG”. Both names exhibit
the abbreviation ”eG”, which represents the legal form ”eingetragene
Genossenschaft” (ELF code AZFE). This would be correct for entity B, however,
entity A represents a ”Stiftung” (ELF code V2YH) (english: foundation) that
belongs to entity B.

We found Bag-of-Words to not classify correctly in this scenario, as it in
general cannot decide which token has more relevance - ”Stiftung” or ”eG”.
Also, we claim that any simplistic rule-based approach will have problems
here. The Transformer however was able to classify this correctly.

Using the python library transformers-
interpret282828https://github.com/cdpierse/transformers-interpret, which
utilizes integrated gradients (Sundararajan et al., 2017), we are able to
compute and visualize in Table LABEL:table:tokensequence the relevance
attribution of each token to the Transformer’s entity legal form prediction.
Please note that the legal name of entity B can be understood as substring of
entity A’s legal name.

Table 4. Consideration of token sequence | Entity A | Entity B |
---|---|---|---
Token | Attribution | Attribution |
[CLS] | 0.00 | 0.00 |
unsere | 0.11 |  |
kinder | 0.12 |  |
, | 0.05 |  |
unsere | 0.10 |  |
zukunft | 0.17 |  |
- | 0.08 |  |
stiftung | 0.82 |  |
der | 0.43 |  |
volksbank | 0.01 | 0.09 |
oden | 0.06 | 0.06 |
##wald | 0.18 | 0.17 |
eg | 0.18 | 0.98 |
[SEP] | 0.00 | 0.00 |
Prediction: |  V2YH ✓ |  AZFE ✓ |

For entity name A, the transformer attributes the highest scores to the tokens
”stiftung” and ”der”, which represent the phrase ”foundation of”, resulting in
a correct classification of the legal form. For entity name B, the highest
attribution is correctly associated with the token ”eg”. This demonstrates how
the transformer model is able to correctly predict the legal form for both
entities due to its ability to consider the positioning and sequence of
individual tokens.

##  5\. Expert review

In order to validate the plausibility of the models’ results, we selected ten
jurisdictions with models having a high F1-score. In these jurisdictions, we
examined entities, for which the model predicted an ELF code different from
the recorded ELF code present in the LEI data at that time. Upon manual
review, we found that many of the model’s predictions were indeed correct,
especially for instances with a high prediction probability value.

To further evaluate our findings we utilized GLEIF’s challenge
facility292929https://www.gleif.org/en/lei-data/gleif-data-quality-
management/challenge-lei-data and requested updates to the LEI data based on
the legal forms suggested by our models. The suggested legal forms were then
reviewed by the respective local LEI issuing organizations, which are
considered experts in entity data management within their accredited
jurisdictions. A detailed view on how many of these requests have been
accepted can be found in Table LABEL:table:expertreview.

In total, we requested updates for 7,256 entities in the ten selected
jurisdictions, and 6,088 of these entities were updated based on the proposed
entity legal form, resulting in an overall acceptance rate of 83.9%. The high
acceptance rate proves that our models produce plausible legal form
classifications. Moreover, this indicates that the vast majority of LEI data
accurately reflects the correct legal form information and therefore serves as
a reliable source for training.

Table 5. Requested reference data updates Jurisdiction |  Requested updates |  Entities updated |  Acceptance rate
---|---|---|---
DE |  3,282 |  3,224 |  98.24%
ES |  1,284 |  1,273 |  99.14%
DK |  750 |  317 |  42.26%
GB |  547 |  420 |  76.78%
LU |  395 |  253 |  64.05%
CH |  324 |  169 |  52.15%
US-DE |  332 |  183 |  55.12%
NL |  163 |  106 |  65.03%
NO |  118 |  87 |  73.72%
SE |  61 |  56 |  91.8%

##  6\. Conclusion

Our study successfully applied Transformer-based language models for
classifying entity legal forms from raw legal entity names. Utilizing various
BERT variants, we compared their performance against traditional baselines on
a substantial subset of LEI data, covering over 1.1 million legal entities
from 30 different jurisdictions.

Our findings revealed that the presented models can effectively learn
statistical relationships that are advantageous, even in cases where legal
entity names lack explicit representation of the corresponding legal form.
This advantage over rule-based approaches reduces development efforts and
enables scalability to multiple jurisdictions.

Furthermore, we observed that language-specific pre-training has a positive
impact on the models’ performance, underscoring the importance of considering
the linguistic nuances inherent in legal names. The ability of language models
to capture the sequential nature of legal names also proved advantageous, as
it enhances accuracy in handling edge cases, where traditional methods may
struggle.

The expert review process played a critical role in validating the reliability
of our models. We sought confirmation from local experts in ten selected
jurisdictions, and the vast majority of the proposed legal forms were
confirmed. This clear indication from the experts reinforces the
trustworthiness and accuracy of our legal form classifications.

The LEI data and ELF codes played a crucial role in our study, providing
valuable ground truth labels for legal form classification. We believe that
broader adoption of such standards will significantly enhance transparency,
while improving data integration tasks in various domains.

In this regard, we contribute our insights and techniques as part of an open
source library called LENU (Legal Entity Name
Understanding)303030https://github.com/Sociovestix/lenu. The library is freely
available and is distributed under Creative Commons Zero 1.0 (CC0-1.0)
Universal license313131https://creativecommons.org/publicdomain/zero/1.0/. We
invite all stakeholders to use it for entity legal form classification.

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