ICCUB-21-005

Imperial-TP-KM-2021-01

Colourful Poincaré symmetry,
gravity and particle actions

Joaquim Gomis1, Euihun Joung2, Axel Kleinschmidt3,4 and Karapet Mkrtchyan5

1Departament de Física Quàntica i Astrofísica
and Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona
Martí i Franquès , ES-08028 Barcelona, Spain 2Department of Physics, Kyung Hee
University
Seoul 02447, Korea 3Max-Planck-Institut für Gravitationsphysik (Albert-
Einstein-Institut)
Am Mühlenberg 1, DE-14476 Potsdam, Germany 4International Solvay Institutes
ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium 5Theoretical Physics Group,
Blackett Laboratory,
Imperial College London SW7 2AZ, U.K

We construct a generalisation of the three-dimensional Poincaré algebra that
also includes a colour symmetry factor. This algebra can be used to define
coloured Poincaré gravity in three space-time dimensions as well as to study
generalisations of massive and massless free particle models. We present
various such generalised particle models that differ in which orbits of the
coloured Poincaré symmetry are described. Our approach can be seen as a
stepping stone towards the description of particles interacting with a non-
abelian background field or as a starting point for a worldline formulation of
an associated quantum field theory.
---

###### Contents

  1. 1 Introduction
  2. 2 Colouring the Poincaré algebra in $3$ space-time dimensions
  3. 3 Coloured gravity in 3d Minkowski space-time
    1. 3.1 Invariant bilinear form
    2. 3.2 Coloured gravity action in 3d Minkowski space
  4. 4 Free coloured particle model
    1. 4.1 Coloured Minkowski space
    2. 4.2 Different types of massive particle actions
    3. 4.3 Coadjoint orbits of ${\mathfrak{su}}(N,N)$
    4. 4.4 Massive coloured particle in component form
    5. 4.5 Constraint structure and degrees of freedom
    6. 4.6 Examples for small $N$
    7. 4.7 Massless coloured particle
    8. 4.8 Reductions to subspaces
  5. 5 Free coloured particle in AdS background
    1. 5.1 Uncoloured particle in AdS
    2. 5.2 Coloured particle in AdS
  6. 6 Conclusions
  7. A Derivation from embedding in AdS algebra
  8. B Free massive Poincaré particle
  9. C General remarks on particle actions
  10. D From the coloured Hamiltonian to the Lagrangian action

##  1 Introduction

Space-time symmetries that are larger than those realised in conventional
gravitational systems, including bosonic generators in non-trivial
representations of isometry algebra are usually ruled out in field theories of
finitely many interacting particles. The Coleman–Mandula theorem establishes
that such symmetries are not compatible with a non-trivial S-matrix [1].
However, there are ways to evade the Coleman–Mandula theorem.

Einstein gravity in three dimensions can be written in Chern–Simons form [2,
3]. This formulation is based on gauging the global symmetry algebra and is
background independent as well as directly related to the Einstein–Hilbert
formulation of gravity, if the dreibein is non degenerate. Such a formulation
can be generalised to higher-spin gravities without matter [4, 5, 6] thus
providing consistent examples of gravitational theories with extended
symmetries. These theories lack bulk propagating degrees of freedom and in
this way avoid no-go theorems based on S-matrix considerations. Other possible
extensions that have been considered are given by Chern–Simons actions based
on relativistic or non-relativistic Maxwell algebra extensions of the Poincaré
algebra, see for example [7, 8, 9, 10, 11, 12].

A different extension of three-dimensional space-time symmetries is realised
in coloured gravity [13]. There, instead of adding generators corresponding to
massless fields of spin $s\geq 3$, one has a colour extension of the isometry
itself, with multiple copies of generators which correspond to Killing vectors
and scalars associated to massless spin-two and spin-one fields. Such an
extension is non-trivial for associative algebras of isometries [13, 14], the
first example was provided in [15]. For the Poincaré algebra in arbitrary
dimensions, such a colouring requires the colour algebra to be commutative and
associative [16, 17, 18], and together with the requirement of positive-
definite bilinear form for the colour algebra (needed for unitarity) is
trivial as shown in [19]: the corresponding multi-gravity is described by a
sum of mutually non-interacting Einstein–Hilbert actions.111In three space-
time dimensions one can also deform the Poincaré algebra which gives a theory
different from Einstein gravity, see [20] where also an extension to a
collection of spin-two fields governed by this deformed algebra was studied.
In three dimensions, the (A)dS3 isometry algebra, being a real form of
$\mathfrak{sl}_{2}\oplus\mathfrak{sl}_{2}$, can be extended to an associative
algebra by a double central extension. Incidentally, the same central
extension allows taking different non-relativistic limits in three dimensions
[21].

It is a generic problem to couple matter to the gravitational systems with
extended symmetries such as the coloured (higher-spin) gravity given by a
Chern–Simons action in three dimensions. Lagrangian formulations for known
examples of higher-spin gravities with matter in three dimensions [22, 23] are
not available yet222In [24] an interesting proposal was made using a higher-
dimensional space. See also [25] for a similar idea., despite some steps in
that direction (see [26] and references therein). In this paper we take a step
back and instead try to understand it at the level of the worldline
formulation of particles, since it is the first step towards an associated
quantum field theory from a worldline approach.

We first note that in the theories with extended space-time symmetries the
very notion of particle has to be reconsidered. Since particles in field
theories as we know them are defined as (unitary) irreducible representations
of the isometry algebra, for extended space-time symmetry algebras, one should
consider the irreducible representations of the extended algebra. In
principle, the representations of the larger symmetry algebra should decompose
in terms of the representations of the original isometry algebra. We just note
here, that in case that the representation of the larger algebra has higher
Gelfand–Kirillov (GK) dimension (see e.g. [27] for the definition) than that
of regular particles, which will be true for the cases of our interest, the
corresponding spectrum should be expected to contain infinite number of
particles.

A natural expectation is that systems with extended symmetries may allow for a
mechanism of (spontaneous) symmetry breaking, that would lead to a regime with
a more conventional gravitational system and massive particles coupled to it,
much like in string theory. If such a scenario is realistic, then the notion
of space and time is emergent and makes sense only for low-energy systems
while at high energies (beyond the Planck scale?) it has to be abandoned or
replaced by a more general notion. On the other hand, the higher symmetries of
gravitational systems and their representations can be studied independently
of the possible space-time interpretations.

We will show in this paper that not only the (A)dS3 but also the Poincaré
algebra in three dimensions can be extended via a colour decoration. This
example of the coloured Poincaré symmetry is interesting for several reasons.

In Minkowski space the motion of a relativistic particle in a fixed electro-
magnetic field is described by the Lorentz force. In the case of constant
electro-magnetic field the symmetry algebra is given by the
Bacry–Combe–Richards algebra [28] that contains four space-time translations
that do not commute, two Lorentz boost transformations and two central
charges. Another possibility is to enlarge the Poincaré algebra with tensorial
non-central charges, leading to the so-called Maxwell algebra [29]. If one
also enlarges Minkowski space with tensorial bosonic coordinates, the particle
moving in a constant electro-magnetic field can be made invariant under the
Maxwell symmetry [30]. In an even further enlarged space-time one can
generalise the Maxwell algebra to a free Maxwell algebra [31] that can
describe the motion of the particle in a general electromagnetic field.

The motion of a relativistic particle with colour-flavour indices in a fixed
Yang–Mills background, with space-time coordinates $x^{a}$ and colour
coordinates $y_{I}$ (in the adjoint of $SU(N)$), was first studied by Wong
[32]. As in the electro-magnetic case it will be interesting to see whether
the motion of a coloured particle in a fixed Yang–Mills background has
additional invariances associated with the colour coordinates, in particular,
whether there exists a non-abelian generalisation of the Maxwell algebra for
the case of covariantly constant Yang–Mills background [33, 34].

In this paper we take some first steps in this direction by constructing the
coloured $({\mathfrak{s}}){\mathfrak{u}}(N)$ Poincaré algebra in three
dimensions.333We comment on generalisations to higher dimensions in the
conclusions. The generators are generalised commuting translations, internal
symmetry transformations and coloured Lorentz transformations. There is a
natural generalisation of Minkowski space to a coloured Minkowski space with
coordinates $(x^{a},y_{I},x^{a}_{I})$ where $x^{a}$ are the ordinary space-
time coordinates, $y_{I}$ are the colour internal coordinates and $x^{a}_{I}$
are coloured space-time coordinates. There is a coloured generalisation of the
Poincaré transformations in this space. Since the space-time is enlarged, the
usual Coleman–Mandula theorem no longer applies directly. Interestingly, one
can restrict to the subspace defined by $x^{a}_{I}=0$ at the expense of
breaking the full coloured Poincaré symmetry to Poincaré and colour
transformations. This subspace has the same coordinates as the Wong particle.

In the coloured Minkowski space we construct massive and massless particle
actions that are invariant under the coloured generalisation of the Poincaré
algebra using a variety of methods. The methods differ in which coadjoint
orbits of the coloured Poincaré algebra we consider and we discuss their
classification in some detail.444For a discussion of coadjoint orbits in the
uncoloured case see for example [35]. The choice of orbit is at the heart of
Wigner’s construction of induced representations and presents the starting
point for a particle interpretation. If we restrict our constructions to the
subspace $x^{a}_{I}=0$ we can recover the free Wong equations.

We also construct the particle action with a coloured AdS3 symmetry [13, 14],
in the flat limit we recover the coloured Poincaré particle found before. We
also notice that subalgebra ${\mathfrak{u}}(N,N)$ of coloured Poincaré, in
three dimensions, with an appropriate identification of the generators of
coloured two-dimensional translations gives the coloured AdS2 algebra. This
algebra could be used to construct the coloured AdS2 particle or the one-
dimensional coloured conformal particle mechanics that generalises to the
coloured case the one of reference [36].

The organisation of the paper is as follows. We first discuss the algebraic
method for colouring the three-dimensional Poincaré algebra in section 2. In
section 3, we show that there is an invariant bilinear form on the
$N$-coloured Poincaré algebra $\mathrm{cPoin}_{3}(N)$ and use it to construct
a Chern–Simons theory corresponding to coloured Poincaré gravity. Section 4
discusses particle actions built from $\mathrm{cPoin}_{3}(N)$ that utilise
coloured Minkowski that is also introduced there. Various types of particle
actions differ by the co-adjoint orbits of $\mathrm{cPoin}_{3}(N)$ they
describe. We also consider the coloured particle in an AdS background in
section 5 before offering some concluding remarks in section 6. Several
appendices contain complementary details.

##  2 Colouring the Poincaré algebra in $3$ space-time dimensions

At the kinematic level, we first address the problem of adding colour indices
to the Poincaré algebra, such that the new generators form a Lie algebra.
Since the tensor product of two Lie algebras is not a Lie algebra, we have to
resort to a different construction. In $D=3$ space-time dimensions a similar
problem has been studied in the case of AdS and higher spin algebras [13, 14]
that we rely on. The construction is based on embedding the Poincaré Lie
algebra into an associative algebra and then tensoring it with an associative
‘colour’ algebra. Since the tensor products of associative algebras is
associative and every associative algebra can be turned into a Lie algebra
using the commutator, the construction yields a Lie algebra that contains a
colouring of the original Poincaré Lie algebra.

Before addressing the embedding the lifting of the Poincaré algebra to an
associative algebra, we first discuss the colour algebra. This we take to be
${\mathfrak{gl}}(N,\mathbb{C})$ with ${\mathfrak{u}}(N)$ as its basis over
$\mathbb{C}$, corresponding to the subspace of all anti-hermitian $(N\times
N)$ matrices. The product in this basis is given by (see, e.g., [37])

$\displaystyle
T^{I}\,T^{J}=-\frac{1}{N}\,\delta^{IJ}\,\mathbf{I}+\frac{1}{2}\,(\text{i}\,d^{IJ}{}_{K}+f^{IJ}{}_{K})\,T^{K}\,,\quad\mathbf{I}\,T^{I}=T^{I}\,\mathbf{I}=T^{I}\,,\quad\mathbf{I}\,\mathbf{I}=\mathbf{I}\,.$
(2.1)

Here, $T^{I}$ are the traceless generators of ${\mathfrak{su}}(N)$ with
$I=1,\ldots,N^{2}-1$ and $\mathbf{I}$ is the $N\times N$ identity matrix. The
structure constants $f^{IJ}{}_{K}$ in
$\left[T^{I},T^{J}\right]=f^{IJ}{}_{K}T^{K}$ are real while the real
$d^{IJ}{}_{K}$ are the invariant tensors mapping the symmetric product of two
${\mathfrak{su}}(N)$ adjoint representations back to the adjoint of
${\mathfrak{su}}(N)$. They exist for $N>2$ and the identities satisfied by the
invariant tensors turn ${\mathfrak{gl}}(N,\mathbb{C})$ into an associative
algebra. Here, $\delta^{IJ}$ is the invariant metric on the adjoint of
${\mathfrak{su}}(N)$. When using this to raise indices, $d^{IJK}$ becomes
totally symmetric and $f^{IJK}$ becomes totally antisymmetric. When colouring
the Poincaré algebra later, we shall use only a real slice
${\mathfrak{u}}(N)\subset{\mathfrak{gl}}(N,\mathbb{C})$, and for this reason
we shall refer the associative algebra ${\mathfrak{gl}}(N,\mathbb{C})$ as
${\mathfrak{u}}(N)$ with an abuse of notation.

To write the algebra in a uniform way, we denote the anti-hermitian element
$T^{0}=\text{i}\,\mathbf{I}$ and use the indices
${\widehat{I}}=0,1,\ldots,N^{2}-1$ to label all the generators of
${\mathfrak{u}}(N)$. The associative product is given by

$\displaystyle
T^{{\widehat{I}}}\,T^{{\widehat{J}}}=\frac{1}{2}\,\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,T^{{\widehat{K}}}+\frac{\text{i}\,}{2}\,\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,T^{{\widehat{K}}}\,,$
(2.2)

with real $\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}$ and
$\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}$. The product
decomposes into its anti-symmetric and symmetric parts according to

$\displaystyle\left[T^{{\widehat{I}}},T^{{\widehat{J}}}\right]=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,T^{{\widehat{K}}}\,,\quad\quad\left\\{T^{{\widehat{I}}},T^{{\widehat{J}}}\right\\}=\text{i}\,\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,T^{{\widehat{K}}}\,.$
(2.3)

The tensors $\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}$ and
$\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}$ satisfy
identities analogous to the usual ${\mathfrak{su}}(N)$ identities. When
splitting the adjoint indices they have components

$\displaystyle\widehat{f}\,^{0{\widehat{I}}}{}_{{\widehat{K}}}$
$\displaystyle=0\,,$ $\displaystyle\widehat{f}\,^{IJ}{}_{0}$
$\displaystyle=0\,,$ $\displaystyle\widehat{f}\,^{IJ}{}_{K}$
$\displaystyle=f^{IJ}{}_{K}\,,$ (2.4a)
$\displaystyle\widehat{d}\,^{0{\widehat{I}}}{}_{{\widehat{K}}}$
$\displaystyle=2\,\delta^{{\widehat{I}}}_{{\widehat{K}}}\,,$
$\displaystyle\widehat{d}\,^{IJ}{}_{0}$
$\displaystyle=\frac{2}{N}\,\delta^{IJ}\,,$
$\displaystyle\widehat{d}\,^{IJ}{}_{K}$ $\displaystyle=d^{IJ}{}_{K}\,.$ (2.4b)

The Poincaré Lie algebra in $(2+1)$ space-time dimensions has Lorentz and
translations generators $M_{a},P_{a}$ and commutation relations555The indices
$a,b,c=0,1,2$ are Minkowski space-time indices. We use $\varepsilon^{012}=+1$
and indices are raised and lowered with the $(-++)$ Minkowski metric.

$\displaystyle\left[M_{a},M_{b}\right]=\varepsilon_{ab}{}^{c}\,M_{c}\,,\quad\quad\left[M_{a},P_{b}\right]=\varepsilon_{ab}{}^{c}\,P_{c}\,,\quad\quad\left[P_{a},P_{b}\right]=0\,.$
(2.5)

Its embedding into an associative algebra can be achieved as follows. The
starting point is to write the generators of the Poincaré algebra as tensor
products

$\displaystyle M_{a}=L_{a}\otimes\mathcal{I}\,,\quad\quad
P_{a}=L_{a}\otimes\mathcal{J}\,,$ (2.6)

where $\mathcal{I}$ and $\mathcal{J}$ are generators of an associative abelian
algebra ${\mathfrak{a}}$ (over $\mathbb{R}$) satisfying

$\displaystyle\mathcal{I}^{2}=\mathcal{I}\,,\quad\quad\mathcal{J}^{2}=0\,,\quad\quad\mathcal{I}\,\mathcal{J}=\mathcal{J}\,\mathcal{I}=\mathcal{J}\,,$
(2.7)

while $L_{a}$ are generators of
${\mathfrak{sl}}(2,\mathbb{R})\cong{\mathfrak{su}}(1,1)$ satisfying
$\left[L_{a},L_{b}\right]=\varepsilon_{ab}{}^{c}\,L_{c}$. For reality of the
${\mathfrak{sl}}(2,\mathbb{R})$ structure constants $\varepsilon^{abc}$ we
require $L_{a}^{\dagger}=-L_{a}$. For the anti-hermiticity of $M_{a}$ and
$P_{a}$, we assign to $\mathcal{I}$ and $\mathcal{J}$ the hermiticity
properties666The associative algebra (2.7) is the same as the semi-group
$S_{E}^{(1)}=\\{\lambda_{0},\lambda_{1},\lambda_{2}\\}$ with
$\lambda_{i}\cdot\lambda_{j}=\lambda_{j}\cdot\lambda_{i}=\lambda_{i+j}$ where
$\lambda_{j}=0$ for $j>1$. One can use the Lie algebra expansion method [38,
39, 40, 41] with this semi-group to obtain a new Lie algebra that is
isomorphic to the Lie algebra we describe below. If we assign the hermiticity
properties $\lambda^{\dagger}_{i}=\lambda_{i}$, then the Lie algebra obtained
by the expansion method is exactly the same as we present in (2.13).

$\displaystyle\mathcal{I}^{\dagger}=\mathcal{I}\,,\quad\mathcal{J}^{\dagger}=\mathcal{J}\,.$
(2.8)

Even though ${\mathfrak{sl}}(2,\mathbb{R})$ is only a Lie algebra and not
associative, the Lie bracket $[x\otimes a,y\otimes
b]=\frac{1}{2}\left(\left[x,y\right]\otimes\left\\{a,b\right\\}+\left\\{x,y\right\\}\otimes\left[a,b\right]\right)=[x,y]\otimes
ab$ on ${\mathfrak{sl}}(2,\mathbb{R})\otimes{\mathfrak{a}}$ is well-defined
since ${\mathfrak{a}}$ is abelian, a fact that was also used in the context of
(conformal) gravity [18, 39]. The Lie algebra
${\mathfrak{sl}}(2,\mathbb{R})\otimes{\mathfrak{a}}$ can now be embedded in an
associative algebra by extending the first factor and defining the eight-
dimensional associative algebra

$\displaystyle{\mathfrak{A}}={\mathfrak{u}}(1,1)\otimes{\mathfrak{a}}=\left\langle
L_{a}\otimes\mathcal{I},\,L_{a}\otimes\mathcal{J},\,\text{i}\,\mathbf{I}_{2}\otimes\mathcal{I}\,,\text{i}\,\mathbf{I}_{2}\otimes\mathcal{J}\right\rangle.$
(2.9)

Here, the associative product on ${\mathfrak{u}}(1,1)$ is given by

$\displaystyle
L_{a}\,L_{b}=\frac{1}{2}\,\varepsilon_{ab}{}^{c}\,L_{c}+\frac{1}{4}\,\eta_{ab}\,\mathbf{I}_{2}\,,$
(2.10)

with $\mathbf{I}_{2}$ being the identity element of the algebra, with
$\mathbf{I}_{2}^{\dagger}=\mathbf{I}_{2}$. This formula is similar to (2.1).

