Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France
Centre de Physique Théorique

This is a collection of notes on the properties of left-invariant metrics on
the eight-dimensional compact Lie group ${\rm SU}(3)$. Among other topics we
investigate the existence of invariant pseudo-Riemannian Einstein metrics on
this manifold. We recover the known examples (Killing metric and Jensen
metric) in the Riemannian case (signature $(8,0)$), as well as a Gibbons et al
example of signature $(6,2)$, and we describe a new example, which is
Lorentzian (i.e., of signature $(7,1)$. In the latter case the associated
metric is left-invariant, with isometry group ${\rm
SU}(3)\times{\mathrm{U}}(1)$, and has positive Einstein constant. It seems to
be the first example of a Lorentzian homogeneous Einstein metric on this
compact manifold.

These notes are arranged into a paper that deals with various other subjects
unrelated with the quest for Einstein metrics but that may be of independent
interest: among other topics we describe the various groups that may arise as
isometry groups of left-invariant metrics on ${\rm SU}(3)$, provide
parametrizations for these metrics, give several explicit results about the
curvatures of the corresponding Levi-Civita connections, discuss modified
Casimir operators (quadratic, but also cubic) and Laplace-Beltrami operators.
In particular we discuss the spectrum of the Laplacian for metrics that are
invariant under ${\rm SU}(3)\times{\mathrm{U}}(2)$, a subject that may be of
interest in particle physics.

##  1 Introduction

This paper is an excursion in the land of left-invariant geometry, more
precisely in the land of left-invariant pseudo-Riemannian metrics on the Lie
group ${\rm SU}(3)$. Its main purpose is to illustrate several known concepts
and methods of Riemannian geometry in this particular case.

The only mathematical result which is probably new is the existence and
description of an homogeneous Einstein metric (actually a family), with
Lorentz signature, i.e., with signature $(7,1)$. Of course, any quadratic form
of signature $(7,1)$ in $R^{8}$ gives rise, by group translations, to a
Lorentzian homogeneous metric on the group ${\rm SU}(3)$, such metrics,
usually, are not Einstein metrics. The example that we present in sect. 3
seems to be the first example of an homogeneous Einstein Lorentzian metric on
this $8$-dimensional compact manifold. Its isometry group is isomorphic with
${\rm SU}(3)\times{\mathrm{U}}(1)$.

It is probably the proper place to mention that, for us, the word “metric”
means “pseudo-Riemannian metric”: it is non-degenerate but the requirement of
positive-definiteness is relaxed; the signature can be arbitrary. For the same
reason a metric for which the curvature of the associated Levi-Civita
connection obeys the Einstein condition will be called an Einstein metric (we
shall not use the terminology “pseudo-Einstein”). The symbol $G$ will denote,
most of the time, the Lie group ${\rm SU}(3)$, but several discussions can
often be generalized to any simple (or even semi-simple) compact Lie group.

Left-invariant metrics are fully characterized by their value at the origin of
the group, and therefore by a non-degenerate bilinear symmetric form on the
Lie algebra, equivalently (after having chosen an appropriate basis), by an
$8\times 8$ non-degenerate symmetric matrix. Such metrics are left-invariant
by construction, but their isometry group can be larger than ${\rm SU}(3)$: as
a rule it is isomorphic with ${\rm SU}(3)\times K$ where $K$, that we call the
right isometry group, is some Lie subgroup of ${\rm SU}(3)$ (more about it
later). We shall give111This was already discussed long ago, with physical
applications in mind, in [4]. a parametrization, in terms of $8\times 8$
matrices, of those left-invariant metrics for which the right isometry group
is $K$, for the various possible choices of $K$.

Then, for every $K$, we shall study the Einstein condition and describe the
metrics for which this condition holds. This occurs (when it occurs) for
specific values of the real parameters that enter our various parametrizations
of the corresponding bilinear forms. In some cases, i.e., for some choices of
$K$, our analysis is complete. Unfortunately, in some other cases we could not
solve the equations in full generality, and we had to assume extra relations
between the otherwise independent parameters in order to complete our study.

We shall discover, along the way, several infinite families of homogenous
Einstein metrics, but once one takes into account the action of the group of
diffeomorphisms on the space of metrics giving rise, in general, to the notion
of (pseudo) Riemannian structures, and in particular to the notion of Einstein
structures, these families reduce, up to scaling, to only four cases, three of
which were already known: the Killing metric (which is properly Riemannian and
for which $K$ is ${\rm SU}(3)$), the so-called Jensen metric [12] (which is
also properly Riemannian and for which $K$ is ${\mathrm{S}O}(3)$), a
particular Lorentz metric (for which $K$ is a member – that we call
${\mathrm{U}}(1)_{I}$ – of a specific conjugacy class of ${\mathrm{U}}(1)$
subgroups), and a metric of signature $(6,2)$, that was already discovered by
[10], for which $K$ is trivial.

This paper grew up from a collection of notes whose purpose was to illustrate
several concepts of (pseudo) Riemannian geometry, in particular left-invariant
geometry, in a case going beyond the three-sphere $S^{3}$, aka ${\rm SU}(2)$,
for which the study of left-invariant metrics has been thoroughly studied,
long ago, in many places, hence the choice of ${\rm SU}(3)$ which is the next
case in the ${\rm SU}(N)$ family. For this reason the reader will find here a
section, entitled “Miscellaneous”, with contents that have to do with left-
invariant geometry, but that is not directly related with the theme of
Einstein metrics: There we shall discuss for instance quadratic and cubic
Casimir elements (possibly modified), Laplace-Beltrami operators, sectional
curvatures, Ricci decompositions, etc. These concepts are of course standard,
and we shall add nothing fundamentally new to the study of their general
properties, but we shall give a number of explicit results that, we hope, will
entertain the reader or trigger the interest of a few students.

As for the classification of Einstein left-invariant metrics on ${\rm SU}(3)$
with given right isometry group, or of homogeneous Einstein structures, what
we can offer is unfortunately incomplete since, in several cases, we could not
solve the Einstein equation in full generality while keeping all the
parameters allowed by the choice of a given right isometry group. We hope that
some courageous readers will take up this study. It is of no surprise that the
system of equations that one needs to solve gets more and more complex, with
more and more parameters, as soon as one chooses a right isometry group that
gets smaller. For this reason, even if one can easily solve these equations by
hand when the right isometry group $K$ is large enough (for instance
${\mathrm{S}O}(3)$, ${\mathrm{U}}(2)$, or ${\rm SU}(3)$ itself), the use of a
computer system becomes almost compulsory for smaller groups; for example our
homogenous Einstein space with Lorentz signature involves parameters that are
algebraic integers of degree $15$ with large coefficients, a manual handling
of such large expressions is inefficient and prone to error. Most calculations
done here were carried out using the Mathematica software system.

The paper ends with a section devoted to possible physical applications. One
of them, in particle physics, using left-invariant metrics with right isometry
group ${\mathrm{U}}(2)$, was described long ago, see [4], and we shall add
almost nothing to this discussion, apart from resetting the problem in a
slightly more general framework. Physical applications, if any, of the
existence of pseudo-Riemannian homogenous Einstein metrics on ${\rm SU}(3)$,
in particular of the one that has Lorentz signature, remain to be found.

##### Reminders.

Every non-compact connected smooth manifold admits a Lorentz metric, and a
compact connected smooth manifold admits a Lorentz metric if and only if its
Euler characteristic is zero ([18], p. 149). A useful corollary arises when
there is a non-vanishing vector field, this implying that the Euler
characteristic is zero. In particular, any compact parallelizable manifold,
including any compact Lie group (they admit many non-vanishing vector fields
!), has Euler characteristic zero. So one a priori knows that the groups ${\rm
SU}(N)$, and in particular the group ${\rm SU}(3)$, admits Lorentz metrics. In
the case of Lie groups however one does not need such general theorems to
establish this property since, as already recalled, any non-degenerate
symmetric bilinear form with Lorentz signature on the Lie algebra gives rise
to a left-invariant Lorentz metric on the group itself, by using group
translations; these metrics are obviously homogeneous, and, in our case, have
an isometry group isomorphic with ${\rm SU}(3)\times K$, where $K$ is some
subgroup of ${\rm SU}(3)$. Usually these metrics are not Einstein metrics:
this is one motivation for the search of those which are such.

##  2 Metrics and curvatures

###  2.1 Killing form, inner product, and renormalized Killing inner product

Since the Lie group ${\rm SU}(3)$ is compact, the Killing form is a negative
definite bilinear symmetric form on the Lie algebra $\mathfrak{su}(3)$. Its
opposite, the Killing inner product, defines an Euclidean structure on
$\mathfrak{su}(3)$ and, using left translations (for instance), a Riemannian
metric on the group itself: the Killing metric.

It is useful and standard to define the renormalized Killing form by dividing
the Killing form by $2g$, where $g$ is the dual Coxeter number. We call $k$
the Killing inner product (so the Killing form is $-k$), and
$\widehat{k}=k/2g$ the renormalized Killing inner product. For ${\rm SU}(3)$,
$g=3$, therefore $\widehat{k}=k/6$. Warning: Through notational abuse we also
call $k$ and $\widehat{k}$ the corresponding bi-invariant metrics on the
manifold $G={\rm SU}(3)$.

###  2.2 Several basis

Let $(e_{a})$ be an arbitrary basis of the tangent space at the identity of
$G$, identified with ${\mathrm{Lie}}(G)$. Through notational abuse we also
call $e_{a}$ the corresponding left-fundamental222According to the present
standard terminology they are also called left-invariant although they commute
with the right-fundamental ones (in some old references left-fundamental
vector fields are called right-invariant). vector fields obtained from the
latter by letting $G$ act on itself by left multiplication. The structure
constants of the global moving frame $(e_{a})$ defined by the equality
$[e_{a},e_{b}]={x_{ab}}^{c}\,e_{c}$ are identified with the structure
constants of the basis $(e_{a})$ in the Lie algebra.

The dual (also called inverse) Killing metric, in the moving frame $(e_{a})$
has components $k^{ab}$ and therefore reads333Here and below we use the first
Einstein summation convention: an index variable that appears twice, once as a
superscript and once as a subscript, must be summed over.
$k^{-1}=k^{ab}\,e_{a}\otimes e_{b}$. The Killing metric itself reads
$k=k_{ab}\,\theta^{a}\otimes\theta^{b}$, where $(\theta^{a})$ is the moving
co-frame dual to $(e_{a})$. Replacing $k$ by $\widehat{k}$ we have similar
expressions for the renormalized Killing metric, with
$\widehat{k}_{ab}=k_{ab}/2g$ and $\widehat{k}^{ab}=2g\,k^{ab}$. Since $g=N$
for ${\rm SU}(N)$, we have $\widehat{k}=k/6$ and $\widehat{k}^{-1}=6k^{-1}$
for ${\rm SU}(3)$.

Let $(X_{a})$ be a basis of ${\mathrm{Lie}}(G)$ which is orthonormal for the
inner product $k$. The ordered set of vectors
$\widehat{X}_{a}=\sqrt{2g}\,X_{a}$, in particular
$\widehat{X}_{a}=\sqrt{6}\,X_{a}$ for $G={\rm SU}(3)$, is then an orthonormal
basis for the inner product $\widehat{k}$. We have
$k^{-1}=\delta^{ab}\,X_{a}\otimes X_{b}$ and
$\widehat{k}^{-1}=2g\,\delta^{ab}\,X_{a}\otimes X_{b}$, where $\delta^{ab}$ is
the Kronecker symbol.

Define $i\,L_{a}\in\,\mathfrak{su}(3)$ by the equality
$i\,L_{a}=2\sqrt{3}\,X_{a}=\sqrt{2}\,\widehat{X}_{a}$. It is traditional444The
hermitian matrices $\lambda_{a}$ are usually called Gell-Mann matrices in the
physics literature. to call $i\,\lambda_{a}$ the $3\times 3$ anti-hermitian
matrices that represent the $i\,L_{a}$ in the defining representation of ${\rm
SU}(3)$.

Let $E^{i}_{j}$ be single-entry $3\times 3$ matrices. One obtains the
$\lambda_{a}$ as follows :
$\lambda_{1}=E^{1}_{2}+E^{2}_{1},\,\lambda_{2}=i(E^{2}_{1}-E^{1}_{2}),\,\lambda_{3}=E^{1}_{1}-E^{2}_{2},\,\lambda_{4}=E^{3}_{1}-E^{1}_{3},\,\lambda_{5}=i(E^{3}_{1}-E^{1}_{3}),\,\lambda_{6}=E^{2}_{3}+E^{3}_{2},\,\lambda_{7}=i(E^{3}_{2}-E^{2}_{3}),\,\lambda_{8}=\tfrac{1}{\sqrt{3}}(E^{1}_{1}+E^{2}_{2}-2E^{3}_{3}).$

distinct

The Lie bracket of two Lie algebra elements can be written as a matrix
commutator in any chosen representation. It is standard to call
$-2\,{f_{ab}}^{c}$ the (real) structure constants of the basis $i\lambda_{a}$,
i.e., $[i\lambda_{a},i\lambda_{b}]=-2{f_{ab}}^{c}(i\lambda_{c})$,
equivalently, $[\lambda_{a},\lambda_{b}]=2i{f_{ab}}^{c}\,\lambda_{c}$. From
$Tr(\lambda_{a}.\lambda_{b})=2\,\delta_{ab}$ one obtains
$4i\,{f_{ab}}^{c}=Tr([\lambda_{a},\lambda_{b}]\,.\,\lambda_{c})$; using
cyclicity of the trace one finds that ${f_{ab}}^{c}$ is antisymmetric in its
last two indices $b,c$.

At the origin of ${\rm SU}(3)$, the left-invariant vector fields $X_{a}$,
identified with Lie algebra elements, are expressed as matrices
$\tfrac{i}{2\sqrt{3}}\,\lambda_{a}$ in the defining representation. The
structure constants of the basis $(X_{a})$, which is orthonormal for $k$, are
therefore equal to $\tfrac{-1}{\sqrt{3}}{f_{ab}}^{c}$. Notice that in the
adjoint representation, the generators $iL_{a}$ are represented by (real
antisymmetric) matrices $2f_{a}$ which have matrix elements ${2\,f_{ab}}^{c}$.

###  2.3 Left-invariant pseudo-Riemannian metrics and isometry groups

##### Isometry groups.

Isometry groups of left-invariant metrics on ${\rm SU}(3)$ are isomorphic with
${\rm SU}(3)\times K$ where $K$ is a subgroup of ${\rm SU}(3)$. Left-invariant
metrics on a Lie group are homogeneous since the isometry group acts
transitively on the manifold. The group $K$ can be ${\rm SU}(3)$ itself (bi-
invariant metrics) or some subgroup of the maximal subgroups of the latter,
which, up to conjugacy, are
${\mathrm{U}}(2)=S({\mathrm{U}}(2)\times{\mathrm{U}}(1))$ (locally ${\rm
SU}(2)\times{\mathrm{U}}(1))$ and ${\mathrm{S}O}(3)$, sometimes called the
“${\mathrm{S}O}(3)$ principal subgroup”. Hence, restricting oneself to closed
connected subgroups, one finds that the candidates for a (right) isometry
group $K$, up to isomorphism, are the members of the following list555The
Hasse diagram of nontrivial Lie subalgebras of ${\mathrm{Lie}}({\rm SU}(3))$,
up to equivalence (conjugacy by an inner automorphism), can be found in [9]. :
$S({\mathrm{U}}(2){\mathrm{U}}(1))\sim{\mathrm{U}}(2)$, ${\rm SU}(2)$,
${\mathrm{S}O}(3)$, ${\mathrm{U}}(1)\times{\mathrm{U}}(1)$, and
${\mathrm{U}}(1)$.

Two remarks are in order here: 1) If, for some left-invariant metric, $K$
contains ${\rm SU}(2)$, it also contains ${\mathrm{U}}(2)$ (see, below, the
paragraph called “Parametrizations”), so ${\rm SU}(2)$ should be removed from
the previous list. 2) In order to discuss left-invariant metrics up to
equivalence (a notion that will be made precise later), specifying $K$ up to
isomorphism is a priori not enough and, in general, one needs to specify $K$
up to conjugacy; however, maximal compact subgroups in a connected Lie group
are all conjugate, and maximal tori are also conjugate, so only the last
possibility of the above list, namely the case $K={\mathrm{U}}(1)$, needs to
be specified further.

