Conformal Galilean Spin-3 Gravity in 3d

I. Lovrekovic∗, K. Schaefer∗

∗ Institute for Theoretical Physics, Technische Universität Wien,
Wiedner Hauptstrasse, 8-10,
1040, Wien, Austria

###### Abstract

We consider Galilean limit of conformal algebra for spin-2 and spin-3 fields,
and study the gauge theory of these algebras. We analyze the equations of
motion and obtain the spectra of the theories.

##  1 Introduction

Galilean theories have been studied in many works in different number of
dimensions [1] [2] [3] [4] [5] [6]. When we say Galilean theory we have to
specify it in the context of gauge/gravity duality. Number of studies were
devoted to Galilean (conformal) field theories [7] [8] [9] [10] [11], which,
together with similar non-relativistic theories, were found to be manifested
in the condensed matter systems. In this work, we will focus on the gravity
side.

In general, Galilean theory is invariant under the Galilean transformations.
In the non-conformal case Galilean symmetries arise after Inönü Wigner
contraction of the Poincare symmetries. This picture can be viewed as a light
cone completely opened, where speed of light goes to infinity, i.e. the light
rays of a light cone are closing 0 and 180 degrees angles. One obtains the
space where exchange of information is possible with each point at any time.
Galilean symmetries hugely depend on the number of dimensions in which we are
considering them. In two dimensions they had additional application because
they can be interchanged with Carrolian symmetries by exchanging the time and
space component.

Galilean theory has been studied in four dimensions in the relation with
Newton Cartan theory. Where it was shown that one cannot obtain Newton-Cartan
gravity by gauging the Galilean algebra, instead, one has to add a central
extension to it, obtaining Bargmann algebra. Gauging Bargmann algebra
reproduces Newton-Cartan gravity. The metric of the Galilean gravity is
degenerate and one can’t solve the equations for all components of the spin
connection. This is solved by the central extension, in Bargmann algebra.
Importance of the Newton-Cartan gravity is in our current understanding of the
universe at the non-relativistic scales. Newton-Cartan theory is obtained in
the non-relativistic limit of Einstein gravity. The study in [12] showed that
one can obtain torsional Newton-Cartan geometry from Galilean gauge theory in
4d.
Interesting suggestion, however much less studied in this context, included
addition of conformal invariance [13]. The analysis of conformal Galielean
theories have so far been focused on electrodynamics [10] and boundary
theories. Boundary theories are here referred to in the context of AdS/CFT
duality, where the conformal field theory at the boundary has symmetries of
non-relativistic groups, among which are also Galilean symmetries [14, 15,
16].
Conformal symmetry was shown to be very important in physics. ’t Hooft has
speculated that conformal symmetries plays a key role at the physics at the
Planck energy scale [17]. He worked on a number of studies analyzing the
conformal generalizations of Einstein gravity [18, 19]. In four dimensions, it
was shown that a theory with conformal symmetry can explain galactic rotation
curves without the addition of dark matter [20], and it can be considered as a
well defined theory [21, 22]. The negative sides are that theory has ghosts
which are usually treated with Pais-Uhlenbeck [24] oscillator, however purely
on theoretical side, scientists often study it as a toy model for its
additional symmetry. In two dimensions the symmetry plays important role in
the field theories dual to higher dimensional bulk gravity theories, in
AdS/CFT duality. In three dimensions conformal gravity theory can be written
as a gauge theory of conformal algebra in terms of Chern-Simons action [23].

Higher spin gravities of Inönü Wigner contracted algebras have been considered
in [25, 26]. [25] studied gravity theories of spin three fields in non- and
ultra-relativistic limit, while [26] focused on the spectrum and construction
of the conformal gravity theory in the ultra-relativistic limit. The
construction of the algebras for the higher spin gravities in these limits was
recently described in [27] using similar methods as in [28].

Here, we study Galilean conformal gravity and its spin-3 generalization in
three dimensions, obtaining it as an Inönü Wigner contraction of the conformal
algebra, and conformal spin-3 algebra respectively. This leads to the spectrum
of the fields in the theory, equations of motion, and constraints. Algebra of
conformal graviton and conformal spin-3 field in our construction is
isomorphic to the $sl(5)$ algebra, so one way to look at the contraction is as
Galilean limit of $sl(5)$ algebra. The paper is structured as follows. In the
second section we overview Chern-Simons theory as a gauge theory of $so(3,2)$
group, and the construction of the algebras that we use as gauge algebras. We
also outline the contraction of algebras previously constructed. In section
three we consider Galilean conformal gravity as a gauge theory of Galilean
conformal algebra (GCA), while in section four we consider spin three Galilean
conformal gravity.

##  2 Chern-Simons theory

Our starting point is the three dimensional Chern-Simons action of the form

$\displaystyle S=\int\langle A\wedge dA+\frac{2}{3}A\wedge A\wedge A\rangle.$
(2.1)

A is Lie algebra valued one form, and depending on the algebra in which we
value the field A, one obtains theories of the different properties. For the
underlying gauge algebra $sl(2,\mathbb{R})\oplus sl(2,\mathbb{R})$ the theory
corresponds to three dimensional Einstein gravity, and it can be generalized
to higher spin algebra in which case one obtains the theory for massless
higher spins in 3d [29]. When the underlying algebra is conformal one obtains
three dimensional conformal gravity [23], which can also be generalized to
higher spin theory [30], [31]. The specific property of the theory in [31] is
that it contains infinite number of theories with finite number of fields of
arbitrary spin. To take the non-relativistic limit of this theory we first
take the limit of the constructed underlying algebras, and then evaluate in
them the field A and corresponding gauge parameter. This leads to the
equations of motion that we can solve to obtain the spectrum of the theory.

###  2.1 Construction

To consider the non-relativistic limit of GCA and its generalization to
arbitrary higher spin case, first we need to know the $so(3,2)$ higher spin
extension, and the algebras that have its nontrivial representation with
$so(3,2)$ as a subalgebra. By taking nontrivial finite-dimensional irreducible
representation $V$ of the $so(3,2)$ one can construct finite spectrum of
higher spin fields. $V$ is irreducible tensor or a spin-tensor, evaluated in
the $U(so(d,2))$. To evaluate $V$ one multiplies the $so(3,2)$ tensors
$T_{AB}=-T_{BA}$ in the representation and determines the generated algebra
[31]. Here, we will denote the indices of $so(3,2)$ with $A,B,..=0,..,4$ and
an invariant metric with $\eta_{AB}$.

Generally this algebra is denoted with $\mathfrak{hs}(V)$. To see how one
obtains spin-2 and spin-3 algebra in this decomposition we take the vector
representation, denoted by Young diagram
\begin{picture}(10.0,10.0){}\end{picture} that has one-cell. The corresponding
algebra has the spectrum,
$\mathfrak{hs}(\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}})$:

$\displaystyle\mathfrak{hs}(\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}})=$
$\displaystyle\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}}\otimes\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}}=\parbox{10.0pt}{\begin{picture}(10.0,20.0){}\end{picture}}\oplus\parbox{20.0pt}{\begin{picture}(20.0,10.0){}\end{picture}}\oplus\bullet.$
(2.2)

Algebra that we consider is a matrix algebra that has $t{}_{A}{}^{B}\,$
generators that are decomposed with respect to $so(3,2)$. From the commutation
relations of the $gl_{3+2}$

$\displaystyle[t{}_{A}{}^{B}\,,t{}_{C}{}^{D}\,]=-\delta{}_{A}{}^{D}\,t{}_{C}{}^{B}\,+\delta{}_{C}{}^{B}\,t{}_{A}{}^{D}\,\,.$
(2.3)

one can read out the irreducible generators in the $so(3,2)$ base $R$,
$S_{AB}=S_{BA}$ and $T_{AB}=-T_{BA}$, as

$\displaystyle R=t{}_{C}{}^{C}\,\,,$ $\displaystyle S_{AB}$
$\displaystyle=t_{A|B}+t_{B|A}-\frac{2}{d+2}t{}_{C}{}^{C}\,\eta_{AB}\,,$
$\displaystyle T_{AB}=t_{A|B}-t_{B|A}\,.$ (2.4)

They close the commutation relations

$\displaystyle[T_{AB},T_{CD}$
$\displaystyle]=\eta_{BC}T_{AD}-\eta_{AC}T_{BD}-\eta_{BD}T_{AC}+\eta_{AD}T_{BC}\,,$
(2.5a) $\displaystyle[T_{AB},S_{CD}$
$\displaystyle]=\eta_{BC}S_{AD}-\eta_{AC}S_{BD}+\eta_{BD}S_{AC}-\eta_{AD}S_{BC}\,,$
(2.5b) $\displaystyle[S_{AB},S_{CD}$
$\displaystyle]=\eta_{BC}T_{AD}+\eta_{AC}T_{BD}+\eta_{BD}T_{AC}+\eta_{AD}T_{BC}\,.$
(2.5c)

Here $R$ is associated with $1$ in $gl(V)$ which make it commute with
everything, and allows us to truncate it away. This leads to a theory with a
the conformal graviton $T_{AB}$ and a field $S_{AB}$ which is similar to the
spin-three partially-massless field:

$\displaystyle\omega=$ $\displaystyle\omega^{A,B}T_{AB}+\omega^{AB}S_{AB}.$
(2.6)

Conformal graviton $\omega^{A,B}$ is the same one that was described in
conformal gravity as a gauge theory in three dimensions, in [23]. To obtain
conformal gravity as a gauge theory of $so(3,2)$ one has to first fix the
commutation relations of the conformal algebra $so(3,2)$

$\displaystyle[D,P^{a}]=$ $\displaystyle-P^{a}\,,$
$\displaystyle[J^{ab},P^{c}]$ $\displaystyle=P^{a}\eta^{bc}-P^{b}\eta^{ac}\,,$
(2.7a) $\displaystyle[D,K^{a}]=$ $\displaystyle K^{a}\,,$
$\displaystyle[J^{ab},K^{c}]$ $\displaystyle=K^{a}\eta^{bc}-K^{b}\eta^{ac}\,,$
(2.7b) $\displaystyle[P^{a},K^{b}]=$ $\displaystyle-J^{ab}+\eta^{ab}D\,,$
$\displaystyle[J^{ab},J^{cd}]$
$\displaystyle=J^{ad}\eta^{bc}-J^{ac}\eta^{bd}-J^{bd}\eta^{ac}+J^{bc}\eta^{ad},,$
(2.7c)

(for $a=0,1,2$) and $P^{a}$ translations, $J^{ab}$ Lorentz boosts, $K^{a}$
special conformal tranformations, and $D$ dilatation. Then it is required to
evaluate the connection $\omega$ and the corresponding gauge parameter in the
$so(3,2)$

$\displaystyle\omega=\frac{1}{2}\varpi^{a,b}J_{ab}+e^{a}P_{a}+f^{a}K_{a}+bD.$
(2.8)

The gauge is then fixed, and solving the equations of motion gives conformal
gravity as a gauge theory of the $so(3,2)$ algebra. In other words, one can
write the Chern-Simons action in the same way as in (2.1) however in latter
case $\overline{\omega}$ is evaluated in the group of Lorentz rotations. Here,
we follow that general procedure, for the non-relativistic limit of our
conformal algebras.