The coloured Poincaré algebra in $D=3$, $\mathrm{cPoin}_{3}(N)$, is then
defined to be the Lie algebra associated with the associative algebra given by
the tensor product

$\displaystyle\mathrm{cPoin}_{3}(N)={\mathfrak{A}}\otimes{\mathfrak{u}}(N)\,.$
(2.11)

A basis of the vector space $\mathrm{cPoin}_{3}$ is given by the generators

$\displaystyle M_{a}^{{\widehat{I}}}$
$\displaystyle=-\text{i}\,L_{a}\otimes\mathcal{I}\otimes T^{\widehat{I}}\,,$
(2.12a) $\displaystyle P_{a}^{{\widehat{I}}}$
$\displaystyle=-\text{i}\,L_{a}\otimes\mathcal{J}\otimes T^{\widehat{I}}\,,$
(2.12b) $\displaystyle\ N^{{\widehat{I}}}$
$\displaystyle=\phantom{\text{i}\,}\mathbf{I}_{2}\otimes\mathcal{I}\otimes
T^{\widehat{I}}\,,$ (2.12c) $\displaystyle Q^{{\widehat{I}}}$
$\displaystyle=\phantom{\text{i}\,}\mathbf{I}_{2}\otimes\mathcal{J}\otimes
T^{\widehat{I}}\,.$ (2.12d)

The factors of i are introduced such that all generators are anti-hermitian.

The coloured Poincaré algebra then has the following commutation relations.

$\displaystyle\left[M_{a}^{{\widehat{I}}},M_{b}^{{\widehat{J}}}\right]$
$\displaystyle=\frac{1}{2}\,\varepsilon_{ab}{}^{c}\,\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,M_{c}^{{\widehat{K}}}-\frac{1}{4}\,\eta_{ab}\,\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,N^{{\widehat{K}}}\,,$
(2.13a) $\displaystyle\left[M_{a}^{{\widehat{I}}},N^{{\widehat{J}}}\right]$
$\displaystyle=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,M_{a}^{{\widehat{K}}}\,,$
(2.13b) $\displaystyle\left[N^{{\widehat{I}}},N^{{\widehat{J}}}\right]$
$\displaystyle=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,N^{{\widehat{K}}}\,,$
(2.13c)
$\displaystyle\left[M_{a}^{{\widehat{I}}},P_{b}^{{\widehat{J}}}\right]$
$\displaystyle=\frac{1}{2}\,\varepsilon_{ab}{}^{c}\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,P_{c}^{{\widehat{K}}}-\frac{1}{4}\,\eta_{ab}\,\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,Q^{{\widehat{K}}}\,,$
(2.13d) $\displaystyle\left[M_{a}^{{\widehat{I}}},Q^{{\widehat{J}}}\right]$
$\displaystyle=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,P_{a}^{{\widehat{K}}}\,,$
(2.13e) $\displaystyle\left[N^{{\widehat{I}}},P_{a}^{{\widehat{J}}}\right]$
$\displaystyle=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,P_{a}^{{\widehat{K}}}\,,$
(2.13f) $\displaystyle\left[N^{{\widehat{I}}},Q^{{\widehat{J}}}\right]$
$\displaystyle=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,Q^{{\widehat{K}}}\,,$
(2.13g)
$\displaystyle\left[P_{a}^{{\widehat{I}}},P_{b}^{{\widehat{J}}}\right]$
$\displaystyle=0\,,$ (2.13h)
$\displaystyle\left[P_{a}^{{\widehat{I}}},Q^{{\widehat{J}}}\right]$
$\displaystyle=0\,,$ (2.13i)
$\displaystyle\left[Q^{{\widehat{I}}},Q^{{\widehat{J}}}\right]$
$\displaystyle=0\,.$ (2.13j)

The first three lines are the algebra ${\mathfrak{u}}(N,N)$ and the remaining
lines can also be understood as the Lie algebra with the semi-group
$S_{E}^{(1)}$ applied to ${\mathfrak{u}}(N,N)$, see footnote 6.

We see from the commutation relations (2.13) that the ${\widehat{I}}=0$
components $M_{a}^{0}$ and $P_{a}^{0}$ satisfy the usual uncoloured Poincaré
algebra and can be identified with these according to

$\displaystyle P_{a}\equiv P_{a}^{0}\,,\quad M_{a}\equiv M_{a}^{0}\,.$ (2.14)

The Lie algebra $\mathrm{cPoin}_{3}(N)$ has a two-dimensional center that can
be quotiented out. This center is spanned by $N^{0}$ and $Q^{0}$. From the
definition (2.11) we see that777We note that the result of the construction is
the same as constructing the co-adjoint extension of ${\mathfrak{u}}(N,N)$
using [42, 43].

$\displaystyle\mathrm{cPoin}_{3}\cong{\mathfrak{u}}(N,N)\inplus_{\mathrm{adj}}{\mathfrak{u}}(N,N)\,,$
(2.15)

where the notation indicates a semi-direct sum of the Lie algebra
${\mathfrak{u}}(N,N)$ acting on another copy of the space
${\mathfrak{u}}(N,N)$, now treated as an abelian algebra, via the adjoint
action.888In this notation, the usual uncoloured Poincaré algebra is written
as the semi-direct sum
${\mathfrak{so}}(1,D-1)\inplus_{\mathrm{vec}}\mathbb{R}^{1,D-1}$ in any space-
time dimension $D$. The space $\mathbb{R}^{1,D-1}$ is the abelian algebra of
translations and acted upon by the Lorentz algebra using the
${\mathfrak{so}}(1,D-1)$ vector representation. The Lie algebra
${\mathfrak{u}}(N,N)$ arises since
${\mathfrak{u}}(1,1)\otimes{\mathfrak{u}}(N)\cong{\mathfrak{u}}(N,N)$ upon
using ${\mathfrak{sl}}(2,\mathbb{R})\cong{\mathfrak{su}}(1,1)$. The non-
abelian first ${\mathfrak{u}}(N,N)$ in (2.15) is due to the idempotent
$\mathcal{I}$ in (2.9) whereas the second abelian ${\mathfrak{u}}(N,N)$ is due
to the nilpotent $\mathcal{J}$. The two-dimensional center in this language
corresponds to the two ${\mathfrak{u}}(1)$ inside each of the two
${\mathfrak{u}}(N,N)$,999The presence of two ${\mathfrak{u}}(1)$ makes it
possible to contract $\mathrm{cPoin}_{3}(N)$ to a non-relativistic colour
algebra with a non-degenerate bilinear form, analogously to the uncolored case
of the extended Bargmann algebra as a contraction of
${\mathfrak{u}}(1,1)\inplus_{\mathrm{adj}}{\mathfrak{u}}(1,1)$ [21].  and we
are left with the semi-direct sum

$\displaystyle{\mathfrak{su}}(N,N)\inplus_{\mathrm{adj}}{\mathfrak{su}}(N,N)\,.$
(2.16)

Because of the structure of (2.15) we shall refer to the generators
$M_{a}^{\widehat{I}}$ and $N^{\widehat{I}}$ as generalised (coloured) Lorentz
generators and to $P_{a}^{\widehat{I}}$ and $Q^{\widehat{I}}$ as generalised
(coloured) translation generators that form an abelian subalgebra and whose
eigenvalues are generalised momenta. As is evident from (2.15) and (2.16), the
coloured Poincaré algebra is not a direct sum of the usual Poincaré algebra
${\mathfrak{iso}}(2,1)$ with an internal symmetry algebra. This is not in
contradiction with the Coleman–Mandula theorem as we shall also enlarge the
space-time beyond three dimensions when we construct particle actions below.

Since, according to (2.16), all objects are elements of
${\mathfrak{su}}(N,N)$, it will be convenient to use an explicit matrix
representation. In the defining representation of ${\mathfrak{su}}(N,N)$ an
element of the Lie algebra ${\mathfrak{su}}(N,N)$ is written in block matrix
form as

$\displaystyle\begin{pmatrix}{\bf A}&{\bf B}\\\ {\bf C}&{\bf D}\end{pmatrix}$
(2.17)

with $N\times N$ blocks satisfying

$\displaystyle{\bf A}^{\dagger}=-{\bf A}\,,\quad{\bf B}^{\dagger}={\bf
C}\,,\quad{\bf C}^{\dagger}={\bf B}\,,\quad{\bf D}^{\dagger}=-{\bf
D}\,,\quad\mathrm{tr}({\bf A}+{\bf D})=0\,.$ (2.18)

We shall use bold letters to denote $(N\times N)$ matrices and ‘$\mathrm{tr}$’
for the corresponding trace over $(N\times N)$-matrices. The identity
$(N\times N)$ matrix is written as ${\bf I}$.

#### Coloured AdS2 algebra

The first three lines of (2.13) are the ${\mathfrak{u}}(N,N)$ subalgebra of
$\mathrm{cPoin}_{3}(N)$, explicitly

$\displaystyle\left[M_{a}^{{\widehat{I}}},M_{b}^{{\widehat{J}}}\right]$
$\displaystyle=\frac{1}{2}\,\varepsilon_{ab}{}^{c}\,\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,M_{c}^{{\widehat{K}}}-\frac{1}{4}\,\eta_{ab}\,\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,N^{{\widehat{K}}}\,,$
(2.19a) $\displaystyle\left[M_{a}^{{\widehat{I}}},N^{{\widehat{J}}}\right]$
$\displaystyle=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,M_{a}^{{\widehat{K}}}\,,$
(2.19b) $\displaystyle\left[N^{{\widehat{I}}},N^{{\widehat{J}}}\right]$
$\displaystyle=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,N^{{\widehat{K}}}\,.$
(2.19c)

It can be understood as the tensor product algebra
${\mathfrak{u}}(1,1)\otimes{\mathfrak{u}}(N)$ :

$\displaystyle M_{a}^{{\widehat{I}}}$ $\displaystyle=-\text{i}\,L_{a}\otimes
T^{\widehat{I}}\,,$ (2.20a) $\displaystyle\ N^{{\widehat{I}}}$
$\displaystyle=\phantom{\text{i}\,}\mathbf{I}_{2}\otimes T^{\widehat{I}}\,.$
(2.20b)

With a proper identification of the generators, this algebra is the coloured
conformal algebra in one dimension or the coloured AdS2 algebra.

The basis of the coloured conformal algebra is given by

$D^{{\widehat{I}}}=M_{0}^{{\widehat{I}}}\,,\qquad
H^{{\widehat{I}}}=M_{1}^{{\widehat{I}}}+M_{2}^{{\widehat{I}}}\,,\qquad
K^{{\widehat{I}}}=M_{1}^{{\widehat{I}}}-M_{2}^{{\widehat{I}}}\,.$ (2.21)

The generators with the zero component in the colour indices give the
uncoloured $D=1$ conformal algebra.

The coloured AdS2 algebra can be written in terms of conformal generators

$P_{1}{}^{{\widehat{I}}}=\frac{1}{\ell}D^{{\widehat{I}}}\,,\qquad
P_{0}{}^{{\widehat{I}}}=\frac{1}{2\ell}(H^{{\widehat{I}}}+K^{{\widehat{I}}})\,,\qquad
M_{0}{}^{{\widehat{I}}}=\frac{1}{2}(H^{{\widehat{I}}}-K^{{\widehat{I}}})\,.$
(2.22)

The parameter $\ell$ denotes the radius of AdS2. The generators with the zero
component in the colour indices give the uncoloured AdS2 algebra. This algebra
can be used in a BF theory [44, 45, 46, 47] to construct a coloured analogue
of Jackiw–Teitelboim gravity (see the upcoming work [48] for a detailed
analysis).

##  3 Coloured gravity in 3d Minkowski space-time

The coloured Poincaré algebra introduced in the previous section can be
understood as a zero cosmological constant limit of the coloured (A)dS3
algebra [13], see also appendix A. Furthermore, one can define a Chern–Simons
action for coloured gravity in 3d Minkowski space.

###  3.1 Invariant bilinear form

Let us recall the structure of the Poincaré algebra in three dimensions. As
usual for Poincaré algebra in any dimensions, one can define a degenerate
bilinear form

$\displaystyle\left\langle P_{a}\,|\,P_{b}\right\rangle=0\,,\quad\left\langle
P_{a}\,|\,M_{b}\right\rangle=0\,,\quad\left\langle
M_{a}\,|\,M_{b}\right\rangle=\eta_{ab}\,.$ (3.1)

In three dimensions, there is a different, non-degenerate bilinear form (see,
e.g., [3]) which is, however, non-diagonal

$\displaystyle\left\langle P_{a}\,|\,P_{b}\right\rangle=0\,,\quad\left\langle
P_{a}\,|\,M_{b}\right\rangle=\eta_{ab}\,,\quad\left\langle
M_{a}\,|\,M_{b}\right\rangle=0\,.$ (3.2)

The latter bilinear form allows us to define the Chern–Simons action for
Poincaré algebra, equivalent to the Einstein–Hilbert action in three
dimensions for an invertible dreibein. This is possible in three dimensions
due to the fact that the Lorentz and translation generators of the Poincaré
algebra carry the same Lorentz representation — they are both vectors of
${\mathfrak{so}}(1,2)$, or, equivalently, are in a representation isomorphic
to the adjoint representation of
${\mathfrak{su}}(1,1)\simeq{\mathfrak{so}}(1,2)$. This property generalises to
the coloured Poincaré algebra, which has an invariant bilinear form with
similar properties as we show below.

First we note that the most general bilinear form for the three-dimensional
Poincaré algebra has a one-parameter freedom up to normalisation (see, e.g.,
[12]). Therefore, the most general Chern–Simons action based on three-
dimensional Poincaré algebra can be given in the form,

$\displaystyle
S=\frac{\kappa_{1}}{4\,\pi}\int\mathrm{Tr}(\,e\wedge(d\omega+\omega\wedge\omega)\,)+\frac{\kappa_{2}}{4\,\pi}\int\mathrm{Tr}(\omega\wedge
d\omega+\frac{2}{3}\omega\wedge\omega\wedge\omega)\,,$ (3.3)

where we do not write the contractions of fiber Lorentz indices for
simplicity. The second term in this action is the Lorentz–Chern–Simons term,
which, supplemented with the torsion constraint describes conformal gravity in
three dimensions. Vanishing of the torsion is a consequence of the equations
of motion of the action (3.3). However, substituting the solution of the
torsion constraint back into the action will change its physical content since
the torsionless constraint is not a consequence of the $\omega$ equations
alone for $\kappa_{2}\neq 0$. This situation is analogous to theories with
“third-way consistency” [49]. If one nevertheless substitutes the solution of
the torsion constraint back in the action, one gets so-called Topologically
Massive Gravity (TMG) [50] with mass value $m\sim\kappa_{1}/\kappa_{2}$.

In order to construct an invariant bilinear form for the coloured Poincaré
algebra, we first start from the coloured Lorentz subalgebra
${\mathfrak{u}}(N,N)\simeq{\mathfrak{u}}(1,1)\otimes{\mathfrak{u}}(N)$. For
the colour group, ${\mathfrak{u}}(N)$, the bilinear form has the following
form

$\displaystyle\left\langle
N^{I}\,|\,N^{J}\right\rangle=\alpha\,\delta^{IJ}\,,\quad\left\langle
N^{I}\,|\,N^{0}\right\rangle=0\,,\quad\left\langle
N^{0}\,|\,N^{0}\right\rangle=\beta\,,$ (3.4)

which is non-degenerate when $\alpha$ and $\beta$ are arbitrary non-zero
numbers. A general ansatz for the rest of the generators,
$M_{a}^{\hat{I}}=(M_{a}^{I}\,,\,M_{a})$, can be given in the form:

$\displaystyle\left\langle
M_{a}^{I}\,|\,M_{b}^{J}\right\rangle=\gamma\,\eta_{ab}\,\delta^{IJ}\,,\quad\left\langle
M_{a}\,|\,M_{b}\right\rangle=\delta\,\eta_{ab}\,.$ (3.5)

The constraint of invariance of the bilinear form implies

$\displaystyle\gamma=\frac{1}{4}\,\alpha\,,\quad\delta=\frac{N}{4}\,\alpha\,.$
(3.6)

As expected, the invariant bilinear form of the algebra ${\mathfrak{u}}(N,N)$
depends on two arbitrary parameters $\alpha\,,\,\beta$. One of these numbers
parametrises the norm of the central element, which we can drop. Then, the
bilinear form of the reduced coloured Lorentz algebra $\mathfrak{su}(N,N)$ is
fixed uniquely up to normalisation:

$\left\langle\mathcal{M}^{A}|\mathcal{M}^{B}\right\rangle=\rho\,\delta^{AB}\,,\qquad\mathcal{M}^{A}\in\mathfrak{su}(N,N)\,,$
(3.7)

where $A,B=1,2,\dots,4\,N^{2}-1$ denote here adjoint indices of
$\mathfrak{su}(N,N)$. The most general invariant bilinear form for the algebra
$\mathfrak{su}(N,N)\inplus_{\mathrm{adj}}\mathfrak{su}(N,N)$ is given in the
following form.

$\displaystyle\left\langle\mathcal{M}^{A}|\mathcal{M}^{B}\right\rangle=\rho\,\delta^{AB}\,,\quad\left\langle\mathcal{M}^{A}|\mathcal{P}^{B}\right\rangle=\lambda\,\delta^{AB}\,,\quad\left\langle\mathcal{P}^{A}|\mathcal{P}^{B}\right\rangle=0\,,$
(3.8)

where $\rho$ and $\lambda$ are arbitrary numbers. This bilinear form is non-
degenerate for $\lambda\neq 0$.

In analogy with (3.3), one can write a Chern–Simons action for coloured
Poincaré gravity in three dimensions in the following schematic form.

$\displaystyle
S=\frac{\kappa}{4\,\pi}\int\mathrm{Tr}(E\wedge(d\Omega+\Omega\wedge\Omega))+\frac{\tilde{\kappa}}{4\,\pi}\int\mathrm{Tr}(\Omega\wedge\,d\Omega+\frac{2}{3}\Omega\wedge\Omega\wedge\Omega)\,,$
(3.9)

where the fields $E,\,\Omega$ constitute the coloured Poincaré connection,

$\displaystyle{\cal A}=E^{A}\,{\cal P}_{A}+\Omega^{A}\,{\cal M}_{A}\,.$ (3.10)

Naively, solving the torsion constraint,

$\displaystyle T^{A}=dE^{A}+\tfrac{1}{2}\,f^{A}{}_{BC}(\Omega^{B}\wedge
E^{C}+E^{B}\wedge\Omega^{C})=0\,,$ (3.11)

for $\Omega=\Omega(E)$ and plugging back into the action (3.9), one will get
the coloured analogue of TMG. This substitution, even if possible technically,
is leading to a different physical theory which we will not study here.