Notice that, with the exception of ${\mathrm{U}}(1)$, specifying the type of
the subgroup $K$ up to isomorphism is also enough to determine the quotient
${\rm SU}(3)/K$ as a smooth manifold. Again, in the case $K={\mathrm{U}}(1)$
one has to be more specific. These quotients666Here we only think of these
quotients as homogeneous spaces defined by the pair $(G,K)$. are well known:
we have the complex projective space $CP^{2}={\rm SU}(3)/{\mathrm{U}}(2)$, the
sphere $S^{5}={\rm SU}(3)/{\rm SU}(2)$, the irreducible symmetric space ${\rm
SU}(3)/{\mathrm{S}O}(3)$ (sometimes called the Wu manifold), the flag manifold
${\rm SU}(3)/({\mathrm{U}}(1)\times{\mathrm{U}}(1)$, and the various Aloff-
Wallach spaces ${\rm SU}(3)/{\mathrm{U}}(1)$.

In order to obtain a parametrization for left-invariant metrics on ${\rm
SU}(3)$ invariant under a given isometry group ${\rm SU}(3)\times K$, the
first step is to specify $K$ itself, or rather its Lie algebra, in terms of
${\rm SU}(3)$ generators; this is conveniently done in the defining
representation, in terms of the matrices $\lambda_{a}$. The second step is to
impose the vanishing of the Lie derivative of an arbitrary left-invariant
metric (a symmetric $8\times 8$ matrix with $8(8+1)/2=36$ arbitrary real
constant parameters) in the direction of the generators of the Lie subalgebra
of the chosen isometry subgroup $K$. Equivalently, one can impose (or check)
the equality ${r}^{T}\,.\,h^{-1}\,.\,r=h^{-1}$ with $r=exp(u)$ for the
generators $u$ of the chosen subgroup $K$ in the adjoint representation of
${\rm SU}(3)$; one takes $u=2f_{a}$, for the one-parameter subgroups generated
by the vectors $i\,L_{a}$.

Our choice777Given the isomorphy types of the right isometry groups $K$, we
make here specific (but of course arbitrary) choices that define $K$ as
concrete subgroups of ${\rm SU}(3)$. of generators for $Lie(K)$, for the
various candidates, is as follows. For ${\mathrm{U}}(2)\sim
S({\mathrm{U}}(2){\mathrm{U}}(1))$ (locally isomorphic with ${\rm
SU}(2)\times{\mathrm{U}}(1)$), we choose the generators
$\\{\lambda_{1},\lambda_{2},\lambda_{3}\\}$ and $\lambda_{8}$. For
${\mathrm{S}O}(3)$ we choose the generators
$\\{\lambda_{2},\lambda_{5},\lambda_{7}\\}$888${\rm SU}(2)$ and
${\mathrm{S}O}(3)$ are of course locally isomorphic, but not isomorphic.. We
identify the Cartan subgroup ${\mathrm{U}}(1)\times{\mathrm{U}}(1)$ with a
fixed maximal torus of ${\rm SU}(3)$, namely the one generated by
$\lambda_{3}$ and $\lambda_{8}$.

Let us take $e^{i\phi}\in{\mathrm{U}}(1)$, $k,\ell\in Z$, and call
${\mathrm{U}}(1)_{k,\ell}$ the subgroup of ${\rm SU}(3)$ defined as the set of
$3\times 3$ diagonal matrices with diagonal
$(e^{ik\phi},e^{i\ell\phi},e^{-i(k+\ell)\phi})$. Any one-dimensional subgroup
of ${\rm SU}(3)$ is conjugate to such an ${\mathrm{U}}(1)_{k,\ell}$. Two
manifolds of the type ${\rm SU}(3)/{{\mathrm{U}}(1)}_{k,\ell}$ (Aloff-Wallach
spaces) are diffeomorphic, and therefore homeomorphic, if the corresponding
${\mathrm{U}}(1)$ subgroups are conjugated in ${\rm SU}(3)$. However Aloff-
Wallach spaces are not necessarily homeomorphic, and even when they are, they
may sometimes be non diffeomorphic. This subtle problem is investigated in
[14]. Consider $S_{3}$ acting on the triple $(k,l,-k-\ell)$, and identify this
finite group with the Weyl group of ${\rm SU}(3)$. Take $\sigma\in S_{3}$. One
can show [19] that the action of the latter on the Cartan torus changes
${\mathrm{U}}(1)_{k,\ell}$ to ${\mathrm{U}}(1)_{\sigma(k),\sigma(\ell)}$. It
is therefore enough to assume that $k\geq\ell\geq 0$, and that $k$ and $\ell$
are co-prime (multiplying $(k,\ell)$ by an integer does not change the
subgroup). One recovers the special cases ${\mathrm{U}}(1)_{Y}$ for
$(k,\ell)=(1,1)$, and ${\mathrm{U}}(1)_{I}$ for $(k,\ell)=(1,-1)\sim(1,0)$.
Another labelling possibility (that hides the roles played by $k$ and $\ell$)
is to introduce a single index $\upsilon$: up to an appropriate scaling of the
generator $\tfrac{k-\ell}{2}\lambda_{3}+\tfrac{k+\ell}{2}\sqrt{3}\lambda_{8}$,
the same one-dimensional subgroup ${\mathrm{U}}(1)_{k,\ell}$ (that one can may
call ${\mathrm{U}}(1)_{\upsilon}$) is generated by
$\upsilon\,\lambda_{3}+\sqrt{3}\,\lambda_{8}$.

##### Notations.

It is traditional in physics to introduce the operators
$I=\tfrac{1}{2}\,L_{3}$ (isospin), $Y=\tfrac{1}{\sqrt{3}}\,L_{8}$
(hypercharge) and $Q=I+Y/2$, (electric charge). We shall use these notations.
In the fundamental representation, where one replaces $L_{a}$ by
$\lambda_{a}$, these operators $I,Y,Q$ are therefore respectively represented
by the diagonal matrices $\text{diag}(1/2,-1/2,0)$,
$\text{diag}(1/3,1/3,-2/3)$ and $\text{diag}(2/3,-1/3,-1/3)$. One also calls
${\mathrm{U}}(1)_{I}$, ${\mathrm{U}}(1)_{Y}$, and ${\mathrm{U}}(1)_{Q}$, the
subgroups respectively generated by $\lambda_{3}$, by $\lambda_{8}$ and by
$Q$. When the right isometry group is a Cartan subgroup, our above specific
choice for $K$ amounts to take it equal to
${\mathrm{U}}(1)_{I}\times{\mathrm{U}}(1)_{Y}$. Notice that the subgroups
${\mathrm{U}}(1)_{Y}=\\{(e^{i\phi/3},e^{i\phi/3},e^{i(-2)\phi/3})\\}$ and
${\mathrm{U}}(1)_{Q}=\\{(e^{i(2)\phi/3},e^{i(-1)\phi/3},e^{i(-1)\phi/3})\\}$,
with $e^{i\phi}\in S^{1}$, equivalently $e^{i\phi/3}\in S^{1}$, respectively
equal to ${\mathrm{U}}(1)_{1,1}$ and
${\mathrm{U}}(1)_{2,-1}={\mathrm{U}}(1)_{-2,1}$, are conjugated in ${\rm
SU}(3)$ by a permutation of the Weyl group $S_{3}$ (the triple $(-2,1,1)$
being equivalent to $(1,1,-2)$), but they are not conjugated to the subgroup
${\mathrm{U}}(1)_{I}$.
Some mathematical readers could ask why physicists prefer to define $Q$ and
$Y$ as before, without incorporating a multiplicative factor equal to $3$ in
the definition, a choice that would indeed look more natural since irreps of
${\mathrm{U}}(1)$ are labelled by integers. The problem is that, by so doing,
quarks (identified with basis vectors of the fundamental representations)
would have charge $\pm 1$ and the proton (identified with a specific vector in
the tensor cube of the defining representation) would have charge $+3$.
However, conventionally, the latter has electric charge $+1$ (minus the charge
of the electron). One could of course suggest to modify the standard
terminology and redefine the notion of electric charge in such a way that the
electron has electric charge $-3$, but this is not going to happen!
For this reason $Y$ and $Q$ are defined as above and quarks turn out to have
”fractional electric charge”: $\pm 1/3$ or $\pm 2/3$.

##### Parametrizations.

We now give parametrizations for the dual metric $h^{-1}$ in the basis
$(X_{a})$ which is orthonormal for the Killing metric, assuming that the
isometry group of $h$ is ${\rm SU}(3)\times K$. Remember that in the defining
representation, the vector fields $X_{a}$, at the origin, are represented by
matrices $\tfrac{i}{2\sqrt{3}}\lambda_{a}$. The reader can obtain the
following results by imposing the vanishing of the Lie derivative of $h^{-1}$
with respect to the generators $\lambda_{k}$ of $K$, i.e., 999As usual, a
summation over repeated indices is understood.

$h^{ij}([\lambda_{k},\lambda_{i}]\otimes\lambda_{j}+\lambda_{i}\otimes[\lambda_{k},\lambda_{j}])=0$
(1)

For the Killing metric, $K$ is ${\rm SU}(3)$, and we have
$k^{-1}=\delta^{ab}\,X_{a}\otimes X_{b}$, in other words, $k^{ab}$ is the unit
matrix $8\times 8$. Bi-invariant metrics are multiples of the Killing metric
$k$ since ${\rm SU}(3)$ is simple, and they have the same isometry group; they
read $h^{-1}=\alpha\,\delta^{ab}\,X_{a}\otimes X_{b}$ where $\alpha$ is some
real constant.

In the same basis $(X_{a})$ the parametrization of $h^{-1}$, with components
$h^{ab}$, for the other choices of $K$ specified in the list (2), reads as in
the following table (3) (the generic parameters appearing in these expressions
are arbitrary real numbers and the dots stand for $0$’s) :

$\begin{array}[]{ccc}&{\mathrm{S}O}(3):\\{\lambda_{2},\lambda_{5},\lambda_{7}\\},\qquad{\mathrm{U}}(2):\;\\{\lambda_{1},\lambda_{2},\lambda_{3};\lambda_{8}\\},\qquad{\mathrm{U}}(1)\times{\mathrm{U}}(1):\;\\{\lambda_{3},\lambda_{8}\\}\\\
&{\mathrm{U}}(1)_{I}:\;\\{\lambda_{3}\\},\qquad\qquad{\mathrm{U}}(1)_{Y}:\;\\{\lambda_{8}\\}.\end{array}$
(2)

$\begin{array}[]{ccc}\left(\begin{array}[]{cccccccc}\alpha&.&.&.&.&.&.&.\\\
.&\beta&.&.&.&.&.&.\\\ .&.&\alpha&.&.&.&.&.\\\ .&.&.&\alpha&.&.&.&.\\\
.&.&.&.&\beta&.&.&.\\\ .&.&.&.&.&\alpha&.&.\\\ .&.&.&.&.&.&\beta&.\\\
.&.&.&.&.&.&.&\alpha\\\
\end{array}\right)&\left(\begin{array}[]{cccccccc}\alpha&.&.&.&.&.&.&.\\\
.&\alpha&.&.&.&.&.&.\\\ .&.&\alpha&.&.&.&.&.\\\ .&.&.&\beta&.&.&.&.\\\
.&.&.&.&\beta&.&.&.\\\ .&.&.&.&.&\beta&.&.\\\ .&.&.&.&.&.&\beta&.\\\
.&.&.&.&.&.&.&\gamma\\\
\end{array}\right)&\left(\begin{array}[]{cccccccc}\alpha&.&.&.&.&.&.&.\\\
.&\alpha&.&.&.&.&.&.\\\ .&.&\beta&.&.&.&.&\zeta\\\ .&.&.&\gamma&.&.&.&.\\\
.&.&.&.&\gamma&.&.&.\\\ .&.&.&.&.&\delta&.&.\\\ .&.&.&.&.&.&\delta&.\\\
.&.&\zeta&.&.&.&.&\varepsilon\\\ \end{array}\right)\\\ &&\\\
K={\mathrm{S}O}(3)&K={\mathrm{U}}(2)&K={\mathrm{U}}(1)\times{\mathrm{U}}(1)\end{array}$
$\begin{array}[]{ccc}\left(\begin{array}[]{cccccccc}\alpha&.&.&.&.&.&.&.\\\
.&\alpha&.&.&.&.&.&.\\\ .&.&\beta&.&.&.&.&\zeta\\\
.&.&.&\gamma&.&\theta&\eta&.\\\ .&.&.&.&\gamma&\eta&-\theta&.\\\
.&.&.&\theta&\eta&\delta&.&.\\\ .&.&.&\eta&-\theta&.&\delta&.\\\
.&.&\zeta&.&.&.&.&\epsilon\\\
\end{array}\right)&\left(\begin{array}[]{cccccccc}\varkappa_{11}&\varkappa_{12}&\varkappa_{13}&.&.&.&.&\epsilon_{1}\\\
\varkappa_{12}&\varkappa_{22}&\varkappa_{23}&.&.&.&.&\epsilon_{2}\\\
\varkappa_{13}&\varkappa_{23}&\varkappa_{33}&.&.&.&.&\epsilon_{3}\\\
.&.&.&\alpha&.&\gamma&\delta&.\\\ .&.&.&.&\alpha&-\delta&\gamma&.\\\
.&.&.&\gamma&-\delta&\beta&.&.\\\ .&.&.&\delta&\gamma&.&\beta&.\\\
\epsilon_{1}&\epsilon_{2}&\epsilon_{3}&.&.&.&.&\epsilon_{8}\\\
\end{array}\right)&{}\hfil\\\ &&\\\
K={\mathrm{U}}(1)_{I},\,\text{or}\,{\mathrm{U}}(1)_{k,-k}&K={\mathrm{U}}(1)_{Y},\,\text{or}\,{\mathrm{U}}(1)_{k,k}&{}\hfil\end{array}$
(3)

The parametrization obtained for the matrices given in table (3), for the
specific subgroups $K$ given in (2) should be understood as generic ones:
obviously, for particular choices of the real parameters entering these
matrices the right isometry group can be larger than $K$ (for instance by
taking all the diagonal coefficients equal to $1$, and by setting to $0$ the
off-diagonal ones, one recover the Killing metric, for which $K={\rm SU}(3)$).
More generally the matrices $h^{-1}$ that obey the Killing equation (1) for a
chosen group $K$, as specified in (2), determine left-invariant metrics for
which the right isometry group is equal either to $K$ or to an over-group of
$K$ that should be equal or conjugated to one member of the list (2).

Remarks (proofs, using (1) and the commutation relations in
$\mathfrak{su}(3)$, are immediate, and left to the reader):
$\bullet$ Invariance of a metric under ${\rm SU}(2)$ implies invariance under
${\mathrm{U}}(2)$.
$\bullet$ Imposing invariance under ${\mathrm{U}}(1)_{k,0}$, with $k>0$
amounts, up to conjugacy, to impose invariance under ${\mathrm{U}}(1)_{k,-k}$,
and therefore gives for $h^{-1}$ the same parametrization as the one obtained
when $K={\mathrm{U}}(1)_{I}$.
$\bullet$ Invariance under any ${\mathrm{U}}(1)_{k,\ell}$, with $k>\ell>0$,
implies invariance under ${\mathrm{U}}(1)\times{\mathrm{U}}(1)$.
We can therefore restrict our attention to the subgroups $K$ given by the list
(2).

The above parametrizations were already obtained and commented in [4] for the
various choices of the subgroup $K$. In the same reference, an application to
particle physics was given, namely the interpretation of the mass operator for
various types of mesons in terms of the Laplacian associated to left-invariant
metrics for which $\text{Lie}(K)=\mathfrak{su}(2)\oplus\mathfrak{u}(1)$. We
shall come back to this discussion at the end of the present article.

The number of free parameters appearing in the previous expressions of
$h^{-1}$ could be a priori determined by considering these metrics as coming
from an $ad(K)$ invariant bilinear form at the origin of the coset space
$({\rm SU}(3)\times K)/K$, with $K$ diagonally embedded, and by reducing the
isotropy action of $K$ in the tangent space at the identity ($\mathbb{R}^{8}$)
into a sum of real irreducible representations (irreps).