First we linearize a theory over the Minkowski vacuum. This can be done by
choosing $\omega_{0}=h^{a}P_{a}$ with $h^{a}=h_{\mu}{}^{a}dx^{\mu}$, and the
background metric $h_{\mu}{}^{a}=\delta_{\mu}{}^{a}$. One can write the
linearized equations and the linearized gauge symmetries as

$\displaystyle d\omega+\omega_{0}\wedge\omega+\omega\wedge\omega_{0}=0\,,$
$\displaystyle\delta\omega=d\xi+[\omega_{0},\xi].$ (2.9)

The background field is $\omega_{0}$, and $\omega$ is the one-form evaluated
in the Lie algebra. This algebra is one of the algebras of the so(3,2)
irreducible modules that we obtain from the decomposition of the
$\mathfrak{hs}(V)$, and by taking the non-relativistic limit of the algebra.
When we insert the field and gauge parameter expansion in the equations (2.9)
we obtain linearized equations of motion and linearized gauge transformations
for the $t^{\Lambda}$ generators, field $\omega=\omega^{\Lambda}t_{\Lambda}$
and gauge parameter $\xi=\xi^{\Lambda}t_{\Lambda}$. That will lead to

$\displaystyle
d\omega^{\Lambda}t_{\Lambda}+h^{a}\wedge\omega^{\Lambda}[P_{a},t_{\Lambda}]=0,$
$\displaystyle\delta\omega^{\Lambda}t_{\Lambda}=d\xi^{\Lambda}t_{\Lambda}+h^{a}\wedge\xi^{\Lambda}[P_{a},t_{\Lambda}].$
(2.10)

$\Lambda$ are indices in the light cone coordinates $\Lambda=\\{a,+,-\\}$,
$\eta^{AB}$ is corresponding metric with $\eta_{+-}=\eta_{-+}=1$, and
$t^{\Lambda}$ are generators of so(3,2) irreducible modules $T_{AB}$, $S_{AB}$
etc..

###  2.2 Inönü Wigner contraction

To consider the Galilean limit we use Inönü Wigner (IW) contraction [32] of
the algebra (2.7). We decompose an algebra $\mathfrak{g}$ in
$\mathfrak{g}=\mathfrak{h}+\mathfrak{i}$ direct sum of vector spaces. For
$\mathfrak{h}$ a subalgebra and $\mathfrak{i}$ an ideal. The generators of the
ideal are rescaled with contraction parameter $\epsilon$, where
$\mathfrak{i}\rightarrow\epsilon\mathfrak{i}$. That gives for commutation
relations

$\displaystyle[\mathfrak{h},\mathfrak{h}]=\mathfrak{h},$
$\displaystyle[\mathfrak{h},\mathfrak{i}]=\frac{1}{\epsilon}\mathfrak{h}+\mathfrak{i},$
$\displaystyle[\mathfrak{i},\mathfrak{i}]=\frac{1}{\epsilon^{2}}\mathfrak{h}+\frac{1}{\epsilon}\mathfrak{i}.$
(2.11)

If we take $\epsilon\to\infty$, we need to obtain well defined limit. For this
limit to hold, one needs to get $\mathfrak{h}$ as a subalgebra of
$\mathfrak{g}$. If we had
$[\mathfrak{h},\mathfrak{h}]=\mathfrak{h}+\epsilon\mathfrak{i}$ we would not
have well defined limit $\epsilon\to\infty$. For the spin-2 case we are able
to compare the GCA with the literature [8] and see that this procedure leads
us to the same algebra.

##  3 Spin-2

The first algebra we consider is the algebra described by the generator
$T_{A,B}=-T_{B,A}$, the Young module \begin{picture}(10.0,20.0){}\end{picture}
, and it was given in (2.7). We have the coordinates $t,x_{i}$ for $i=1,2$ and
we refer to the coordinate t as the 0-th coordinate. The rescaled generators
in the non-relativistic limit are those that are along the $x_{i}$
coordinates. If we rename the generators of $so(3,2)$ algebra such that
$T_{i+}=P_{i},T_{0+}=H,T_{i-}=K_{i},T_{0-}=K_{0},T_{ij}=J_{ij},T_{0i}=B_{i}$,
and $T_{+-}=-D$ we can decompose an algebra into

$\displaystyle\mathfrak{i}=\\{P_{a},K_{a},B_{a}\\},$
$\displaystyle\mathfrak{h}=\\{J_{ab},D,H,K_{0}\\}.$ (3.1)

When we take into account this decomposition, (2.7) and (2.11) we obtain GCA

$\displaystyle[D,P_{a}]$ $\displaystyle=-P_{a}$ $\displaystyle[D,H]=-H$
$\displaystyle[D,K_{a}]=K_{a}$ (3.2a) $\displaystyle[D,K_{0}]$
$\displaystyle=K_{0}$ $\displaystyle[P_{a},K_{0}]=B_{a}$
$\displaystyle[H,K_{b}]=-B_{b}$ (3.2b) $\displaystyle[H,K_{0}]$
$\displaystyle=\eta_{00}D$ $\displaystyle[B_{b},H]=-\eta_{00}P_{b}$
$\displaystyle[B_{b},K_{0}]=-\eta_{00}K_{b}$ (3.2c)
$\displaystyle[J_{ab},P_{c}]$ $\displaystyle=P_{a}\eta_{bc}-P_{b}\eta_{ac}$
$\displaystyle[J_{ab},K_{c}]=K_{a}\eta_{bc}-K_{b}\eta_{ac}$
$\displaystyle[J_{ab},B_{d}]=B_{a}\eta_{bd}-B_{b}\eta_{ad}$ (3.2d)
$\displaystyle[J_{ab},J_{cd}]=J_{ad}\eta_{bc}-J_{ac}\eta_{bd}-J_{bd}\eta_{ac}+J_{bc}\eta_{ad}$
(3.3)

in which we wrote only non-vanishing commutation relations. We write the field
decomposed in the generators (3.1) of the algebra (3.2) and (3.3)

$\displaystyle\omega$ $\displaystyle=e^{a}P_{a}+\tau
H+\tfrac{1}{2}\omega^{a,b}J_{ab}+\beta^{a}B_{a}-bD+f^{a}K_{a}+\kappa K_{0}$
(3.4)

as well as the gauge parameter $\xi$

$\displaystyle\xi$
$\displaystyle=\xi^{a+}P_{a}+\xi^{0+}H+\tfrac{1}{2}\xi^{a,b}J_{ab}+\xi^{0a}B_{a}-\xi^{+-}D+\xi^{a-}K_{a}+\xi^{0-}K_{0}.$
(3.5)

Inserting the gauge field $\omega$ and gauge parameter $\xi$ in the linearized
equations (2.10) gives

$\displaystyle D:$ $\displaystyle-db-h^{0}\wedge\kappa=0$ $\displaystyle\delta
b$ $\displaystyle=d\xi^{+-}+h^{0}\xi^{0-}$ (3.6a) $\displaystyle P_{i}:$
$\displaystyle de^{i}-h^{j}\wedge\omega^{i}{}_{j}-h^{i}\wedge
b-h^{0}\wedge\beta^{i}=0$ $\displaystyle\delta e^{i}$
$\displaystyle=d\xi^{i+}-h^{j}\xi^{i}{}_{j}-h^{i}\xi^{+-}-h^{0}\xi^{0i}$
(3.6b) $\displaystyle H:$ $\displaystyle d\tau-h^{0}\wedge b=0$
$\displaystyle\delta\tau$ $\displaystyle=d\xi^{0+}-h^{0}\xi^{+-}$ (3.6c)
$\displaystyle J_{ij}:$ $\displaystyle d\omega^{ij}J_{ij}=0$
$\displaystyle\delta\omega^{ij}$ $\displaystyle=d\xi^{ij}$ (3.6d)
$\displaystyle B_{j}:$ $\displaystyle d\beta^{i}-h^{0}\wedge
f^{i}+h^{i}\wedge\kappa=0$ $\displaystyle\delta\beta^{i}$
$\displaystyle=d\xi^{0i}-h^{0}\wedge\xi^{i-}+h^{i}\wedge\xi^{0-}$ (3.6e)
$\displaystyle K_{i}:$ $\displaystyle df^{i}=0$ $\displaystyle\delta f^{i}$
$\displaystyle=d\xi^{i-}$ (3.6f) $\displaystyle K_{0}:$ $\displaystyle
d\kappa=0$ $\displaystyle\delta\kappa$ $\displaystyle=d\xi^{0-}$ (3.6g)

Considering relations (1.14a), the gauge transformation condition for m and
for 0 components leads to $\delta b_{m}=\partial_{m}\xi^{+-}$, a gauge
transformation for the field $b_{m}$ since $h_{m}{}^{0}=0$. And we can fix a
field $b_{0}$ to be zero using the parameter $\xi^{0-}=-\partial_{0}\xi^{+-}$.
From the equation (1.14a) we consider mn and m0 components respectively

$\displaystyle-\partial_{m}b_{n}+\partial_{n}b_{m}=0$ (3.7a)
$\displaystyle\partial_{0}b_{m}+\kappa_{m}=0.$ (3.7b)

and obtain condition on the $b_{m}$ field, and definition of the field
$\kappa_{m}$ in terms of the field $b_{m}$.
The following relations we consider are (1.14b). The gauge condition we write
for the m and 0 components. The m component gauge transformation consists of
symmetric part with trace and antisymmetric part. The symmetric part gives
$\delta
e_{(mi)}=\frac{1}{2}(\partial_{m}\xi_{i}{}^{+}+\partial_{i}\xi_{m}{}^{+})-h_{mi}\xi^{+-}.$
If we choose to fix the gauge parameter $\xi^{+-}$, we can set the trace of
the $e_{mi}$ field to zero $e_{m}{}^{m}=0$ when
$\xi^{+-}=\frac{1}{2}\partial_{m}\xi^{m+}$. This gives for linearized gauge
transformation of the field $e_{(mi)}$

$\displaystyle\delta e_{(mi)}$
$\displaystyle=\frac{1}{2}(\partial_{m}\xi_{i}{}^{+}+\partial_{i}\xi_{m}{}^{+})-\frac{1}{2}h_{mi}\partial_{l}\xi^{l+}.$
(3.8)

The antisymmetric part $\delta e_{[mi]}$ can be fixed to zero by
$\xi_{im}=\frac{1}{2}(\partial_{m}\xi_{i}{}^{+}-\partial_{i}\xi_{m}{}^{+}).$
To remember $e_{mi}$ is symmetric traceless field we define
$e_{mi}\equiv\phi_{mi}$. The 0 component of gauge transformation allows us to
fix $e_{0}{}^{i}=0$ using the gauge fixing $\xi^{0i}=\partial_{0}\xi^{i+}$.