The space-time parity transformation multiplies odd space-time forms by $-1$
and transforms the coloured Poincaré algebra generators as $({\cal
P}_{A},{\cal M}_{A})\to(-{\cal P}_{A},{\cal M}_{A})$. Consequently it acts on
the one-form fields as $(E,\Omega)\to(E,-\Omega)$ while
$(dE,d\Omega)\to(-dE,d\Omega)$. Under this action the first term of (3.9) is
invariant, while the second term changes sign.

For constructing the theory that is even under space-time parity, one makes
use of the non-diagonal bilinear form ($\rho=0$ case), which can be given in
the following form (we write only non-zero scalar products).

$\displaystyle\left\langle
M_{a}^{I}\,|\,P_{b}^{J}\right\rangle=\tfrac{1}{4}\,\lambda\,\eta_{ab}\,\delta^{IJ}\,,\quad\left\langle
N^{I}\,|\,Q^{J}\right\rangle=\lambda\,\delta^{IJ}\,,\quad\left\langle
M_{a}\,|\,P_{b}\right\rangle=\tfrac{N}{4}\,\lambda\,\eta_{ab}\,.$ (3.12)

Using this bilinear form, one can write a Chern–Simons action with coloured
Poincaré symmetry, that will be equivalent to the flat limit of coloured
gravity in (A)dS3, studied in [13]. It is natural to choose
$\lambda=\frac{4}{N}$ to recover $\left\langle
M_{a}\,|\,P_{b}\right\rangle=\eta_{ab}$ from [3] (the overall constant is
incorporated in the Chern–Simons level).

###  3.2 Coloured gravity action in 3d Minkowski space

As discussed above, the action for coloured gravity in three-dimensional
Minkowski space-time can be given in the Chern–Simons form with coloured
Poincaré gauge symmetry, using the non-degenerate bilinear form. There is one
choice of the bilinear form, (3.12), that reproduces (3.2) for the uncoloured
Poincaré subalgebra. We will use (3.12) to reproduce Einstein–Hilbert gravity
for the $\mathfrak{su}(N)$ singlet spin-two field.

Coloured Poincaré gravity can be defined by a Chern–Simons action,

$\displaystyle
S[\mathcal{A}]=\frac{\kappa}{4\,\pi}\int\mathrm{Tr}\left(\mathcal{A}\wedge
d\mathcal{A}+\frac{2}{3}\,\mathcal{A}\wedge\mathcal{A}\wedge\mathcal{A}\right)\,,$
(3.13)

where
$\mathcal{A}\in{\mathfrak{su}}(N,N)\inplus_{\mathrm{adj}}{\mathfrak{su}}(N,N)$
(we factor by the center since it decouples in the action). In analogy with
[13], we will further decompose $\mathcal{A}$ as

$\displaystyle\mathcal{A}=\mathcal{B}+\mathcal{C}\,,$ (3.14)

where $\mathcal{B}$ incorporates the uncoloured isometry and colour
group:101010Here the one-form field $\omega$ is related to the standard spin-
connection as $\omega^{a}=\frac{1}{2}\epsilon^{abc}\omega_{bc}$. Similarly for
its $\mathfrak{su}(N)$-valued analogue, $\omega^{a}_{I}$.

$\displaystyle\mathcal{B}=\omega^{a}\,M_{a}+e^{a}\,P_{a}+\varphi_{I}\,N^{I}+\tilde{\varphi}_{I}\,Q^{I}\,,$
(3.15)

while $\mathcal{C}$ corresponds to the coloured spin-two sector:

$\displaystyle\mathcal{C}=\omega^{a}_{I}\,M^{I}_{a}+e^{a}_{I}\,P^{I}_{a}\,.$
(3.16)

The action then can be written in the form,

$\displaystyle
S[\mathcal{B},\mathcal{C}]=\frac{\kappa}{4\,\pi}\,\int\mathrm{Tr}\left(\mathcal{B}\wedge
d\mathcal{B}+\frac{2}{3}\,\mathcal{B}\wedge\mathcal{B}\wedge\mathcal{B}+\mathcal{C}\wedge
D_{\mathcal{B}}\,\mathcal{C}+\frac{2}{3}\,\mathcal{C}\wedge\mathcal{C}\wedge\mathcal{C}\right),$
(3.17)

with

$\displaystyle
D_{\mathcal{B}}\,\mathcal{C}=d\mathcal{C}+\mathcal{B}\wedge\mathcal{C}+\mathcal{C}\wedge\mathcal{B}\,.$
(3.18)

The sector that depends only on $\mathcal{B}$ can be explicitly written in the
form,

$S_{\mathcal{B}}=\frac{\kappa}{4\,\pi}\,\int\mathrm{Tr}\left(\mathcal{B}\wedge
d\mathcal{B}+\frac{2}{3}\,\mathcal{B}\wedge\mathcal{B}\wedge\mathcal{B}\right)=S_{\rm
GR}+S_{\rm Vector}\,,$ (3.19)

with

$\displaystyle S_{\rm GR}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{\kappa}{2\,\pi}\int
e^{a}\wedge(d\omega^{b}+\tfrac{1}{2}\,\epsilon^{bcd}\,\omega_{c}\wedge\omega_{d})\,\eta_{ab}\,,$
(3.20) $\displaystyle S_{\rm Vector}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{2\,\kappa}{N\,\pi}\int\tilde{\varphi}^{I}\wedge(d\varphi^{J}+f^{J}{}_{KL}\,\varphi^{K}\wedge\varphi^{L})\delta_{IJ}\,.$
(3.21)

For zero cosmological constant, as opposed to the (A)dS3 case, the rewriting
of the action in terms of chiral and anti-chiral pieces is not possible
neither in the pure gravity sector, $S_{\rm GR}$, nor the vector field sector,
$S_{\rm Vector}$. The equations of motion for the vector sector (neglecting
contribution from interactions with coloured gravitons) can be given as

$\displaystyle F(\varphi)=d\varphi+\varphi\wedge\varphi=0\,,$ (3.22)
$\displaystyle
D_{\varphi}\tilde{\varphi}=d\tilde{\varphi}+\varphi\wedge\tilde{\varphi}+\tilde{\varphi}\wedge\varphi=0\,,$
(3.23)

and corresponds to a BF theory (see, e.g., [51, 52, 53]). For non-zero
cosmological constant, $\Lambda\neq 0$, the Lagrangian of the GR sector would
get an additional piece $\sim\Lambda\,e\wedge e\wedge e$, and the gauge vector
field sector would get an additional piece
$\sim\Lambda\,\tilde{\varphi}\wedge\tilde{\varphi}\wedge\tilde{\varphi}$, both
essential for diagonalisation of the theory and deformation of the algebra.

The action of the matter sector has the following form,

$\displaystyle
S_{\mathcal{C}}=\frac{\kappa}{4\,\pi}\,\int\mathrm{Tr}\left(\mathcal{C}\wedge
D_{\mathcal{B}}\,\mathcal{C}+\frac{2}{3}\,\mathcal{C}\wedge\mathcal{C}\wedge\mathcal{C}\right)\,,$
(3.24)

or, in a more explicit form,

$\displaystyle S_{{\cal C}}$
$\displaystyle=\frac{\kappa}{2\,N\,\pi}\int\Big{(}\delta^{IJ}\eta_{ab}\,e^{a}_{I}\wedge
d\omega^{b}_{J}+\frac{1}{2}\,\delta^{IJ}\,\epsilon_{abc}\,\omega^{a}_{I}\wedge\omega^{b}_{J}\wedge
e^{c}$ $\displaystyle\hskip
56.9055pt+\delta^{IJ}\epsilon_{abc}\,\omega^{a}_{I}\wedge
e^{b}_{J}\wedge\omega^{c}+\frac{1}{4}d^{IJK}\,\epsilon_{abc}\,\omega^{a}_{I}\wedge\omega^{b}_{J}\wedge
e^{c}_{K}$ $\displaystyle\hskip
56.9055pt-f^{IJK}\eta_{ab}\,\omega^{a}_{I}\wedge
e^{b}_{J}\wedge\varphi_{K}-\frac{1}{2}\,f^{IJK}\,\eta_{ab}\,\omega^{a}_{I}\wedge\omega^{b}_{J}\wedge\tilde{\varphi}_{K}\Big{)}\,.$
(3.25)

It is interesting to note that the multiple background solutions with
different cosmological constants of coloured gravity in (A)dS3 [13] all go to
the Minkowski solution in the $\Lambda\to 0$ limit.

The second term of (3.9), the coloured Lorentz–Chern–Simons term, is also
straightforward to compute:

$\displaystyle S_{\text{CLCS}}$
$\displaystyle=\frac{\tilde{\kappa}}{4\,\pi}\int(\omega^{a}\wedge\,d\,\omega^{b}\,\eta_{ab}+\tfrac{1}{3}\,\epsilon_{abc}\,\omega^{a}\wedge\omega^{b}\wedge\omega^{c})$
$\displaystyle\quad+\frac{\tilde{\kappa}}{N\,\pi}\int(\varphi^{I}\wedge\,d\varphi^{J}\,\delta_{IJ}+\tfrac{2}{3}\,f_{IJK}\,\varphi^{I}\,\wedge\varphi^{J}\wedge\varphi^{K})$
$\displaystyle\quad+\frac{\tilde{\kappa}}{4\,N\,\pi}\int\Big{(}\delta^{IJ}\eta_{ab}\,\omega^{a}_{I}\wedge
d\omega^{b}_{J}+\tfrac{1}{6}d^{IJK}\,\epsilon_{abc}\,\omega^{a}_{I}\wedge\omega^{b}_{J}\wedge\omega^{c}_{K}$
$\displaystyle\quad+\delta^{IJ}\,\epsilon_{abc}\,\omega^{a}_{I}\wedge\omega^{b}_{J}\wedge\omega^{c}-f^{IJK}\eta_{ab}\,\omega^{a}_{I}\wedge\omega^{b}_{J}\wedge\varphi_{K}\Big{)}\,.$
(3.26)

##  4 Free coloured particle model

In this section, we are considering massive and massless particle actions
invariant under the coloured Poincaré algebra. First we introduce the coloured
generalisation of Minkowski space and then discuss how the different ways of
constructing Poincaré particle actions become coloured. In our analysis we
shall consistently quotient by the two-dimensional center generated by $N^{0}$
and $Q^{0}$ and thus restrict to
${\mathfrak{su}}(N,N)\inplus_{\mathrm{adj}}{\mathfrak{su}}(N,N)$ as given in
(2.16).

###  4.1 Coloured Minkowski space

The algebra $\mathrm{cPoin}_{3}(N)$ is a vast extension of the Poincaré
algebra (2.5). Its centerless version (2.16) includes an abelian part spanned
by $P_{a}$, $P_{a}^{I}$ and $Q^{I}$. This should be thought of as coloured
translations and there is an associated coloured Minkowski space. Introducing
dual coordinates for each of the coloured translation generators we obtain
coordinates $x^{a}$, $x^{a}_{I}$ and $y_{I}$, where the first $x^{a}$ is to be
thought of as the usual Minkowski coordinate (invariant under
${\mathfrak{su}}(N)$ and the others are the coloured extensions. Similarly,
the coloured Lorentz algebra ${\mathfrak{su}}(N,N)$ has generators $M_{a}$,
$M_{a}^{I}$ and $N^{I}$, where the first generator is to be thought of as the
usual Lorentz generator and the others as coloured generalisations.

On a coloured Minkowski point $(x^{a},x^{a}_{I},y_{I})$ represented by

$\mathbb{X}=x^{a}\,P_{a}+x^{a}_{I}\,P_{a}^{I}+y_{I}\,Q^{I}\,,$ (4.1)

an infinitesimal coloured Poincaré transformation is obtained from the Lie
algebra element
$(\mathbb{O},\mathbb{A})\in{\mathfrak{su}}(N,N)\inplus_{\mathrm{adj}}{\mathfrak{su}}(N,N)$
with111111The notation $\omega$ for the parameter here should not be confused
with the gauge fields $\omega$ from section 3.

$\displaystyle\mathbb{O}=\underbrace{\omega^{a}\,M_{a}+\omega^{a}_{I}\,M_{a}^{I}+\sigma_{I}\,N^{I}}_{\text{coloured
Lorentz}}\,,\qquad\mathbb{A}=\underbrace{\alpha^{a}\,P_{a}+\alpha^{a}_{I}\,P_{a}^{I}+\beta_{I}\,Q^{I}}_{\text{coloured
translations}}\ $ (4.2)

and produces the linear transformation,

$\delta\mathbb{X}=[\mathbb{O},\mathbb{X}]+\mathbb{A}\,.$ (4.3)

More explicitly, the transformation rule reads

$\displaystyle\delta x^{a}$
$\displaystyle=\varepsilon_{bc}{}^{a}\,\omega^{b}\,x^{c}+\frac{1}{N}\,\delta^{IJ}\,\varepsilon_{bc}{}^{a}\,\omega^{b}_{I}\,x^{c}_{J}+\alpha^{a}\,,$
(4.4a) $\displaystyle\delta x^{a}_{I}$
$\displaystyle=\varepsilon_{bc}{}^{a}\,\omega^{b}x^{c}_{I}+\varepsilon_{bc}{}^{a}\,\omega^{b}_{I}\,x^{c}+\frac{1}{2}\,\varepsilon_{bc}{}^{a}\,d^{JK}{}_{I}\,\omega^{b}_{J}\,x^{c}_{K}+f^{JK}{}_{I}\,\omega^{a}_{J}\,y_{K}+f^{JK}{}_{I}\,\sigma_{J}\,x^{a}_{K}+\alpha^{a}_{I}\,,$
(4.4b) $\displaystyle\delta y_{I}$
$\displaystyle=-\frac{1}{4}\,\eta_{ab}\,f^{JK}{}_{I}\,\omega^{a}_{J}\,x^{b}_{K}+f^{JK}{}_{I}\,\sigma_{J}y_{K}+\beta_{I}\,.$
(4.4c)

Here, we have used (2.13).

The usual Lorentz part $\omega^{a}$ acts on the usual coordinate $x^{a}$ in
the standard way. The coloured Lorentz generators $N^{I}$, that satisfy the
${\mathfrak{su}}(N)$ algebra according to (2.13c), generate colour rotations
on the colour-extended coordinates $x^{a}_{I}$ and $y_{I}$. Note that these
coordinates are in the adjoint of ${\mathfrak{su}}(N)$.121212It is amusing to
observe that the transformation of the usual Minkowski coordinates $x^{a}$
does not depend on the remaining coordinates in the limit $N\to\infty$.

The generators of coloured Lorentz group are obtained by tensoring such
matrices with $\mathcal{I}$ and the generators of the coloured translations by
tensoring with $\mathcal{J}$, see (2.7) and (2.12). We shall suppress the
tensoring with these elements consistently. The action of the generalised
Lorentz group on generalised Minkowski space can just be written as matrix
operations.

In the representation (2.17), a point on coloured Minkowski space is written
as

$\displaystyle\mathbb{X}=\begin{pmatrix}\text{i}\,{\bf X}^{0}+\text{i}\,{\bf
Y}&{\bf X}^{+}\\\ {\bf X}^{-}&-\text{i}\,{\bf X}^{0}+\text{i}\,{\bf
Y}\end{pmatrix}$ (4.5)

with ${\bf X}^{\pm}={\bf X}^{1}\pm\text{i}\,{\bf X}^{2}$ and

$\displaystyle{\bf X}^{a}$ $\displaystyle=x^{a}\,{\bf
I}+\text{i}\,x^{a}_{I}\,T^{I}\,,$ $\displaystyle{\bf Y}$
$\displaystyle=\text{i}\,y_{I}\,T^{I}\,.$ (4.6)

All coefficients are real-valued, thus $\left({\bf
X}^{\pm}\right)^{\dagger}={\bf X}^{\mp}$. Writing the parameter of a
transformation (4.2) as

$\displaystyle\mathbb{O}=\begin{pmatrix}\text{i}\,\bm{\omega}^{0}+\text{i}\,\bm{\sigma}&\bm{\omega}^{+}\\\
\bm{\omega}^{-}&-\text{i}\,\bm{\omega}^{0}+\text{i}\,\bm{\sigma}\end{pmatrix}\,,\qquad\mathbb{A}=\begin{pmatrix}\text{i}\,\bm{\alpha}^{0}+\text{i}\,\bm{\beta}&\bm{\alpha}^{+}\\\
\bm{\alpha}^{-}&-\text{i}\,\bm{\alpha}^{0}+\text{i}\,\bm{\beta}\end{pmatrix},$
(4.7)

with component expansions similar to above, the transformation in (4.3)
corresponds exactly to the matrix commutator.

A metric on coloured Minkowski space that is invariant under coloured Lorentz
transformations is given by

$\displaystyle\frac{1}{2N}\mathrm{Tr}(\mathbb{X}^{2})=x^{a}x^{b}\eta_{ab}+\frac{1}{N}x^{a}_{I}x^{b}_{J}\eta_{ab}\delta^{IJ}-\frac{1}{N}y_{I}y_{J}\delta^{IJ}\,,$
(4.8)

where $\eta_{ab}$ is the $(-++)$ Minkowski metric and $\delta^{IJ}$ the
$SU(N)$ invariant metric. We see that it consists of the usual Minkowski
metric for the coordinates $x^{a}$ and the other coordinates are suppressed by
factors of $1/N$. Using the invariant metric (4.8) one can define the notion
of coloured null, time-like and space-like vectors in the usual way depending
on whether the norm is zero, negative or positive, respectively.

For writing particle actions, we shall also make use of the matrix notation
(4.5) and moreover require a matrix representation of the (conjugate) momentum
that we write as

$\displaystyle\mathbb{P}=\begin{pmatrix}\text{i}\,{\bf
P}^{0}+\text{i}\,{\bf\Pi}&{\bf P}^{+}\\\ {\bf P}^{-}&-\text{i}\,{\bf
P}^{0}+\text{i}\,{\bf\Pi}\end{pmatrix}\,.$ (4.9)

The momentum vector $\mathbb{P}$ transforms in the coadjoint of the coloured
Lorentz algebra ${\mathfrak{su}}(N,N)$, thus
$\mathbb{P}\in{\mathfrak{su}}(N,N)^{*}$:
$\delta_{\mathbb{O}}\mathbb{P}=[\mathbb{O},\mathbb{P}]$. The components
satisfy $\left({\bf P}^{\pm}\right)^{\dagger}={\bf P}^{\mp}$ and ${\bf\Pi}$ is
traceless and hermitian. Expanding out these matrices explicitly we write

$\displaystyle{\bf
P}^{a}=p^{a}\,\mathbf{I}+\text{i}\,p^{a}_{I}\,T^{I}\,,\quad{\bf\Pi}=\text{i}\,\pi_{I}\,T^{I}\,.$
(4.10)

We note that in the uncoloured case $N=1$ we have

$\displaystyle N=1:\hskip
28.45274pt\mathbb{P}^{2}=\begin{pmatrix}p^{a}p_{a}&0\\\
0&p^{a}p_{a}\end{pmatrix}=p^{a}p_{a}\mathbb{I}\,.$ (4.11)

That $\mathbb{P}^{2}$ is proportional to the identity matrix is unique to the
case $N=1$.