##### Pseudo-Riemannian structures.

In view of using the above parametrizations to explicitly determine various
curvature tensors, one wants to have as few free coefficients as possible. It
is therefore useful to consider pseudo-Riemannian structures, rather than
pseudo-Riemannian metrics. The group of diffeomorphisms of a manifold acts by
pullback on its space of (pseudo) Riemannian metrics. The quotient space is,
by definition, the space of Riemannian structures. The stabilizer of this
action at a given point, i.e., at a given metric, is the isometry group of
this metric. Two metrics belonging to the same orbit have conjugated
stabilizers, i.e., conjugated isometry groups, and each stratum (that maybe
contains distinct orbits) of the obtained stratification is characterized by
an isometry group, up to conjugacy. It may also happen that distinct metrics
belonging to the same orbit have the same isometry group —we shall meet one
such example in what follows.

Left-invariant metrics of signature $(p,q)$ on ${\rm SU}(3)$ can be associated
with elements of $GL(8,\mathbb{R})/O(p,q)$ since they can defined by arbitrary
symmetric bilinear forms of prescribed signature on the tangent space at the
origin of ${\rm SU}(3)$, i.e., in $\mathbb{R}^{8}$, but the associated
Riemannian structures are associated with points of the orbit space of the
latter under the action of $Ad({\rm SU}(3))\subset{\mathrm{S}O}(8)$.

Equivalence under this action generically (i.e., when the right isometry group
$K$ is trivial) reduces the number of free parameters from $36=(8\times 9)/2$
to $28=36-8$. For ${\rm SU}(3)\times K$ invariant metrics, with $K$ non
trivial, one may use rotations defined by elements of $Ad({\rm SU}(3))$ that
commute with the action of $K$ to decrease the number of parameters entering
the matrices of table (3) determined by solving equation (1).

For instance, if $K={\mathrm{U}}(1)_{I}$, this number is reduced from $8$ to
$7$: setting $(h^{-1})^{\prime}={r}^{T}.h^{-1}.r$ with $h^{-1}$ as in table
(3), and using $r=\exp(x\ f_{8})$ with $x=\arctan(\theta/\eta)$, one obtains a
new matrix $(h^{-1})^{\prime}$ of the same family that can be directly
obtained from $(h^{-1})$ by replacing only $\eta$ by
$\eta^{\prime}=\sqrt{\theta^{2}+\eta^{2}}$ and the coefficient
$\theta=h^{-1}_{(6,4)}=h^{-1}_{(4,6)}=-h^{-1}_{(5,7)}=-h^{-1}_{(7,5)}$, in
table (3), by $0$. Since $\lambda_{3}$ and $\lambda_{8}$ commute, these two
matrices $(h^{-1})$ and $(h^{-1})^{\prime}$ define left-invariant metrics that
have the same right isometry group.
In a similar way the number of parameters, if $K={\mathrm{U}}(1)_{Y}$, can be
reduced from $14$ to $11$: the $3\times 3$ symmetric sub-matrix $\varkappa$,
in the upper left corner of $h^{-1}$, can be assumed to be diagonal.

In this way one obtains respectively $1,2,3,6,7,11,28$ parameters (instead of
$1,2,3,6,8,14,36$) for the choices $K={\rm
SU}(3),{\mathrm{S}O}(3),{\mathrm{U}}(2),{\mathrm{U}}(1)\times{\mathrm{U}}(1),{\mathrm{U}}(1)_{I},{\mathrm{U}}(1)_{Y},\\{e\\}$.

A last simplification, further reducing by one the number of parameters, is to
consider metrics only up to scale, i.e., metrics that differ by a constant
conformal transformation (this changes the obtained curvatures by an overall
multiplicative constant). The number of parameters for the previous choices of
$K$, once we identify metrics that differ by equivalence and scaling, becomes
$0,1,2,5,6,10,27$.

##### Decomposition of a bilinear symmetric form of rank $8$ on ${\rm SU}(3)$
irreps

The group $G={\rm SU}(3)$ acts on the vector space of $8\times 8$ symmetric
matrices —the symmetric subspace of the tensor square of the adjoint
representation. This action is not irreducible and, denoting the irreps that
appear in this symmetric subspace by their dimension, we have the direct sum
decomposition: $36=1\oplus 8\oplus 27$, with three terms respectively
associated with the irreps of highest weights $(0,0)$, $(1,1)$, and $(2,2)$.
Let us call ${{}_{1}h^{-1}}$, ${{}_{8}h^{-1}}$, ${{}_{27}h^{-1}}$, the
projections of the dual metric $h^{-1}$ on these three vector subspaces.
Calling $d_{a,b,c}=\tfrac{1}{4}\ Tr(\lambda_{a}[\lambda_{b},\lambda_{c}]_{+})$
where
$[\lambda_{b},\lambda_{c}]_{+}=\lambda_{b}\lambda_{c}+\lambda_{c}\lambda_{b}$
is the anti-commutator, and $d_{a}$ the (symmetric) matrices with elements
$(d_{a})_{b,c}=d_{a,b,c}$, it is easy to show that
${{}_{1}h^{-1}}=\tfrac{1}{8}\,Tr(h^{-1})\,\,\mathrm{l}\\!\\!\\!1$ and that
${{}_{8}h^{-1}}=\tfrac{3}{5}\,\sum_{a=1\ldots 8}\,Tr(h^{-1}d_{a})\,d_{a}$; the
last projection, ${{}_{27}h^{-1}}$, can be obtained by difference. Let us
illustrate this decomposition by assuming that the metric $h$ belongs to the
family of metrics for which the right isometry group $K$ is (at least)
${\mathrm{U}}(2)$, with the parametrization given in (2). One obtains
immediately $h^{-1}={{}_{1}h^{-1}}+{{}_{8}h^{-1}}+{{}_{27}h^{-1}}$, with

$\begin{split}{{}_{1}h^{-1}}=&A\,\,\,\mathrm{l}\\!\\!\\!1\\\
{{}_{8}h^{-1}}=&{B\sqrt{3}}\,d_{8}=B\,m_{8}\quad\text{with}\quad
m_{8}=\text{diag}\left(1,1,1,-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-1\right)\\\
{{}_{27}h^{-1}}=&C\,m_{27}\quad\text{with}\quad
m_{27}=\text{diag}\left(1,1,1,-3,-3,-3,-3,9\right)\\\
\text{where}\;&A=\frac{1}{8}(3\alpha+4\beta+\gamma),\quad
B=\frac{1}{5}(3\alpha-2\beta-\gamma),\quad
C=\frac{1}{40}(\alpha-4\beta+3\gamma)\end{split}$ (4)

We chose to illustrate this decomposition of $h$ (actually of $h^{-1}$) in the
case $K={\mathrm{U}}(2)$, but one can do it as well for the other
cases101010The Killing metric has projection onto $\,\,\mathrm{l}\\!\\!\\!1$
only. For the $K={\mathrm{S}O}(3)$ family (see (2)), the decomposition is as
in (4) but with $A=\frac{5\alpha}{8}+\frac{3\beta}{8}$, $B=0$ and
$C=\frac{3\alpha}{8}-\frac{3\beta}{8}$, with the same $m_{8}$ but with
$m_{27}=\text{diag}(1,-(5/3),1,1,-(5/3),1,-(5/3),1)$. For the Jensen sub-
family (Einstein metrics, see sect. 3), one has $A=\frac{19\alpha}{4}$, $B=0$
and $C=-\frac{15\alpha}{4}$.. One may notice, however, that such
decompositions (that do not seem to be much used) have no reason to be
compatible with the signature, or even with the non-degenerateness, of the
chosen bilinear form. Nevertheless one can consider families, or subfamilies,
of bilinear forms for which one or several of the above projections vanish. We
shall come back to this possibility in the last section.

###  2.4 Curvature tensors

Expressions for curvature tensors of the Levi-Civita connection (the
torsionless metric connection) defined by an invariant metric on a Lie group
can be found in various places in the literature. Unfortunately these
expressions are often written in a a basis (a moving frame) for which the
chosen metric is orthonormal. Here we want to study various metrics while
keeping the same basis. For this reason we shall give expressions of the
various curvature tensors in a basis $(e_{a})$ made of arbitrary left-
invariant vector fields111111These formulae can be found in [5].; we call
${x_{ab}}^{c}$ the corresponding structure constants:
$[e_{a},e_{b}]={x_{ab}}^{c}\,e_{c}$.

The chosen metric (call it $h$) defines musical isomorphisms between a vector
space and its dual; in particular, using the structure constants
${x_{ab}}^{c}$ and the metric coefficients $h_{ab}$ or $h^{ab}$, one can
define new symbols such as $x_{abc}={x_{ab}}^{d}\,h_{dc}$,
${x^{a}}_{bc}=h^{ae}\,{x_{eb}}^{d}\,h_{dc}$, etc. The term
${x^{m}}_{ik}x_{mjl}$, for instance, extracted from (5) below, actually means
$\sum_{m^{\prime},k^{\prime},l^{\prime}}\,{x_{m^{\prime}i}}^{k^{\prime}}{x_{mj}}^{l^{\prime}}\,h^{m^{\prime}m}\,h_{k^{\prime}k}\,h_{l^{\prime}l}$
when expressed in terms of structure constants and metric (or inverse metric)
coefficients121212One could write such expressions with all the indices at the
same level (writing for instance $x_{mik}x_{mjl}$) provided one uses the
second Einstein summation convention, which supposes chosen a fixed metric
$h$: an index variable that appears twice at the same level, i.e., twice as a
superscript or twice as a subscript, should be summed over using the chosen
metric or its dual.. Observe that the symbols $x_{abc}$ are not, in general,
antisymmetric with respect to the last two indices since the metric $h$ is not
assumed to be bi-invariant.

Call $R^{a}_{\;bcd}$ the components of the Riemann curvature tensor. The last
two indices ($c$ and $d$) are the form indices, and the first two ($a$ and
$b$) are the fiber indices. Using the metric $h$, one defines
$R_{abcd}=h_{aa^{\prime}}R^{a^{\prime}}_{\;bcd}$. The components of the Ricci
tensor are $\varrho_{bd}=R^{a}_{\;bad}$ and the scalar curvature is
$\tau=\varrho^{d}_{\;d}:=h^{db}\varrho_{bd}$. One can also define the Einstein
tensor $\mathtt{G}=\varrho-\frac{1}{2}\,\tau\,h$. One has:

$\begin{split}R_{abcd}=\frac{1}{4}&(x_{acm}\,{x_{bd}}^{m}+2x_{abm}\,{x_{cd}}^{m}-x_{bcm}\,{x_{ad}}^{m}-{x_{ab}}^{m}\,x_{mcd}+{x_{ab}}^{m}\,x_{mdc}-{x_{cd}}^{m}\,x_{mab}\\\
&+{x_{cd}}^{m}\,x_{{mba}}+({x^{m}}_{ac}+{x^{m}}_{ca})(x_{mbd}+x_{mdb})-({x^{m}}_{bc}+{x^{m}}_{cb})(x_{mad}+x_{mda})\end{split}$
(5)

$\varrho_{bd}=-\frac{1}{2}x_{mbn}\,{x_{nd}}^{m}-\frac{1}{2}x_{mbn}\,{x_{md}}^{n}+\frac{1}{4}{x_{mnb}}\,{x^{mn}}_{d}-\frac{1}{2}(x_{mbd}+x_{mdb}){{x^{m}}_{n}}^{n}$
(6)

$\tau=-\frac{1}{4}{x^{mk}}_{n}\,{x_{mk}}^{n}-\frac{1}{2}{{x}_{m}}^{kn}\,{x_{nk}}^{m}-{{x_{m}}_{k}}^{k}\,{{x^{m}}_{n}}^{n}$
(7)

Notice that in order to calculate the Ricci tensor for a specific left-
invariant metric, one does not need to evaluate the Riemann tensor first.

In the following we shall always express the components of the curvature
tensors in the basis $(X_{a})$ for which the Killing metric is orthonormal: we
shall take $(e_{a})=(X_{a})$, hence
${x_{ab}}^{c}=\tfrac{-1}{\sqrt{3}}{f_{ab}}^{c}$ in all cases.

Notice that the last term of (6) and (7), a trace, vanishes for unimodular
groups, in particular for ${\rm SU}(3)$, so we can safely drop it in the
practical calculations that come next.

##  3 Pseudo-Riemannian homogeneous Einstein metrics on ${\rm SU}(3)$

As before, the isometry group of a left-invariant metrics $h$ on ${\rm SU}(3)$
is denoted ${\rm SU}(3)\times K$. It is clear that any subgroup of $K$ is also
a group of isometries of such a metric $h$. The inverse metrics $h^{-1}$ are
parametrized as in sect. 2.3 but we can also incorporate an overall (constant)
real scaling factor in their definition. The Einstein condition for the metric
$h$ reads

$\varrho=\kappa\,h$

the real number $\kappa$ being called the Einstein constant. Equivalently, one
can solve the Einstein equation $\mathtt{G}+\Lambda\,h=0$, where $\mathtt{G}$
is the Einstein tensor; $\Lambda$ is the so-called cosmological constant
(although there is no cosmological interpretation in the present context!).
For Einstein metrics one has obviously $\tau=8\kappa$ since $\text{dim}({\rm
SU}(3))=8$, moreover $\Lambda=\kappa-\tau/2$, therefore $\Lambda=3\kappa$.

Remark. A pseudo-Riemannian metric on ${\rm SU}(3)$ which is left invariant
and $K$-right invariant, with $K$ a Lie subgroup, is therefore $ad_{K}$
invariant and passes to an ${\rm SU}(3)$-invariant pseudo-Riemannian metric on
the quotient ${\rm SU}(3)/K$, but even if the metric one starts from is an
Einstein metric, the metric on the homogenous space ${\rm SU}(3)/K$ has no
reason to be Einstein (and in general it is not). For instance the homogeneous
metrics induced on Aloff-Wallach spaces from the Killing metric on ${\rm
SU}(3)$ are not Einstein (and the so-called Aloff-Wallach metrics [1] – that
are ${\rm SU}(3)$ invariant and have positive sectional curvature – are not
Einstein either), although each of these spaces admits an homogeneous Einstein
metric and even a Lorentz-Einstein metric (see [22]). The aim of the previous
brief comment is only to stress the fact that our purpose in the present
section is to study the Einstein condition for left-invariant metrics on ${\rm
SU}(3)$ itself: we shall not study what happens on its quotients. By way of
contrast, however, notice that the calculations performed in this section are
the same for any Lie group with Lie algebra ${\mathrm{Lie}}({\rm SU}(3))$, in
particular for ${\rm SU}(3)/Z_{3}$, which is not homotopically trivial.

We now study the Einstein condition on ${\rm SU}(3)$ for the various
parametrizations of the metrics for which the right isometry group is $K$, as
in (2), (3), or an over-group of the latter.

#### $K={\rm SU}(3)$

These are the bi-invariant metrics $h=k/\alpha$, where $k$ is the Killing
metric. For a simple Lie group $G$, the Ricci tensor of $k$ is
$\varrho=\frac{1}{4}\,k$. It therefore defines an Einstein space with Einstein
constant $\kappa=1/4$. Its scalar curvature is $\tau=\text{dim}(G)/4$.

The Ricci tensor is invariant under constant scaling of the metric (a general
property), the Einstein condition is therefore also satisfied when $k$ is
scaled by $1/\alpha$, the Einstein constant becoming $\kappa=\alpha/4$, with
$\tau=\alpha\,\text{dim}(G)/4$; therefore $\tau=2\alpha$ for $G={\rm SU}(3)$.
Moreover $\Lambda=3\alpha/4$.

#### $K={\mathrm{S}O}(3)$

For these metrics, the Ricci tensor is diagonal, with diagonal

$\left\\{\frac{1}{2}-\frac{\alpha}{4\beta},\frac{1}{24}\left(\frac{5\alpha^{2}}{\beta^{2}}+1\right),\frac{1}{2}-\frac{\alpha}{4\beta},\frac{1}{2}-\frac{\alpha}{4\beta},\frac{1}{24}\left(\frac{5\alpha^{2}}{\beta^{2}}+1\right),\frac{1}{2}-\frac{\alpha}{4\beta},\frac{1}{24}\left(\frac{5\alpha^{2}}{\beta^{2}}+1\right),\frac{1}{2}-\frac{\alpha}{4\beta}\right\\}$

The scalar curvature, for this family, is
$\tau=\frac{-5\alpha^{2}+20\alpha\beta+\beta^{2}}{8\beta}$. The Einstein
condition gives a second degree equation, with tho real solutions,
$\alpha=\beta,\kappa=\alpha/4$, the already obtained Killing metric, and
another solution, the Jensen metric [12]: $\beta=11\alpha$, with Einstein
constant $\kappa=\tfrac{21}{44}\,\alpha$. Both are properly Riemannian
(signature $(8,0)$). We recover the scalar curvature $\tau=2\alpha$ in the
first case, and find $\tau=42\,\alpha/11$ in the second.