Equation of motion (3.6b) inserting the gauge fixings $e_{mi}=\phi_{mi}$,
$e_{0}{}^{i}=0$ and $b_{0}=0$ is in components

$\displaystyle mn:$
$\displaystyle\partial_{m}\phi_{ni}-\partial_{n}\phi_{mi}-\omega_{nim}+\omega_{min}-h_{mi}b_{n}+h_{ni}b_{m}=0$
(3.9a) $\displaystyle m0:$
$\displaystyle-\partial_{0}\phi_{mi}-\omega_{0im}+\beta_{mi}=0.$ (3.9b)

First of these equations, (1.26a) has symmetric, antisymmetric and a hook
part. If we consider totally symmetric part we obtain the equality 0=0, which
is reasonable because we know that the field $\omega_{min}$ is antisymmetric
in the last two indices while entire equation already has antisymmetry in m
and n. Antisymmetric part of the equation gives us

$\displaystyle\omega^{imn}-\omega^{\min}+\omega^{nim}=0.$ (3.10)

The hook part of the equation can using the equation (3.10), and the property
$\omega_{min}=-\omega_{mni}$ let us express

$\displaystyle\omega^{nim}=\frac{1}{2}(b^{n}h^{im}-b^{m}h^{in}-\partial^{m}\phi^{in}+\partial^{n}\phi^{im}).$
(3.11)

The equation (3.9b) can be split in symmetric and antisymmetric part.
Symmetric part is going to give

$\displaystyle\beta_{(mi)}=\partial_{0}\phi_{(mi)}.$ (3.12)

Since $e_{(mi)}$ is traceless we notice that $\beta_{m}{}^{m}$ is zero as
well. The antisymmetric part is going to define the antisymmetric part of
$\beta_{mi}$ in terms of $\omega_{0im}$ such that $\beta_{[mi]}=\omega_{0im}.$
In the instance of conformal Galilean algebra, as it is already known there is
not enough constraints to determine spin connection in terms of the remaining
fields.

From the gauge transformation (3.6c) we obtain that from m component
$\delta\tau_{m}=\partial_{m}\xi^{0+}$, while the parameter that is in 0
component we fixed already. That means
$\delta\tau_{0}=\partial_{0}\xi^{0+}-\frac{1}{2}\partial_{m}\xi^{m+}$. The
equation of motion gives in components
$\partial_{m}\tau_{n}-\partial_{n}\tau_{m}=0$ and
$\partial_{m}\tau_{0}-\partial_{0}\tau_{m}+b_{m}=0$ and they allow us to
define $b_{m}$ as $b_{m}=\partial_{0}\tau_{m}-\partial_{m}\tau_{0}$. Here
$\tau_{0}$ remains as undetermined component.

The gauge condition with $J_{ab}$ generator leads for the components m and 0
to $\delta\omega_{mab}=\partial_{m}\xi_{ab}$ and
$\delta\omega_{0ab}=\partial_{0}\xi_{ab}$ respectively. The gauge condition
for m component has symmetric part which is equal to zero, antisymmetric part
$\delta\omega^{A}_{mij}=\tfrac{1}{3}(\partial_{i}\xi_{jm}-\partial_{j}\xi_{im}+\partial_{m}\xi_{ij})=0$
which is zero upon inserting the gauge fixing for $\xi_{ij}$, and a hook part
$\delta\omega^{H}_{mij}=\tfrac{1}{3}(-2\partial_{i}\xi_{jm}-\partial_{j}\xi_{im}+\partial_{m}\xi_{ij})$.
The gauge transformation for the 0 component has only antisymmetric part by
construction
$\delta\omega_{0ab}=\tfrac{1}{2}(\partial_{0}\xi_{ab}-\partial_{0}\xi_{ba})$.
The equation from $J_{ab}$ in the mn components has symmetric, hook and
Riemann parts which are equally 0 because of $\omega_{mab}$ is antisymmetric
in the last two indices, while the antisymmetric part of the equation is
satisfied when the value for $\omega_{mab}$ (obtained from expressions with
$P_{i}$) is inserted. The m0 components give the symmetric and hook equations
which give 0=0 and antisymmetric part which defines equation

$\displaystyle\partial_{a}\omega_{0bm}-\partial_{b}\omega_{0am}+\partial_{m}\omega_{0ab}=0$
(3.13)

for $\beta_{ab}$ taking into account that $\omega_{0ab}=\beta_{[ab]}$.

For $B_{j}$ we obtain the gauge transformations

$\displaystyle m:$ $\displaystyle\delta\beta_{mi}$
$\displaystyle=\partial_{m}\xi^{0}{}_{i}+h_{mi}\xi^{0-}$ (3.14) $\displaystyle
0:$ $\displaystyle\delta\beta_{0i}$
$\displaystyle=\partial_{0}\xi^{0}{}_{i}-\xi_{i}{}^{-}$ (3.15)

The first defines symmetric
$\delta\beta^{S}_{mi}=\tfrac{1}{2}(\partial_{m}\xi^{0}{}_{i}+\partial_{i}\xi^{0}{}_{m})+h_{mi}\xi^{0-}$
and antisymmetric part
$\delta\beta^{A}_{mi}=\tfrac{1}{2}(\partial_{m}\xi^{0}{}_{i}-\partial_{i}\xi^{0}{}_{m})$
of the gauge transformation for the field $\beta_{mi}$. Inserting the value
for the $\xi^{0-}$ and $\xi^{0i}$ we obtain the gauge transformation for the
symmetric traceless part of the field
$\delta\beta^{S}_{(mi)}=\tfrac{1}{2}(\partial_{m}\partial_{0}\xi_{i}{}^{+}+\partial_{i}\partial_{0}\xi_{m}{}^{+})-\tfrac{1}{2}h_{mi}\partial_{0}\partial_{l}\xi^{l+}$
field. We can also verify that the trace of the field $\beta_{mi}$ vanishes.
The 0 component allows to fix gauge parameter $\xi^{i-}=\partial_{0}\xi^{0i}$
and set $\beta_{0}^{i}=0$. The equations with the $B_{i}$ generator can for mn
components lead to the symmetric part and hook part that lead to 0=0 while the
antisymmetric part of the equation gives the equation for the $\beta_{mn}$
field which is equal to the equation (3.13).

The m0 component of the equation will fix the symmetric, antisymmetric parts
of the field $f_{ma}$ and the trace. The symmetric part of the field is

$\displaystyle
f^{S}_{(ma)}=\partial_{0}\beta_{(ma)}-h_{ma}\kappa_{0}=\partial_{0}\partial_{0}\phi_{(ma)}-h_{ma}\kappa_{0}$
(3.16)

while the antisymmetric part is defined by the antisymmetric part of the
$\beta_{ij}$ field $f_{[ma]}^{A}=\partial_{0}\beta_{[ma]}.$

The following gauge generator that we consider, $K_{a}$ gives gauge
transformations of the symmetric traceless and antisymmetric part of $f_{mn}$
field such that $\delta
f_{(ma)}=\tfrac{1}{2}\partial_{m}\xi_{a}{}^{-}+\partial_{a}\xi_{m}{}^{-}-\frac{1}{2}h_{ma}\partial_{l}\xi^{l-}$.
While the antisymmetric part reads $\delta
f_{[ma]}=\tfrac{1}{2}(\partial_{m}\partial_{0}\partial_{0}\xi_{a}{}^{+}-\partial_{a}\partial_{0}\partial_{0}\xi_{m}{}^{+}).$
The mn equation consists of the symmetric and hook parts which give 0=0, and
the antisymmetric part which gives the same equation as (3.13). The m0
equation consists of the symmetric and antisymmetric parts. Symmetric part
gives equation that relates $f_{0i}$ and $\kappa_{0}$

$\displaystyle\frac{1}{2}(\partial_{m}f_{0a}+\partial_{a}f_{0a})+h_{ma}\partial_{0}\kappa_{0}=0$
(3.17)

while the antisymmetric part gives the equation that relates $f_{0a}$ and the
antisymmetric part of the $\beta_{ij}$ field

$\displaystyle\frac{1}{2}(\partial_{m}f_{0a}-\partial_{a}f_{0m})-\partial_{0}\partial_{0}\beta_{ma}=0.$
(3.18)

The gauge transformations with the following generator $K_{0}$ do not have any
more gauge parameter that we can fix, so they only define the transformation
rules of the fields, and give equation for $\tau_{m}$ and $\tau_{0}$ fields.

After we have fixed the gauge, we notice that the undetermined fields that
remain are $\tau_{0},\phi_{mi},\tau_{m},\beta_{[mi]},\kappa_{0},f_{0i}$. We
summarize the fields in the Table 1.

Generator | Field
---|---
D | $b_{0}=0$
| $b_{m}=\partial_{0}\tau_{m}-\partial_{m}\tau_{0}$
$P_{i}$ | $e_{(mi)}\equiv\phi_{mi},e_{[mi]}=0,e_{m}{}^{m}=0$
| $e_{0i}=0$
H | $\tau_{m}$
| $\tau_{0}$
$J_{ij}$ | $\omega^{nim}=\frac{1}{2}(b^{n}h^{im}-b^{m}h^{in}-\partial^{m}\phi^{in}+\partial^{n}\phi^{im})$
| $\omega_{0im}$
$B_{j}$ | $\beta_{(mi)}=\partial_{0}\phi_{mi},\beta_{[mi]}=\omega_{0im}$
| $\beta_{0i}=0$
$K_{i}$ | $f_{(mi)}=\partial_{0}\partial_{0}\phi_{mi}-h_{mi}\kappa_{0}$
| $f_{[mi]}=\partial_{0}\beta_{[mi]}$
| $f_{0i}$
$K_{0}$ | $\kappa_{0}$
| $\kappa_{n}=\partial_{0}b_{m}$
Table 1: Spectrum of fields in Galilean conformal gravity in 3d.

The gauge transformations of the fields are listed in the Appendix in Table 3.
One has to notice that $\beta_{[mi]}$ is equal to 0 component of the spin
connection which shows that due to degeneracy of the metric in this case, we
cannot determine entire spin connection in terms of the other fields, as it is
known for Galilean conformal gravity. One can speculate about adding the
central extension, however GCA does not admit the central extension as
Galilean algebra. However it does admit different kind of central extension.
There is also semi-GCA, which in 3d corresponds to $BMS_{4}$ algebra [7].
Analysis of the $BMS_{4}$ was done in [33] and it would be interesting to see
relation from the side of semi-GCA in 3d. Here, we first want to see the
contribution from the spin-3 field.

##  4 Spin-3

The algebra that describes spin-3 field is defined by the $S_{AB}=S_{BA}$
generator, and a Young module \begin{picture}(20.0,10.0){}\end{picture} . We
define the names of the generators in the analogous way as in the previous
scenario, with a difference that here we add a prefix S- to the generators and
s- to the fields that appeared es well above. This way we additionally point
out that they are defined in relation with $S_{AB}$ generator. We keep the
notation for the field $\omega_{\mu ab}$. The generators that appear here and
do not have antisymmetric analog in spin-2 case are $t_{++}$, $t_{--}$,
$t_{00}\equiv SC$ and $SJ_{kj}$ for when $j=k$. We decompose the generators
analogously as above by setting the generators with space components in the
ideal $\mathfrak{i}$, and the remaining components in $\mathfrak{h}$.