We shall moreover need a matrix Lagrange multiplier $\mathbb{L}$ whose matrix
form we take as

$\displaystyle\mathbb{L}=-\text{i}\,\begin{pmatrix}\text{i}\,{\bf
K}^{0}+\text{i}\,{\bf\Lambda}&{\bf K}^{+}\\\ {\bf K}^{-}&-\text{i}\,{\bf
K}^{0}+\text{i}\,{\bf\Lambda}\end{pmatrix}\,,$ (4.12)

that satisfies $\left({\bf K}^{\pm}\right)^{\dagger}={\bf K}^{\mp}$ and where
$\bf\Lambda$ is now traceful and hermitian. In other words
$\text{i}\,\mathbb{L}\in{\mathfrak{u}}(N,N)$ and we have multiplied by a
convenient $-\text{i}\,$ for simplifying certain expressions later. Expanding
out the matrices we write

$\displaystyle{\bf K}^{a}=f^{a}\,{\bf
I}+\text{i}\,k_{I}^{a}\,T^{I}\,,\quad{\bf\Lambda}=e\,{\bf
I}+\text{i}\,\lambda_{I}\,T^{I}\,.$ (4.13)

The extra ${\mathfrak{u}}(1)$ component corresponds to the parameter $e$.

###  4.2 Different types of massive particle actions

Before discussing particle actions for coloured Minkowski space we briefly
review different particle actions in uncoloured Minkoswki space and how they
are related.

####  4.2.1 Uncoloured Minkowski space

In ordinary Minkowski space with coordinates $x^{a}$, there are several ways
of writing the action for a massive particle of mass $m>0$.

  1. 1.

Geometric action:

$\displaystyle S_{\text{geo}}^{\text{unc.}}[x^{a}]=-m\int
d\tau\sqrt{-\dot{x}^{a}\dot{x}^{b}\eta_{ab}}\,.$ (4.14)

This action describes the proper length of the time-like world-line
($\dot{x}^{a}\dot{x}^{b}\eta_{ab}<0$), where the dot denotes differentiation
with respect to the arbitrary world-line parameter $\tau$. The action is
invariant under general world-line reparametrisations. We note that the
momentum conjugate to $x^{a}$ that follows from this action is

$\displaystyle
p_{a}=\frac{m\dot{x}_{a}}{\sqrt{-\dot{x}^{b}\dot{x}^{c}\eta_{bc}}}\hskip
28.45274pt\Longrightarrow\quad p_{a}p^{a}+m^{2}=0\,.$ (4.15)

The last equation is the usual mass-shell constraint for a massive particle.
In proper time gauge $\dot{x}^{a}\dot{x}_{a}=-1$, the equations of motion are
simply $\ddot{x}^{a}=0$.

  2. 2.

Hamiltonian action:

$\displaystyle S_{\text{Ham}}^{\text{unc.}}[x^{a},p_{a},e]=\int
d\tau\left(\dot{x}^{a}p_{a}+e(p^{a}p_{a}+m^{2})\right).$ (4.16)

This action utilises the einbein $e$ and the (conjugate) momentum $p_{a}$,
both of which appear algebraically and are fully independent variables. By
first integrating out $p^{a}$ and then $e$ by using their equations of motion
one re-obtains (4.14). Since the equation for $e$ is quadratic there are two
solutions, corresponding to changing the overall sign of (4.14). The mass-
shell constraint,

$\displaystyle p^{a}p_{a}+m^{2}=0\,,$ (4.17)

obtained by varying (4.16) with respect to $e$ has two independent solutions
(up to the action of the orthochronous Lorentz group) corresponding to
positive and negative energy particles that are distinguished by the sign of
$p^{0}$. More abstractly, there are two independent semi-simple orbits of the
action of $SO(2,1)$ on the space of momenta $p^{a}$ that satisfy the
uncoloured mass-shell constraint (4.17).

If we write the uncoloured Minkowski coordinates in matrix form using (4.5),
we can also write this action as

$\displaystyle
S_{\text{Ham}}^{\text{unc.}}[\mathbb{X},\mathbb{P},e]=\frac{1}{2}\int
d\tau\mathrm{Tr}\left[\mathbb{P}\,\dot{\mathbb{X}}+e(\mathbb{P}^{2}+m^{2}\mathbb{I})\right],$
(4.18)

where we have also used (4.9) and (4.11). Since for $N=1$,
$\mathbb{P}^{2}=p^{a}p_{a}\mathbb{I}$ we can even write the very same action
in the form

$\displaystyle
S_{\text{Ham}}^{\text{unc.}}[\mathbb{X},\mathbb{P},\mathbb{L}]=\frac{1}{2}\int
d\tau\mathrm{Tr}\left[\mathbb{P}\,\dot{\mathbb{X}}+\mathbb{L}(\mathbb{P}^{2}+m^{2}\mathbb{I})\right],$
(4.19)

where $\mathbb{L}$ is now a matrix of Lagrange multipliers as introduced in
(4.12) with $\frac{1}{2}\mathrm{Tr}\,\mathbb{L}=e$.

  3. 3.

Orbit action:

$\displaystyle S_{\text{orb}}^{\text{unc.}}[x^{a},\vec{p}\,]=\int
d\tau\,\dot{x}^{a}\,p_{a}\,,\quad\quad p_{0}=\pm\sqrt{\vec{p}^{\,2}+m^{2}}\,,$
(4.20)

where the condition $p_{0}=\pm\sqrt{\vec{p}^{\,2}+m^{2}}$ selects only one
massive orbit, depending on the sign one chooses. This action is in a sense
the simplest action one can write for any given co-adjoint orbit, here again
restricted to semi-simple orbits since $p^{a}p_{a}+m^{2}=0$ by construction.
This action also follows from the method of non-linear realisation as we
discuss in more detail in appendix B.

The number of degrees of freedom can be determined from the actions in all
three cases and one finds two degrees of freedom (in configuration space).
This number of degrees of freedom agrees also with the dimension of the orbit
of a massive momentum under $SO(1,2)$. Even though there are two distinct
orbits, corresponding to hyperboloids in the forward and backward light-cone
or positive and negative energy particles, respectively, the dimension of all
these orbits is the same.131313All these actions take exactly the same form in
$D$-dimensional space-time, where the massive particle has $D-1$ degrees of
freedom.

For the case of the massless particle one can most easily proceed from the
Hamiltonian action (4.16) which is perfectly well-behaved in the limit $m\to
0$. The mass-shell constraint $p^{a}p_{a}=0$ now selects nilpotent orbits in
momentum space, corresponding to the light-cones themselves. Imposing the
corresponding orbit condition $p_{0}=\pm\sqrt{\vec{p}^{\,2}}$ in (4.20)
describes the same physics.

Thinking of an (irreducible) particle as corresponding to a single momentum
orbit, the most natural action is (4.20) as it only contains a single orbit
whereas both (4.14) and (4.16) collect several orbits together. While for the
uncoloured Poincaré algebra, there are only two distinct massive orbits (of
positive and negative energy) and the distinction may seem innocuous, we shall
see next that for the coloured Poincaré algebra the distinction is more
important.

####  4.2.2 Coloured Minkowski space

  1. 1.

Geometric action: We have determined the invariant line element in (4.8) and
can therefore immediately write a world-line reparametrisation invariant
action for a massive particle

$\displaystyle S_{\text{geo}}[\mathbb{X}]]=-m\int
d\tau\sqrt{-\tfrac{1}{2N}\mathrm{Tr}\big{[}\dot{\mathbb{X}}^{2}\big{]}}\,.$
(4.21)

The conjugate momentum is given by

$\displaystyle\mathbb{P}=\frac{1}{2N}m\frac{\dot{\mathbb{X}}^{T}}{\sqrt{-\tfrac{1}{2N}\mathrm{Tr}\big{[}{\dot{\mathbb{X}}}^{2}}]}$
(4.22)

that satisfies the constraint

$\displaystyle\mathrm{Tr}\big{[}{\mathbb{P}}^{2}+m^{2}\mathbb{I}]=0\,,$ (4.23)

where $\mathbb{I}$ is the $2N\times 2N$ unit matrix. This scalar constraint is
the generator of world-line diffeomporphisms. The time-like nature of the
coloured velocity corresponds to
$-\mathrm{Tr}\big{[}\dot{\mathbb{X}}^{2}\big{]}>0$. The equations of motion
implied by this geometric action in proper time gauge are simply

$\displaystyle\ddot{\mathbb{X}}=0\,.$ (4.24)

The proper time gauge condition implies
$-\tfrac{1}{2N}\mathrm{Tr}\big{[}\dot{\mathbb{X}}^{2}\big{]}=1$.

  2. 2.

Hamiltonian action: In order to write the Hamiltonian action we now have
several expressions that could be generalised. The canonical action that
corresponds directly to the geometric action above is

$\displaystyle\tilde{S}_{\text{Ham}}[\mathbb{X},\mathbb{P},e]=\frac{1}{2N}\int
d\tau\,\mathrm{Tr}\big{[}\mathbb{P}\,\dot{\mathbb{X}}+e\,(\mathbb{P}^{2}+m^{2}\,\mathbb{I})\big{]}\,,$
(4.25)

and it is the direct generalisation of (4.18). Integrating out $\mathbb{P}$
and $e$ from this action is not straight-forward and we discuss some aspects
of this in appendix D. By varying (4.25) with respect to $e$, we re-obtain the
scalar constraint (4.23). There are many solutions to this scalar constraint,
corresponding to a continuum of different semi-simple orbits. These orbits can
even have different dimensions and thus describe particles with different
numbers of degrees of freedom. We discuss aspects of these orbits in more
detail below in section 4.3.

In order to always have massive particles with the same number of degrees of
freedom, we can alternatively consider the generalisation of (4.19) that
becomes

$\displaystyle
S_{\text{Ham}}[\mathbb{X},\mathbb{P},\mathbb{L}]=\frac{1}{2N}\int
d\tau\,\mathrm{Tr}\big{[}\mathbb{P}\,\dot{\mathbb{X}}+\mathbb{L}\,(\mathbb{P}^{2}+m^{2}\,\mathbb{I})\big{]}\,,$
(4.26)

now with a matrix-valued Lagrangian multiplier $\mathbb{L}$ as written out in
(4.12). Note that $\text{i}\,\,\mathbb{L}\in{\mathfrak{u}}(N,N)$ and therefore
there is a non-vanishing component
$\tfrac{1}{2N}\mathrm{Tr}\big{[}\mathbb{L}\big{]}=e$. The constraint that the
matrix-valued Lagrange multipliers imposes is

$\displaystyle\mathbb{P}^{2}+m^{2}\,\mathbb{I}=0$ (4.27)

which is a $(2N)\times(2N)$ matrix condition. The scalar constraint (4.23) is
its trace and therefore a weaker condition; only in the case $N=1$ the
conditions are equivalent by virtue of (4.11).
The matrix constraint (4.27) restricts the possible momentum orbit more
strongly than the trace constraint and, in particular, all semi-simple orbits
satisfying (4.27) have the same dimension as we shall review in more detail in
section 4.3.

  3. 3.

Orbit action: In analogy with (4.20), we can also consider the simple action
associated with a given orbit. This corresponds to considering

$\displaystyle S_{\text{orb}}=\frac{1}{2N}\int
d\tau\mathrm{Tr}\big{[}\mathbb{P}\,\dot{\mathbb{X}}\big{]}=\int
d\tau\langle\mathbb{P},\dot{\mathbb{X}}\rangle\,,\quad\quad\text{$\mathbb{P}$
belonging to a given massive orbit.}$ (4.28)

In the second step we have introduced the canonical pairing between
${\mathfrak{su}}(N,N)$ and its dual: for any element ${\cal
B}\in{\mathfrak{su}}(N,N)^{*}$ , there is a unique element $H_{{\cal
B}}\in{\mathfrak{su}}(N,N)$ satisfying

$\left\langle{\cal
B},\,A\right\rangle=\frac{1}{2N}\,\mathrm{Tr}\left[A\,H_{{\cal
B}}\right],\qquad\text{for all }A\in{\mathfrak{su}}(N,N)\,,$ (4.29)

where the trace is for the fundamental representation of
${\mathfrak{su}}(N,N)$. By abuse of notation, we write $\mathbb{P}$ both for
the adjoint and the coadjoint representative.

We briefly explain in this case why the orbit action agrees with the action
one would derive from a standard non-linear realisation [54, 55, 56, 57] for a
semi-direct product. We do this for simplicity by focussing on the case when a
representative of the massive orbit is given by

$\displaystyle\mathbb{P}=m\mathcal{P}_{0}^{0}=m\begin{pmatrix}\text{i}\,\mathbf{I}&0\\\
0&-\text{i}\,\mathbf{I}\end{pmatrix},$ (4.30)

where $\mathcal{P}_{0}^{0}\in{\mathfrak{su}}(N,N)^{*}$ is the generator dual
to $P_{0}^{0}$, the usual momentum. The stabiliser of this momentum is
generated by

$\displaystyle M_{0}\equiv M_{0}^{0}\,,\quad M_{0}^{I}\,,\quad N^{I}$ (4.31)

as can be checked using the algebra (2.11). The algebra generated by (4.31) is
${\mathfrak{u}}(1)\oplus{\mathfrak{su}}(N)\oplus{\mathfrak{su}}(N)$. We thus
take $H=\left(U(1)\times SU(N)\times SU(N)\right)$ and write a local
representative of the coset in the form

$\displaystyle g=e^{\mathbb{X}}\,b\,h\,,$ (4.32)

with

$\displaystyle\mathbb{X}$
$\displaystyle=x^{a}P_{a}+x^{a}_{I}P_{a}^{I}+y_{I}Q^{I}\quad$
$\displaystyle\textrm{(generalised coordinate of coloured Minkowski
space)}\,,$ (4.33a) $\displaystyle b$
$\displaystyle=\exp\left[v^{i}M_{i}+w^{i}_{I}M_{i}^{I}\right]\quad$
$\displaystyle\textrm{(generalised boost)}\,,$ (4.33b)

and $h$ belonging to the generalised little group $H$ generated by (4.31). The
indices $a=0,1,2$ are covariant while $i=1,2$ are purely spatial indices. The
generalised boost $b$ represents the broken generalised Lorentz generators
acting on the generalised coordinate $\mathbb{X}$. We think of $x^{a}$ as the
uncoloured position variable and $x^{a}_{I}$ and $y_{I}$ as coloured
generalisation required by the algebra in the way described in section 4.1.
All coordinates here are real.

The Maurer–Cartan form associated with the coset representative (4.32) is

$\displaystyle\Omega=g^{-1}dg=b^{-1}d\mathbb{X}b+b^{-1}db$ (4.34)

where we have chosen the gauge $h=1$. The component of $\Omega$ along $P_{0}$
is invariant under the little group and its pullback to the world-line can be
used as a Lagrangian that by construction then will be invariant under the
full coloured Poincaré group. Using the pairing with the dual space of the
coloured translation

$\displaystyle L_{\text{nlr}}\,dt=m\Big{[}\Omega^{*}\Big{]}_{P_{0}}=\langle
m\,\mathcal{P}_{0},b^{-1}\,d\mathbb{X}\,b\rangle\,,$ (4.35)

where $\langle\cdot,\cdot\rangle$ is the pairing of the Lie algebra with its
dual and $\mathcal{P}_{0}$ is the dual of the generator $P_{0}$. Now, using
the invariance of the pairing, we can rewrite this as

$\displaystyle
L_{\text{nlr}}=\langle\mathrm{Ad}^{*}_{b}(m\,\mathcal{P}_{0}),\tfrac{d\mathbb{X}}{dt}\rangle=\langle\mathbb{P},\dot{\mathbb{X}}\rangle\,,$
(4.36)

where now $\mathbb{P}$ is any element of the orbit of the reference
momentum.141414Note that $\mathbb{P}=\mathrm{Ad}^{*}_{b}(m\,\mathcal{P}_{0})$
belongs to a ‘momentum’ orbit, which is different from the coadjoint orbit,
$\\{\mathrm{Ad}^{*}_{g}(m\,\mathcal{P}_{0})\,|\,\forall
g\in\mathrm{cPoin}_{3}(N)\\}$, of coloured Poincaré symmetry
$\mathrm{cPoin}_{3}(N)$. The coadjoint orbit can be viewed as the ‘phase
space’, and hence the momentum orbit is its Lagrangian subspace. Because the
(coloured) Poincaré symmetry is not semi-simple, the coadjoint orbits are not
isomorphic to the adjoint orbits. In fact the momentum orbit can be viewed as
either the adjoint orbit of $\mathrm{cPoin}_{3}(N)$ or the coadjoint orbit of
the coloured Lorentz group $SU(N,N)$.  We thus have shown that the free
particle Lagrangian obtained from the non-linear realisation is equal to that
of the orbit construction (4.28), see also appendix B.

###  4.3 Coadjoint orbits of ${\mathfrak{su}}(N,N)$

The coloured Lorentz group $SU(N,N)$ acts on the momentum
$\mathbb{P}\in{\mathfrak{su}}(N,N)^{*}$ using the coadjoint action. In the
above actions we have seen that the matrix constraint

$\displaystyle\mathbb{P}^{2}+m^{2}\mathbb{I}=0$ (4.37)

and the scalar constraint

$\displaystyle\mathrm{Tr}\big{[}\mathbb{P}^{2}+m^{2}\mathbb{I}\big{]}=0$
(4.38)

arose. These constraints are both $SU(N,N)$ covariant and the solutions can
therefore be classified in terms of coadjoint orbits. Since
${\mathfrak{su}}(N,N)$ is simple we can identity coadjoint orbits with adjoint
orbits using the non-degenerate trace pairing (4.29). For a general discussion
of orbits see [58].

####  4.3.1 Massive particles: semi-simple orbits

For $m>0$, the relevant orbits are of semi-simple type and have
representatives of the form

$\displaystyle\mathbb{P}=\text{diag}(a_{1},\ldots,a_{N},b_{1}\ldots,b_{N})\,.$
(4.39)

The condition $\mathbb{P}\in{\mathfrak{su}}(N,N)^{*}$ implies that
$\sum_{j=1}^{n}(a_{j}+b_{j})=0$ and our hermiticity conditions (2.18) imply
that the $a_{j}$ and $b_{j}$ are pure imaginary. Interchanging
$a_{j}\leftrightarrow a_{k}$ or $b_{j}\leftrightarrow b_{k}$ for the
representative does not change the orbit.

Imposing the matrix constraint implies that $a_{j}=b_{j}=\pm\text{i}\,m$ for
all $j=1,\ldots,N$ while the scalar constraint implies the much weaker
condition $\sum_{j=1}^{N}(a_{j}^{2}+b_{j}^{2})=-2Nm^{2}$.

As a representative solution of the matrix constraint we have already given
(4.30). Its stabiliser was determined to have the Lie algebra
${\mathfrak{u}}(1)\oplus{\mathfrak{su}}(N)\oplus{\mathfrak{su}}(N)$ and thus
the size of the corresponding orbit $\mathcal{O}_{\eqref{eq:rep1}}$ is

$\displaystyle\dim\mathcal{O}_{\eqref{eq:rep1}}$
$\displaystyle=\dim{\mathfrak{su}}(N,N)-\dim({\mathfrak{u}}(1)\oplus{\mathfrak{su}}(N)\oplus{\mathfrak{su}}(N))=(2N)^{2}-1-(1+2(N^{2}-1))$
$\displaystyle=2N^{2}$ (4.40)

by applying the orbit-stabiliser theorem. The same is true for any other orbit
satisfying the matrix constraint (4.27).