$h^{-1}=\left(\begin{array}[]{cccccccc}1&.&.&.&.&.&.&.\\\ .&11&.&.&.&.&.&.\\\
.&.&1&.&.&.&.&.\\\ .&.&.&1&.&.&.&.\\\ .&.&.&.&11&.&.&.\\\ .&.&.&.&.&1&.&.\\\
.&.&.&.&.&.&11&.\\\ .&.&.&.&.&.&.&1\\\ \end{array}\right)$ (8)

One can recover these solutions as follows, without calculating the Ricci
tensor: write ${\rm SU}(3)$ as a principal bundle with typical fiber
${\mathrm{S}O}(3)$ over the irreducible symmetric space ${\rm
SU}(3)/{\mathrm{S}O}(3)$, consider a first family of metrics $h(t)$ obtained
by dilating the Killing metric in the direction of fibers by an arbitrary
coefficient $t^{2}$, their scalar curvature is
$\tau(h(t))=-\frac{5t^{2}}{8}+\frac{1}{8t^{2}}+\frac{5}{2}$, then define a
second family $\widehat{h}(t)=(1/t^{2})^{3/8}\,h(t)$, the overall scaling
coefficient being chosen in such a way that the Riemannian volume stays
constant when $t$ varies (the determinant of $h(t)$ is $(t^{2})^{3}$). The
stationary points, with respect to $t$, of the scalar curvature
$\tau(\widehat{h}(t))=(t^{2})^{3/8}\,\tau(h(t))$ of the metrics
$\widehat{h}(t)$ are Einstein metrics [12]; one obtains the equation
$\tfrac{d}{dt}\tau(\widehat{h}(t)))=\text{coeff}\times(t^{2}-1)(t^{2}-1/11)$,
hence the solutions.

The above is a particular case of a general construction ([6], [23], see also
[5]). Assuming that both $G$ and $K$ are simple, writing $G$ as a $K$
principal bundle over $G/K$, and dilating the Killing metric of $G$ by $t^{2}$
in the direction of fibers, one first obtains the following formula for the
scalar curvature of the metrics $h(t)$ on $G$: $\tau(h(t))=\tfrac{s}{2}+c\
\tfrac{k}{4}\tfrac{1}{t^{2}}-k(1-c)\tfrac{t^{2}}{4}$, where $n=\text{dim}\,G$,
$k=\text{dim}\,K$, $s=\text{dim}\,G/K$ and $c$ is the embedding coefficient of
$K$ in $G$. This result is immediately obtained by Kaluza-Klein dimensional
reduction, see for instance [5], applied to this particular fibration (in this
simple case one can use O’Neill formulae for Riemannian submersions with
totally geodesic fibers, see [11], [17]). The stationary points of the scalar
curvature $\tau(\widehat{h}(t))=(t^{2})^{k/n}\,\tau(h(t))$ of the metrics
$\widehat{h}(t)=(t^{2})^{-k/n}\,h(t)$ are Einstein metrics [12]. For an
irreducible symmetric pair $(G,K)$ one has $c=1-\tfrac{s}{2k}$; in that case
$\tfrac{d}{dt}\tau(\widehat{h}(t)))=\text{coeff}\times(t^{2}-1)(t^{2}-(2k-s)/(2k+s))$.
The previous results are recovered for $G={\rm SU}(3)$, $K={\mathrm{S}O}(3)$,
using $n=8$, $k=3$ (hence $s=5$ and $c=1/6$).

#### $K={\mathrm{U}}(2)$

The Ricci tensor is diagonal, with non-zero coefficients
$\varrho_{11}=\varrho_{22}=\varrho_{33}$,
$\varrho_{44}=\varrho_{55}=\varrho_{66}=\varrho_{77}$, $\varrho_{88}$,
respectively given by

$\frac{1}{12}\left(\frac{\beta^{2}}{\alpha^{2}}+2\right),\quad\frac{1}{8}\left(-\frac{\beta}{\alpha}-\frac{\beta}{\gamma}+4\right),\quad\frac{\beta^{2}}{4\gamma^{2}}$

The Einstein condition gives only one real solution, $\alpha=\beta=\gamma$,
i.e., the family of bi-invariant metrics (proportional to the Killing metric).

#### $K={\mathrm{U}}(1)\times{\mathrm{U}}(1)$

The non-zero components of the Ricci tensor are $\varrho_{11}=\varrho_{22}$,
$\varrho_{33}$, $\varrho_{44}=\varrho_{55}=\varrho_{66}=\varrho_{77}$,
$\varrho_{88}$, and $\varrho_{38}=\varrho_{83}$, respectively equal to

$\begin{split}&\frac{1}{12}\left(\frac{\gamma\delta}{\alpha^{2}}+\frac{2\alpha\epsilon}{\zeta^{2}-\beta\epsilon}-\frac{\gamma}{\delta}-\frac{\delta}{\gamma}+6\right),\quad\frac{\epsilon^{2}\left(4\alpha^{2}+\gamma^{2}+\delta^{2}\right)+3\zeta^{2}\left(\gamma^{2}+\delta^{2}\right)+2\sqrt{3}\zeta\epsilon\left(\delta^{2}-\gamma^{2}\right)}{24\left(\zeta^{2}-\beta\epsilon\right)^{2}}\\\
&\frac{1}{24}\left(\frac{2\alpha\delta}{\gamma^{2}}-\frac{2\alpha}{\delta}-\frac{2\delta}{\alpha}+\frac{\gamma\left(3\beta-2\sqrt{3}\zeta+\epsilon\right)}{\zeta^{2}-\beta\epsilon}+12\right),\quad\frac{\zeta^{2}\left(4\alpha^{2}+\gamma^{2}+\delta^{2}\right)+3\beta^{2}\left(\gamma^{2}+\delta^{2}\right)+2\sqrt{3}\beta\zeta\left(\delta^{2}-\gamma^{2}\right)}{24\left(\zeta^{2}-\beta\epsilon\right)^{2}}\\\
&\frac{-\zeta\epsilon\left(4\alpha^{2}+\gamma^{2}+\delta^{2}\right)+\beta\gamma^{2}\left(\sqrt{3}\epsilon-3\zeta\right)-\beta\delta^{2}\left(3\zeta+\sqrt{3}\epsilon\right)+\sqrt{3}\zeta^{2}(\gamma-\delta)(\gamma+\delta)}{24\left(\zeta^{2}-\beta\epsilon\right)^{2}}\end{split}$

The Einstein condition gives only one real solution,
$\alpha=\beta=\gamma=\delta=\epsilon$, $\zeta=0$, i.e., the known family of
bi-invariant metrics.

#### $K={\mathrm{U}}(1)_{I}$

The parametrization of a generic left-invariant metric, with
$K={\mathrm{U}}(1)_{I}$, involves the eight parameters
${\alpha,\beta,\gamma,\delta,\epsilon,\zeta,\eta,\theta}$ but we know that we
can fix the scale $\alpha=1$, and set the parameter $\theta$ to $0$ since
different choices for $\theta$ give metrics corresponding to the same
Riemannian structure (see our discussion at the end of sect. 2.3). We are left
with six parameters. The Einstein condition involves one parameter more, the
Einstein constant $\kappa$. We did not solve this set of equations in full
generality: we restricted our attention to the family of metrics obtained by
imposing the further constraint $\gamma=\delta$; in that case, one of the
equations implies that $\zeta$ should vanish.

There are five solutions (only three if one imposes $\eta\geq 0$). The first
is the Killing metric —as expected. The second and third solutions only differ
by a sign flip in the value of the parameter $\eta$, they are properly
Riemannian Einstein metrics and they are equivalent to the Jensen solution.
The last two solutions (again, they only differ by the sign of $\eta$) are
Einstein metrics with a Lorentzian signature.

The non-zero components of the Ricci tensor are $\varrho_{11}=\varrho_{22}$,
$\varrho_{33}$, $\varrho_{44}=\varrho_{55}$, $\varrho_{66}=\varrho_{77}$,
$\varrho_{7,4}=\varrho_{6,5}=\varrho_{5,6}=\varrho_{4,7}$,
$\varrho_{3,8}=\varrho_{8,3}$, $\varrho_{88}$. These seven expressions are
rather huge to be displayed in an article, even after setting $\theta=0$. As
already mentioned, one can show (it is almost straightforward but cumbersome!)
that the hypothesis $\gamma=\delta$, on top of the the Einstein condition,
implies that $\zeta$ should vanish; we shall therefore only display the non-
zero components of the Ricci tensor and of the metric in this simpler case,
which also implies that $\varrho_{44}=\varrho_{55}$ should be equal to
$\varrho_{66}=\varrho_{77}$ and that $\varrho_{3,8}=\varrho_{8,3}$ is $0$.
Removing duplicates, we are left with five non-zero distinct components of the
Ricci tensor:

$\begin{split}\varrho_{11}=\varrho_{22}=&\frac{1}{12}\left(\frac{(\gamma-\eta)(\gamma+\eta)}{\alpha^{2}}-\frac{2\alpha}{\beta}+\frac{4\gamma^{2}}{\eta^{2}-\gamma^{2}}+8\right),\qquad\varrho_{33}=\frac{2\alpha^{2}+\gamma^{2}+\eta^{2}}{12\beta^{2}},\\\
\varrho_{44}=\varrho_{55}=\varrho_{66}=\varrho_{77}=&\frac{1}{24}\left(\gamma\left(\frac{8\alpha\eta^{2}}{\left(\gamma^{2}-\eta^{2}\right)^{2}}-\frac{2}{\alpha}-\frac{1}{\beta}+\frac{12\eta^{2}\epsilon}{\left(\gamma^{2}-\eta^{2}\right)^{2}}-\frac{3}{\epsilon}\right)+12\right),\\\
\varrho_{7,4}=\varrho_{6,5}=\varrho_{5,6}=\varrho_{4,7}=&\frac{1}{24}\eta\left(-\frac{4\gamma^{2}(2\alpha+3\epsilon)}{\left(\gamma^{2}-\eta^{2}\right)^{2}}+\frac{2}{\alpha}-\frac{1}{\beta}+\frac{3}{\epsilon}\right),\\\
\varrho_{88}=&\frac{-2\eta^{2}\left(\gamma^{2}+2\epsilon^{2}\right)+\gamma^{4}+\eta^{4}}{4\epsilon^{2}(\gamma-\eta)(\gamma+\eta)}\end{split}$

The dual metric $h^{-1}$ is specified by the matrix $h^{ij}$ given in (9):

$h^{-1}=\left(\begin{array}[]{cccccccc}\alpha&.&.&.&.&.&.&.\\\
.&\alpha&.&.&.&.&.&.\\\ .&.&\beta&.&.&.&.&.\\\ .&.&.&\gamma&.&.&\eta&.\\\
.&.&.&.&\gamma&\eta&.&.\\\ .&.&.&.&\eta&\gamma&.&.\\\
.&.&.&\eta&.&.&\gamma&.\\\ .&.&.&.&.&.&.&\epsilon\\\ \end{array}\right)$ (9)

The Einstein condition reads $\varrho_{ij}=\kappa\,h_{i,j}$ where the non-zero
components of the matrix $h$ are as follows:

$h_{1,1}=h_{2,2}=\frac{1}{\alpha},h_{3,3}=\frac{1}{\beta},h_{44}=h_{55}=h_{66}=h_{77}=\frac{\gamma}{\gamma^{2}-\eta^{2}},h_{7,4}=h_{6,5}=h_{5,6}=h_{4,7}=\frac{\eta}{\eta^{2}-\gamma^{2}},h_{8,8}=\frac{1}{\epsilon}$

We have five non-linear equations and five unknowns: the five parameters
$\alpha,\beta,\gamma,\epsilon,\eta$ (but one can take $\alpha=1$), and the
Einstein constant $\kappa$.

$\bullet$ One obvious solution of the Einstein condition is obtained by
setting $\eta=0$ and by taking all the other parameters equal: one recover the
bi-invariant metrics.

$\bullet$ Another solution, up to scale, is obtained by setting
${\alpha=1,\beta=11,\gamma=\delta=6,\epsilon=1,\zeta=0,\eta=\pm 5,\theta=0}$.
See (10). The Einstein constant is $\kappa=21/44$. The metric has signature
$(8,0)$.

$h^{-1}=\left(\begin{array}[]{cccccccc}1&.&.&.&.&.&.&.\\\ .&1&.&.&.&.&.&.\\\
.&.&11&.&.&.&.&.\\\ .&.&.&6&.&.&5&.\\\ .&.&.&.&6&5&.&.\\\ .&.&.&.&5&6&.&.\\\
.&.&.&5&.&.&6&.\\\ .&.&.&.&.&.&.&1\\\ \end{array}\right)$ (10)

From the metric defined by (10), and using the remarks at the end of sect.
2.3, one can obtain a one-parameter family of Einstein metrics (all defining
the same Einstein structure), for the same parameters $\alpha_{0}=1$,
$\beta_{0}=11$, $\gamma_{0}=6$, $\epsilon_{0}=1$, $\zeta_{0}=0$, as in (10),
but for arbitrary values of $\theta$, $|\theta|\leq 5$, (remember that we had
imposed a priori the conditions $\theta=0$ and $\delta=\gamma$) while also
setting $\eta=\sqrt{\eta_{0}^{2}-\theta^{2}}=\sqrt{25-\theta^{2}}$ in the
matrix $h^{-1}$ given in table 3 for the subgroup $K={\mathrm{U}}(1)_{I}$. All
these metrics have an isometry group a priori equal or conjugated to an over-
group of this particular subgroup.

The solution $h^{-1}$ is reminiscent of the Jensen metric: it is easy to see
that the two matrices (8) and (10) are congruent; moreover, the value of
$\kappa$ is the same. One is therefore tempted to think that both131313They
are distinct since the symmetric bilinear forms defined by these two matrices,
written in the same basis, are distinct. metrics define the same Riemannian
structure. One could nevertheless be puzzled by the fact that the specific
group ${\mathrm{S}O}(3)$ specified in the list (2) does not leave invariant
the metric (10): only the group ${\mathrm{U}}(1)_{I}$ of the list (2), leaves
it invariant (setting $r_{a}=exp(f_{a})$, the reader can indeed check that,
for $h^{-1}$ given by 8, the equation
${r_{a}}^{T}\,.\,h^{-1}\,.\,r_{a}=h^{-1}$ holds for $a=2,5,7$, whereas, for
$h^{-1}$ given by (10), this equation holds only for $a=3$. The right isometry
group of the latter can be obtained from the same equation by taking linear
combinations of the $r_{a}$ with arbitrary coefficients; one finds that this
group, of type ${\mathrm{S}O}(3)$, is generated by
$\\{\lambda_{3},\tfrac{\lambda_{4}+\lambda_{7}}{\sqrt{2}},\tfrac{\lambda_{5}+\lambda_{6}}{\sqrt{2}}\\}$.
Although distinct from the one specified in (2), it is conjugated to the
latter (because ${\mathrm{S}O}(3)$ is maximal in ${\rm SU}(3)$), and it
contains ${\mathrm{U}}(1)_{I}$, as it should.

$\bullet$ The third solution is a Lorentz metric (signature $(7,1)$).