$\displaystyle\mathfrak{i}=\\{SP_{a},SK_{a},SB_{a}\\}$
$\displaystyle\mathfrak{h}$
$\displaystyle=\\{SH,SK_{0},t_{++},t_{--},SJ_{jk},SC\\}$ (4.1)

To achieve the closure of the algebra $\mathfrak{h}$ we set the trace
components of the $SJ_{ij}$ in $\mathfrak{h}$ as well. One can notice that due
to the trace condition there is no generator $SD$ appearing, but only
generators $SJ_{ii}$ for $i=0,1,2$. The Lie algebra valued one and zero forms
after decomposition in the generators (4.1) read

$\displaystyle\omega$ $\displaystyle=se^{a}SP_{a}+s\tau
SH+\tfrac{1}{2}\omega^{ab}SJ_{ab}+s\beta^{a}SB_{a}+\gamma
SC+\frac{1}{2}\omega^{++}t_{++}+\frac{1}{2}\omega^{--}t_{--}+sf^{a}SK_{a}+s\kappa
SK_{0}\,,$ (4.2) $\displaystyle\xi$
$\displaystyle=\xi_{a+}SP_{a}+\xi^{0+}SH+\tfrac{1}{2}\xi^{ab}SJ_{ab}+\xi^{0a}SB_{a}+\xi^{00}SC+\frac{1}{2}\xi^{++}t_{++}+\frac{1}{2}\xi^{--}t_{--}+\xi^{a-}SK_{a}+\xi^{0-}SK_{0}\,.$
(4.3)

From (2.5) one can read out the commutators of the spin-3 generators and
$P_{\mu}$

$\displaystyle[P_{i},t_{j+}]$ $\displaystyle=-\eta_{ij}t_{++}\,,$
$\displaystyle[P_{i},t_{j-}]$
$\displaystyle=t_{ij}+\frac{1}{2}t{}_{m}{}^{m}\,\eta_{ij}\,,$
$\displaystyle[P_{i},t_{--}]$ $\displaystyle=2t_{i-}\,,$ (4.4)
$\displaystyle[P_{i},t_{++}]$ $\displaystyle=0\,,$
$\displaystyle[P_{i},t_{jk}]$
$\displaystyle=-\eta_{ij}t_{k+}-\eta_{ik}t_{j+}\,.$ (4.5)

and insert the decomposition (4.1) and take IW contraction (2.11). That will
give non-vanishing commutation relations

$\displaystyle[H,SK_{b}]$ $\displaystyle=SB_{b}$
$\displaystyle[H,SC]=-2\eta_{00}SH$ $\displaystyle[H,t_{--}]=2SK_{0}$ (4.6a)
$\displaystyle[P_{a},SJ_{cd}]$
$\displaystyle=-\eta_{ac}SP_{d}-\eta_{ad}SP_{c}$
$\displaystyle[P_{a},t_{--}]=2SK_{a}$ $\displaystyle[P_{a},SK_{0}]=SB_{a}$
(4.6b) $\displaystyle[H,SK_{0}]$
$\displaystyle=SC+\frac{1}{2}(SJ_{a}^{a}+SC)\eta_{00}$
$\displaystyle[H,SH]=-\eta_{00}t_{++}$
$\displaystyle[H,SB_{d}]=-\eta_{00}SP_{d}.$ (4.6c)

When we insert these commutation relations in the linearized gauge equations
and gauge transformations (2.10) we can write the equations

$\displaystyle SP_{a}:$ $\displaystyle
dse^{a}-h^{l}\wedge\omega^{a}{}_{l}+h^{0}\wedge s\beta^{a}=0$
$\displaystyle\delta se^{a}$
$\displaystyle=d\xi^{a+}-h^{l}\wedge\xi^{a}{}_{l}+h^{0}\wedge\xi^{a0}$ (4.7a)
$\displaystyle SH:$ $\displaystyle ds\tau+2h^{0}\wedge\gamma=0$
$\displaystyle\delta s\tau$ $\displaystyle=d\xi^{0+}+2h^{0}\wedge\xi^{00}$
(4.7b) $\displaystyle SJ^{ab}:$ $\displaystyle d\omega^{ab}-h^{0}\wedge
s\kappa h^{ab}=0$ $\displaystyle\delta\omega^{ab}$
$\displaystyle=d\xi^{ab}-h^{0}\wedge\xi^{0-}\eta^{ab}$ (4.7c) $\displaystyle
SB^{a}:$ $\displaystyle ds\beta^{a}+h^{a}\wedge s\kappa+h^{0}\wedge sf^{a}=0$
$\displaystyle\delta s\beta^{a}$
$\displaystyle=d\xi^{a0}+h^{a}\wedge\xi^{0-}+h^{0}\wedge\xi^{a-}$ (4.7d)
$\displaystyle SC:$ $\displaystyle d\gamma+\frac{1}{2}h^{0}\wedge s\kappa=0$
$\displaystyle\delta\gamma$
$\displaystyle=d\xi^{00}+\frac{1}{2}h^{0}\wedge\xi^{0-}$ (4.7e) $\displaystyle
t_{++}:$ $\displaystyle\frac{1}{2}d\omega^{++}+h^{0}\wedge s\tau=0$
$\displaystyle\frac{1}{2}\delta\omega^{++}$
$\displaystyle=\frac{1}{2}d\xi^{++}+h^{0}\wedge\xi^{0+}$ (4.7f) $\displaystyle
t^{--}:$ $\displaystyle d\omega^{--}t_{--}=0$ $\displaystyle\delta\omega^{--}$
$\displaystyle=d\xi^{--}$ (4.7g) $\displaystyle SK_{a}:$ $\displaystyle
dsf^{a}+h^{a}\wedge\omega^{--}=0$ $\displaystyle\delta sf^{a}$
$\displaystyle=d\xi^{a-}+h^{a}\wedge\xi^{--}$ (4.7h) $\displaystyle SK_{0}:$
$\displaystyle ds\kappa+h^{0}\wedge\omega^{--}=0$ $\displaystyle\delta
s\kappa$ $\displaystyle=d\xi^{0-}+h^{0}\wedge\xi^{--}$ (4.7i)

To solve the system of equations (4.7) we start from $t^{++}$ gauge
transformations

$\displaystyle m:$ $\displaystyle\delta\omega_{m}{}^{++}$
$\displaystyle=\partial_{m}\xi^{++}$ (4.8) $\displaystyle 0:$
$\displaystyle\delta\omega_{0}{}^{++}$
$\displaystyle=\partial_{0}\xi^{++}+2\xi^{0+}$ (4.9)

where in the second equation we can fix $\omega_{0}{}^{++}=0$ with
$\xi^{0+}=-\frac{1}{2}\partial_{0}\xi^{++}$. The corresponding equation in
components leads to equation
$\partial_{m}\omega_{n}{}^{++}=\partial_{n}\omega_{m}{}^{++}$ for mn
components, and for m0 components to
$s\tau_{m}=-\frac{1}{2}\partial_{0}\omega_{m}{}^{++}$. That expresses the
$s\tau_{m}$ field in terms of $\omega_{m}^{++}$.

The gauge transformations with the SH generator lead to $\delta
s\tau_{m}=\partial_{m}\xi^{0+}$ for the m component, and allow for the fixing
$s\tau_{0}=0$ when $\xi^{00}=\tfrac{1}{2}\partial_{0}\xi^{0+}$. The
corresponding equation then gives
$\partial_{m}s\tau_{n}=\partial_{n}s\tau_{m}$ and
$\gamma_{m}=\frac{1}{2}\partial_{0}s\tau_{m}$. Where the latter equation
expresses the field $\gamma_{m}$ in terms of the $s\tau_{m}$ field, which we
already expressed in terms of $\omega_{m}^{++}$. $s\tau_{m}$ reads
$s\tau_{m}=-\tfrac{1}{4}\partial_{0}\partial_{0}\omega_{m}^{++}$. So far we
have one undetermined field, $\omega_{m}^{++}$.

The gauge transformations with the $SP_{a}$ generator give the symmetric and
antisymmetric part of the $se_{ma}$ field. The symmetric part can be gauged
away so that $se_{(ma)}=0$ when
$\xi_{dm}=\tfrac{1}{2}(\partial_{m}\xi_{d}^{+}+\partial_{d}\xi_{m}^{+})$. The
antisymmetric part remains unfixed. While the linearized gauge transformations
of the field $se_{md}$ read $\delta
se_{[md]}=\tfrac{1}{2}(\partial_{m}\xi_{d}^{+}-\partial_{d}\xi_{m}^{+})$.

The mn components of the equation have standardly symmetric, antisymmetric and
a hook part. Symmetric part is equally 0, antisymmetric part reads

$\displaystyle\omega^{imn}-\omega^{min}+\omega^{nim}+\partial^{i}se^{mn}-\partial^{m}se^{in}+\partial^{n}se^{im}=0$
(4.10)

here we took into account that $se_{im}$ is an antisymmetric field. When we
take into account that $\omega^{imn}$ is symmetric in last two indices the
equation reduces to equation for $se^{mn}$ field
$\partial^{i}se^{mn}-\partial^{m}se^{in}+\partial^{n}se^{im}=0$. The hook part
of the equation gives

$\displaystyle-\omega^{imn}-\omega^{min}+\partial^{i}se^{nm}+\partial^{m}se^{ni}=0$
(4.11)

taking that $se_{ma}$ is antisymmetric. Taking into account that
$\omega^{imn}$ is symmetric in last two indices gives that symmetric part of
the $\omega^{imn}$ field remains undetermined, while the hook part is defined
by $\omega^{(H)imn}=-\frac{1}{3}(\partial^{i}se^{mn}+\partial^{m}se^{in})$.

The component 0 of the gauge transformation allows to fix
$\xi^{a0}=\partial_{0}\xi^{a+}$ and $se_{0}{}^{a}=0$. The equation of motion
for the component m0 contains the gauge fixings and reads

$\displaystyle\partial_{0}se_{ma}+\omega_{0am}+s\beta_{ma}=0.$ (4.12)

The equation consists of the symmetric and antisymmetric parts. Since
$s\omega_{0am}$ is symmetric in last two indices and $se_{ma}$ is gauge fixed
to be a antisymmetric field, equation defines $s\beta_{(ma)}=-\omega_{0am}$
and the antisymmetric part of the $s\beta_{[am]}=\partial_{0}se_{ma}$.