By contrast, if only the scalar constraint (4.23) is satisfied, there are many
more solutions. Consider for example

$\displaystyle\mathbb{P}=\sqrt{N}\,\text{diag}(\text{i}\,m,0,0,\ldots,0,-\text{i}\,m)\,,$
(4.41)

which solves the trace but not the matrix constraint. The stabiliser is in
this case generated by ${\mathfrak{u}}(1)\oplus{\mathfrak{u}}(N-1,N-1)$, so
that the orbit is of dimension

$\displaystyle\dim\mathcal{O}_{\eqref{eq:rep2}}$
$\displaystyle=\dim{\mathfrak{su}}(N,N)-\dim({\mathfrak{u}}(1)\oplus{\mathfrak{u}}(N-1,N-1))=8N-6$
(4.42)

and is a smaller orbit than the one dictated by the matrix constraint for
$N>3$.

As a final example we consider

$\displaystyle\mathbb{P}=\sqrt{\frac{6}{(2N+1)(N+1)}}\,\text{diag}(\text{i}\,m,2\text{i}\,m,\ldots,N\text{i}\,m,-N\text{i}\,m,\ldots,-2\text{i}\,m,-\text{i}\,m)\,,$
(4.43)

which solves the trace but not the matrix constraint. Its stabiliser is
${\mathfrak{u}}(1)^{2N-1}$ and the orbit has dimension

$\displaystyle\dim\mathcal{O}_{\eqref{eq:rep3}}$
$\displaystyle=\dim{\mathfrak{su}}(N,N)-\dim({\mathfrak{u}}(1)^{2N-1})=2N(2N-1)\,.$
(4.44)

This is the maximal dimension of a semi-simple orbit since the stabiliser is
the minimal one: just the Cartan subalgebra. The dimension of the orbit equals
the number of roots of ${\mathfrak{su}}(N,N)$ and is much larger than the
orbits corresponding to the matrix constraint for $N>1$.

Generally, (complex) semi-simple orbits are related to representative elements
in the closure of the fundamental Weyl chamber (within the Cartan subalgebra)
[58, §2]. Fully generic elements in the interior of the fundamental Weyl
chamber correspond to elements such as the example (4.43) and the
corresponding semi-simple orbits are of maximal dimension. If the semi-simple
element is non-generic and has a larger stabiliser, it lies on boundaries of
the fundamental Weyl chamber and the dimension of the orbit shrinks. The
matrix constraint (4.27) selects a very specific set of such boundaries where
all orbits have the same dimension $(2N)^{2}$. As is clear from the examples
above, the trace constraint (4.23) allows for a wide variety of orbits. In
fact, from a projective point of view all possible semi-simple orbits are
allowed.

Thinking of different semi-simple orbits as different types of massive
particles, just like positive and negative energy particles in the uncoloured
case, we conclude that the geometric action (4.21) and the canonical action
(4.25) contain all possible types of massive particles, where even the number
of degrees of freedom can change.

By contrast, the canonical action (4.26) only contains massive particles with
$(2N)^{2}$ degrees of freedom and contains all orbits of this dimension.

The orbit action (4.28) describes only a single orbit and the choice of a type
of massive particle corresponds to a choice of semi-simple orbit.

For completeness, we give the general form of all representatives of solutions
of the matrix constraint (4.27). By using $SU(N,N)$ transformations, the
diagonal solution for $\mathbb{P}^{2}+m^{2}\,\mathbb{I}=0$ can be brought into
the form

$\widehat{\mathbb{P}}_{N-2\ell}=m\,{\rm
diag}(\underbrace{\text{i}\,,\ldots,\text{i}\,}_{\ell},\underbrace{-\text{i}\,,\ldots,-\text{i}\,}_{N-\ell},\underbrace{\text{i}\,,\ldots\text{i}\,}_{N-\ell},\underbrace{-\text{i}\,,\ldots,-\text{i}\,}_{\ell})\,,$
(4.45)

where we have taken into account the equivalence of interchanging diagonal
elements among the first and the second $N$ entries respectively, since such
interchanges can be made by an $SU(N,N)$ action. Here, the number $\ell$ is
the number of the element $+\text{i}\,$ in the first $N$ entries. Hence,
$\ell$ can take $N+1$ possible values ($\ell=0,1,\ldots,N$) and there exist
$N+1$ distinct orbits satisfying the equation
$\mathbb{P}^{2}+m^{2}\,\mathbb{I}=0$ . The dimension of the orbit ${\cal
O}_{N-2\ell}$ generated by $\widehat{\mathbb{P}}_{N-2\ell}$ is simply
$2N^{2}$, and it corresponds to the homogeneous space,

$\mathcal{O}_{N-2\ell}\simeq U(N,N)/(U(N-\ell,\ell)\times U(N-\ell,\ell))\,,$
(4.46)

which is a pseudo-Euclidean flag variety. The stabiliser is compact only for
$\ell=0$ and $\ell=N$. Note that the dimension does not depend on $\ell$. As
mentioned previously, the orbits that we describe here correspond to semi-
simple coadjoint orbits of $\mathfrak{su}(N,N)$ (and they are related by Weyl
reflections).

####  4.3.2 Massless particles: nilpotent orbits

For the case $m=0$, both the matrix and the trace constraint require the orbit
to have a nilpotent representative. Nilpotent orbits of $SU(N,N)$ are
described by signed Young tableaux [58, Thm. 9.3.3]. These are Young diagrams
with $2N$ boxes filled with $N$ positive and $N$ negative signs in such a way
that signs alternate across rows. Two signed Young diagrams are regarded as
equivalent if and only if one can be obtained from the other by interchanging
rows of equal length. Say, the height of the $k$-th column is $h_{k}$, then
the dimension of the orbit corresponding to the Young diagram
${\bf\lambda}=[h_{1},\ldots,h_{2N}]$ is

$\displaystyle\dim\,{\cal
O}_{\bf\lambda}=4\,N^{2}-\sum_{k=1}^{2N}\,h_{k}^{2}\,.$ (4.47)

The orbits with dimension $2N^{2}$ correspond to the Young diagram
${\bf\lambda}=[N,N]$ and there are $N+1$ different signed Young diagrams:

$\displaystyle\young(+-,::,+-,-+,::,-+)\,.$ (4.48)

These are the nilpotent orbits which are the $m\to 0$ limits of the semi-
simple orbits described by the matrix constraint
$\mathbb{P}^{2}=-m^{2}\,\mathbb{I}$ , and they obviously satisfy the orbit
equation $\mathbb{P}^{2}=0$ . However, the algebraic variety given by the
latter equation does not contain only the $2N^{2}$ dimensional nilpotent
orbits, but also other nilpotent orbits of smaller dimensions. Such orbits
correspond to all signed Young diagrams of the type ${\bm{\lambda}}=[2N-k,k]$
(which can be obtained from the Young diagram (4.48) by moving boxes from the
second to the first column). The number of such signed Young diagrams is $k+1$
and its dimension is $4N^{2}-(2N-k)^{2}-k^{2}=4Nk-2k^{2}$ . The number $k$
takes value in $\\{0,\ldots,N\\}$ including the $2N^{2}$ dimensional one with
$k=N$, so the equation $\mathbb{P}^{2}=0$ describes
$\sum_{k=0}^{N}(k+1)=(N+1)(N+2)/2$ distinct orbits including the trivial one
$\mathbb{P}=0$. The massless particles associated to these orbits are
discussed in section 4.7.

###  4.4 Massive coloured particle in component form

We now work out the Lagrangians in more detail, beginning with the more
general one (4.26) that contains all massive orbits satisfying the matrix-
valued ‘mass-shell’ constraint (4.27).

Plugging the parametrisations (4.5), (4.9) and (4.12) into the action (4.26)
leads to the Lagrangian

$\displaystyle L=\frac{1}{N}\mathrm{tr}\bigg{[}\dot{\bf X}^{a}\,{\bf
P}_{a}-\dot{\bf Y}\,{\bf\Pi}+{\bf\Lambda}\Big{(}{\bf P}_{a}{\bf
P}^{a}-{\bf\Pi}^{2}+m^{2}\mathbf{I}\Big{)}+{\bf K}_{a}\Big{(}\\{{\bf
P}^{a},{\bf\Pi}\\}+\frac{\text{i}\,}{2}\varepsilon^{abc}[{\bf P}_{b},{\bf
P}_{c}]\Big{)}\bigg{]}\,,$ (4.49)

where our space-time conventions were given in footnote 5.

Using the parametrisations (4.6), (4.10) and (4.13), the Lagrangian (4.49)
becomes

$\displaystyle L$
$\displaystyle=\dot{x}^{a}\,p_{a}+\frac{1}{N}\,\dot{x}^{a}_{I}\,p_{a}^{I}-\frac{1}{N}\,\dot{y}^{I}\,\pi_{I}+e\Big{(}p^{a}\,p_{a}+\frac{1}{N}\,p^{a}_{I}\,p_{aI}-\frac{1}{N}\,\pi^{I}\,\pi_{I}+m^{2}\Big{)}$
$\displaystyle\quad+\frac{1}{N}\,\lambda_{I}\Big{(}2\,p^{a}\,p_{a}^{I}-\frac{1}{2}\,d^{JKI}p^{a}_{J}\,p_{aK}-\frac{1}{2}d^{JKI}\,\pi_{J}\,\pi_{K}\Big{)}$
$\displaystyle\quad+\frac{2}{N}\,f_{a}\,p^{a}_{I}\,\pi^{I}+\frac{1}{N}\,k_{aI}\Big{(}2\,p^{a}\,\pi^{I}-d^{JKI}\,p_{aJ}\,\pi_{K}+\frac{1}{2}\,\varepsilon^{abc}\,f^{JKI}\,p_{bJ}\,p_{cK}\Big{)}\,,$
(4.50)

when we also use the algebra (2.1). The adjoint ${\mathfrak{su}}(N)$ indices
$I$ are raised and lowered with the invariant $\delta_{IJ}$.

Focussing on the terms that do not depend on $N$, we recognise immediately the
usual uncoloured (4.16). The terms proportional to $1/N$ represent the effects
of colouring the Poincaré algebra.

We record the equations of motion implied by (4.4)

$\displaystyle\dot{p}_{a}=0\,,\quad\quad\dot{p}_{a}^{I}=0\,,\quad\quad\dot{\pi}_{I}$
$\displaystyle=0\,,$ (4.51a)
$\displaystyle\dot{x}^{a}+2ep^{a}+\frac{2}{N}\lambda_{I}p^{aI}+\frac{2}{N}k_{aI}\pi^{I}$
$\displaystyle=0\,,$ (4.51b)
$\displaystyle\dot{x}^{a}_{I}+2ep^{a}_{I}+2\lambda_{I}p^{a}-d^{IJK}\lambda_{J}p_{aK}+2f_{a}\pi^{I}-d^{IJK}k_{aJ}\pi_{K}+\varepsilon_{abc}f^{IJK}k^{b}_{J}p^{c}_{K}$
$\displaystyle=0\,,$ (4.51c)
$\displaystyle-\dot{y}^{I}-2e\pi^{I}-d^{IJK}\pi_{J}\pi_{K}+2f_{a}p^{aI}+2k_{a}^{I}p^{a}-d^{IJK}k_{aJ}p^{a}_{K}$
$\displaystyle=0\,,$ (4.51d) $\displaystyle
p^{a}p_{a}+m^{2}+\frac{1}{N}p^{a}_{I}p_{a}^{I}-\frac{1}{N}\pi^{I}\pi_{I}$
$\displaystyle=0\,,$ (4.51e) $\displaystyle
2p_{a}^{I}p^{a}-\frac{1}{2}d^{IJK}p_{aJ}p^{a}_{K}-\frac{1}{2}d^{IJK}\pi_{J}\pi_{K}$
$\displaystyle=0\,,$ (4.51f) $\displaystyle p_{a}^{I}\pi_{I}$
$\displaystyle=0\,,$ (4.51g) $\displaystyle
2p^{a}\pi^{I}-d^{IJK}p_{J}^{a}\pi_{K}+\frac{1}{2}\varepsilon^{abc}f^{IJK}p_{bJ}p_{cK}$
$\displaystyle=0\,.$ (4.51h)

The constraints that are enforced by the Lagrange multipliers are given in
(4.51e)–(4.51h). The total number of constraints is
$1+(N^{2}{-}1)+3+3(N^{2}{-}1)=(2N)^{2}$ since the Lagrange multiplier
$\mathbb{L}$ belongs to ${\mathfrak{u}}(N,N)$. We note that these constraints
are not all independent and we are dealing with a reducible system of
constraints. This will be further investigated in section 4.5.

While the Lagrangian (4.4) is valid for any choice of $\mathbb{P}$ satisfying
$\mathbb{P}^{2}=-m^{2}\,\mathbb{I}$, the orbit Lagrangian (4.28) depends on
the choice of a given orbit where this condition is satisfied. In order to
investigate this we take a representative momentum and repeat the analysis
from appendix B in connection with the non-linear realisation discussed above.

Written in matrix form, the representative momentum (4.30) takes the form

$\displaystyle\widehat{\mathbb{P}}=m\begin{pmatrix}\text{i}\,{\bf I}&0\\\
0&-\text{i}\,{\bf I}\end{pmatrix}$ (4.52)

while the boost element $b$ from (4.33b) can be written as

$\displaystyle b=\exp\left[\frac{1}{2}\begin{pmatrix}0&{\bf V}^{+}\\\ {\bf
V}^{-}&0\end{pmatrix}\right]\,,\quad\text{with}\quad({\bf
V}^{+})^{\dagger}={\bf V}^{-}\,.$ (4.53)

Other elements of ${\mathfrak{su}}(N,N)$ stabilise $\widehat{\mathbb{P}}$
under the adjoint action and belong to the subalgebra
${\mathfrak{u}}(1)\oplus{\mathfrak{su}}(N)\oplus{\mathfrak{su}}(N)$.

The orbit of the reference momentum is then given by

$\displaystyle
b\,\widehat{\mathbb{P}}\,b^{-1}=m\begin{pmatrix}\text{i}\,\cosh\sqrt{{\bf
V}^{+}{\bf V}^{-}}&-\text{i}\,\frac{\sinh\sqrt{{\bf V}^{+}{\bf
V}^{-}}}{\sqrt{{\bf V}^{+}{\bf V}^{-}}}{\bf V}^{+}\\\ \text{i}\,{\bf
V}^{-}\frac{\sinh\sqrt{{\bf V}^{+}{\bf V}^{-}}}{\sqrt{{\bf V}^{+}{\bf
V}^{-}}}&-\text{i}\,\cosh\sqrt{{\bf V}^{-}{\bf V}^{+}}\end{pmatrix}\,,$ (4.54)

where the hyperbolic trigonometric functions of the hermitian matrices ${\bf
V}^{+}{\bf V}^{-}$ and ${\bf V}^{-}{\bf V}^{+}$ are defined by their power
series expansions. The formula represents the action of a general coloured
Lorentz boost on the reference momentum (4.52). We rewrite this more compactly
by defining

$\displaystyle{\bf P}^{+}=-m\,\text{i}\,\frac{\sinh\sqrt{{\bf V}^{+}{\bf
V}^{-}}}{\sqrt{{\bf V}^{+}{\bf V}^{-}}}{\bf V}^{+}\,,\quad\quad{\bf
P}^{-}=\left({\bf P}^{+}\right)^{\dagger}\,,$ (4.55)

leading to

$\displaystyle
b\,\widehat{\mathbb{P}}\,b^{-1}=\begin{pmatrix}\text{i}\,\sqrt{m^{2}\,{\bf
I}+{\bf P}^{+}{\bf P}^{-}}&{\bf P}^{+}\\\ {\bf
P}^{-}&-\text{i}\,\sqrt{m^{2}\,{\bf I}+{\bf P}^{-}{\bf
P}^{+}}\end{pmatrix}\,,$ (4.56)

where the square root of matrices is again defined by its Taylor series around
${\bf P}^{\pm}{\bf P}^{\mp}=0$. We note that all elements in the orbit of the
reference momentum (4.52) satisfy

$\displaystyle\left(b\,\widehat{\mathbb{P}}\,b^{-1}\right)^{2}=\widehat{\mathbb{P}}^{2}=-m^{2}\mathbb{I}\,.$
(4.57)

This equation has more solutions than (4.56) and the Lagrangian (4.26)
captures all of them.

Considering the Lagrangian (4.28) for the reference momentum (4.52) we need to
pair (4.56) with the velocity vector $\dot{\mathbb{X}}$ given in (4.5), which
leads to

$\displaystyle L_{\text{orb}}=\frac{1}{2N}\mathrm{tr}\bigg{[}(\dot{{\bf
X}}_{0}-\dot{{\bf Y}})\sqrt{m^{2}\,{\bf I}+{\bf P}^{+}{\bf P}^{-}}+(\dot{{\bf
X}}_{0}+\dot{{\bf Y}})\sqrt{m^{2}\,{\bf I}+{\bf P}^{-}{\bf P}^{+}}+\dot{{\bf
X}}^{+}{\bf P}^{-}+\dot{{\bf X}}^{-}{\bf P}^{+}\bigg{]}\,.$ (4.58)

This is the coloured generalisation of the Lagrangian (4.20). Setting $N=1$
and $\dot{{\bf X}}^{a}$, ${\bf P}^{a}$ to 3-component Lorentz vectors and
$\dot{{\bf Y}}=0$ reduces it manifestly to the standard uncoloured massive
particle.

The Lagrangian (4.58) has been deduced from the representative momentum
(4.52). We shall now explore the relation to the previously derived Lagrangian
(4.49) and try to follow a route similar to that of appendix B that starts
from the constraints implied by (4.58). Varying (4.58) with respect to ${\bf
X}^{0}\pm{\bf Y}$ leads to the constraints

$\displaystyle{\bf C}_{\pm}={\bf P}^{0}\pm{\bf\Pi}-\sqrt{m^{2}{\bf I}+{\bf
P}^{\pm}{\bf P}^{\mp}}=0$ (4.59)

that are $(N\times N)$-matrix valued and where we have introduced new
$(N\times N)$-matrix valued variables ${\bf P}^{0}=-{\bf P}_{0}$ and
${\bf\Pi}$ as integration constants. These constraints are the generalisations
of the positive-energy mass-shell constraint of an uncoloured particle as in
(4.20). In order to allow for generalisations of negative energy, we shall
consider as a generalisation the constraints

$\displaystyle{\bf C}_{\pm}^{2}=\left({\bf
P}^{0}\pm{\bf\Pi}\right)^{2}-\left(m^{2}{\bf I}+{\bf P}^{\pm}{\bf
P}^{\mp}\right)=0\,.$ (4.60)

These are also $(N\times N)$-matrix valued and will be enforced using the
matrix-valued Lagrange multipliers ${\bf\Lambda}\pm{\bf K}$. Enforcing only
these squared constraints allows for more solutions than implied by the
original Lagrangian in the same way as squaring the positive mass condition
for going from (4.20) to (4.16) allows for more solutions, see also appendix
B.