Let $\epsilon$ be the (unique) real root of the 15-th degree polynomial

$\begin{split}&157464000\,x^{15}+403632720\,x^{14}-612290016\,x^{13}-1011752856\,x^{12}+2420977896\,x^{11}-160395147\,x^{10}+8214701211\,x^{9}\\\
&+22205850480\,x^{8}+25959494541\,x^{7}+13520748157\,x^{6}+6727192848\,x^{5}+3545761995\,x^{4}-307092303\,x^{3}+775200861\,x^{2}+1476112248\,x+416419380\end{split}$

Let $\gamma$ be the (unique) real root of the 15-th degree polynomial

$\begin{split}&1203125\,x^{15}-5947500\,x^{14}+27668175\,x^{13}-91826280\,x^{12}+247552546\,x^{11}-578539560\,x^{10}+1139842990\,x^{9}\\\
&-1943457696\,x^{8}+2859080697\,x^{7}-3567181452\,x^{6}+3705721907\,x^{5}-3090965208\,x^{4}+1958091648\,x^{3}-862410240\,x^{2}+238768128\,x-26542080\end{split}$

For these values of $\gamma$ and $\epsilon$, the cubic polynomial with one
indeterminate $x$

$\begin{split}&x^{3}(3\gamma\epsilon+3\gamma-12\epsilon)+\\\
&x^{2}\left(\gamma^{3}(-\epsilon)+12\gamma^{2}\epsilon-3\gamma^{3}-12\gamma\epsilon^{2}-4\gamma\epsilon+12\gamma-48\epsilon\right)+\\\
&x\left(-12\gamma^{3}\epsilon^{2}-7\gamma^{5}\epsilon+12\gamma^{4}\epsilon-52\gamma^{3}\epsilon+96\gamma^{2}\epsilon-3\gamma^{5}-24\gamma^{3}-48\gamma\epsilon^{2}-64\gamma\epsilon\right)\\\
&+(5\gamma^{7}\epsilon-12\gamma^{6}\epsilon+24\gamma^{5}\epsilon-48\gamma^{4}\epsilon+16\gamma^{3}\epsilon+3\gamma^{7}+12\gamma^{5})\end{split}$

has three real roots, two are negative and one is positive; call $\eta^{2}$
its positive root, and call $\eta$ the positive141414One can choose the
negative square root as well because $\eta$ appears only in even powers and in
products $(\gamma-\eta)(\gamma+\eta)$. square root of $\eta^{2}$. Then

$\beta=\frac{(\gamma-\eta)(\gamma+\eta)\left(\gamma^{2}+\eta^{2}+4\right)}{-2\left(\gamma^{2}+4\right)\eta^{2}+\gamma^{2}\left(\gamma^{2}+4\right)+\eta^{4}}$

$\kappa=\frac{\left(\gamma^{2}+\eta^{2}+2\right)\left(-2\left(\gamma^{2}+4\right)\eta^{2}+\gamma^{2}\left(\gamma^{2}+4\right)+\eta^{4}\right)}{12(\gamma-\eta)(\gamma+\eta)\left(\gamma^{2}+\eta^{2}+4\right)}$

Like $\gamma$ and $\epsilon$, the parameter $\beta$, as well as the Einstein
constant $\kappa$, can be expressed as roots of polynomials of degree 15 with
integer coefficients.

$\beta$ is the (unique) real root of the polynomial

$\begin{split}&420959000000\,x^{15}-1864887536000\,x^{14}+3473091156700\,x^{13}-3742325355930\,x^{12}+2779023618983\,x^{11}-1598512715722\,x^{10}+738336195619\,x^{9}\\\
&-286057154856\,x^{8}+100590932418\,x^{7}-32232937198\,x^{6}+8922748831\,x^{5}-2060272970\,x^{4}+375594480\,x^{3}-51335104\,x^{2}+4940624\,x-297440\end{split}$

The Einstein constant $\kappa$ is the (unique) real root of the polynomial

$\begin{split}&75874469299200000000\,x^{15}-194337331275110400000\,x^{14}+301355277599416320000\,x^{13}-332561544757530624000\,x^{12}+282171231781966252800\,x^{11}\\\
&-191136024361902738240\,x^{10}+105464748331948650048\,x^{9}-47804548501070787024\,x^{8}+17858543123347792128\,x^{7}-5477519217851980920\,x^{6}\\\
&+1363429678619072700\,x^{5}-269374969407033333\,x^{4}+40612859877938577\,x^{3}-4362120554579953\,x^{2}+293255347774576\,x-9061971967716\end{split}$

The real $\eta^{2}$ is the (unique) real root of the polynomial

$\begin{split}&7237548828125\,x^{15}+70864769531250\,x^{14}+314655757840625\,x^{13}+889027170133500\,x^{12}+1845686712291930\,x^{11}+2969194934204748\,x^{10}\\\
&+6007481883873834\,x^{9}+14368049748482976\,x^{8}+23991657392689833\,x^{7}+23305737247777970\,x^{6}+9939040159739877\,x^{5}\\\
&-2269867978871308\,x^{4}-3190456836365280\,x^{3}+2429318649600\,x^{2}+508754442240000\,x-6234734592000\end{split}$

Both square roots of $\eta^{2}$ solve the equations and therefore give rise to
two distinct solutions, for the same values of the other parameters.

This Lorentzian Einstein solution is therefore obtained for a dual metric
specified by the matrix $h^{ij}$ given in (9), with the above values of the
parameters. Numerically, $\eta^{2}\simeq 0.0122658$, and

$\\{\epsilon\simeq-0.491148,\,\gamma\simeq 0.233098,\,\eta\simeq\pm
0.110751,\,\beta\simeq
1.41407,\,\zeta=0,\,\alpha=1\\}\quad\text{and}\quad\kappa\simeq 0.121788$ (11)

One can restore the $\alpha$ dependence by scaling the parameters
$\eta,\gamma,\epsilon,\beta$, by $\alpha$. In that case, the Einstein constant
$\kappa$ is also multiplied by $\alpha$. Remember that the Ricci tensor is
invariant under a (constant) rescaling of the metric.

The scalar curvature $\tau$, for the general family of metrics specified by
(9), is

$\frac{8\alpha^{2}\beta\epsilon\left(\gamma^{2}-2\eta^{2}\right)+2\alpha^{3}\epsilon\left(\eta^{2}-\gamma^{2}\right)+\alpha\left(6\beta\eta^{2}\left(\gamma^{2}-4\gamma\epsilon-2\epsilon^{2}\right)-\gamma^{3}(3\beta(\gamma-8\epsilon)+\gamma\epsilon)+\eta^{4}(\epsilon-3\beta)\right)-2\beta\epsilon\left(\gamma^{2}-\eta^{2}\right)^{2}}{12\alpha\beta\epsilon(\gamma-\eta)(\gamma+\eta)}$
(12)

Using the previous values of parameters, one finds that $\tau$, for the
Lorentz-Einstein metric, is equal to $8\kappa$, as it should. Numerically,
$\tau\simeq 0.974303$. Moreover, $\Lambda$, the “cosmological” constant, is
equal to $3\kappa\simeq 0.365363$.

Other properties of the obtained Lorentzian Einstein metric:

  1. 1.

The matrix $h$ has seven positive eigenvalues, and one negative: its signature
is Lorentzian $(7,1)$. Using $\alpha=1$ these numerically sorted eigenvalues
are $(8.17347,8.17347,2.90825,2.90825,1.,1.,0.707178,-2.03605)$.

  2. 2.

The Einstein condition gives two solutions differing from one another by
flipping the sign of $\eta$.

  3. 3.

One can calculate the eight principal Ricci curvatures, check that they are
constant (Einstein manifolds have constant Ricci curvature), all equal to
$\tau/8$. As $\tau>0$, the Ricci signature (the signature of the Ricci
quadratic form) is $(8,0)$.

  4. 4.

We already know, from the chosen parametrization, that the right isometry
group of this metric is ${\mathrm{U}}(1)_{I}$, the vector field $e_{3}$
defined by the basis vector $X_{3}$ being its associated Killing vector field.

  5. 5.

This Lorentzian manifold has, at every point, a cone of time-like directions.
The underlying manifold, being a Lie group, is parallelizable, orientable, and
it is time-orientable for this Lorentz metric. Numerically,
$h(X_{8},X_{8})=-2.03605<0$, the vector field $e_{8}$ (which is not Killing)
is therefore time-like. Notice that the Killing vector field $e_{3}$ is space-
like. The integral curve of the left-invariant vector field $e_{8}$ is a
closed time-like curve. Moreover, it is a geodesic (it is easy to show that
the covariant derivative $\nabla_{e_{8}}\,e_{8}$ vanishes). The integral curve
of $e_{3}$ is also a geodesic.

  6. 6.

One can check that this Lorentzian Einstein metric is a stationary point of
the scalar curvature, when one varies the parameters while keeping the volume
fixed. This provides another way to obtain the above solution. For the metrics
specified by (9), the determinant of $h^{-1}$ is ${\sl
d}=\alpha^{2}\beta\epsilon\left(-2\gamma^{2}\eta^{2}+\gamma^{4}+\eta^{4}\right)$,
and the scalar curvature of the family of metrics ${\sl d}^{1/8}\times h$ (for
which the determinant stays equal to $1$ when the parameters vary) is
$\tau/{\sl d}^{1/8}$, where the expression of $\tau$ in terms of the
parameters $\eta,\gamma,\epsilon,\beta,\alpha$ was given in (12). We shall
only display a few curves that illustrate the stationarity property by giving
plots of $\tfrac{\partial}{\partial u}\tfrac{\tau}{{(-\sl d)}^{1/8}}$, for
$u\in\\{\eta,\gamma,\epsilon,\beta\\}$, in a neighborhood of the found
solution151515The determinant being negative around the extremum that
corresponds to the obtained Einstein metric (because $\epsilon<0$), we
introduce a minus sign in front of ${\sl d}$ in ${\sl d}^{1/8}$..

Figure 1:  Derivative of $\tfrac{\tau}{{(-\sl d)}^{1/8}}$ with respect to
$\beta$, for $\beta$ in $[0,5]$ and in $[1.41,1.42]$.

Figure 2:  Derivative of $\tfrac{\tau}{{(-\sl d)}^{1/8}}$ with respect to
$\gamma$, for $\gamma$ in $[-0.5,0.5]$, $[0.2,0.26]$ and in $[0.231,0.235]$.

Figure 3:  Derivative of $\tfrac{\tau}{{(-\sl d)}^{1/8}}$ with respect to
$\epsilon$, for $\epsilon$ in $[-1,1]$ and in $[-0.5,-0.48]$.

Figure 4:  Derivative of $\tfrac{\tau}{{(-\sl d)}^{1/8}}$ with respect to
$\eta$, for $\eta$ in $[-0.5,0.5]$, $[0,0.15]$, and in $[0.1102,0.1114]$.

  7. 7.

From the previous Lorentz Einstein metric, defined by parameters values that
we now call $\alpha_{0},\beta_{0},\gamma_{0}$,
$\epsilon_{0},\eta_{0},\zeta_{0}$ (remember that we had imposed a priori the
conditions $\theta=0$ and $\delta=\gamma$), one obtains a one-parameter family
of distinct Lorentz Einstein metrics, with the same Einstein constant, for the
same values $\alpha_{0},\beta_{0},\gamma_{0},\epsilon_{0},\zeta_{0}$, but for
arbitrary values of $\theta$ (obeying $\theta^{2}\leq\eta_{0}^{2}$), while
setting $\eta=\sqrt{\eta_{0}^{2}-\theta^{2}}$ in the matrix $h^{-1}$ given in
table 3 for $K={\mathrm{U}}(1)_{I}$ (see our discussion at the end of sect.
2.3). In particular we could trade $\eta$ for $\theta$ by taking
$\theta=\eta_{0}$, then $\eta$ vanishes. All these metrics define the same
pseudo-Riemannian structure. They have the same isometry group
${\mathrm{U}}(1)_{I}$. Notice that the calculations presented in the present
subsection ($K={\mathrm{U}}(1)_{I}$) do not exclude the fact that the right
isometry group could be equal or conjugated to a group larger than this
particular ${\mathrm{U}}(1)$, but it cannot be so, otherwise we would have
already found this left-invariant Einstein Lorentzian metric in one of the
previous subsections.

#### $K={\mathrm{U}}(1)_{Y}$

Such left-invariant metrics are parametrized by the last entry of table 3.
Even after taking into account isometries and scaling, there are too many free
parameters left ($10$ of them) and we could not solve the Einstein condition
for this family in full generality. For this reason we looked at several
subfamilies obtained by imposing conditions on the parameters, but, even then,
we could not find a single example of an Einstein metric in this family,
except, of course, the Killing metric, for which the right isometry group is
${\rm SU}(3)$ itself.

#### $K=\\{e\\}$

Solving explicitly the system of equations coming from the Einstein condition
for this family seems to be a formidable task, even after reducing the number
of parameters from $36$ to $28$ by considering metrics only up to equivalence.
So we shall not have much to say in that case.

We should nevertheless mention one pseudo-Riemannian Einstein metric, of
signature $(6,2)$, for which $K=\\{e\\}$, and that was found in [10], using
other notations. We shall describe it below. Consider first the family of
metrics defined by taking $h^{-1}$ equal to

$\left(\begin{array}[]{cccccccc}\alpha&.&.&.&.&.&.&.\\\ .&\beta&.&.&.&.&.&.\\\
.&.&\gamma&.&.&.&.&.\\\ .&.&.&\alpha&.&.&.&.\\\ .&.&.&.&\beta&.&.&.\\\
.&.&.&.&.&\alpha&.&.\\\ .&.&.&.&.&.&\beta&.\\\ .&.&.&.&.&.&.&\gamma\\\
\end{array}\right)$

For generic values of the parameters $\alpha,\beta,\gamma$, these left-
invariant metrics have a trivial right-isometry group $K$ (the equation for
Lie derivatives stemming from (1) has no non-trivial solution) even though the
same parameter $\beta$ occurs in positions $(2,5,7)$ which are those
corresponding to the generators $\lambda_{a}$ of the ${\mathrm{S}O}(3)$
subgroup defined in (2). For $\alpha=\beta=\gamma$ the right isometry group
$K$ is ${\rm SU}(3)$, and for $\alpha=\gamma$ one recovers the cases already
described in (3) for which $K={\mathrm{S}O}(3)$.

The non-zero components of the Ricci tensor are
$\varrho_{11}=\varrho_{44}=\varrho_{66}$,
$\varrho_{22}=\varrho_{55}=\varrho_{77}$, $\varrho_{33}=\varrho_{88}$, they
are respectively equal to:

$\left\\{\frac{1}{12}\left(\frac{2\beta\gamma}{\alpha^{2}}-\frac{\alpha}{\beta}-\frac{2\gamma}{\beta}-\frac{2\beta}{\gamma}+6\right),\frac{1}{24}\left(\frac{\alpha^{2}}{\beta^{2}}+\frac{4\alpha\gamma}{\beta^{2}}-\frac{4\alpha}{\gamma}-\frac{4\gamma}{\alpha}+9\right),\frac{1}{4}\left(\frac{\alpha\beta}{\gamma^{2}}-\frac{\alpha}{\beta}-\frac{\beta}{\alpha}+2\right)\right\\}$

The Einstein condition is obtained by setting the previous triple equal to
$\\{\kappa/\alpha,\kappa/\beta,\kappa/\gamma\\}$. This system of equations has
three real solutions: one first recovers the multiples of the Killing metric
by taking $\alpha=\beta=\gamma$, with Einstein constant $\kappa=\alpha/4$,
then one recovers the multiples of the Jensen metric, for which
$\alpha=\gamma$, $\beta=11\alpha$, and $\kappa=\tfrac{21}{44}\,\alpha$;
finally one obtains a third solution that we describe now.

Let $P$ be a cubic polynomial with one indeterminate $x$ and real
coefficients, call ${\mathfrak{r}}(P)$ its smallest real root. Then, taking
$\beta=\alpha\,{\mathfrak{r}}(85x^{3}-29x^{2}+27x-3)$ and
$\gamma=\alpha\,{\mathfrak{r}}(768x^{3}+128x^{2}+204x+45)$ defines an Einstein
metric with Einstein constant
$\kappa=\alpha\,{\mathfrak{r}}(14400x^{3}-5520x^{2}+1044x-101)$ and scalar
curvature $\tau=\alpha\,{\mathfrak{r}}(-808+1044x-690x^{2}+225x^{3})$.
Numerically $\beta/\alpha\simeq 0.121$, $\gamma/\alpha\simeq-0.213$,
$\kappa/\alpha\simeq 0.196$, $\tau/\alpha\simeq 1.568$. This left-invariant
Einstein metric has signature $(6,2)$ and its right isometry group $K$ is
trivial.

As already mentioned this solution was already found in [10] where the authors
give the matrix elements of $h$ (not of its inverse $h^{-1}$), using a
different scaling, in terms of two reals $x_{1},x_{2}$. Their values can be
compared to the above ones by writing
$h=({1}/{\beta})\text{diag}(\\{x_{1},1,x_{2},x_{1},1,x_{1},1,x_{2}\\})$; one
finds $x_{1}=\beta/\alpha$ (given above) and
$x_{2}=\beta/\gamma={\mathfrak{r}}(768-128x-1860x^{2}+1275x^{3})\simeq-0.570$.
Notice that $x_{2}=-\tfrac{(1-x_{1})(1-5x_{1})}{5x_{1}}$. The Einstein
constant for the metric $\beta\,h$ is $\kappa/\beta\simeq 1.616$, and can be
written161616The Einstein constants given in reference [10] differ from ours
by two overall multiplicative factors: one comes from the fact that their
matrix expression of $h$, compared to ours, is rescaled by $\beta$, and the
other (equal to $3$) comes from the fact that the basis vectors used by these
authors to define their metrics differ from our basis vectors $(X_{i})$ by a
factor $\sqrt{3}$. $\frac{(1-x_{1})(10x_{1}-1)}{20(1-5x_{1})x_{1}^{2}}$.