The gauge transformation with the generator $SJ^{ab}$ reads
$\delta\omega_{mab}=\partial_{m}\xi_{ab}$ and it has symmetric
$\delta\omega^{(mij)}=\tfrac{1}{3}(\partial^{i}\partial^{j}\xi^{+}{}^{m}+\partial^{i}\partial^{m}\xi^{+}{}^{j}+\partial^{m}\partial^{j}\xi^{+}{}^{i})$,
antisymmetric $\delta\omega^{[mij]}=0$ and hook part
$\delta\omega^{(H)mij}=\tfrac{1}{6}(-2\partial^{j}\partial^{i}\xi^{+}{}^{m}+\partial^{m}\partial^{i}\xi^{+}{}^{j}+\partial^{m}\partial^{j}\xi^{+}{}^{i})$.
The 0 component of the transformation gives
$\delta\omega_{0}{}^{ab}=\partial_{0}\xi^{ab}-\xi^{0-}h_{ab}$. Using the
$\xi^{0-}$ we can fix a trace of $\omega_{0}^{ab}$. Equations of motion in the
mn component give equation
$\partial_{m}\omega_{nab}-\partial_{n}\omega_{mab}=0$ which can be divided in
components to give equations for the field $\omega_{(mab)}$ and for the hook
part of the field $\omega^{(H)}_{mab}$. Symmetric, antisymmetric, and Riemann
component are equally zero, while hook component reads

$\displaystyle\partial^{i}\omega^{jmn}-\partial^{j}\omega^{imn}-\partial^{j}\omega^{min}-\partial^{j}\omega^{nim}+\partial^{m}\omega^{jin}+\partial^{n}\omega^{jim}=0$
(4.13)

The symmetric equation from m0 component of equations of motion gives

$\displaystyle\partial_{0}{}\omega^{abm}+\partial_{0}{}\omega^{bam}-2\partial_{0}{}\omega^{mab}+2sk^{m}h^{ab}-sk^{b}h^{am}-sk^{a}h^{bm}-$
(4.14)
$\displaystyle\partial^{a}\omega_{0}{}^{bm}-\partial^{b}\omega_{0}{}^{am}+2\partial^{m}\omega_{0}{}^{ab}=0$

antisymmetric part is 0, and the hook part leads to

$\displaystyle\partial_{0}{}\omega^{abm}+\partial_{0}{}\omega^{bam}-2\partial_{0}{}\omega^{mab}+2sk^{m}h^{ab}-sk^{b}h^{am}-sk^{a}h^{bm}-$
(4.15)
$\displaystyle\partial^{a}\omega_{0}{}^{bm}-\partial^{b}\omega_{0}{}^{am}+2\partial^{m}\omega_{0}{}^{ab}=0.$

The combination of these equations gives that
$sk^{m}=\partial_{0}{}\omega^{am}{}_{a}-\partial_{a}\omega_{0}{}^{ma}.$

The gauge transformation with $SB^{a}$ generator defines $\delta
s\beta_{(ma)}=\frac{1}{2}(\partial_{m}\xi^{0}{}_{a}+\partial_{a}\xi^{0}{}_{m})+h_{ma}\xi^{0-}$
and $\delta
s\beta_{[ma]}=\frac{1}{2}(\partial_{m}\xi^{0}{}_{a}-\partial_{a}\xi^{0}{}_{m})$
for m component. The trace of the $s\beta_{ma}$ is also vanishing since trace
of $\omega_{0am}$ is gauge fixed to zero with $\xi^{0-}$. The 0 component
allows to fix $s\beta_{0}{}^{a}=0$ when $\xi^{a-}=-\partial_{0}\xi^{0a}$. The
equations, for mn components, relate $s\kappa_{a}$ and $s\beta_{ab}$. The
symmetric part of the equation is equally zero, while the antisymmetric part
gives equation for the antisymmetric part of $s\beta^{ab}$

$\displaystyle\partial^{j}s\beta^{[mn]}-\partial^{m}s\beta^{[jn]}+\partial^{n}s\beta^{[jm]}=0.$
(4.16)

The hook component

$\displaystyle\tfrac{1}{3}(-s\kappa^{n}h^{jm}-s\kappa^{m}h^{jn}+2s\kappa^{j}h^{mn}-\partial^{j}s\beta^{mn}-\partial^{j}s\beta^{nm}+\partial^{m}s\beta^{jn}+\partial^{n}s\beta^{jm})=0$
(4.17)

can be contracted with $h^{ab}$ to give $s\kappa^{m}$ in terms of
$s\beta_{ab}$

$\displaystyle\tfrac{1}{3}(-s\kappa^{m}-\partial_{a}s\beta^{ma}+\partial^{m}s\beta^{a}{}_{a})=0.$
(4.18)

The m0 equation of $SB^{a}$ gives
$s\kappa_{0}=\frac{1}{2}\partial_{0}s\beta_{m}{}^{m}=0$ taking into
consideration that $s\beta_{0}{}^{a}$ is gauge fixed to zero, as well as
$\omega_{0a}{}^{a}$, and that we will be able to fix a trace of $sf_{a}{}^{a}$
field. This means that one has $sf_{am}=\partial_{0}s\beta_{am}$.

The SC gauge transformation gives the transformations for
$\delta\gamma_{m}=\partial_{m}\xi^{00}$ and
$\delta\gamma_{0}=\partial_{0}\xi^{00}+\frac{1}{2}\xi^{0-}$. The equations
define $\partial_{[m}\gamma_{n]}=0$ and $s\kappa_{m}$ in terms of $\gamma_{m}$
and $\gamma_{0}$, such that
$s\kappa_{m}=\partial_{m}\gamma_{0}-\partial_{0}\gamma_{m}$.

The transformation with $SK_{a}$ defines $\delta
sf_{(ma)}=\frac{1}{2}(\partial_{m}\xi_{a-}+\partial_{a}\xi_{m-})+h_{ma}\xi^{--}$
and $\delta sf_{[ma]}=\frac{1}{2}(\partial_{m}\xi_{a-}-\partial_{a}\xi_{m-})$.
The component 0 defines $\delta sf_{0a}=\partial_{0}\xi^{a-}$. The equations
of motion in mn component, vanish for the symmetric part, for antisymmetric
they give equation for antisymmetric part of $sf_{[mn]}$ so that
$\partial^{j}sf^{[mn]}-\partial^{m}sf^{[jn]}+\partial^{n}sf^{[jm]}=0$. The
hook component reads
$\tfrac{1}{3}(-\omega^{--}{}^{n}h^{jm}-\omega^{--}{}^{m}h^{jn}+2\omega^{--}{}^{j}h^{mn}-\partial^{j}sf^{mn}-\partial^{j}sf^{nm}+\partial^{m}sf^{jn}+\partial^{n}sf^{jm})=0$
and after contracting with $h^{jm}$ it defines
$\omega^{n--}=\partial^{n}sf_{a}{}^{a}-\partial_{a}sf^{na}$. The 0 component
defines the field $\omega_{0}^{--}=-\frac{1}{2}\partial_{m}sf^{m}{}_{0}$.

For the $SK_{0}$ the transformation in the m component gives $\delta
s\kappa_{m}=\partial_{m}\xi^{0-}$ and in 0 component one obtains $\delta
s\kappa_{0}=\partial_{0}\xi^{0-}+\xi^{--}$. The equations of motion give
$\partial_{[n}s\kappa_{m]}=0$ and
$\omega_{m}^{--}=\partial_{m}s\kappa_{0}-\partial_{0}s\kappa_{m}$ for mn and
m0 components respectively. The remaining fields are
$se_{[am]},\omega_{m}^{++},\omega^{(mab)}\equiv\phi^{mab},\omega_{0}{}^{ab},sf_{0i}$,
and $\gamma_{0}$. The summary of the fields is given in Table 2, while their
gauge transformations are written in the Appendix, in Table 4.

Generator | Field
---|---
$t_{++}$ | $\omega_{0}{}^{++}=0$
| $\omega_{m}{}^{++}$
$SP_{i}$ | $se_{(mi)}=0$
| $se_{[mi]}$
| $se_{0i}=0$
SH | $s\tau_{m}=-\frac{1}{2}\partial_{0}\partial_{0}\omega_{m}{}^{++}$
| $s\tau_{0}=0$
$SJ_{ij}$ | $\omega^{(H)imn}=-\frac{1}{3}(\partial^{i}se^{mn}+\partial^{m}se^{in}),\omega^{[imn]}=0$
| $\omega^{(imn)}\equiv\phi^{imn}$
| $\omega_{0im}$
$SC$ | $\gamma_{0}$
| $\gamma_{m}=-\frac{1}{4}\partial_{0}\partial_{0}\omega_{m}^{++}$
$SB_{j}$ | $s\beta_{(mi)}=-\omega_{0im},s\beta_{[im]}=\partial_{0}se_{mi}$
| $s\beta_{0i}=0$
$SK_{i}$ | $sf_{(mi)}=\partial_{0}s\beta_{im}$
| $sf_{0i}$
$SK_{0}$ | $s\kappa_{0}=0$
| $s\kappa^{m}=-\partial_{a}s\beta^{ma}$
$t_{--}$ | $\omega_{0}{}^{--}=-\frac{1}{2}\partial_{m}sf_{0}{}^{m}$
| $\omega_{m}{}^{++}=-\partial_{0}s\kappa_{m}$
Table 2: Spectrum of fields in spin-3 Galilean conformal gravity.

It is clear that this number of fields remains due to degeneracy of metric. If
we compare the number of fields in this non-relativistic limit of the
conformal spin-3 case and the number of remaining fields of conformal spin-3
in ultra-relativistic limit in [26], we can see that here there is more
undetermined fields remaining. There we had
$\phi_{mni},\omega^{0++},s\beta_{(mi)},\gamma_{m}$ and $\gamma_{0}$. Here we
have $se_{[am]}$ and $sf_{0m}$ that do not have analog in ultra-relativistic
limit, while there we had $\omega_{0}^{++}$ which was here gauge fixed to
zero. The other fields are either obtained from the same relativistic field,
or are determined in terms of each other. If we wanted to compare the ultra-
relativistic and non-relativistic limit of the spin-3 fields we would have to
consider semi-GCA. In that case we would have similar situation as in the two
dimensions when one can interchange the time and space component and relate
these two limits.

##  5 Conclusion

We have considered Galilean limit of conformal algebra for spin-2 and spin-3
fields, and studied the gauge theory of these algebras. We obtained the
spectrum of the theories. In spin-2 the remaining fields were
$\tau_{0},\phi_{mi},\tau_{m},\beta_{[mi]},\kappa_{0},f_{0i}$. While in the
spin-3 case these were
$se_{[am]},\omega_{m}^{++},\phi_{mab},\omega_{0ab},sf_{0i}$, and $\gamma_{0}$.
The action that is obtained in these theories does not include all the fields
that appear in the spectrum. This is due to the degeneracy of the metric, and
it is similar to the situation that happens for the non-conformal case [34].
Interesting thing to consider would be to see if in the non-relativistic case
of the gauged theory of the s-GCA one can obtain the results that modify
Newton-Cartan gravity in the analogous way that relativistic conformal gravity
modifies Einstein gravity. One of the aspects in which relativistic conformal
gravity has been studied [35] showed that one can obtain galactic rotation
curves without the addition of the dark matter. In the non-conformal case,
Galilean algebra had to be supplemented by the central extension, so that
Bargmann algebra leads to the Newton-Cartan gravity when the theory is a gauge
theory. When one tries to apply the analogous thinking to the conformal case,
this is not possible, because GCA does not admit the same kind of central
extension as Galilean algebra [7]. The reason for this can be seen if one
tries to calculate Jacobi identities for the generators ($B_{i},P_{i},D$)
where $B_{i}$ generators are the Inönü Wigner contracted generators of the
Lorentz generators. One of the interesting things for the future research
would be to study s-GCA or certain extension of the GCA to obtain the algebra
gauging which gives the analogy of the non-relativistic conformal gravity.
Using the same type of extension one could then implement to obtain also
higher spin extension of the type [31] of GCA.