The resulting generalised Lagrangian is then

$\displaystyle\tilde{L}$
$\displaystyle=\frac{1}{2N}\mathrm{tr}\bigg{[}-(\dot{\bf X}^{0}+\dot{\bf
Y})\left({\bf P}^{0}+{\bf\Pi}\right)-(\dot{\bf X}^{0}-\dot{\bf Y})\left({\bf
P}^{0}-{\bf\Pi}\right)+{\bf P}^{+}\dot{\bf X}^{-}+{\bf P}^{-}\dot{\bf X}^{+}$
$\displaystyle\quad-\left({\bf\Lambda}+{\bf K}\right)\left(\left({\bf
P}^{0}+{\bf\Pi}\right)^{2}-\left(m^{2}{\bf I}+{\bf P}^{+}{\bf
P}^{-}\right)\right)-\left({\bf\Lambda}-{\bf K}\right)\left(\left({\bf
P}^{0}-{\bf\Pi}\right)^{2}-\left(m^{2}{\bf I}+{\bf P}^{-}{\bf
P}^{+}\right)\right)\bigg{]}$
$\displaystyle=\frac{1}{N}\mathrm{tr}\bigg{[}\dot{\bf X}^{a}{\bf
P}_{a}-\dot{\bf Y}{\bf\Pi}+{\bf\Lambda}\left({\bf P}^{a}{\bf
P}_{a}-{\bf\Pi}^{2}+m^{2}{\bf I}\right)-{\bf K}\left(\left\\{{\bf
P}^{0},{\bf\Pi}\right\\}+\text{i}\,\left[{\bf P}^{1},{\bf
P}^{2}\right]\right)\bigg{]}$ (4.61)

with ${\bf P}^{\pm}={\bf P}_{1}\pm\text{i}\,{\bf P}_{2}={\bf
P}^{1}\pm\text{i}\,{\bf P}^{2}$ as well as ${\bf P}_{0}=-{\bf P}^{0}$ due to
the $(-++)$ signature.

The Lagrangian (4.4) does not manifestly agree with (4.49) that we deduced
from the natural squared constraint (4.27) in ${\mathfrak{u}}(N,N)$, in
particular, it does not exhibit manifest Lorentz invariance under the usual
$SO(1,2)$ group. However, the transformation of the Lagrange multipliers under
the coloured Lorentz group is exactly such that it is opposite to that of the
constraints so that the Lagrangian is invariant by construction.

The matrix mass-shell Lagrangian (4.49) also has two extra Lagrange
multipliers ${\bf K}_{1}$ and ${\bf K}_{2}$ that are not present in (4.4). The
constraints they enforce from (4.49) are

$\displaystyle\left\\{{\bf P}^{1},{\bf\Pi}\right\\}+\text{i}\,\left[{\bf
P}_{2},{\bf P}_{0}\right]=0\,,\quad\left\\{{\bf
P}^{2},{\bf\Pi}\right\\}+\text{i}\,\left[{\bf P}_{0},{\bf P}_{1}\right]=0$
(4.62)

that are equivalent to

$\displaystyle\left({\bf P}_{0}\pm{\bf\Pi}\right){\bf P}^{\pm}={\bf
P}^{\pm}\left({\bf P}_{0}\mp{\bf\Pi}\right)\,.$ (4.63)

When the original constraint (4.59) is satisfied, meaning ${\bf
P}_{0}\pm{\bf\Pi}=\sqrt{m^{2}{\bf I}+{\bf P}^{\mp}{\bf P}^{\pm}}$, the above
constraints follow. However, they need not follow from the squared constraint
${\bf C}_{\pm}^{2}=0$ in (4.60) and thus the manifestly covariant Lagrangian
(4.49) is not equivalent to (4.4). We therefore consider the manifestly
covariant Lagrangian (4.49) as the appropriate generalisation of the massive
particle (4.52) to all possible massive orbits and shall work with (4.49) in
the following.

###  4.5 Constraint structure and degrees of freedom

In this section we revisit the discussion of the dimension of semi-simple
orbits of section 4.3 and relate it to the reducibility of the constraints
implied by the matrix constraint (4.27). The reducibility of the constraint
means that not all of the $(2N)^{2}$ components of (4.27) can be counted
independently. Rather, we are interested in the dimension of the solution
space of $\mathbb{P}^{2}=-m^{2}\mathbb{I}$.

From the discussion in section 4.3 we know that the dimension of the solution
space is given by the size of an orbit of solutions and that these can be
characterised by the stabilisers of the semi-simple representative of the
orbit. For the case of the massive momentum given in (4.30) the stabiliser was
$S(U(N)\times U(N))\subset SU(N,N)$. For any other orbit of solutions of
(4.27) the stabiliser is a similar real form of the same dimension. The
dimension of the stabiliser equals the number of independent constraints
contained in (4.27) since this is the number by which the number of variables
is reduced to obtain an orbit of solutions.

A canonical analysis then reveals the number of degrees of freedom. It is
important to consider the configuration space variables of (4.26) as
$\mathbb{X}$, $\mathbb{P}$ and $\mathbb{L}$, where the first two have
$(2N)^{2}-1$ components while $\mathbb{L}$ has $(2N)^{2}$ components.
Canonically, the conjugate momenta of $\mathbb{P}$ and $\mathbb{L}$ vanish and
there is another constraint relating the conjugate momentum of $\mathbb{X}$ to
$\mathbb{P}$. These are all the primary constraints and their numbers are
$(2N)^{2}-1$, $(2N)^{2}$ and $(2N)^{2}-1$, respectively.

The constraint (4.27) appears as a secondary constraint after time evolution
of the primary constraint that the momentum $\pi_{\mathbb{L}}$ conjugate to
$\mathbb{L}$ vanishes. As we argued above, there are only $2N^{2}-1$
independent constraints contained in (4.27) in agreement with the dimension of
the stabiliser of a solution. These $2N^{2}-1$ constraints and the $(2N)^{2}$
conditions $\pi_{\mathbb{L}}=0$ are first-class constraints, the remaining two
sets of constraints are second-class. The count of degrees of freedom in phase
space is therefore

$\displaystyle\underbrace{2{\times}((2N)^{2}-1)}_{(\mathbb{X},\pi_{\mathbb{X}})}+\underbrace{2{\times}((2N)^{2}-1)}_{(\mathbb{P},\pi_{\mathbb{P}})}+\underbrace{2{\times}(2N)^{2}}_{(\mathbb{L},\pi_{\mathbb{L}})}-2\underbrace{((2N)^{2}+2N^{2}-1)}_{\text{first-
class}}-\underbrace{2((2N)^{2}-1)}_{\text{second-class}}=2{\times}2N^{2}\,.$
(4.64)

Since this counting is in phase space, the number of degrees of freedom in
configuration space is $2N^{2}$ which agrees with the dimension of the orbit
momentum: $(2N)^{2}-1-(2N^{2}-1)=2N^{2}$ studied in section 4.3.

Let us also compare the Hamiltonian action (4.26) that contains the matrix
Lagrange multiplier to the geometric action (4.21).

The equations of motion following from the Hamiltonian action (4.26) written
in matrix form are

$\displaystyle\dot{\mathbb{P}}$ $\displaystyle=0\,,$ (4.65)
$\displaystyle\dot{\mathbb{X}}+\mathbb{P}\mathbb{L}+\mathbb{L}\mathbb{P}$
$\displaystyle=0\,,$ (4.66) $\displaystyle\mathbb{P}^{2}+m^{2}\mathbb{I}$
$\displaystyle=0\,.$ (4.67)

The derivative here is with respect to the Lagrangian time variable. The
second equation (4.66) is uniquely solvable for $\mathbb{P}$ for generic
$\mathbb{L}$; a closed form expression is not available but a formal series
expansion can be obtained. Instead of solving for $\mathbb{P}$ we verify that
the equations of motion agree with those of the geometric action (4.21). We
deduce from (4.66)

$\displaystyle\dot{\mathbb{X}}+\frac{1}{m^{2}}\mathbb{P}\dot{\mathbb{X}}\mathbb{P}=0\,,$
(4.68)

and since $\mathbb{P}$ is constant, the same equation also holds for
$\ddot{\mathbb{X}}$

$\displaystyle\ddot{\mathbb{X}}+\frac{1}{m^{2}}\mathbb{P}\ddot{\mathbb{X}}\mathbb{P}=0\,.$
(4.69)

This equation can be understood as the projection of $\ddot{\mathbb{X}}$ to
what in the uncoloured case would be the transversal directions to
$\mathbb{P}$. If we can also argue that the ‘longitudinal’ components

$\displaystyle\ddot{\mathbb{X}}-\frac{1}{m^{2}}\mathbb{P}\ddot{\mathbb{X}}\mathbb{P}$
(4.70)

vanish, then the Hamiltonian equations would imply $\ddot{\mathbb{X}}=0$ in
agreement with the equations of the geometric Lagrangian (4.21) (in proper
time gauge). In the uncoloured case the longitudinal combinations are gauge-
variant and can be set to zero by a choice of gauge and for this reason we now
study the gauge symmetries of the system (4.26).

The gauge transformations entailed by the matrix constraint (4.67) are for
matrix parameter $\text{i}\,\mathbb{E}\in{\mathfrak{u}}(N,N)$

$\displaystyle\delta_{\mathbb{E}}\mathbb{P}=0\,,\quad\delta_{\mathbb{E}}\mathbb{X}=\left\\{\mathbb{E},\mathbb{P}\right\\}\,,\quad\delta_{\mathbb{E}}\mathbb{L}=-\dot{\mathbb{E}}\,.$
(4.71)

These gauge transformations are reducible since $\mathbb{X}$ is left invariant
by any gauge parameter of the form
$\mathbb{E}=\big{[}\mathbb{F},\mathbb{P}\big{]}$ for an arbitrary matrix
$\mathbb{F}$ when (4.67) is used. If this $\mathbb{F}$ is itself of the form
$\big{\\{}\mathbb{G},\mathbb{P}\big{\\}}$, it leads to a vanishing
$\mathbb{E}$ when (4.67) is used. This reflects the fact that the constraints
form a reducible system of gauge transformations. Intuitively, the total
effective number of gauge parameters is given by
$(2N)^{2}\left[\sum_{n=0}^{\infty}(-1)^{n}\right]-1=2N^{2}-1$ by geometric sum
regularisation and where the $-1$ is due to the identity component that has to
be treated differently. The reducibility of the gauge transformations is tied
to the reducibility of the constraint (4.67)

The longitudinal component (4.70) transforms as follows under gauge
transformations

$\displaystyle\delta_{\mathbb{E}}\left[\ddot{\mathbb{X}}-\frac{1}{m^{2}}\mathbb{P}\ddot{\mathbb{X}}\mathbb{P}\right]=2\left\\{\ddot{\mathbb{E}},\mathbb{P}\right\\}$
(4.72)

and is clearly gauge-variant. Moreover, it is in the kernel of the
‘transversal’ projection (4.68). We therefore expect to be able to gauge the
longitudinal part to zero fully. For this to be correct, we need that any
element in the kernel of the transversal projection is of the from
$\\{\ddot{\mathbb{E}},\mathbb{P}\\}$ for some $\ddot{\mathbb{E}}$ with
$\mathbb{P}$ satisfying (4.67). This can be checked to be true.

We therefore conclude that the geometric action (4.21) and the Hamiltonian
action (4.26) with matrix constraint both imply the equation of motion

$\displaystyle\ddot{\mathbb{X}}=0$ (4.73)

upon fully fixing the gauge invariances. In the geometric case this only
requires the choice of proper time gauge
$\frac{1}{2N}\mathrm{Tr}\big{[}\dot{\mathbb{X}}^{2}\big{]}=-1$ while for the
Hamiltonian action this requires fixing all longitudinal components (4.70) to
zero which is a larger set of gauge constraints. The two systems are therefore
not equivalent and this agrees also with the analysis of which orbits they
support that we carried out in section 4.3.

###  4.6 Examples for small $N$

We shall now analyse the Lagrangian (4.49) for small values of $N$, beginning
with the uncoloured $N=1$ case as a quick consistency check. It is useful to
start with the component expression (4.4).

#### The uncoloured $N=1$ case

For the case $N=1$, there are no ${\mathfrak{su}}(N)$ generators $T^{I}$ and
the symmetry group is just the usual Poincaré group in $2+1$ dimensions.
Without $T^{I}$ generators there are no coordinates $x^{a}_{I}$, $y^{I}$ or
associated momentum or Lagrange multiplier components. The Lagrangian (4.4)
reduces simply to

$\displaystyle L=\dot{x}^{a}p_{a}+e(p^{a}p_{a}+m^{2})\,.$ (4.74)

This agrees with the well-known free massive particle (4.16). One can check
similarly that the ‘orbit Lagrangian’ (4.58) reduces to (4.20).

The only Lagrange multiplier is $e$ and the corresponding scalar constraint
$p^{a}p_{a}+m^{2}=0$ has two branches of solution up to Lorentz rotation of
the momentum, corresponding to positive and negative energy particles.

#### The case $N=2$

For the case $N=2$ we can rewrite (4.4) by using
${\mathfrak{su}}(2)\cong{\mathfrak{so}}(3)$ to replace ${\mathfrak{su}}(2)$
adjoint indices $I$ by ${\mathfrak{so}}(3)$ vectors, e.g.
$p_{a}^{I}\to\vec{p}_{a}$. Then the trace $\delta_{IJ}$ becomes the scalar
product of vectors and $f^{IJK}$ becomes the cross-product. Note that
$d^{IJK}=0$ for $N=2$. We will continue to write out the
${\mathfrak{so}}(1,2)$ Lorentz indices that are raised and lowered with the
Minkowski metric.

Then (4.4) becomes

$\displaystyle L$
$\displaystyle=\dot{x}^{a}p_{a}+\frac{1}{2}\dot{\vec{x}}^{\,a}\cdot\vec{p}_{a}-\frac{1}{2}\dot{\vec{y}}\cdot\vec{\pi}+e\left(p^{a}p_{a}+\frac{1}{2}\vec{p}^{\,a}\cdot\vec{p}_{a}-\frac{1}{2}\vec{\pi}\cdot\vec{\pi}+m^{2}\right)+\vec{\lambda}\cdot\vec{p}^{\,a}p_{a}$
$\displaystyle\quad+f_{a}\,\vec{p}^{\,a}\cdot\vec{\pi}+\frac{1}{2}\vec{k}_{a}\cdot\left(2\,\vec{\pi}\,p^{a}+\frac{1}{2}\varepsilon^{abc}\vec{p}_{b}\times\vec{p}_{c}\right)\,.$
(4.75)

The constraints implied by this Lagrangian from varying the Lagrange
multipliers $e$, $\vec{\lambda}$, $f_{a}$ and $\vec{k}_{a}$ are

(1 constraint) $\displaystyle
p^{a}p_{a}+\frac{1}{2}\vec{p}^{\,a}\cdot\vec{p}_{a}-\frac{1}{2}\vec{\pi}\cdot\vec{\pi}+m^{2}$
$\displaystyle=0\,,$ (4.76a) (3 constraints) $\displaystyle\vec{p}^{\,a}p_{a}$
$\displaystyle=0\,,$ (4.76b) (3 constraints)
$\displaystyle\vec{p}^{\,a}\cdot\vec{\pi}$ $\displaystyle=0\,,$ (4.76c)
(3$\times$3 constraints) $\displaystyle
4\vec{\pi}\,p^{a}+\varepsilon^{abc}\vec{p}_{b}\times\vec{p}_{c}$
$\displaystyle=0\,.$ (4.76d)

We have also written out the number of equations. The total of $16$
constraints is reducible. There are combinations that vanish identically
without the use of the equations of motion and the simplest example is given
by $\vec{\pi}\cdot(\vec{p}^{\,a}p_{a})-p_{a}(\vec{\pi}\,\vec{p}^{\,a})$ which
is a single constraint. According to the general discussion in section 4.3,
the solution space to the constraints is eight-dimensional for $N=2$ and we
shall analyse this by solving them explicitly.

We first consider the case when the particle has non-vanishing ordinary
energy, so that $p^{0}\neq 0$. Then we can use all three equations of (4.76b)
to solve for $\vec{p}^{\,0}$ by

$\displaystyle\vec{p}^{\,0}=\frac{1}{p^{0}}\left(\vec{p}^{\,1}p_{1}+\vec{p}^{\,2}p_{2}\right)\,.$
(4.77)

Similarly, the three $a=0$ components of (4.76d) can be used to solve for
$\vec{\pi}$ as

$\displaystyle\vec{\pi}=-\frac{1}{2p^{0}}\,\vec{p}_{1}\times\vec{p}_{2}$
(4.78)

Substituting these two solutions into (4.76c) shows that they are satisfied
automatically, therefore the constraints are 1-stage reducible. Moreover,
substituting them into the $a=1$ and $a=2$ components of (4.76d) imposes no
further constraints either, reflecting again the reducibility. The only
remaining constraint is (4.76a) which yields the scalar constraint

$\displaystyle
p^{a}p_{a}+m^{2}+\frac{1}{2p^{0}p_{0}}\left(\vec{p}_{1}p_{1}+\vec{p}_{2}p_{2}\right)^{2}+\frac{1}{2}\vec{p}_{1}\cdot\vec{p}_{1}+\frac{1}{2}\vec{p}_{2}\cdot\vec{p}_{2}+\frac{1}{8p^{0}p_{0}}\left(\vec{p}_{1}\times\vec{p}_{2}\right)\cdot\left(\vec{p}_{1}\times\vec{p}_{2}\right)=0\,.$
(4.79)

This is one equation for the nine variables $p^{a}$, $\vec{p}_{1}$ and
$\vec{p}_{2}$, leading in general to an eight-dimensional solution space. This
can be checked explicitly for generic values by reading the above equation as
a quadratic equation for $p^{0}p_{0}$ and studying its discriminant. Among the
four solutions of
$p_{0}=\pm\omega_{1}(p_{1},p_{2},\vec{p}_{1},\vec{p}_{2}),\pm\omega_{2}(p_{1},p_{2},\vec{p}_{1},\vec{p}_{2})$
with

$\omega_{1}(p_{1},p_{2},\vec{p}_{1},\vec{p}_{2})>p_{1}^{2}+p_{2}^{2}+\frac{1}{2}\,\vec{p}_{1}^{\,2}+\frac{1}{2}\,\vec{p}_{2}^{\,2}+m^{2}>\omega_{2}(p_{1},p_{2},\vec{p}_{1},\vec{p}_{2})>0\,.$
(4.80)

The two surfaces $p_{0}=\pm\omega_{1}(p_{1},p_{2},\vec{p}_{1},\vec{p}_{2})$
correspond to the “momentum” shells or orbits where the energy is bounded from
either below or above. These orbits, denoted by $\mathcal{O}_{+2}$ and
$\mathcal{O}_{-2}$, can be viewed as the colour extension of the positive and
negative energy momentum shells. The other two surfaces
$p_{0}=\pm\omega_{2}(p_{1},p_{2},\vec{p}_{1},\vec{p}_{2})$ rejoin at
the“equator” with $p_{0}=0$,151515By solving the system (4.76) with $p_{0}=0$,
one can obtain a seven dimensional surface corresponding to this “equator”.
and form a single orbit, denoted by $\mathcal{O}_{0}$. The orbits
$\mathcal{O}_{+2},\mathcal{O}_{0}$ and $\mathcal{O}_{-2}$ can be obtained from
the representative momenta,

$\widehat{\mathbb{P}}_{+2}=m\,\begin{pmatrix}\text{i}\,&0&0&0\\\
0&\text{i}\,&0&0\\\ 0&0&-\text{i}\,&0\\\
0&0&0&-\text{i}\,\end{pmatrix}\,,\quad\widehat{\mathbb{P}}_{0}=m\,\begin{pmatrix}\text{i}\,&0&0&0\\\
0&-\text{i}\,&0&0\\\ 0&0&\text{i}\,&0\\\
0&0&0&-\text{i}\,\end{pmatrix}\,,\quad\widehat{\mathbb{P}}_{-2}=m\,\begin{pmatrix}-\text{i}\,&0&0&0\\\
0&-\text{i}\,&0&0\\\ 0&0&\text{i}\,&0\\\ 0&0&0&\text{i}\,\end{pmatrix}\,,$
(4.81)

and they correspond to the homogeneous spaces,

$\mathcal{O}_{\pm 2}\simeq U(2,2)/(U(2)\times
U(2))\,,\qquad\mathcal{O}_{0}\simeq U(2,2)/(U(1,1)\times U(1,1))\,.$ (4.82)

The first orbit $\widehat{\mathbb{P}}_{+2}$ is exactly the one analysed in
(4.56) for general $N$ and the representative momentum (4.30). From (4.56) one
can see that $p^{0}$ is bounded from below. The representative
$\widehat{\mathbb{P}}_{-2}$ of the third orbit is the negative of
$\widehat{\mathbb{P}}_{+2}$ and its orbit therefore has $p^{0}$ bounded from
above. The middle orbit has points with $p^{0}=0$ and is a genuine new type of
orbit for coloured Poincaré symmetry.