##  4 Miscellaneous

###  4.1 The quadratic Casimir operator

The quadratic Casimir element of the simple Lie group $G$ for the renormalized
Killing form (resp. for the Killing form) is the element of the universal
enveloping algebra defined171717We remind the reader that the Killing inner
product $k$ is the opposite of the Killing form, hence the minus sign in front
of the expressions defining $\Omega_{2}^{k}$ and ${\widehat{\Omega}_{2}}$,
since $(X_{a})$ is an orthonormal basis for $k$. See sect. 2.2. by
${\widehat{\Omega}_{2}}=-\sum_{a}\widehat{X}_{a}.\widehat{X}_{a}$ (resp.
${\Omega_{2}}=-\sum_{a}X_{a}.X_{a}$). Casimir elements can be evaluated in any
representation, and, in an irreducible representation,
${\widehat{\Omega}_{2}}$ (resp. ${\Omega_{2}}$) is a multiple of the identity
matrix, with eigenvalue ${\widehat{C}_{2}}$ (resp. ${C_{2}}$). The definition
of Casimir operators involves the inverse Killing inner product, so, using
$\widehat{k}=k/2g$, one obtains the relation181818We also remind the reader
that $g$ is the dual Coxeter number, which is equal to $N$ for ${\rm SU}(N)$.:

${C_{2}}={\widehat{C}_{2}}/2g.$ (13)

Explicitly, for an irreducible representation of highest weight $\mathpzc{w}$,
one obtains

${\widehat{C}_{2}}=\langle\mathpzc{w}+\rho,\mathpzc{w}+\rho\rangle-\langle\rho,\rho\rangle=\langle\mathpzc{w},\mathpzc{w}+2\rho\rangle$
(14)

where $\rho$ is the Weyl vector and $\langle.,.\rangle$ is the Cartan inner
product in the space of roots, normalized in such a way that the length square
of long roots is equal to $2$. One has also:

${C_{2}}=\sum_{\alpha}\langle\mathpzc{w}+\rho,\alpha\rangle^{2}-\langle\rho,\alpha\rangle^{2}$
(15)

where $\alpha$ runs over the set of all roots (use the identity
$\sum_{\alpha}|\alpha\rangle\langle\alpha|=2g$ to relate (14) and (15) as in
(13)).

For ${\rm SU}(N)$ in the defining representation one obtains
${\widehat{C}_{2}}=(N^{2}-1)/N$ and ${C_{2}}=(N^{2}-1)/2$. In the adjoint
representation one obtains ${\widehat{C}_{2}}=2N$ and ${C_{2}}=1$.

In the case of ${\rm SU}(3)$, one can use for instance (14) to show that, for
an irreducible representation of highest weight $\mathpzc{w}$ with (Dynkin)
components $(o_{1},o_{2})$ in the basis of fundamental weights,

${\widehat{C}_{2}}=\frac{2}{3}(o_{1}^{2}+o_{1}o_{2}+o_{2}^{2})+2(o_{1}+o_{2})$
(16)

Equivalently, one can evaluate
${\widehat{\Omega}_{2}}=-\sum_{a}\tfrac{iL_{a}}{\sqrt{2}}.\tfrac{iL_{a}}{\sqrt{2}}$
and
${\Omega_{2}}=-\sum_{a}\tfrac{iL_{a}}{2\sqrt{3}}.\tfrac{iL_{a}}{2\sqrt{3}}$ in
the chosen representations. The above general relations, in the case of ${\rm
SU}(3)$, give: ${\widehat{C}_{2}}=6$, ${C_{2}}=1$ in the adjoint
representation, and ${\widehat{C}_{2}}=8/3$, ${C_{2}}=4/9$ in the defining
representation (these values can also be directly calculated by representing
the $iL_{a}$ generators by matrices $2f_{a}$ in the former case and by
matrices $i\lambda_{a}$ in the latter).

For the group ${\rm SU}(2)$, and for an irreducible representation of highest
weight $2j$ (where the “spin” variable $j$ is an integer or a half-integer),
of dimension $2j+1$, the value $j(j+1)$ presented in the majority of quantum
physics textbooks as eigenvalue of “the Casimir operator” corresponds to a
Casimir element neither associated with the Killing form on ${\rm SU}(2)$
(${C_{2}}=j(j+1)/2$) nor with the renormalized Killing form
(${\widehat{C}_{2}}=2j(j+1)$). Details: the unique long root, which is also
the highest weight $\sigma=2$ of the vector representation (of dimension 3),
obeys $<2,2>=2$, so $<1,1>=1/2$, and (14), using $\rho=1$, indeed gives
${\widehat{C}_{2}}\,=\,<2j+1,2j+1>-<1,1>=((2j+1)^{2}-1)<1,1>=4j(j+1)<1,1>=2j(j+1)$.
In order to obtain $j(j+1)$ one has to use another rescaled Killing form,
namely $k/2=2{\widehat{k}}$, in which case the associated Casimir can still
formally be given by the rhs of 14, provided one normalizes the Cartan inner
product in such a way that the length square of long root is equal to $1$, a
choice that is also often made in the same quantum physics textbooks (but
remember that for us this length square is equal to $2$).

##### Dynkin index.

In an arbitrary basis $(e_{a})$, we have
$Tr(\mathpzc{w}(e_{a})\mathpzc{w}(e_{b}))=-2\iota{w}\;{\widehat{k}}_{ab}=-\iota_{\mathpzc{w}}/g\,k_{ab}$.
Here $\iota_{\mathpzc{w}}$ denotes the Dynkin index191919Some authors
incorporate a pre-factor $2$ in the definition of the Dynkin index. of the
representation $\mathpzc{w}$ of the Lie group $G$. If $\mathpzc{w}$ is the
defining representation of ${\rm SU}(N)$, one has $\iota{w}=1/2$. If
$\mathpzc{w}$ is the adjoint representation of $G$, one has $\iota{w}=g$; in
particular, for $G={\rm SU}(N)$, $\iota{w}=N$. More generally, one has the
relation:
${\widehat{C}_{2}}=2\,\iota_{\mathpzc{w}}\times{\text{dim}(Lie(G))}/{\text{dim}(\mathpzc{w})}$.

###  4.2 Restriction to subgroups: branching

We consider the Lie algebra embedding
${\mathrm{Lie}}({\mathrm{U}}(2))\subset{\mathrm{Lie}}({\rm SU}(3))$ i.e.,
$\mathfrak{su}(2)\oplus\mathfrak{u}(1)\subset\mathfrak{su}(3)$, and we take
$\mathfrak{u}(1)$ as the Lie algebra of the subgroup called
${\mathrm{U}}(1)_{Y}$ in previous sections. This is a Levi type subalgebra :
the set of simple roots of the semi-simple component of the subalgebra can be
chosen as a subset of the set of simple roots of the given Lie algebra. Call
$\alpha_{1},\alpha_{2}$ the simple roots of $\mathfrak{su}(3)$ and
$\omega_{1},\omega_{2}$ its fundamental weights. We take $v=\alpha_{1}$ as the
simple root of $\mathfrak{su}(2)$ (the “$v$” stands for “vector” since the
$\mathfrak{su}(2)$ irrep of highest weight $v$ is the vector representation)
and $t$ the fundamental $\mathfrak{u}(1)$ weight. The ${\mathrm{U}}(1)_{Y}$
generator is $3\,Y=\sqrt{3}\,L_{8}$ and reads
$\sqrt{3}\,\lambda_{8}=\text{diag}(1,1,-2)$ in the defining representation;
its eigenvalues are integers, as they should. Notice that
$k(\sqrt{3}\,iL_{8},\sqrt{3}\,iL_{8})=3\times 12$, so
$\widehat{k}(\sqrt{3}\,iL_{8},\sqrt{3}\,iL_{8})=3\times 12/6=6$ and
$\widehat{k}^{-1}(t,t)=1/6$. Notice also that $v=2\sigma$, where $\sigma$
denotes the $\mathfrak{su}(2)$ fundamental weight202020The component along
$\sigma$ of each weight of an irrep of $\mathfrak{su}(2)$ is equal to twice
the “(iso-)spin”. For instance those of the spinorial irrep (highest weight
$\sigma$) are twice $\pm 1/2$, those of the vectorial irrep (highest weight
$v$) are twice $(1,0,-1)$..

The simple root $\alpha_{2}$ of $\mathfrak{su}(3)$ is a priori a linear
combination of $v$ and $t$: we have $\alpha_{2}=a\,v+b\,t$. We determine $a$
and $b$ from the inner products of roots and weights calculated using the
Cartan matrix or its inverse. As usual, all roots have length $2$ both for
$\mathfrak{su}(3)$ and for $\mathfrak{su}(2)$ (we have only long roots here),
so $\langle\,\alpha_{1},\alpha_{1}\rangle=\langle\,v,v\rangle=2$. From the
Cartan matrix of $\mathfrak{su}(3)$, namely $\begin{pmatrix}2&-1\\\
-1&2\end{pmatrix}$, we get $\langle\,\alpha_{1},\alpha_{2}\rangle=-1$,
moreover $\mathfrak{su}(2)$ and $\mathfrak{u}(1)$ are orthogonal subspaces for
$\langle\,,\,\rangle$, so $\langle\,v,\,t\rangle=0$, therefore
$a\langle\,v,\,v\rangle=-1$, and we obtain $a=-1/2$. We have also
$\langle\,\alpha_{2},\alpha_{2}\rangle=2$, therefore
$a^{2}\langle\,v,v\rangle+b^{2}\langle\,t,t\rangle=2$. Using
$\langle\,t,t\rangle=1/6$ one gets $b=3$. Therefore $\alpha_{1}=v$, and
$\alpha_{2}=-v/2+3t$.

The restriction matrix defining the embedding in terms of fundamental weights
(which also gives the ${\mathrm{U}}(2)$ weight components $2I$ and $3Y$ from
the Dynkin components $(o_{1},o_{2})$ of the highest weight $\mathpzc{w}$ of
any irreducible ${\rm SU}(3)$ representation) reads:

$\left(\begin{array}[]{c}\omega_{1}\\\ \omega_{2}\\\
\end{array}\right)=\left(\begin{array}[]{cc}1&1\\\ 0&2\\\
\end{array}\right)\left(\begin{array}[]{c}\sigma\\\ t\\\
\end{array}\right)\qquad(2I,3Y)=\left(o_{1},o_{2}\right)\,\left(\begin{array}[]{cc}1&1\\\
0&2\\\ \end{array}\right)$ (17)

Examples.
Consider the basic (fundamental) irrep of ${\rm SU}(3)$ with highest weight
$\mathpzc{w}=(1,0)$, of dimension $3$. Using the restriction matrix (eq 17) on
the weight system of $\mathpzc{w}$, namely $\\{(1,0),(-1,1),(0,-1)\\}$ we
obtain the weights appearing in the branching from $\mathfrak{su}(3)$ to
$\mathfrak{su}(2)\oplus\mathfrak{u}(1)$, namely $\\{(1,1),(-1,1),(0,-2)\\}$;
the associated decomposition of irreps, in terms of highest weights, reads
$(1,0)\rightarrow(1,1)\oplus(0,-2)$ where, on the right hand side, the first
member ($2I$) of each pair is the component along $\sigma$ of the ${\rm
SU}(2)$ highest weight and where the second member ($3Y$) is the component of
the ${\mathrm{U}}(1)$ weight along $t$. Equivalently, in terms of
dimensions212121Remember that an ${\rm SU}(2)$ irrep with highest weight $2I$
(i.e., spin $I$) has dimension $2I+1$, and that an ${\rm SU}(3)$ irrep with
highest weight components $(o_{1},o_{2})$ has dimension
$(o_{1}+1)(o_{2}+1)(o_{1}+o_{2}+2)/2$.: $[3]\rightarrow[2]_{1}\oplus[1]_{-2}$
where the subindex of $[2I+1]_{3Y}$ refers to the component of the
${\mathrm{U}}(1)$ weight. Conservation of the ${\mathrm{U}}(1)$ (hyper) charge
reads $2\times(1)+1\times(-2)=0$.
For the adjoint representation (highest weight $\mathpzc{w}=(1,1)$, of
dimension $8$), the branching rule can be obtained in the same way and reads,
when written in terms of dimensions (no confusion can arise in this case):
$[8]\rightarrow[3]_{0}\oplus[2]_{3}\oplus[2]_{-3}\oplus[1]_{0}$.
Let us conclude this section with a slightly more involved example: we
consider the ${\rm SU}(3)$ representation of highest weight
$\mathpzc{w}=(2,1)$, which is of dimension $[15]$. Using the restriction
matrix on the weight
system222222$\\{(2,1),(3,-1),(0,2),(1,0),(1,0),(-2,3),(2,-2),(-1,1),(-1,1),(0,-1),(0,-1),(-3,2),(1,-3),(-2,0),(-1,-2)\\}$.
of this highest weight of ${\rm SU}(3)$, we obtain the
weights232323$\\{(2,4),(3,1),(0,4),(1,1),(1,1),(-2,4),(2,-2),(-1,1),(-1,1),(0,-2),(0,-2),(-3,1),(1,-5),(-2,-2),(-1,-5)\\}$.
appearing in the branching to ${\mathrm{U}}(2)$. The associated decomposition
reads
$(2,1)\rightarrow(3,1)\oplus(2,4)\oplus(2,-2)\oplus(1,1)\oplus(1,-5)\oplus(0,-2)$
where, again, on the right hand side, the first member ($2I$) of each pair is
the component along $\sigma$ of the ${\rm SU}(2)$ highest weight and where the
second member ($3Y$) is the component along $t$ of the ${\mathrm{U}}(1)$
weight. In terms of dimensions, this rhs reads
$[4]_{1}\oplus[3]_{4}\oplus[3]_{-2}\oplus[2]_{1}\oplus[2]_{-5}\oplus[1]_{-2}$
and we can check the conservation of the ${\mathrm{U}}(1)$ (hyper) charge:
$4\times(1)+3\times(4)+3\times(-2)+2\times(1)+2\times(-5)+1\times(-2)=0$.

###  4.3 Laplacian

Let $h$ be a Riemannian or pseudo-Riemannian metric on the Lie group $G$.
Assuming that $h$ is left-invariant (hence homogeneous), we can write
$h=h_{ab}\,\theta^{a}\otimes\theta^{b}$ where $h_{ab}$ are constants (real
numbers) and $(\theta^{a})$ is the global moving co-frame dual to the
arbitrary moving frame $(e_{a})$ defined from an arbitrary basis, also called
$(e_{a})$, in the Lie algebra of $G$ identified with the tangent space to $G$
at the identity. The dual (i.e., inverse) metric reads
$h^{-1}=h^{ab}\,e_{a}\otimes e_{b}$. With the usual convention, the rough
metric Laplacian (or Laplace-Beltrami operator) on functions on the manifold
has negative spectrum —so it is the opposite of the De Rham Laplacian on
$0$-forms— and can be written as the second-order differential operator
$\Delta=h^{ab}\,e_{a}\circ e_{b}$ where the vector fields $e_{a}$ act on
functions on $G$. More generally, when studying the action of the Laplacian on
sections of vector bundles over $G$, the $e_{a}$ would act as a Lie derivative
of sections in the direction $a$.

##### Laplacian of bi-invariant metrics.

We call $\Delta_{0}$ the Laplacian associated with the Killing metric $k$; its
eigenstates are labelled by irreducible representations $\mathpzc{w}$ of $G$,
the eigenvalues of $-\Delta_{0}$ are equal to the Casimir eigenvalues
${C_{2}}$ (see 15) evaluated in the representation $\mathpzc{w}$, and the
degeneracy is $dim(\mathpzc{w})^{2}$, see [2] and [8].