##  6 Acknowledgements

This work was supported by the Hertha Firnberg grant T 1269-N of the Austrian
Science Fund FWF.

##  7 Appendix

Table of gauge transformations and fixed gauge parameters for the spin-2 case
of GCA.

D | $\delta b_{0}=0$ | $\xi^{0-}=-\partial_{0}\xi^{+-}$
---|---|---
| $\delta b_{m}=\partial_{m}\partial_{0}\xi^{0+}$ |
$P_{i}$ | $\delta e_{[mi]}=0$ | $\xi_{i,m}=\frac{1}{2}(\partial_{m}\xi_{i}{}^{+}-\partial_{i}\xi_{m}{}^{+})$
| $\delta e_{(mi)}=\frac{1}{2}(\partial_{m}\xi_{i}{}^{+}+\partial_{i}\xi_{m}{}^{+})-\frac{1}{2}h_{mi}\partial_{l}\xi^{l+}$ |
| $\delta e_{0i}=0$ | $\xi_{0,i}=\partial_{o}\xi_{i}{}^{+}$
H | $\delta\tau_{m}=\partial_{m}\xi^{0+}$ |
| $\delta\tau_{0}=0$ | $\xi^{+-}=\partial_{0}\xi^{0+}$
$J_{ij}$ | $\delta\omega_{m}{}_{ij}=\frac{1}{2}\partial_{m}(\partial_{j}\xi_{i}{}^{+}-\partial_{i}\xi_{j}{}^{+})-\frac{1}{4}h_{m}{}^{i}\partial_{l}(\partial_{j}\xi^{l+}-\partial^{l}\xi_{j}{}^{+})$ |
| $\delta\omega_{0ij}=\frac{1}{2}\partial_{0}(\partial_{j}\xi_{i}{}^{+}-\partial_{i}\xi_{j}{}^{+})$ |
$B_{j}$ | $\delta\beta_{(mj)}=\frac{1}{2}(\partial_{m}\partial_{0}\xi_{j}{}^{+}+\partial_{j}\partial_{0}\xi_{m}{}^{+})-h_{mj}\partial_{l}\partial_{0}\xi^{l+}$ |
| $\delta\beta_{[mj]}=\frac{1}{2}(\partial_{m}\partial_{0}\xi_{j}{}^{+}-\partial_{j}\partial_{0}\xi_{m}{}^{+})$ |
| $\delta\beta_{0}{}^{j}=0$ | $\xi^{j-}=-\partial_{0}\xi_{0}{}^{j}$
$K_{i}$ | $\delta f_{(mi)}=\frac{1}{2}(-\partial_{m}\partial_{0}\partial_{0}\xi_{i}{}^{+}-\partial_{i}\partial_{0}\partial_{0}\xi_{m}{}^{+})+\frac{1}{2}h_{mi}\partial_{l}\partial_{0}\partial_{0}\xi^{l+}$ |
| $\delta f_{[mi]}=\frac{1}{2}(-\partial_{m}\partial_{0}\partial_{0}\xi_{i}{}^{+}+\partial_{i}\partial_{0}\partial_{0}\xi_{m}{}^{+})$ |
| $\delta f_{m}{}^{m}=-\partial_{m}\partial_{0}\partial_{0}\xi^{m+}$ |
| $\delta f_{0}{}^{i}=-\partial_{0}\partial_{0}\partial_{0}\xi^{i+}$ |
$K_{0}$ | $\delta\kappa_{m}=-\partial_{m}\partial_{0}\partial_{0}\xi^{0+}$ |
| $\delta\kappa_{0}=-\partial_{0}\partial_{0}\partial_{0}\xi^{0+}$ |
Table 3: Gauge transformations and gauge parameters for the spin-2 case.

Table of gauge transformations and fixed gauge parameters for the spin-3 case.

$t^{++}$ | $\delta\omega_{0}^{++}=0$ | $\xi^{0+}=-\frac{1}{2}\partial_{0}\xi^{++}$
---|---|---
| $\delta\omega_{m}^{++}=\partial_{m}\xi^{++}+2s\tau_{m}\xi^{0+}$ |
$SP_{i}$ | $\delta se_{(mi)}=0$ | $\xi_{im}=\frac{1}{2}(\partial_{m}\xi_{i}{}^{+}+\partial_{i}\xi_{m}{}^{+})$
| $\delta se_{[mi]}=\frac{1}{2}(\partial_{m}\xi_{i}{}^{+}-\partial_{i}\xi_{m}{}^{+})-\frac{1}{2}h_{mi}\partial_{l}\xi^{l+}$ |
| $\delta se_{0i}=0$ | $\xi_{0i}=\partial_{0}\xi_{i}{}^{+}$
SH | $\delta s\tau_{m}=\partial_{m}\xi^{0+}$ |
| $\delta s\tau_{0}=0$ | $\xi^{00}=\frac{1}{2}\partial_{0}\xi^{0+}$
$SJ_{ij}$ | $\delta\omega_{(mij)}=\frac{1}{2}(\partial_{i}\partial_{j}\xi_{m+}+\partial_{i}\partial_{m}\xi_{j+}+\partial_{m}\partial_{j}\xi_{i+})$ |
| $\delta\omega_{(H)mij}=\frac{1}{6}(-2\partial_{i}\partial_{j}\xi_{m+}+\partial_{i}\partial_{m}\xi_{j+}+\partial_{m}\partial_{j}\xi_{i+})$ |
| $\delta\omega_{[mij]}=0$ |
| $\delta\omega_{0ab}=\partial_{0}\xi_{ab}-h_{ab}\xi^{0-}$ | $\xi^{0-}=-\partial_{0}\xi_{a}{}^{a}$
$SB_{j}$ | $\delta s\beta_{(mj)}=\frac{1}{2}(\partial_{m}\partial_{0}\xi_{j}{}^{+}+\partial_{j}\partial_{0}\xi_{m}{}^{+})-h_{mj}\partial_{l}\partial_{0}\xi^{l+}$ |
| $\delta s\beta_{[mj]}=\frac{1}{2}(\partial_{m}\partial_{0}\xi_{j}{}^{+}-\partial_{j}\partial_{0}\xi_{m}{}^{+})$ |
| $\delta s\beta_{0}{}^{j}=0$ | $\xi^{j-}=-\partial_{0}\xi_{0}{}^{j}$
SC | $\delta\gamma_{m}=\partial_{m}\xi^{00}$ |
| $\delta\gamma_{0}=\partial_{0}\xi^{00}+\frac{1}{2}\xi^{0-}$ |
$SK_{i}$ | $\delta sf_{(mi)}=\frac{1}{2}(-\partial_{m}\partial_{0}\partial_{0}\xi_{i}{}^{+}-\partial_{i}\partial_{0}\partial_{0}\xi_{m}{}^{+})+h_{ma}\xi^{--}$ |
| $\delta sf_{[mi]}=\frac{1}{2}(-\partial_{m}\partial_{0}\partial_{0}\xi_{i}{}^{+}+\partial_{i}\partial_{0}\partial_{0}\xi_{m}{}^{+})$ |
| $\delta sf_{m}{}^{m}=0$ | $\xi^{--}=\frac{1}{2}\partial_{m}\partial_{0}\partial_{0}\xi^{m+}$
| $\delta sf_{0}{}^{i}=-\partial_{0}\partial_{0}\partial_{0}\xi^{i+}$ |
$SK_{0}$ | $\delta s\kappa_{m}=-\partial_{m}\partial_{0}\partial_{l}\xi^{l+}$ |
| $\delta s\kappa_{0}=-\partial_{0}\partial_{0}\partial_{l}\xi^{l+}+\frac{1}{2}\partial_{m}\partial_{0}\partial_{0}\xi^{m+}$ |
Table 4: Gauge transformations and gauge parameters for the spin-3 case.

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# LEARNING TO JOINTLY TRANSCRIBE AND SUBTITLE FOR END-TO-END SPONTANEOUS
SPEECH RECOGNITION

###### Abstract

TV subtitles are a rich source of transcriptions of many types of speech,
ranging from read speech in news reports to conversational and spontaneous
speech in talk shows and soaps. However, subtitles are not verbatim (i.e.
exact) transcriptions of speech, so they cannot be used directly to improve an
Automatic Speech Recognition (ASR) model. We propose a multitask dual-decoder
Transformer model that jointly performs ASR and automatic subtitling. The ASR
decoder (possibly pre-trained) predicts the verbatim output and the subtitle
decoder generates a subtitle, while sharing the encoder. The two decoders can
be independent or connected. The model is trained to perform both tasks
jointly, and is able to effectively use subtitle data. We show improvements on
regular ASR and on spontaneous and conversational ASR by incorporating the
additional subtitle decoder. The method does not require preprocessing
(aligning, filtering, pseudo-labeling, …) of the subtitles.

Index Terms—  speech recognition, multitask learning, end-to-end, subtitles,
spontaneous speech

##  1 Introduction

Speech recognition aims to identify the words in a spoken audio fragment and
transcribe what has been said into written text. Traditionally, Automatic
Speech Recognition (ASR) systems made use of a limited vocabulary of words and
pronunciations, and consisted of several components to model speech
statistically. More recently, data-driven end-to-end ASR systems have been
able to directly convert a sequence of audio features into a sequence of
(sub)words or characters.

ASR requires a large dataset with careful transcriptions of spoken utterances.
While TV subtitles seem the obvious candidate, as they are so abundant and
manually annotated, they are not verbatim (i.e. exact) transcriptions of what
was said. Subtitles are optimised for readability on screen, and are therefore
often shorter and cleaner (e.g. no repetitions, no hesitations). Consequently,
the timings of subtitles on screen can be inaccurate compared to the spoken
sentences. On top of that, subtitles are “clean”, with edits to correct non-
grammatical sentences and poor or dialectal word choice.

Nevertheless, being able to use TV subtitles would have many advantages.
Subtitles can cover a very broad domain, from read speech in broadcast news to
spontaneous and conversational speech in talk shows, interviews, soaps,
sitcoms, etc. This is in contrast to the labeled speech datasets which often
contain prepared speech (e.g. audiobooks in LibriSpeech [1]). Subtitles cover
many languages, and they have been manually annotated by human annotators.
Additionally, accented speech and strong dialects are more common among free
speech on TV, and less covered in current speech datasets.

Reducing the need for well-labeled speech data is a long-standing problem. The
performance of ASR has seen tremendous improvements from pre-training on
unlabeled data in a self-supervised manner [2, 3, 4]. Unlabeled speech data is
plentifully available, and can even be (pseudo-)labeled by pre-trained models
to iteratively improve models by self-training [5]. Furthermore, pre-training
reduces the need for large amounts of labeled data and has significantly
advanced performance in low resourced languages, on account of cross- and
multilingual positive transfer [6] from learned high-level acoustic feature
representations to the target language.