To recapitulate, the “momentum” space of the massive $N=2$ coloured particle
consists of three distinct orbits of dimension eight, in agreement with the
general discussion in section 4.3.

###  4.7 Massless coloured particle

We now consider the massless particle. Again, we make the simplest choice for
its representative ‘momentum’ assigning zero eigenvalues to all colour
associated ‘momenta’:

$P_{0}=P_{1}=E\,,\qquad P_{2}=0\,,\qquad P_{a}^{I}=0\,,\qquad Q^{I}=0\,,$
(4.83)

where $E$ is the energy of this reference ‘momentum’ state. The little group
is generated by $M_{0}+M_{1},N_{0}^{I}+N_{1}^{I}$ and $T^{I}$, hence, the
remaining ‘broken’ generators of the coloured Lorentz ${\mathfrak{su}}(N,N)$
are

$M_{0}-M_{1}\,,\qquad M_{2},\qquad M_{0}^{I}-M_{1}^{I}\,,\qquad M_{2}^{I}\,.$
(4.84)

Again the eigenvalues (4.83) define an element ${\cal K}$ in the coadjoint
space ${\mathfrak{su}}(N,N)^{*}$:

$\left\langle P_{a},{\cal
K}\right\rangle=E\left(\delta_{a}^{0}+\delta_{a}^{1}\right),\qquad\left\langle
P_{a}^{I},{\cal K}\right\rangle=0\,,\qquad\left\langle Q^{I},{\cal
K}\right\rangle=0\,.$ (4.85)

Through the isomorphism (4.29), ${\cal K}$ is mapped to

$H_{\cal K}=E\left(P_{0}+P_{1}\right).$ (4.86)

The massless orbit Lagrangian is given by

$L_{\text{orb}}=\kappa\,\left\langle\,\dot{\mathbb{X}}\,,\,{\rm
Ad}^{*}_{b}\,{\cal
K}\right\rangle=\frac{\kappa}{2N}\,\mathrm{Tr}\left[\dot{\mathbb{X}}\,b\,E\,(P_{0}+P_{1})\,b^{-1}\right],$
(4.87)

where $b$ is the exponentiation of a linear combination of the generators in
(4.84):

$b=\exp(\mathbb{V})\,,\qquad\mathbb{V}=\begin{pmatrix}\mathbf{V}&-\mathbf{V}+\text{i}\,\,\mathbf{W}\\\
\mathbf{V}+\text{i}\,\,\mathbf{W}&-\mathbf{V}\end{pmatrix}\,.$ (4.88)

Here, we packaged arbitrary linear combinations of the generators in (4.84)
into $N\times N$ anti-hermitian matrices $\mathbf{V}$ and $\mathbf{W}$. The
parameter $\kappa$ in (4.87) is a dimensionful scale that has no further role
at the classical level.

We first notice that

$b\,E\,(P_{0}+P_{1})\,b^{-1}=\mathbb{R}\,\mathbb{R}^{\dagger}\,\begin{pmatrix}\mathbf{I}&\mathbf{0}\\\
\mathbf{0}&-\mathbf{I}\end{pmatrix}\,,$ (4.89)

where $\mathbb{R}$ is an $N\times 2N$ rectangular matrix,

$\mathbb{R}=\sqrt{E}\,\exp(\mathbb{V})\,\begin{pmatrix}\mathbf{I}\\\
-\mathbf{I}\end{pmatrix}\,.$ (4.90)

Since $\mathbb{V}$ can be decomposed as $\mathbb{V}=\mathbf{V}\otimes
A+\text{i}\,\,\mathbf{W}\otimes B$ and where

$A=\begin{pmatrix}1&-1\\\ 1&-1\end{pmatrix},\qquad B=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},$ (4.91)

satisfy

$A^{2}=0\,,\qquad B^{2}=I\,,\qquad A\,B=-A\,,\qquad B\,A=A\,,$ (4.92)

we find

$\displaystyle\exp(\mathbb{V})$ $\displaystyle\\!=\\!$ $\displaystyle
e^{\text{i}\,\mathbf{W}\otimes
B}+\int_{0}^{1}dt\,e^{\text{i}\,t\,\mathbf{W}\otimes B}\,(\mathbf{V}\otimes
A)\,e^{\text{i}\,(1-t)\,\mathbf{W}\otimes B}$ (4.93) $\displaystyle\\!=\\!$
$\displaystyle\cos(\mathbf{W})\otimes I+\text{i}\,\sin(\mathbf{W})\otimes
B+\int_{0}^{1}dt\,e^{\text{i}\,t\,\mathbf{W}\otimes I}\,(\mathbf{V}\otimes
A)\,e^{-\text{i}\,(1-t)\,\mathbf{W}\otimes I}\,.$

From the above, the matrix $\mathbb{R}$ is obtained as

$\mathbb{R}=\begin{pmatrix}\mathbf{R}+\text{i}\,\mathbf{S}\,\mathbf{R}\\\
-\mathbf{R}+\text{i}\,\mathbf{S}\,\mathbf{R}\end{pmatrix}=\begin{pmatrix}\mathbf{I}+\text{i}\,\mathbf{S}\\\
-(\mathbf{I}-\text{i}\,\mathbf{S})\end{pmatrix}\,\mathbf{R}\,,$ (4.94)

with two $N\times N$ hermitian matrices $\mathbf{R}$ and $\mathbf{S}$,

$\mathbf{R}=\sqrt{E}\,e^{-\text{i}\,\mathbf{W}}\,,\qquad\mathbf{S}=-2\,\text{i}\,\int_{0}^{1}dt\,e^{\text{i}\,t\,\mathbf{W}}\,\mathbf{V}\,e^{\text{i}\,t\,\mathbf{W}}\,.$
(4.95)

In terms of $\mathbf{R}$ and $\mathbf{S}$, the orbit is parametrised by

$\mathbb{P}=b\,E\,(P_{0}+P_{1})\,b^{-1}=\begin{pmatrix}\mathbf{I}+\text{i}\,\mathbf{S}\\\
-(\mathbf{I}-\text{i}\,\mathbf{S})\end{pmatrix}\,\mathbf{R}^{2}\,\begin{pmatrix}\mathbf{I}-\text{i}\,\mathbf{S}&\mathbf{I}+\text{i}\,\mathbf{S}\end{pmatrix}\,.$
(4.96)

Note however that for a one-to-one correspondence between $\mathbf{R}$ and
$\mathbf{W}$, the former matrix should be invertible. Foregoing the
invertibility of $\mathbf{R}$, the parametrisation (4.96) contains other
orbits besides the one generated by $E\,(P_{0}+P_{1})$ .

Now, let us look for algebraic constraints for $\mathbb{P}$, which lead to the
parametrisation (4.96). From its explicit form of the parametrisation, one can
verify

$\mathbb{P}^{2}=0\,,$ (4.97)

which is already obvious from its definition and the fact that
$(P_{0}+P_{1})^{2}=0$.

A Lagrangian that contains all possible solutions of $\mathbb{P}^{2}=0$ is
given by

$\displaystyle
L=\frac{1}{2N}\mathrm{Tr}\left[\dot{\mathbb{X}}\,\mathbb{P}+\mathbb{L}\,\mathbb{P}^{2}\right]\,,$
(4.98)

which agrees with the Lagrangian (4.26) restricted to $m^{2}=0$. This is
completely analogous to what happens in the uncoloured case.

Similarly to the massive coloured particle, the massless Lagrangian above
contains several orbits, but this time the structure is richer: in the limit
$m\to 0$, there appear new orbits of lower dimensions on the boundary of the
$N+1$ massive orbits of dimension $2N^{2}$. More precisely, the massive orbits
develop open regions in the massless limit and the closure of these orbits
contain such sub-orbits. The uncoloured analogue of this situation is as
follows: the massive momentum orbit, which is an upper hyperboloid, becomes a
cone in the massless limit. Since this cone is open near the origin, its
closure contains the origin, which is the new (trivial) orbit.

While in the uncoloured case the new sub-orbit is only the trivial orbit, the
situation is more interesting in the coloured case. Here, the sub-orbits are
non-trivial and they may even contain sub-sub-orbits. The sub-sub-orbits, in
turn, may contain sub-sub-sub-orbits, etc, exhibiting an interesting inclusion
structure that is familiar from the study of co-adjoint nilpotent orbits of
${\mathfrak{su}}(N,N)$ that can be arranged in a Hasse (or closure) diagram
[58]. To quote the results directly, the massless coloured particle contains
$k+1$ orbits of dimension $4Nk-2k^{2}$ where $k$ varies from $0$ to $N$. The
orbits with $k=N$ are the massless limit of the massive orbits, whereas the
other orbits with $k<N$ are newly appeared sub-orbits. The orbit with $k=0$ is
the origin, corresponding to the zero-dimensional trivial orbit, see also
section 4.3 for further discussions.

###  4.8 Reductions to subspaces

The full coloured Minkowski space (4.1) has coordinates
$\mathbb{X}=(x^{a},x^{a}_{I},y_{I})$ that transform under the coloured
Poincaré algebra according to (4.4). We now consider the restriction to
subspaces of coloured Minkowski space. These will necessarily break some of
the coloured Poincaré symmetries.

The most extreme subspace to be considered is ordinary Minkowski space
obtained by setting $x^{a}_{I}=y_{I}=0$. Inspection of (4.4) shows that this
breaks the coloured Poincaré symmetry to the usual Poincaré symmetry.

A more interesting restriction is obtained by only setting $x^{a}_{I}=0$. In
this case we still have the ordinary Minkowski coordinates $x^{a}$ and the
coloured coordinate $y_{I}$ that we would like to think of as the space of an
internal degree of freedom of a particle. Setting $x_{I}^{a}=0$ breaks the
coloured Poincaré algebra down to the usual Poincaré symmetry and only
coloured transformations generated by $N^{I}$ and $Q^{I}$, where we recall
that the $N^{I}$ form an ${\mathfrak{su}}(N)$ algebra. With only these
generations, the equations (4.4) reduce to

$\displaystyle\delta x^{a}$
$\displaystyle=\varepsilon_{bc}{}^{a}\omega^{b}x^{c}+\alpha^{a}\,,$
$\displaystyle\delta y_{I}$
$\displaystyle=-f^{JK}{}_{I}\sigma_{J}y_{K}+\beta_{I}\,,$ (4.99)

so that the two coordinates decouple with $x^{a}$ transforming under
${\mathfrak{iso}}(2,1)$ and $y_{I}$ transforming under
${\mathfrak{su}}(N)\inplus_{\text{adj}}{\mathfrak{su}}(N)$.

Let us also consider what happens for the equations of motion (4.51) under the
assumption that $x^{a}_{I}=0$ along with a vanishing of the conjugate momentum
$p_{a}^{I}=0$. We keep, however, the Euler–Lagrange equation (4.51h) for
$k_{aI}$. The equations of motion simplify considerably in this reduction. We
begin with (4.51h) that becomes simply

$\displaystyle p^{a}\pi^{I}=0$ (4.100)

and there are two branches of solutions: $(i)$ with $\pi^{I}=0$ and $(ii)$
with $p^{a}=0$. Since we wish to describe a massive particle in ordinary
Minkowski space, we focus on case $(i)$ where $p^{a}\neq 0$. Then all the
remaining equations imply $\lambda^{I}=0$

$\displaystyle\dot{x}^{a}$ $\displaystyle=-2ep^{a}\,,$ (4.101a)
$\displaystyle\dot{y}^{I}$ $\displaystyle=2k_{aI}p^{a}\,,$ (4.101b)
$\displaystyle p^{a}p_{a}+m^{2}$ $\displaystyle=0\,.$ (4.101c)

We see that the first equation is the usual propagation of a particle in
Minkowski space and the last equation puts it on the standard mass-shell. The
middle equation determines the propagation of the colour coordinate $y^{I}$
and is a bit reminiscent of the Wong equation [32, 59] that was developed to
describe the coupling of a coloured particle to a Yang–Mills background.
However, the equation contains the Lagrange multiplier $k_{aI}$ that should
not be interpreted as the background field but takes the value zero. Therefore
the colour component $y^{I}$ propagates freely. This is analogous to the
coupling of an ordinary charged particle to an electro-magnetic background:
The presence of a non-trivial, constant background requires the generalisation
of the Poincaré algebra to the so-called Maxwell algebra where
$[P_{a},P_{b}]=Z_{ab}$, where the non-vanishing commutator of translations
corresponds to the electro-magnetic field [28, 29]. Non-constant backgrounds
can be incorporated by further generalising the algebra [30, 31]. We will
comment more on the possible extension of the coloured Poincaré algebra to
have non-trivial commutators of coloured translations in the conclusions.

##  5 Free coloured particle in AdS background

In this section, we study the AdS3 analog of the previous construction of the
coloured particle in three-dimensional Minkowski space-time. For that, we
begin with the coloured AdS3 isometry algebra,

${\mathfrak{su}}(N,N)\oplus{\mathfrak{su}}(N,N)\,,$ (5.1)

which has been considered in [13, 14], see also appendix A. One can view this
algebra as the colour extension of
${\mathfrak{so}}(2,2)\simeq{\mathfrak{su}}(1,1)\oplus{\mathfrak{su}}(1,1)$,
and the flat space algebra (2.16) is related to the above by a Inönü–Wigner
contraction. To identify the corresponding particle action, we follow the
method of non-linear realisation (or coadjoint action). We are putting the AdS
radius $\ell=1$ in this section. Let us first consider the uncoloured case.

###  5.1 Uncoloured particle in AdS

One can obtain an action for a particle in AdS${}_{d}\cong
SO(d-1,2)/SO(d-1,1)$ simply by parametrising an $SO(d-1,2)$ element as

$g=t\,b\,h\,,$ (5.2)

with an $SO(d-1)$ element $h$ and

$t=\exp(x^{0}\,P_{0})\,\exp(x^{i}\,P_{i})\,,\qquad b=\exp(v^{j}\,M_{0j})\,,$
(5.3)

and following the method of non-linear realisation, see for example [54, 55,
56, 57]. The elements $b$ and $h$ together form an element of the Lorentz
group $SO(d-1,1)$. It leads to the particle action with the coordinate system,

$ds^{2}=\cosh^{2}r\left(-dt^{2}+dr^{2}+\sinh
r^{2}\,d\Omega^{2}_{d-2}\right)\,,$ (5.4)

where $r=\sqrt{x^{i}x_{i}}$ and $t=x^{0}$. This metric does describe AdSd
space-time, but the Lorentz symmetry ${\mathfrak{so}}(d-1,1)$ is not manifest
(or linearly realised). Since the colour extension is based on the extension
of the 3d Lorentz ${\mathfrak{so}}(1,2)\simeq{\mathfrak{su}}(1,1)$ to
${\mathfrak{su}}(N,N)$, this choice of group element is not convenient and we
need to find another parametrisation with manifest Lorentz symmetry.

For Lorentz covariance, we can replace
$t=\exp(x^{0}\,P_{0})\,\exp(x^{i}\,P_{i})$ in (5.2) by $t=\exp(x^{a}\,P_{a})$.
Or more generally, we can consider161616Note however that the Lie group
element $t$ depends not only on the function $f$ but in general also on the
representation of $P_{a}$. Here, we take the defining representation.

$t=f(x^{a}\,P_{a})\,.$ (5.5)

In order that $t$ becomes a group element, we need to require

$f(z)\,f(-z)=1\,.$ (5.6)

Any real function $f(z)$ with (5.6) would result in good AdSd particle actions
but with different coordinate systems. For instance, in three dimensions the
choice $f(z)=e^{z}$ corresponding to $t=e^{x^{a}\,P_{a}}$ gives the
Lagrangian171717A similar construction can be found e.g. in [60] where
curvature corrections to flat space-time were studied.

$L=\dot{x}^{\mu}\,e_{\mu}^{a}\,p_{a}+e\,(p^{2}+m^{2})\,,$ (5.7)

with the dreibein,

$e_{\mu}^{a}=\frac{\sinh x}{x}\,\delta_{\mu}^{a}+\left(1-\frac{\sinh
x}{x}\right)\frac{x^{a}\,x_{\mu}}{x^{2}}\,.$ (5.8)

Here, we used the notation, $x^{2}=x^{a}\,x_{a}$ and $x=\sqrt{x^{2}}$. Another
convenient choice which we shall adopt for the colour extension is of Cayley
transform type $f(z)=\frac{1+2z}{1-2z}$, that is

$t=\frac{1+2\,x\cdot P}{1-2\,x\cdot P}\,,$ (5.9)

with $x\cdot P=x^{a}\,P_{a}$ . Using identities of Pauli matrices, one can
find

$t^{-1}\,\dot{t}=\frac{2}{(1-x^{2})^{2}}\left[\left((1+x^{2})\,\dot{x}^{a}-2\,x\cdot\dot{x}\,x^{a}\right)P_{a}-2\epsilon_{abc}\,x^{a}\,\dot{x}^{b}\,M^{c}\right],$
(5.10)

and the corresponding particle action,

$\displaystyle S$ $\displaystyle\\!=\\!$ $\displaystyle\int dt\,\langle
m\,{\cal P}^{0},g^{-1}\dot{g}\rangle=\int dt\,\langle{\rm
Adj}^{*}_{b}(m\,{\cal P}^{0}),t^{-1}\dot{t}\,\rangle$ (5.11)
$\displaystyle\\!=\\!$ $\displaystyle\int
dt\left(\sqrt{p_{i}^{2}+m^{2}}\,e^{0}_{\mu}(x)+p_{i}\,e^{i}_{\mu}(x)\right)\dot{x}^{\mu}\,,$

with the dreibein of the form,

$e^{a}_{\mu}=2\,\frac{(1+x^{2})\,\delta^{a}_{\mu}-2\,x_{\mu}\,x^{a}}{(1-x^{2})^{2}}\,.$
(5.12)

Combining with the negative energy sector corresponding to the coadjoint
element $-m\,{\cal P}^{0}$, we can rewrite the above action with a Lagrange
multiplier $e$ as

$S[x,p]=\int
dt\left(\dot{x}^{\mu}\,e^{a}_{\mu}(x)\,p_{a}+e(p^{2}+m^{2})\right),$ (5.13)

analogously to the flat space case. The dreibein (5.12) gives the metric,

$ds^{2}=4\,\frac{(1+x^{2})^{2}\,dx^{2}-4\,(x\cdot dx)^{2}}{(1-x^{2})^{4}}\,,$
(5.14)

and one can check that it describes AdS3: with the redefinition,

$y^{\mu}=\frac{2\,x^{\mu}}{1+x^{2}}\,,$ (5.15)

we can bring it to a more standard AdS metric,

$ds^{2}=\frac{dy^{2}}{(1-y^{2})^{2}}\,.$ (5.16)

The AdS boundary is located at $y^{2}=1$ or equivalently $x^{2}=1$ .