##### Laplacian of left-invariant metrics.

Let $h$ be an arbitrary left-invariant metric on the Lie group $G$, the
spectrum of the corresponding Laplacian $\Delta$ is discussed in a number of
places (see for instance [13], [15], and references therein). Using the Peter-
Weyl theorem together with left-invariance of the metric one can replace a
difficult problem of analysis on manifolds by a simpler algebraic problem: as
in the bi-invariant case, the eigenvalues of the Laplace operator can be
obtained, up to sign, as eigenvalues of some appropriate metric-dependent
modified Casimir operator (consider for instance the expression (19) below)
evaluated in irreducible representations of $G$. One should be careful with
this terminology because the associated modified Casimir elements (that can be
defined in the enveloping algebra of $\text{Lie}(G)$) are not, in general,
central.

For left-invariant metrics $h$ with isometry group $G\times K$ and more
generally for naturally reductive metrics on Lie groups one can certainly
write general results but here we only want to focus on the $K$ dependence of
the eigenvalues in a few specific cases, and we shall be happy with some
elementary calculations.

So we return to the case $G={\rm SU}(3)$, call $X_{a}$ the vectors of an
orthonormal basis for the Killing metric, $h^{ab}$ the covariant components of
some chosen left-invariant metric $h$ in the same basis, and $C(\mathpzc{w})$
the list of eigenvalues of the operator $\Delta=h^{ab}\,X_{a}.X_{b}$ evaluated
in some chosen non trivial representation $\mathpzc{w}$ of $G$. The degeneracy
of each eigenvalue is at least $dim(\mathpzc{w})$. With the notations of sect.
2.2, namely setting $X_{a}=\frac{i}{2\sqrt{3}}L_{a}$, we can also write
$\Delta=-\tfrac{1}{12}h^{ab}\,L_{a}.L_{b}$. Taking for instance $h=k$, the
Killing metric, we have

$\begin{split}\Delta_{0}=&\left(X_{1}.X_{1}+X_{2}.X_{2}+X_{3}.X_{3}\right)+\left(X_{4}.X_{4}+X_{5}.X_{5}+X_{6}.X_{6}+X_{7}.X_{7}\right)+X_{8}.X_{8}\\\
{}=&\frac{-1}{12}\left(\left(L_{1}.L_{1}+L_{2}.L_{2}+L_{3}.L_{3}\right)+\left(L_{4}.L_{4}+L_{5}.L_{5}+L_{6}.L_{6}+L_{7}.L_{7}\right)+L_{8}.L_{8}\right)\\\
\end{split}$ (18)

and replacing $L_{a}$ by $\lambda_{a}$ (in the defining representation), or by
$-2if_{a}$ (in the adjoint), one recovers the known Casimir eigenvalues.

Let us now choose a metric $h$ for which $K={\mathrm{U}}(2)$, with parameters
$\alpha,\beta,\gamma$ as in (3). The Laplacian reads as follows, and we may
introduce the notation ${\Omega_{2}^{{\mathrm{U}}(2)}}$ to denote the
“modified Casimir operator” defined as $-\Delta$.

$\begin{split}\Delta=&\frac{-1}{12}\left(\alpha\left(L_{1}.L_{1}+L_{2}.L_{2}+L_{3}.L_{3}\right)+\beta\left(L_{4}.L_{4}+L_{5}.L_{5}+L_{6}.L_{6}+L_{7}.L_{7}\right)+\gamma
L_{8}.L_{8}\right)\end{split}$ (19)

In the fundamental representation $\mathpzc{w}=(1,0)$ of ${\rm SU}(3)$,
$-\Delta$ is a $3\times 3$ diagonal matrix with diagonal:

$C(1,0)=\left\\{\frac{1}{36}(9\alpha+6\beta+\gamma),\frac{1}{36}(9\alpha+6\beta+\gamma),\frac{1}{9}(3\beta+\gamma)\right\\}$

In the adjoint representation $\mathpzc{w}=(1,1)$, $-\Delta$ is an $8\times 8$
diagonal matrix with diagonal:

$C(1,1)=\left\\{\frac{1}{3}(2\alpha+\beta),\frac{1}{3}(2\alpha+\beta),\frac{1}{3}(2\alpha+\beta),\frac{1}{4}(\alpha+2\beta+\gamma),\frac{1}{4}(\alpha+2\beta+\gamma),\frac{1}{4}(\alpha+2\beta+\gamma),\frac{1}{4}(\alpha+2\beta+\gamma),\beta\right\\}$

More generally, consider the difference $\Delta-\beta\Delta_{0}$, this makes
the term $L_{4}.L_{4}+L_{5}.L_{5}+L_{6}.L_{6}+L_{7}.L_{7}$ disappear:

$\begin{split}\Delta-\beta\Delta_{0}=&\frac{-1}{12}\left((\alpha-\beta)\left(L_{1}.L_{1}+L_{2}.L_{2}+L_{3}.L_{3}\right)+(\gamma-\beta)L_{8}.L_{8}\right)\end{split}$

From the discussion in sect. 4.1, we identify
$L_{1}.L_{1}+L_{2}.L_{2}+L_{3}.L_{3}$ with the ${\rm SU}(2)$ quadratic
Casimir, of eigenvalue $4I(I+1)$ in the irreducible representation of
isospin242424In particle physics applications, $I$ is called the isospin, and
$Y$ the hypercharge, see our paragraph on notations in sect. 2.3 $I$ (the
highest weight component is $2I$), and $L_{8}.L_{8}$ with the
remaining252525The restriction of $h$ to ${\mathrm{U}}(2)$ is a bi-invariant
metric on this subgroup. quadratic ${\mathrm{U}}(1)$ operator, of eigenvalue
$3Y^{2}$. For an arbitrary representation $\mathpzc{w}=(o_{1},o_{2})$ of ${\rm
SU}(3)$, and for a representation that appears in the branching from
$\mathpzc{w}$ to the subgroup ${\mathrm{U}}(2)$ (remember, see sect. 4.2, that
such a term is characterized by the highest weight $2I$ of ${\rm SU}(2)$ and a
weight $3Y$ of ${\mathrm{U}}(1)$), the eigenvalues of the modified quadratic
Casimir operator ${\Omega_{2}^{{\mathrm{U}}(2)}}=-\Delta$, with $\Delta$ given
by (19), are therefore:

$C(o_{1},o_{2};I,Y)=\beta\,{C_{2}(o_{1},o_{2})}+((\alpha-\beta)\,\frac{1}{3}\,I(I+1)+(\gamma-\beta)\,\frac{1}{4}\,Y^{2})$
(20)

where $C_{2}$ is the eigenvalue of the Casimir element associated with the
Killing form of ${\rm SU}(3)$.
If $\mu^{2}\in\mathbb{R}^{+}$, then, upon scaling of $h^{-1}$ by $\mu^{2}$,
the rhs of the previous equation gets multiplied by the same factor.

Examples.
Consider the fundamental irrep of ${\rm SU}(3)$ with highest weight
$\mathpzc{w}=(1,0)$, of dimension $3$. The eigenvalue of $-\Delta_{0}$ given
by (16) is $4/9$ whereas those of $-\Delta$, given by (20), for each of the
terms appearing in the branching rule obtained at the end of sect. 4.2, are
sums of three contributions: the irrep $[2]_{1}$ in the branching of $[3]$ has
$I=1/2$ and $Y=1/3$ since $[2]_{1}\equiv[2I+1]_{3Y}$, and it is such that
$\left\\{\beta\,{{C_{2}}},\frac{1}{3}I(I+1)(\alpha-\beta),\frac{1}{4}Y^{2}(\gamma-\beta)\right\\}=\left\\{\frac{4\beta}{9},\frac{\alpha-\beta}{4},\frac{\gamma-\beta}{36}\right\\}$,
whose sum is $\frac{1}{36}(9\alpha+6\beta+\gamma)$; in the same way, the three
contributions for the irrep $[1]_{-2}$ sum to $\frac{1}{9}(3\beta+\gamma)$.
Both values were expected since we had to recover the eigenvalues of $-\Delta$
given by the diagonal $3\times 3$ matrix $C(1,0)$ obtained previously.
For the adjoint representation, whose branching to ${\mathrm{U}}(2)$ was also
given in sect. 4.2, the eigenvalue of $-\Delta_{0}$ is $1$, and those of
$-\Delta$, calculated using (20), coincide, of course, for each of the four
terms of the branching decomposition, with the components of the diagonal
$8\times 8$ matrix $C(1,1)$ obtained previously.
Let us finally consider the ${\rm SU}(3)$ representation of highest weight
$\mathpzc{w}=(2,1)$, of dimension $[15]$, whose branching to ${\mathrm{U}}(2)$
was also considered in sect. 4.2. The eigenvalue of $-\Delta_{0}$ given by
(16) is $32/3$ and those of $-\Delta$, given by (20), read respectively
$[4]_{1}:\frac{5\alpha}{12}+10\beta+\frac{\gamma}{4}$,
$[3]_{4}:\frac{2}{3}(\alpha+16\beta-\gamma)$,
$[3]_{-2}:\frac{2\alpha}{3}+9\beta+\gamma$,
$[2]_{1}:\frac{1}{36}(-\alpha+376\beta+9\gamma)$,
$[2]_{-5}:\frac{1}{12}(3\alpha+50\beta+75\gamma)$,
$[1]_{-2}:\frac{29\beta}{3}+\gamma$, for the six different terms that appear
in the branching rule. The multiplicity, which is $15\times 15$ for a bi-
invariant metric (right isometry group ${\rm SU}(3)$), becomes
$15\times(2I+1)$, where the values of $(2I+1)$ are the consecutive members of
the list $\left(4,3,3,2,2,1\right)$, when the right isometry group is
${\mathrm{U}}(2)$.

Eigenvalues of $\Delta$ for left-invariant metrics with other right isometry
groups $K$ can be obtained and discussed along similar lines, this study is
left to the reader.

###  4.4 The cubic Casimir operator

From the commutation relations of $Lie(G)$, when $G={\rm SU}(3)$, it is
straightforward (but cumbersome) to show that the most general central cubic
element of the enveloping algebra of $\mathfrak{su}(3)$ is proportional to
$\Omega_{3}+x\,\frac{3}{2}\,{\widehat{\Omega}_{2}}$, with

$\begin{split}\Omega_{3}&=h_{1}.(\overline{L_{12}}.L_{12}+\overline{L_{45}}.L_{45}-2\overline{L_{67}}.L_{67})/3+h_{2}.(2\overline{L_{12}}.L_{12}-\overline{L_{45}}.L_{45}-\overline{L_{67}}.L_{67})/3+\overline{L_{45}}.L_{67}.L_{12}+\overline{L_{67}}.\overline{L_{12}}.L_{45}+\\\
&\overline{L_{12}}.L_{12}-\overline{L_{67}}.L_{67}+\frac{1}{3}\left(\frac{1}{3}(h_{2}.h_{1}.h_{1}-h_{2}.h_{2}.h_{1})+\frac{2}{9}(h_{1}.h_{1}.h_{1}-h_{2}.h_{2}.h_{2})+(h_{1}.h_{1}-h_{2}.h_{2})+(h_{1}-h_{2})\right)\end{split}$
(21)

where $L_{ij}=(L_{i}+i\,L_{j})/2$, $h_{1}=L_{3}$,
$h_{2}=(-L_{3}+\sqrt{3}L_{8})/2$, and $x$ is an arbitrary real parameter.
Setting $x=0$ i.e., discarding the quadratic Casimir term
${\widehat{\Omega}_{2}}$, we are left with an essentially cubic262626The terms
appearing in 21 (taking $x=0$) are indeed purely cubic when written in terms
of the chosen generators $L_{ij}$ and $h_{j}$, although the same expression,
when written in terms of the generators $L_{a}$, as in 22, contains terms
linear in $L_{3}$ and $L_{8}$. term that we can write as

$\begin{split}\Omega_{3}&=\frac{\left(L_{1}.L_{1}+L_{2}.L_{2}+L_{3}.L_{3}\right).L_{8}}{4\sqrt{3}}-\frac{\left(L_{4}.L_{4}+L_{5}.L_{5}+L_{6}.L_{6}+L_{7}.L_{7}\right).L_{8}}{8\sqrt{3}}+\\\
&\frac{1}{8}L_{3}.\left(L_{4}.L_{4}+L_{5}.L_{5}-L_{6}.L_{6}-L_{7}.L_{7}\right)+\frac{1}{4}\left(L_{1}.L_{4}.L_{6}+L_{1}.L_{5}.L_{7}-L_{2}.L_{4}.L_{7}+L_{2}.L_{5}.L_{6}\right)-\frac{L_{8}.L_{8}.L_{8}}{12\sqrt{3}}+\frac{L_{3}}{2}-\frac{L_{8}}{2\sqrt{3}}\end{split}$
(22)

We can now evaluate this expression in an irreducible representation of
highest weight $\mathpzc{w}$, with components $(o_{1},o_{2})$ in the basis of
fundamental weights; calling $C_{3}$ the eigenvalue of $\Omega_{3}$, one finds

$C_{3}=\frac{1}{27}(o_{1}-o_{2})(3+2o_{1}+o_{2})(3+o_{1}+2o_{2})$ (23)

Notice that $C_{3}$ is equal to $20/27$ in the defining representation, and it
vanishes in the adjoint, as in all real representations since it is
proportional to $(o_{1}-o_{2}$).

One can play with the idea of relaxing the centrality requirement and look for
the most general cubic element commuting with the generators of some Lie
subgroup, in particular some right isometry group $K$. For instance, choosing
$K={\mathrm{U}}(2)$ we impose the vanishing of Lie derivatives with respect to
$L_{1},L_{2},L_{3}$ and $L_{8}$, in which case one finds (again the proof is
straightforward) that the most general such cubic element, up to scale, can be
written as
${\Omega_{3}^{{\mathrm{U}}(2)}}+9\,x\,{\Omega_{2}^{{\mathrm{U}}(2)}}$, where
$x$ is an arbitrary real parameter, where
${\Omega_{2}^{{\mathrm{U}}(2)}}=-\Delta$, with $\Delta$ the Laplacian given by
(19), and where the remaining operator, ${\Omega_{3}^{{\mathrm{U}}(2)}}$, is

$\begin{split}{}&\frac{1}{72}\left(6\sqrt{3}\,A(L_{1}.L_{1}.L_{8}+L_{2}.L_{2}.L_{8}+L_{3}.L_{3}.L_{8}\right)-3\sqrt{3}\,B\left(L_{4}.L_{4}.L_{8}+L_{5}.L_{5}.L_{8}+L_{6}.L_{6}.L_{8}+L_{7}.L_{7}.L_{8}\right)-2\sqrt{3}\,CL_{8}.L_{8}.L_{8}+\\\
&9\,U\left(2L_{1}.L_{4}.L_{6}+2L_{1}.L_{5}.L_{7}-2L_{2}.L_{4}.L_{7}+2L_{2}.L_{5}.L_{6}+L_{3}.L_{4}.L_{4}+L_{3}.L_{5}.L_{5}-L_{3}.L_{6}.L_{6}-L_{3}.L_{7}.L_{7}+4L_{3}\right)-12\,V\sqrt{3}L_{8})\end{split}$
(24)

In this expression $A,B,C,U,V$ denote arbitrary real parameters, and by
setting all of them equal to $1$, one recovers the essentially cubic Casimir
element $\Omega_{3}$ given previously. Using ${\rm SU}(3)$ left translations,
the element ${\Omega_{3}^{{\mathrm{U}}(2)}}$ of the universal enveloping
algebra defines a cubic differential operator on the group which is ${\rm
SU}(3)$ left-invariant by construction, but also right invariant under $K$.

The interested reader will show that, for an arbitrary representation
$\mathpzc{w}=(o_{1},o_{2})$ of ${\rm SU}(3)$, and for a representation that
appears in the branching from $\mathpzc{w}$ to the subgroup ${\mathrm{U}}(2)$
(a representation characterized, as in the previous section, by a pair of
integers $(2I,3Y)$), the eigenvalue of the operator
${\Omega_{3}^{{\mathrm{U}}(2)}}$ is equal to

$U*C_{3}+(U_{B}\,*{C_{2}}+U_{AB}\,*I(I+1)+U_{BC}\,*Y^{2})*Y+U_{V}*Y$ (25)

where $C_{3}$ is given by (23), $C_{2}={\widehat{C}_{2}}/6$ is obtained from
(16), and $U_{B}=\frac{3}{2}(U-B)$, $U_{AB}=\frac{1}{2}(2A+B-3U)$,
$U_{BC}=\frac{1}{8}(3B-2C-U)$, $U_{V}=\frac{1}{2}(U-V)$. This is the cubic
analog of formula 20.