In this work, we propose a model that jointly performs automatic (verbatim)
speech recognition and automatic subtitling. Besides solving two useful tasks
at once, the model is able to make use of the subtitle data. We apply a
multitask learning (MTL) framework [7, 8] with a shared encoder and two
decoders, inspired by end-to-end speech translation where machine translation
and ASR are additional subtasks with independent [9] or interconnected
decoders [10, 11] for each task. We show that the introduction of the subtitle
decoder and subtitle data improves the performance of the end-to-end ASR model
on both a regular ASR benchmark and on a spontaneous speech dataset, without
construction of a spontaneous ASR dataset. Experiments are performed on
Belgian Dutch (Flemish), a medium-resourced language with strong regional
dialects. Finally, we note that parallel work on pre-training with unlabeled
data for learning representative features (e.g. for Belgian Dutch [12]), is
complementary to this work, and these pre-trained feature representations can
be plugged into the front-end of an ASR system.

##  2 Related work

Most research on the use of subtitle data can be divided into two strands. On
the one hand, subtitles (either detected on screen in video’s [13] or gathered
separately) are refined and strongly filtered to build a corpus for ASR [14,
15], e.g. based on matching alignment [16, 17, 18] or thresholding on
posterior probability [19]. On the other hand, the output of ASR systems can
be adapted with models into readable subtitles, by compression [20] or with a
(genre-dependent) language model [21], or directly translated into subtitles
in a different language [22]. The Multi-Genre Broadcast (MGB) challenge
inspired some of these works, with the MGB-1 challenge [23] providing English
subtitles and the MGB-2 challenge [24] Arabic subtitles, and related
competitions, e.g. the IberSpeech-RTVE challenge on Spanish Broadcast media
[25]. Finally, there has also been research on using weak supervision from
contextual text information in social media video’s to improve ASR [26],
however these annotations are further away from verbatim transcriptions than
subtitles.

The work in this paper distinguishes itself by concurrently solving both the
ASR and subtitling task, and, without having to adapt or refine the data
(which can be done complementary to our method in a pre-processing stage, e.g.
subtitle alignment [27]), improving the current ASR system in all domains. No
parallel data is used, i.e. data for which there are both verbatim and
subtitle transcriptions, as it is not generally available in most languages.

##  3 Method

The proposed model adapts the end-to-end hybrid CTC-Attention ASR model,
introduced in [28], to the multitask learning framework, by attaching a second
decoder for subtitle generation and sharing the encoder.

(a) E2E ASR model

(b) Multitask model - ASR + Subtitles

Independent decoders

(c) Multitask model - ASR + Subtitles

Cross-connected decoders

Fig. 1: Model architecture: comparison of (a) a general end-to-end
CTC/Attention hybrid ASR model, and (b) the proposed multitask model with two
decoders. In (c) the proposed multitask model has additional cross-connections
between both decoders.

###  3.1 End-to-end ASR

End-to-end (E2E) models solve the ASR problem as a sequence-to-sequence
problem. First of all, an encoder computes feature representations from the
input sequence. The feature extractor in this case is a Transformer [29],
relying on self-attention [30] to compute feature representations.

In general, end-to-end architectures either use Connectionist Temporal
Classification (CTC) [31] to generate an output sequence from the encoder
states, or an attention-based mechanism that can attend to different parts of
the encoder output to align the input features with the output sequence. Both
methods have their strengths and drawbacks. CTC, although efficient, relies on
independent Markov assumptions and can only predict monotonic alignments. Pure
attention-based methods are hard to train and optimise as there is no
constraint on possible alignments.

In this work, we adopt the hybrid CTC/attention architecture [28], that
combines both methods, in the baseline ASR system. Figure 1(a) shows the
hybrid model. CTC regularises the encoder by constraining a monotonic
alignment and predicting an aligned sequence of output tokens including blanks
from the encoder outputs. The encoder outputs could resemble small acoustic
units because of the regularisation. The attention-based decoder [29] learns
the alignment between the encoder states and the elements of the output
sequence. It can attend to the entire encoder output and during decoding the
predicted tokens are generated autoregressively, i.e. one-by-one.

The training objective is a weighted sum of the label smoothed cross-entropy
classification loss of the tokens generated by the attention decoder and the
CTC objective, respectively denoted by $\mathcal{L}_{att,asr}$ and
$\mathcal{L}_{ctc}$.

$\mathcal{L}_{asr}=(1-\lambda_{ctc})\mathcal{L}_{att,asr}+\lambda_{ctc}\mathcal{L}_{ctc}$
(1)

During decoding, the CTC and attention scores are combined in a joint beam
search where the produced hypotheses are pre-scored by CTC.

###  3.2 Dual decoder - MTL

The proposed model is depicted in Figure 1(b). An additional attention-based
decoder branch is added to the encoder. In this multitask model, the encoder
is shared but the decoders are completely independent and they both attend to
the encoder’s states (as the multitask model in [9]). When an auxiliary task
is imposed to a model, the performance on the original task might increase if
the additional task is useful. The inclusion of a second independent decoder
could benefit the learned representation in the shared encoder.

The first decoder, the ASR Decoder, is the same as in the previous section and
generates a verbatim transcription. The second decoder, the Subtitle Decoder,
generates a subtitle for the input features. Due to summarisation and
compression, subtitles align differently than verbatim transcriptions. Hence
we use an attention-based loss without CTC for the subtitles. The final
training objective $\mathcal{L}_{tot}$ is now a weighted sum of both decoders’
attention losses and the CTC objective, with $\mathcal{L}_{asr}$ as defined in
Equation 1.

$\mathcal{L}_{tot}=\lambda_{asr}\mathcal{L}_{asr}+\lambda_{subs}\mathcal{L}_{att,subs}$
(2)

During training, batches are filled with utterances from the ASR dataset (i.e.
verbatim targets) and utterances from the subtitle dataset, equally mixed per
batch. The ASR decoder and CTC losses are backpropagated for the verbatim
utterances and masked for the subtitled utterances. Likewise, the subtitle
decoder loss is backpropagated for the subtitled data and masked for the ASR
data.

###  3.3 Connected decoders

As ASR and subtitling are related tasks, it would seem useful to have some
kind of connection between both decoders. Knowing what was said is useful to
generate a subtitle, and vice versa. Chaining or cascading the decoders one
after another, e.g. an ASR decoder followed by a subtitle decoder, is
difficult with autoregressive decoders. Therefore, we investigate the Cross
Dual-decoder Transformer proposed in [10]. In this model, there are additional
cross-attention blocks added between the decoders, where the decoders can also
attend to the previous outputs of the other decoder. Both autoregressive
decoders process the data simultaneously and work in a synchronous way. The
output of the cross-attention with the other decoder is summed together with
or concatenated to the outputs of the decoder’s own attention operations [10].

The model with cross-connected decoders, depicted in Figure 1(c), requires
parallel data for both the ASR and the subtitle decoder to learn useful cross-
attention weights. Therefore, we first generate pseudo-labels with a pre-
trained multitask model with independent decoders, i.e. verbatim labels for
the subtitle data and subtitles for the ASR data. Second, we finetune the
model with additional cross-connections between the decoders on the parallel
(pseudo-)labeled data. This is an extension of iterative self-training [5]
with pseudo-labels.

During decoding, the ASR and subtitle hypotheses are expanded together as
tuples of (ASR, subtitle) combinations in one joint beam. Only valid
combinations for the next tokens are considered. The scores per tuple are
computed as a weighted combination of the ASR and subtitle hypotheses’ scores
during the beam search. Equal weights for the ASR and subtitles appear to give
the best results.

##  4 Experimental setup

###  4.1 Data

The multitask model is trained on both verbatim ASR data, from an official
database, and subtitled data gathered in-house.

Verbatim. The Corpus Gesproken Nederlands (CGN) [32], or Spoken Dutch Corpus,
is a speech database containing 270 hours of Belgian Dutch (Flemish) speech.
The complete dataset includes read speech, news reports, lectures,
commentaries, and also narrowband telephone speech (8 kHz resampled to 16 kHz)
and spontaneous conversational speech (VL-train-all in [12]). For evaluation,
we use the dev-other set of 15h of speech, named cgn-dev, which does not
contain telephone and conversational speech, and has no overlap in speakers
with the training set.

Subtitled. The subtitled dataset contains 720 hours of speech, segmented based
on the subtitle timings on screen. The data consists of 190 hours of broadcast
TV news, 230 hours of soap TV series, and the remaining 300 hours of speech
originate from various talk shows about news, politics, entertainment, etc.
For evaluation, we manually annotated a set of 6 hours of speech (around 6500
utterances), named subs-annot, from a mixture of show types, and keep it held
out of the training set. Finally, for one experiment we have extended the
subtitled dataset to 1700 hours of speech gathered from the same and similar
sources.

###  4.2 Model configuration

The models are implemented with the ESPnet2 library [33] and follow the
standard encoder-decoder architecture of end-to-end ASR models. The encoder is
a 12-layered Transformer [29] with a Conv2D input layer to transform the input
features, and an output dimension of 256. The ASR decoder and the subtitle
decoder are both 6-layered Transformer decoders. Every Transformer block has 4
attention heads and 2048 linear units.

The input features are 80-dimensional Mel filterbank features, concatenated
with 3-dimensional pitch features, extracted from windows of 25ms and with a
frame shift of 10ms. Utterance-level mean-variance normalisation and
SpecAugment [34] augmentations are applied to the inputs. The targets are 5000
unigram BPE’s, and the BPE (Byte-Pair Encoding) model is trained on both the
ASR transcriptions and the subtitles. No language model is applied, not during
and not after decoding.

###  4.3 Training details

All models are trained for 100 epochs with a batch size of 32 and gradient
accumulation over 8 steps. We use the Adam optimizer [35] with a learning rate
schedule for Transformers [29] with 25000 warm-up steps and a peak learning
rate of 0.004 for models trained from scratch and 0.0004 for ASR-initialised
models. For evaluation, we average the 10 models with the best ASR accuracy on
the validation set. The CTC weight $\lambda_{ctc}$ is set to 0.3, both during
training and decoding. The cross-entropy label smoothing weight is set to 0.1.
Decoding is implemented as beam search with beam size 20.

##  5 Results

In this section we will describe the experiments conducted to evaluate the
proposed model and discuss the results. We examine the performance of the
multitask model on a speech recognition task, and compare it to the ASR model
without subtitle decoder. The models are evaluated based on the Word Error
Rate (WER) on a regular ASR test set cgn-dev and on the spontaneous test set
subs-annot. Furthermore, the capabilities of automatic subtitling of the
multitask model are explored by comparing the output hypothesis of the
subtitle decoder to the real subtitle on screen. As metric, we utilise the
smoothed BLEU-4 score [36] with uniform weights. Finally, we investigate if
finetuning with generated pseudo-labels can yield further improvements. We
compare the multitask model with independent decoders to the multitask model
with cross-attention connected decoders on ASR and the subtitling task.

###  5.1 Verbatim Speech Recognition

####  5.1.1 Multitask model from scratch

First of all, we examine the effect of adding the additional subtitle decoder
to the ASR model. The importance of both decoders can be weighted in the loss
function (Equation 2) with the hyperparameters $\lambda_{subs}$ and
$\lambda_{asr}$. Table 1 shows the results for different combinations of both
weights. The first line, i.e. the model with $\lambda_{subs}=0$, is the
baseline ASR model.