###  5.2 Coloured particle in AdS

Let us colour decorate the action (5.11). For that, we decompose an element of
$SU(N,N)\times SU(N,N)$ as

$g=t\,l\,$ (5.17)

where $l$ is an element of the diagonal $SU(N,N)$, that is the colour
extension of Lorentz subgroup. The other element $t$ belongs to the off-
diagonal part which can be interpreted as the colour extension of translation,
and we use the parametrisation analogous to (5.9)

$t=\frac{1+\mathbb{X}}{1-\mathbb{X}}\,.$ (5.18)

Here, $\mathbb{X}=x^{a}_{\widehat{I}}\,P_{a}^{\widehat{I}}+y_{I}\,Q^{I}$ is an
$2N\times 2N$ matrix with the condition (2.18). In principle, we can take any
function $f(\mathbb{X})$ with (5.6), but the expression of $\dot{h}$ becomes
complicated or implicit for a generic $f(z)$. Different choices of
$f(\mathbb{X})$ corresponds to a field redefinition of $\mathbb{X}$, so we
consider only the simple case above.

Choosing the same representative of the coadjoint orbit, $m\,{\cal
P}^{0}_{0}$, as in flat space case, we find

$S=m\int dt\,\langle{\rm Ad}^{*}_{l}(m\,{\cal
P}^{0}_{0}),t^{-1}\dot{t}\rangle\,,$ (5.19)

as the action for coloured particle in AdS3 . Using the isomorphism between
${\mathfrak{su}}(N,N)$ and ${\mathfrak{su}}(N,N)^{*}$, we can rewrite the
above as

$S=\frac{1}{2N}\int
dt\,\mathrm{Tr}\left[l\,\widehat{\mathbb{P}}\,l^{-1}\,t^{-1}\dot{t}\right],$
(5.20)

where each element is expressed in the fundamental representation of $SU(N,N)$
and $\widehat{\mathbb{P}}$ is as given in (4.52). The $2N\times 2N$ matrix

$\mathbb{P}=l\,\widehat{\mathbb{P}}\,l^{-1}\,,$ (5.21)

satisfies the condition $\mathbb{P}^{2}+m^{2}\,\mathbb{I}=0$, but the latter
contains other orbits besides the one determined by $m\,{\cal P}^{0}_{0}$. The
issue of algebraically describing the orbit at the cost of including other
ones is the same as in the flat space case. In the end, we find the AdS3
analog of the coloured action (4.26) as

$S=\frac{1}{N}\int
dt\,\mathrm{Tr}\left[\mathbb{P}\left(\frac{1}{1+\mathbb{X}}\,\dot{\mathbb{X}}\,\frac{1}{1-\mathbb{X}}\right)-\mathbb{L}\left(\mathbb{P}^{2}+m^{2}\,\mathbb{I}\right)\right],$
(5.22)

where $\mathbb{X},\mathbb{P}\in{\mathfrak{su}}(N,N)$ and
$\mathbb{L}\in{\mathfrak{u}}(N,N)$. By a simple field redefinition, the action
above can be written as

$S=\frac{1}{N}\int
dt\,\mathrm{Tr}\left[\mathbb{P}\,\dot{\mathbb{X}}-\mathbb{L}\,\Big{(}\big{[}(1-\mathbb{X})\mathbb{P}(1+\mathbb{X})\big{]}^{2}+m^{2}\,\mathbb{I}\Big{)}\right],$
(5.23)

employing the standard Liouville form.

##  6 Conclusions

In this paper, we have defined a coloured extension of the three-dimensional
Poincaré algebra. This algebra, after removing its center, is isomorphic to
${\mathfrak{su}}(N,N)\inplus_{\mathrm{adj}}{\mathfrak{su}}(N,N)$, where the
first ${\mathfrak{su}}(N,N)$ is the colouring of the Lorentz symmetry that
acts on the coloured translations in the adjoint representation.

We have defined Poincaré gravity based on this algebra using a Chern–Simons
formulation where an interesting interplay between the colour degrees of
freedom and the ordinary gravity degrees of freedom is implied by the
structure of the algebra (and choice of bilinear form).181818For the coloured
AdS3 case, see [13, 14].

The coloured Poincaré algebra can also be used to define coloured massive and
massless free particles. There are different types of coloured particles
depending on the choice of co-adjoint orbit of the coloured Poincaré algebra,
generalising positive and negative energy particles. There are various
different Lagrangians depending on how many orbits one wishes to describe at
the same time. We have also considered the extension to particles in coloured
AdS space based on the algebra originally constructed in [13].

There are several interesting avenues to explore in order to further
investigate the structures introduced here. One question, raised in the
introduction, is that of relating our analysis to the classical description of
coloured particles in terms of the Wong equation [32, 59] and we have taken
some first steps in this direction in section 4.8, where we considered
subspaces of coloured Minkowski space. Even more interesting would be to
consider a further extension of the coloured Poincaré algebra to a coloured
Maxwell algebra. The uncoloured Maxwell algebra allows for the description of
a charged particle in an electro-magnetic background [28, 29, 30, 31] and we
expect the coloured Maxwell algebra to be the right framework for (constant)
Yang–Mills backgrounds.

The coloured Maxwell algebra has further generators
$Z^{{\widehat{K}}},Z_{a}^{{\widehat{K}}}$ on top of those of the Poincaré
algebra. The commutation relations of the coloured Maxwell algebra can be
obtained via semi-group expansion (see footnote 6) of the algebra
${\mathfrak{u}}(N,N)$ with semi-group
$S_{E}^{(2)}=\\{\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3}\\}$

$\displaystyle\left[M_{a}^{{\widehat{I}}},M_{b}^{{\widehat{J}}}\right]$
$\displaystyle=\frac{1}{2}\,\varepsilon_{ab}{}^{c}\,\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,M_{c}^{{\widehat{K}}}-\frac{1}{4}\,\eta_{ab}\,\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,N^{{\widehat{K}}}\,,$
(6.1a) $\displaystyle\left[M_{a}^{{\widehat{I}}},N^{{\widehat{J}}}\right]$
$\displaystyle=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,M_{a}^{{\widehat{K}}}\,,\hskip
28.45274pt\left[N^{{\widehat{I}}},N^{{\widehat{J}}}\right]=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,N^{{\widehat{K}}}\,,$
(6.1b) $\displaystyle\left[M_{a}^{{\widehat{I}}},P_{b}^{{\widehat{J}}}\right]$
$\displaystyle=\frac{1}{2}\,\varepsilon_{ab}{}^{c}\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,P_{c}^{{\widehat{K}}}-\frac{1}{4}\,\eta_{ab}\,\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,Q^{{\widehat{K}}}\,,$
(6.1c) $\displaystyle\left[M_{a}^{{\widehat{I}}},Q^{{\widehat{J}}}\right]$
$\displaystyle=\left[N^{{\widehat{I}}},P_{a}^{{\widehat{J}}}\right]=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,P_{a}^{{\widehat{K}}}\,,$
(6.1d) $\displaystyle\left[N^{{\widehat{I}}},Q^{{\widehat{J}}}\right]$
$\displaystyle=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,Q^{{\widehat{K}}}\,,$
(6.1e) $\displaystyle\left[P_{a}^{{\widehat{I}}},P_{b}^{{\widehat{J}}}\right]$
$\displaystyle=\left[M_{a}^{{\widehat{I}}},Z_{b}^{{\widehat{J}}}\right]=\frac{1}{2}\,\varepsilon_{ab}{}^{c}\,\widehat{d}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,Z_{c}^{{\widehat{K}}}-\frac{1}{4}\,\eta_{ab}\,\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,Z^{{\widehat{K}}}\,,$
(6.1f) $\displaystyle\left[P_{a}^{{\widehat{I}}},Q^{{\widehat{J}}}\right]$
$\displaystyle=\left[M_{a}^{{\widehat{I}}},Z^{{\widehat{J}}}\right]=\left[N^{{\widehat{I}}},Z_{a}^{{\widehat{J}}}\right]=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,Z_{a}^{{\widehat{K}}}\,,$
(6.1g) $\displaystyle\left[Q^{{\widehat{I}}},Q^{{\widehat{J}}}\right]$
$\displaystyle=\left[N^{{\widehat{I}}},Z^{{\widehat{J}}}\right]=\widehat{f}\,^{{\widehat{I}}{\widehat{J}}}{}_{{\widehat{K}}}\,Z^{{\widehat{K}}}\,.$
(6.1h)

All generators here are anti-hermitian and the generators of the semi-group
$\lambda_{i}$ are hermitian. This algebra can also be embedded in a free Lie
algebra construction of the type considered in [31]. The algebra (6.1) is a
quotient of the semi-direct sum of the ${\mathfrak{u}}(N,N)$ with the free Lie
algebra generated by the translations
$(P_{a}^{{\widehat{I}}},Q^{{\widehat{I}}})$ that transform in the adjoint of
${\mathfrak{u}}(N,N)$. One could similarly consider other quotients leading to
generalisations of the $\mathcal{B}_{\infty}$ algebra studied in [61, 62] or
to half a Kac–Moody algebra [62].

Besides describing particles one may wonder whether there are coloured
extensions of the actions of extended objects such as strings or branes (see,
e.g., [57]). In the non-relativistic limit one might also have to change the
algebra, see for example [63].

Our analysis was restricted to classical particle dynamics and it would be
interesting to extend the models to (free or interacting) field theories. A
first question is which of our particle actions to quantise. As the different
actions in section 4 comprise different orbits, this will correspond to
including different irreducible representations at the quantum
level.191919When an action contains several orbits the classical phase space
is in general a Poisson rather than a symplectic manifold and it would be
interesting to apply Kontsevich’s deformation quantisation methods [64] in
this context. The constraints (4.23) and (4.27) are quadratic in the canonical
variables and will lead to quadratic wave equations on the space of
wavefunctions $\Psi(\mathbb{P})$. However, as is evident from the detailed
analysis of the case with $SU(2)$ colour in section 4.6, the wave equation on
the reduced phase space can be of higher order. After solving most of the
matrix-valued constraints, we arrived at the final constraint (4.79) arising
in the case $N=2$ for the massive particle. This scalar equation can be read
as a quartic condition on the momenta and therefore should translate into a
fourth order Klein–Gordon equation in canonical quantisation of this mass-
shell constraint. We note the general constraint (4.27) is matrix-valued and
therefore would correspond to a matrix-valued Klein–Gordon operator whose
direct interpretation is as an orbit condition. This situation is similar to
what happens in $Sp(2N)$-invariant (higher-spin) field theory [65] (see [66]
for a review), where the matrix-valued field-equation can be interpreted as
the condition of the minimal nilpotent orbit of $Sp(2N)$ (see [67] for
discussions about the minimal nilpotent orbits of classical Lie algebras in
the context of higher spin field theory).

If the coloured particle is properly quantised, it will correspond to a
representation of the group $\mathrm{cPoin}_{3}(N)$. This representation can
be constructed analogously to the induced scalar representation of the
ordinary Poincaré algebra: the representation space is the space of square
integrable functions on the orbit with a proper $SU(N,N)$ invariant measure on
it. The invariant measure can be obtained by solving the matrix-valued
“momentum-shell constraint” $\mathbb{P}^{2}+m^{2}\,\mathbb{I}=0$, and hence
the hermitian product can be defined as

$\langle\psi_{1},\psi_{2}\rangle=\int
d\mathbb{P}\,\delta(\mathbb{P}^{2}+m^{2}\,\mathbb{I})\,\psi_{1}(\mathbb{P})^{*}\,\psi_{2}(\mathbb{P})\,,$
(6.2)

where $d\mathbb{P}$ is the $((2N)^{2}-1)$-dimensional measure on
${\mathfrak{su}}(N,N)$ , and $\delta(\mathbb{P}^{2}+m^{2}\,\mathbb{I})$ is the
$(2N^{2}-1)$-dimensional delta distribution supported on the momentum orbits
that we have described above. The action of an element of the coloured
Poincaré group, $(\mathbb{\Lambda},\mathbb{A})\in SU(N,N)\ltimes SU(N,N)$ , is
given by

$(U(\mathbb{\Lambda},\mathbb{A})\,\psi)(\mathbb{P})=e^{\text{i}\,\,\mathrm{tr}(\mathbb{A}\,\mathbb{\Lambda}^{-1}\,\mathbb{P}\,\mathbb{\Lambda})}\,\psi(\mathbb{\Lambda}^{-1}\,\mathbb{P}\,\mathbb{\Lambda})\,,$
(6.3)

and the unitarity follows from the invariance of the measure and the delta
distribution,

$d(\mathbb{\Lambda}^{-1}\,\mathbb{P}\,\mathbb{\Lambda})=d\mathbb{P}\,,\qquad\delta(\mathbb{\Lambda}^{-1}(\mathbb{P}^{2}+m^{2}\,\mathbb{I})\,\mathbb{\Lambda})=\delta(\mathbb{P}^{2}+m^{2}\,\mathbb{I})\,.$
(6.4)

A similar construction can be done in the AdS3 case.202020More specifically,
the colour symmetry algebra for AdS3 is
${\mathfrak{su}}(N,N)\oplus{\mathfrak{su}}(N,N)$, and the unitary
representation would be of the form $\pi\otimes\pi$ where $\pi$ is a unitary
irreducible representation of ${\mathfrak{su}}(N,N)$ (we take the same
representation for both sides of ${\mathfrak{su}}(N,N)$ because we are
describing scalar particles). To identify $\pi$, we can rely on the GK
dimension, which is the same in both flat and AdS3 space. Since GK dimension
is additive under tensor product, the GK dimension of $\pi$ should be $N^{2}$,
i.e., half of the dimension $2N^{2}$ of the momentum orbit. In the massless
case, we have seen that the constraint $\mathbb{P}^{2}=0$ contains momentum
orbits of dimensions $4Nk-2k^{2}$ with $k=1,\ldots,N$ (except for the trivial
orbit $k=0$). Therefore, the representations $\pi_{k}$ corresponding to
massless coloured particle in AdS3 should have GK dimension $2Nk-k^{2}$ .
Interestingly, such representations can be realised by using the reductive
dual pair $(U(N,N),U(k))$, where we take the trivial representation for the
$U(k)$ . For a review of dual pairs see [68]. Disregarding the diagonal $U(1)$
of $U(N,N)$, the corresponding $SU(N,N)$ representation is the generalised
Verma module induced from the one dimensional representation of the maximal
compact subgroup $U(1)\times SU(N)\times SU(N)\subset SU(N,N)$ with the
${\mathfrak{u}}(1)$ eigenvalue $k\,N$ . Turning back to the massive case, the
representation is again given by the dual pair $(U(N,N),U(N))$ where we take
other one-dimensional representations with eigenvalue $n$ for $U(1)\subset
U(N)$ while keeping the trivial representation for $SU(N)\subset U(N)$ . Then,
the corresponding $SU(N,N)$ representation is the generalised Verma module
induced from the one dimensional representation of $U(1)\times SU(N)\times
SU(N)\subset SU(N,N)$ with the ${\mathfrak{u}}(1)$ eigenvalue $n+N^{2}$ . In
the flat limit where the AdS radius $\ell$ tends to $\infty$, we take the
limit $n\to\infty$ while keeping $n/\ell$ fixed as a constant proportional to
the mass $m$.

We also note that we are here dealing with an extension of a space-time
symmetry and may wonder how this is compatible with the Coleman–Mandula
theorem. Although we have not explicitly constructed a field theory with
coloured Poincaré invariance, we do not expect any contradiction to the
theorem since the space-time has been vastly extended from Minkowski space-
time to coloured Minkowski space. We consider this extension to be a bit
similar in spirit to what happens in supersymmetry (or higher spin theory).

Finally, our construction was strongly based on three-dimensional space-time
and one big challenge is to generalise the construction to arbitrary space-
time dimensions. As discussed here and in [13, 14], non-abelian colour
decorations require associative extensions of the algebra of isometries to
start with. Such extensions usually involve generators in non-trivial
representations of the isometry algebra (see, e.g., [69]). Exceptions, i.e.
extensions of isometry algebras involving only generators in trivial
representations (that is, corresponding to symmetries of vector gauge fields),
are possible in two dimensions (as discussed in section 2) and three
dimensions (as we studied in this paper) as well as in five dimensions: the
AdS5 isometry algebra $\mathfrak{so}(2,4)\simeq\mathfrak{su}(2,2)$ (which is
the same as conformal algebra in four dimensions) can be extended by a single
generator to $\mathfrak{u}(2,2)$ allowing for colour generalisation to
$\mathfrak{u}(2M,2M)\cong{\mathfrak{u}}(2,2)\otimes{\mathfrak{u}}(M)$.
Interpreting the latter as the coloured AdS5 isometry algebra, the momentum
orbits that we have considered in this paper can be viewed as the phase spaces
of coloured particles in AdS5. It is intriguing to realise that the $N=2$
coloured AdS3 particle, having the phase space ${\cal O}_{\pm 2}\times{\cal
O}_{\pm 2}$, can be interpreted as a particle on two copies of AdS5 since the
phase of the latter is simply one factor ${\cal O}_{\pm 2}$. Here, ${\cal
O}_{\pm 2}$ refer to the (momentum) orbits discussed around (4.81).

We also note that our analysis was restricted to scalar particles and it would
be interesting to consider particles with spin as well. Spin corresponds to a
non-trivial representation of the stability group which is here also enlarged
compared to the uncoloured case. For instance, in the case of the
representative momentum (4.52), the stability algebra was
${\mathfrak{u}}(1)\oplus{\mathfrak{su}}(N)\oplus{\mathfrak{su}}(N)$ and taking
a non-trivial representation of ${\mathfrak{u}}(1)$ would be analogous to
ordinary spin but one could also envisage more colourful versions of spin by
taking non-trivial representations of the ${\mathfrak{su}}(N)$ parts.
Extensions to superalgebras could also be interesting to explore.

### Acknowledgements

We would like to thank Thomas Basile for useful discussions. JG acknowledges
the hospitality at Max-Planck-Institut für Gravitationsphysik (Albert-
Einstein-Institut), Potsdam where this work was started. EJ and KM are
grateful to the Erwin Schrödinger International Institute for Mathematics and
Physics for the hospitality during the program on “Higher Spins and
Holography”. The work of JG has been supported in part by MINECO
FPA2016-76005-C2-1-P and PID2019-105614GB-C21 and from the State Agency for
Research of the Spanish Ministry of Science and Innovation through the Unit of
Excellence Maria de Maeztu 2020-203 award to the Institute of Cosmos Sciences
(CEX2019-000918-M). The work of EJ was supported by National Research
Foundation (Korea) through the grant NRF-2019R1F1A1044065. KM is supported by
the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant number 844265. Part of this work was carried out
while KM was at Max-Planck Institut für Gravitationsphysik and Scuola Normale
Superiore di Pisa.

##  Appendix A Derivation from embedding in AdS algebra

The uplift of the Poincaré algebra to an associative algebra proceeds in two
steps. First we embed the $(2+1)$-dimensional Poincaré algebra in the
corresponding AdS algebra
${\mathfrak{so}}(2,2)\cong{\mathfrak{sl}}(2,\mathbb{R})\oplus{\mathfrak{sl}}(2,\mathbb{R})$
that we then extend to the associative algebra
${\mathfrak{a}}={\mathfrak{u}}(1,1)\oplus{\mathfrak{u}}(1,1)$. The coloured
Poincaré algebra in $D=3$ is then obtained in the contracting limit where the
AdS radius is sent to infinity.

To see this in more detail consider first the AdS algebra