It is important to note that training on the joint set of ASR and subtitle
data, as if it were all verbatim data, always diverges, irrespective of the
learning rate. We postulate that the inexact transcripts could impair the CTC
alignment and the classification, and the inaccurate timings are difficult to
incorporate in verbatim speech recognition. This is the key advantage of the
proposed multitask model.

Table 1 shows that the additional decoder can slightly improve the WER on the
regular cgn-dev and strongly improve the WER on the spontaneous subs-annot,
because the encoder is improved by incorporating the subtitle data for the
subtitling task. Using equal weights in the loss function seems most
beneficial in this setup. If the weight of the subtitle decoder
$\lambda_{subs}$ is too large, the model tends to diverge. It is a
hyperparameter that needs some tuning, but always improves results when chosen
between a wide range.

Decoder | WER (%)
---|---
$\mathbf{\lambda_{asr}}$ | $\mathbf{\lambda_{subs}}$ | cgn-dev | subs-annot
1.0 | 0.0 | 15.90 | 37.61
0.9 | 0.1 | 16.54 | 36.76
0.7 | 0.3 | 14.50 | 32.47
0.5 | 0.5 | 14.29 | 30.48
0.3 | 0.7 | 79.98 (div) | 83.18 (div)
0.1 | 0.9 | 103.57 (div) | 109.56 (div)
Table 1: WER for multitask models trained from scratch with varying weights in
the objective function. The first row shows the single decoder ASR model as
baseline. Diverged models are annotated with (div).

To get a sense of the difference between the ASR and subtitle decoder output,
Table 2 shows a sample utterance transcription from the spontaneous test set
and the generated hypotheses from both decoders, with the model in Table 1
using $\lambda_{subs}=0.5$. Although for this utterance the reference subtitle
is much shorter and extremely hard to predict, it is clear that the subtitle
decoder output is cleaner, e.g. no apostrophies for abbreviating words and no
hesitations (‘uh’). We found this to be a recurring theme during analysis of
the subtitle output, with additional normalisation of dialectal words (e.g.
‘gij’ becomes ‘jij’) and the decoder leaving out repetitions due to
stammering.

REF | en ’t is net op dat moment dat het virus eigenlijk toeslaat da’s een uh eerste feit
---|---
(verbatim) | and it’s right at that moment that the virus actually hits that’s an uh first fact
REF | dan slaat het virus toe dat is een eerste feit
(subtitle) | then the virus hits that is a first fact
ASR | ’t is net op dat moment dat het virus eigenlijk toelaat was een uh feit
hypothesis | it’s right at that moment that the virus actually permits was an uh fact
SUB | het is net op dat moment dat het virus eigenlijk toelaat als een feit
hypothesis | it is right at that moment that the virus actually permits as a fact
Table 2: Comparison of the reference ASR transcription and the reference
subtitle transcription to the hypotheses generated by the outputs of the ASR
decoder (ASR) and the subtitle decoder (SUB). The utterance is part of the
subs-annot test set. Translation in English is added for the reader.

####  5.1.2 Adapting pre-trained ASR models to multitask models

Second, we examine if a pre-trained single-decoder ASR model can be improved
with the additional decoder, or if it would harm the result on the ASR task.
The ASR model is a state-of-the-art E2E ASR model [28]. Table 3 shows the
results when the encoder is initialised with a pre-trained ASR encoder, and
the ASR and/or the subtitle decoder branch are initialised with the parameters
from the decoder of a pre-trained ASR model. The baseline model of the
previous section is used as pre-trained ASR model to initialise the weights.

The WER on both sets strongly improves by extending a pre-trained ASR model
with the additional subtitle decoder and using the subtitle data. Both the ASR
decoder and subtitle decoder branch can be initialised with an ASR model.

Decoder | Initialisation | WER (%)
---|---|---
$\mathbf{\lambda_{asr}}$ | $\mathbf{\lambda_{subs}}$ | ASR | Subs | cgn-dev | subs-annot
1.0 | 0.0 | / | / | 15.90 | 37.61
0.5 | 0.5 | / | / | 14.29 | 30.48
0.5 | 0.5 | ASR | / | 12.27 | 28.33
0.5 | 0.5 | ASR | ASR | 12.22 | 28.38
Table 3: WER for multitask models with initialisation of the ASR decoder (ASR)
and/or subtitle decoder (Subs) with the weights of a pre-trained ASR decoder.
The models in the first two rows are not initialised and trained from scratch.
The models in the bottom two rows both use a pre-trained ASR encoder. The
subtitle dataset contains 700h of data.

####  5.1.3 Using more subtitled data

Finally, we examine if using more subtitled data would additionally improve
the results. We compare the ASR performance of a model using 700 hours of
subtitled data in Table 3 to a model using 1700 hours of subtitled data in
Table 4. The verbatim ASR dataset remains the same.

The results on 1700 hours (using the same settings as in the previous
experiments) are comparable to the results on 700 hours of subtitled data, but
not better. We notice the same trend of improvements across the board for the
different models. The first line of the table suggests that further finetuning
of $\lambda_{asr}$ might yield further improvements.

Decoder | Initialisation | WER (%)
---|---|---
$\mathbf{\lambda_{asr}}$ | $\mathbf{\lambda_{subs}}$ | ASR | Subs | cgn-dev | subs-annot
0.7 | 0.3 | / | / | 14.37 | 32.84
0.5 | 0.5 | / | / | 16.26 | 33.45
0.5 | 0.5 | ASR | / | 12.56 | 30.04
0.5 | 0.5 | ASR | ASR | 12.23 | 29.86
Table 4: WER for multitask models trained on the extended 1700h subtitle
dataset, with or without pre-trained initialisation.

###  5.2 Subtitling

We evaluate the output of the subtitle decoder with a BLEU score against the
reference on-screen subtitle in the test set. Table 5 summarises the results
of all previous models. For the baseline (i.e. the first row), we use the
output of the ASR decoder. This gives a notice of how close the subtitles are
to verbatim transcriptions.

For almost all cases, the improvements in BLEU score follow the improvements
in WER as in the previous section. The double-initialised dual-decoder model
achieves the highest BLEU scores. Additionally, we notice that for most cases
there is no improvement with the model using 1700 hours of subtitle data over
the model using 700 hours of subtitle data. This might suggest that the
additional 1000 hours of data, mostly coming from different sources, is
slightly mismatched with the test set or contains more non-Dutch
transcriptions.

# Hours | Decoder | Initialisation | BLEU (%)
---|---|---|---
Subs | $\mathbf{\lambda_{asr}}$ | $\mathbf{\lambda_{subs}}$ | ASR | Subs | subs-annot
0 | 1.0 | 0.0 | / | / | 30.84
700 | 0.9 | 0.1 | / | / | 40.86
0.7 | 0.3 | / | / | 43.63
0.5 | 0.5 | / | / | 45.65
0.3 | 0.7 | / | / | 15.50 (div)
0.1 | 0.9 | / | / | 8.08 (div)
0.5 | 0.5 | ASR | / | 36.51
0.5 | 0.5 | ASR | ASR | 46.81
1700 | 0.7 | 0.3 | / | / | 41.66
0.5 | 0.5 | / | / | 41.41
0.5 | 0.5 | ASR | / | 43.60
0.5 | 0.5 | ASR | ASR | 43.85
Table 5: BLEU scores for multitask models with varying weights in the
objective function and possible initialisation. The first row shows the
single-decoder ASR model. Diverged models are annotated with (div).

###  5.3 Connected decoders

Section 3.3 showed a method to connect both decoders of the multitask model.
Because the connected decoders require parallel data, we generate pseudo-
labels with the ASR decoder for the subtitled data and with the subtitle
decoder for the verbatim data. For pseudo-labeling we decode with the best-
performing model, i.e. the model of the last row in Table 3, which is also
used for initialisation of all weights except for the weights in the cross-
attention part between the decoders. The loss weights $\lambda_{asr}$ and
$\lambda_{subs}$ are both kept as 0.5. We compare the multitask model with
cross-connected decoders to the multitask model with independent decoders (as
in previous sections), starting from the same initialisation and further
trained with pseudo-labels. Table 6 shows the ASR results and Table 7 shows
the subtitling results.

Further finetuning of the multitask model with independent decoders on the
generated pseudo-labels (row 2 in Table 6), starting from the pre-trained
multitask model (row 1 in Table 6), slightly improves the WER on both sets, at
the expense of a drop in BLEU score.

Finetuning the pre-trained multitask model (row 1 in Table 6) with additional
cross-attention connections between the decoders is not able to improve the
WER on the regular cgn-dev test set. This might be because the ASR decoder
cross-attention and the Subtitle-to-ASR cross-attention are merged (either
summed up or concatenated and linearly projected), which could implicitly give
an extra weight to the subtitles in the ASR decoder branch, which might have a
slight negative impact on the ASR decoder in very clean settings due to the
domain mismatch. However, when the encoder is kept frozen during finetuning
and only the decoders are finetuned, we do obtain the best ASR performance on
the spontaneous subs-annot test set and by far the best BLEU score.

Model | PL | WER (%)
---|---|---
Cross | Method | Freeze |  | cgn-dev | subs-annot
No | / | / | No | 12.22 | 28.38
No | / | / | Yes | 12.04 | 28.33
Yes | Sum | / | Yes | 12.32 | 28.44
Yes | Concat | / | Yes | 12.37 | 28.39
Yes | Concat | Enc. | Yes | 13.46 | 27.69
Table 6: WER for multitask models with independent decoders and with cross-connected decoders. The decoder attention outputs and the inter-decoder cross-attention outputs can be concatenated or summed. The models can be further trained with pseudo-labels (PL) generated by the decoders for missing targets. The encoder can be kept frozen during finetuning with the pseudo-labels. Model | PL | BLEU
---|---|---
Cross | Method | Freeze |  | subs-annot
No | / | / | No | 46.81
No | / | / | Yes | 36.44
Yes | Sum | / | Yes | 45.97
Yes | Concat | / | Yes | 45.38
Yes | Concat | Encoder | Yes | 51.11
Table 7: BLEU scores for multitask models with independent decoders and with
cross-connected decoders. The decoder attention outputs and the inter-decoder
cross-attention outputs can be concatenated or summed. The models can be
further trained with pseudo-labels (PL) generated by the decoders for missing
targets. The encoder can be kept frozen during finetuning with the pseudo-
labels.

##  6 Conclusion

We have proposed a model for joint automatic speech recognition and subtitle
generation, that is able to leverage subtitled data to improve ASR performance
on both regular and spontaneous speech through multitask learning of both
tasks. A pre-trained ASR model can easily be adapted with a second decoder,
and its performance can be improved even with the inaccurate subtitle
transcriptions.

In this work, we use a dual-decoder model with independent autoregressive
decoders and a dual-decoder model with synchronously cross-connected
autoregressive decoders.

In future work, we will investigate non-autoregressive decoders to chain the
decoders in a cascaded way. We will also look at the MGB challenge for
evaluation of the models. The multitask models might also be improved through
better alignment of the subtitles’ endpoints with the input speech.

##  7 Acknowledgement

This research received funding from the Flemish Government under the
“Onderzoeksprogramma Artificiële Intelligentie (AI) Vlaanderen” programme and
was supported by Research Foundation Flanders (FWO) under grant S004923N of
the SBO programme.

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