Example: Choose again the fundamental representation $(1,0)$, so ${C_{2}}=4/9$
and $C_{3}=20/27$. The branching of this ${\rm SU}(3)$ irrep to
${\mathrm{U}}(2)$ reads $[3]\mapsto[2]_{1}\oplus[1]_{-2}$ (the notation on the
rhs is $[2I+1]_{3Y}$, like in sect. 4.2). Equation 24 with $L_{a}$ replaced by
$\lambda_{a}$ gives a $3\times 3$ diagonal matrix with diagonal
$\left(\frac{A}{4}-\frac{B}{12}-\frac{C}{108}+\frac{3U}{4}-\frac{V}{6},\frac{A}{4}-\frac{B}{12}-\frac{C}{108}+\frac{3U}{4}-\frac{V}{6},\frac{B}{3}+\frac{2C}{27}+\frac{V}{3}\right)$;
its matrix elements can be also be obtained from (25) by setting $I=1/2,Y=1/3$
for the first two, and $I=0,Y=-2/3$ for the last. Finally, one recovers the
same value $20/27$ of the undeformed cubic Casimir $C_{3}$ by setting
$A=B=C=U=V=1$.

###  4.5 Sectional curvatures

Sectional curvatures $\chi$ are associated with the choice of a two-
dimensional linear subspace of the tangent space at some point of the manifold
under consideration. Here the manifold is a Lie group, and the chosen metrics
are left-invariant, so it is enough to consider sectional curvatures at the
origin. There is a general formula, due to Milnor [16], that expresses these
quantities in terms of the structure constants of an orthonormal basis (for
the chosen metric) of left-invariant vector fields. However, we remind the
reader that, in these notes, for all choices of the right isometry group $K$
we have chosen the same basis $(X_{a})$ to perform curvature calculations,
namely the one that is orthonormal for the Killing metric, and which is
therefore not orthonormal for a general left-invariant metric metric $h$. For
this reason we could not use the Milnor formula (with the exception of the
Killing metric) and had to rely on the general expression

$\chi(u,v)=\frac{\langle{\mathcal{R}}(v,u)u,v\rangle}{\langle
u,u\rangle\langle v,v\rangle-\langle u,v\rangle^{2}}$

where $u$ and $v$ are two linearly independent vectors at the origin, where
the inner product $\langle\,,\,\rangle$ is defined by $h$, and272727The signs
in the definition of ${\mathcal{R}}$ are opposite to [16], this explains the
order of arguments in the expression of $\chi(u,v)$. where
${\mathcal{R}}(u,v)=[\nabla_{u},\nabla_{v}]-\nabla_{[u,v]}$. From the
calculated Riemann tensors one can determine the sectional curvatures
$\chi(X_{a},X_{b})=R_{abab}/(h_{aa}h_{bb}-h_{ab}^{2})$. We list them below,
for the parametrizations of $h$ given in (3) that correspond to right isometry
groups $K={\rm SU}(3)$, ${\mathrm{U}}(2)$ or ${\mathrm{S}O}(3)$. For the
groups $K={\mathrm{U}}(1)\times{\mathrm{U}}(1)$, ${\mathrm{U}}(1)_{Y}$ and
$\\{e\\}$ (the trivial subgroup), we have also explicit results for sectional
curvatures in terms of the parameters specifying the metric, but these
expressions are too large to be displayed on paper. For the
$K={\mathrm{U}}(1)_{I}$ family, the results are also too large to be displayed
but we shall nevertheless give explicit sectional curvatures for a subfamily.
For the special case of the Lorentzian Einstein metric obtained in sect. 3 we
only give numerical values (exact values typically involve specific roots of
$15^{th}$ degree polynomials).
Warning (again): the choice of a pair $X_{a},X_{b}$ with $a\neq b$ determines
a two-dimensional linear subspace at the origin but we remind the reader that
our vectors $X_{a}$ are usually not orthonormal for the chosen metric. By
definition $\chi(X_{a},X_{b})$ is symmetric in $a$ and $b$, and it is not
defined (one can set it equal to $0$) for $a=b$. We give tables of $\chi$ for
$a<b$ running from $1$ to $8$. In some cases we also give the Ricci principal
curvatures (one can check that their sum is the already given scalar
curvature).

$K={\rm SU}(3)$
$\begin{array}[]{cccccccc}.&\frac{\alpha}{12}&\frac{\alpha}{12}&\frac{\alpha}{48}&\frac{\alpha}{48}&\frac{\alpha}{48}&\frac{\alpha}{48}&0\\\
.&.&\frac{\alpha}{12}&\frac{\alpha}{48}&\frac{\alpha}{48}&\frac{\alpha}{48}&\frac{\alpha}{48}&0\\\
.&.&.&\frac{\alpha}{48}&\frac{\alpha}{48}&\frac{\alpha}{48}&\frac{\alpha}{48}&0\\\
.&.&.&.&\frac{\alpha}{12}&\frac{\alpha}{48}&\frac{\alpha}{48}&\frac{\alpha}{16}\\\
.&.&.&.&.&\frac{\alpha}{48}&\frac{\alpha}{48}&\frac{\alpha}{16}\\\
.&.&.&.&.&.&\frac{\alpha}{12}&\frac{\alpha}{16}\\\
.&.&.&.&.&.&.&\frac{\alpha}{16}\\\ \end{array}$

All sectional curvatures are non negative, as expected282828Any compact Lie
group admits a bi-invariant metric with non-negative sectional curvatures [16]
and there is only one bi-invariant metric (up to scale) on ${\rm SU}(3)$..
Some of them vanish, also as expected since the $3$-sphere group ${\rm SU}(2)$
is the only simply connected Lie group which admits a left invariant metric of
strictly positive sectional curvature [21]. All Ricci principal curvatures are
equal to $\alpha/4$.

$K={\mathrm{U}}(2)$
$\begin{array}[]{cccccccc}.&\frac{\alpha}{12}&\frac{\alpha}{12}&\frac{\beta^{2}}{48\alpha}&\frac{\beta^{2}}{48\alpha}&\frac{\beta^{2}}{48\alpha}&\frac{\beta^{2}}{48\alpha}&0\\\
.&.&\frac{\alpha}{12}&\frac{\beta^{2}}{48\alpha}&\frac{\beta^{2}}{48\alpha}&\frac{\beta^{2}}{48\alpha}&\frac{\beta^{2}}{48\alpha}&0\\\
.&.&.&\frac{\beta^{2}}{48\alpha}&\frac{\beta^{2}}{48\alpha}&\frac{\beta^{2}}{48\alpha}&\frac{\beta^{2}}{48\alpha}&0\\\
.&.&.&.&-\frac{\beta(9\alpha\beta-16\alpha\gamma+3\beta\gamma)}{48\alpha\gamma}&\frac{\beta(4\alpha-3\beta)}{48\alpha}&\frac{\beta(4\alpha-3\beta)}{48\alpha}&\frac{\beta^{2}}{16\gamma}\\\
.&.&.&.&.&\frac{\beta(4\alpha-3\beta)}{48\alpha}&\frac{\beta(4\alpha-3\beta)}{48\alpha}&\frac{\beta^{2}}{16\gamma}\\\
.&.&.&.&.&.&-\frac{\beta(9\alpha\beta-16\alpha\gamma+3\beta\gamma)}{48\alpha\gamma}&\frac{\beta^{2}}{16\gamma}\\\
.&.&.&.&.&.&.&\frac{\beta^{2}}{16\gamma}\\\ \end{array}$

Ricci principal curvatures:
$\left\\{\frac{\beta^{2}}{12\alpha}+\frac{\alpha}{6},\frac{\beta^{2}}{12\alpha}+\frac{\alpha}{6},\frac{\beta^{2}}{12\alpha}+\frac{\alpha}{6},-\frac{\beta^{2}}{8\alpha}-\frac{\beta^{2}}{8\gamma}+\frac{\beta}{2},-\frac{\beta^{2}}{8\alpha}-\frac{\beta^{2}}{8\gamma}+\frac{\beta}{2},-\frac{\beta^{2}}{8\alpha}-\frac{\beta^{2}}{8\gamma}+\frac{\beta}{2},-\frac{\beta^{2}}{8\alpha}-\frac{\beta^{2}}{8\gamma}+\frac{\beta}{2},\frac{\beta^{2}}{4\gamma}\right\\}$.

$K={\mathrm{S}O}(3)$
$\begin{array}[]{cccccccc}.&\frac{\alpha^{2}}{12\beta}&\frac{\alpha(4\beta-3\alpha)}{12\beta}&\frac{\alpha(4\beta-3\alpha)}{48\beta}&\frac{\alpha^{2}}{48\beta}&\frac{\alpha(4\beta-3\alpha)}{48\beta}&\frac{\alpha^{2}}{48\beta}&0\\\
.&.&\frac{\alpha^{2}}{12\beta}&\frac{\alpha^{2}}{48\beta}&\frac{\beta}{48}&\frac{\alpha^{2}}{48\beta}&\frac{\beta}{48}&0\\\
.&.&.&\frac{\alpha(4\beta-3\alpha)}{48\beta}&\frac{\alpha^{2}}{48\beta}&\frac{\alpha(4\beta-3\alpha)}{48\beta}&\frac{\alpha^{2}}{48\beta}&0\\\
.&.&.&.&\frac{\alpha^{2}}{12\beta}&\frac{\alpha(4\beta-3\alpha)}{48\beta}&\frac{\alpha^{2}}{48\beta}&\frac{\alpha(4\beta-3\alpha)}{16\beta}\\\
.&.&.&.&.&\frac{\alpha^{2}}{48\beta}&\frac{\beta}{48}&\frac{\alpha^{2}}{16\beta}\\\
.&.&.&.&.&.&\frac{\alpha^{2}}{12\beta}&\frac{\alpha(4\beta-3\alpha)}{16\beta}\\\
.&.&.&.&.&.&.&\frac{\alpha^{2}}{16\beta}\\\ \end{array}$

In particular, sectional curvatures for the (Jensen) Einstein metric read:
$\quad\begin{array}[]{cccccccc}.&\frac{1}{132}&\frac{411}{132}&\frac{411}{528}&\frac{1}{528}&\frac{411}{528}&\frac{1}{528}&0\\\
.&.&\frac{1}{132}&\frac{1}{528}&\frac{111}{48}&\frac{1}{528}&\frac{111}{48}&0\\\
.&.&.&\frac{411}{528}&\frac{1}{528}&\frac{411}{528}&\frac{1}{528}&0\\\
.&.&.&.&\frac{1}{132}&\frac{411}{528}&\frac{1}{528}&\frac{411}{176}\\\
.&.&.&.&.&\frac{1}{528}&\frac{111}{48}&\frac{1}{176}\\\
.&.&.&.&.&.&\frac{1}{132}&\frac{411}{176}\\\ .&.&.&.&.&.&.&\frac{1}{176}\\\
\end{array}$

Ricci principal curvatures:
$\left\\{-\frac{\alpha(\alpha-2\beta)}{4\beta},-\frac{\alpha(\alpha-2\beta)}{4\beta},-\frac{\alpha(\alpha-2\beta)}{4\beta},-\frac{\alpha(\alpha-2\beta)}{4\beta},-\frac{\alpha(\alpha-2\beta)}{4\beta},\frac{5\alpha^{2}+\beta^{2}}{24\beta},\frac{5\alpha^{2}+\beta^{2}}{24\beta},\frac{5\alpha^{2}+\beta^{2}}{24\beta}\right\\}$.

For the Jensen metric they are all equal to $21\alpha/44$.

$K={\mathrm{U}}(1)_{I}$.

The general case, with its $8$ parameters, is too large to be displayed. We
can nevertheless (using tiny fonts (!)) exhibit the sectional curvatures
obtained for the subfamily $\theta=0,\zeta=0$, $\delta=\gamma$.

$\begin{split}&\left\\{\begin{split}&\frac{\cdot,\alpha(4\beta-3\alpha)}{12\beta},\frac{\alpha^{2}}{12\beta},\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\alpha\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\alpha^{2}\eta^{2}\gamma+4\alpha\eta^{4}}{48\alpha\gamma^{3}-48\alpha\gamma\eta^{2}},\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\alpha\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\alpha^{2}\eta^{2}\gamma+4\alpha\eta^{4}}{48\alpha\gamma^{3}-48\alpha\gamma\eta^{2}},\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\alpha\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\alpha^{2}\eta^{2}\gamma+4\alpha\eta^{4}}{48\alpha\gamma^{3}-48\alpha\gamma\eta^{2}},\\\
&\qquad\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\alpha\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\alpha^{2}\eta^{2}\gamma+4\alpha\eta^{4}}{48\alpha\gamma^{3}-48\alpha\gamma\eta^{2}},0\end{split}\right\\},\\\
&\left\\{\begin{split}&\cdot,\cdot,\frac{\alpha^{2}}{12\beta},\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\alpha\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\alpha^{2}\eta^{2}\gamma+4\alpha\eta^{4}}{48\alpha\gamma^{3}-48\alpha\gamma\eta^{2}},\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\alpha\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\alpha^{2}\eta^{2}\gamma+4\alpha\eta^{4}}{48\alpha\gamma^{3}-48\alpha\gamma\eta^{2}},\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\alpha\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\alpha^{2}\eta^{2}\gamma+4\alpha\eta^{4}}{48\alpha\gamma^{3}-48\alpha\gamma\eta^{2}},\\\
&\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\alpha\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\alpha^{2}\eta^{2}\gamma+4\alpha\eta^{4}}{48\alpha\gamma^{3}-48\alpha\gamma\eta^{2}},0\end{split}\right\\},\\\
&\left\\{\cdot,\cdot,\cdot,\frac{\gamma^{2}-\eta^{2}}{48\beta},\frac{\gamma^{2}-\eta^{2}}{48\beta},\frac{\gamma^{2}-\eta^{2}}{48\beta},\frac{\gamma^{2}-\eta^{2}}{48\beta},0\right\\},\\\
&\left\\{\begin{split}&\cdot,\cdot,\cdot,\cdot,\frac{\beta\left(-9\gamma^{4}+16\epsilon\gamma^{3}+18\eta^{2}\gamma^{2}-16\epsilon\eta^{2}\gamma-9\eta^{4}+8\alpha\epsilon\eta^{2}\right)-3\epsilon\left(\gamma^{2}-\eta^{2}\right)^{2}}{48\beta\gamma^{2}\epsilon},\frac{-4\alpha^{2}\eta^{2}-3\left(\gamma^{2}-\eta^{2}\right)^{2}+4\alpha\left(\gamma^{3}-\eta^{2}\gamma+3\epsilon\eta^{2}\right)}{48\alpha\gamma^{2}},\frac{4\alpha^{2}\eta^{2}-3\left(\gamma^{2}-\eta^{2}\right)^{2}+4\alpha\left(\gamma^{3}-\gamma\eta^{2}\right)}{48\alpha\left(\gamma^{2}-\eta^{2}\right)},\\\
&\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\epsilon\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\epsilon^{2}\eta^{2}\gamma+4\epsilon\eta^{4}}{16\gamma^{3}\epsilon-16\gamma\epsilon\eta^{2}}\end{split}\right\\},\\\
&\left\\{\cdot,\cdot,\cdot,\cdot,\cdot,\frac{4\alpha^{2}\eta^{2}-3\left(\gamma^{2}-\eta^{2}\right)^{2}+4\alpha\left(\gamma^{3}-\gamma\eta^{2}\right)}{48\alpha\left(\gamma^{2}-\eta^{2}\right)},\frac{-4\alpha^{2}\eta^{2}-3\left(\gamma^{2}-\eta^{2}\right)^{2}+4\alpha\left(\gamma^{3}-\eta^{2}\gamma+3\epsilon\eta^{2}\right)}{48\alpha\gamma^{2}},\frac{\gamma^{5}-2\eta^{2}\gamma^{3}-4\epsilon\eta^{2}\gamma^{2}+\eta^{4}\gamma+4\epsilon^{2}\eta^{2}\gamma+4\epsilon\eta^{4}}{16\gamma^{3}\epsilon-16\gamma\epsilon\eta^{2}}\right\\},\\\
&\left\\{\cdot,\cdot,\cdot,\cdot,\cdot,\cdot,\frac{\beta\left(-9\gamma^{4}+16\epsilon\gamma^{3}+18\eta^{2}\gamma^{2}-16\epsilon\eta^{2}\gamma-9\eta^{4}+8\alpha\epsilon\eta^{2}\right)-3\epsilon\left(\gamma^{2}-\eta^{2}\right)^{2}}{48\beta\gamma^{2}\epsilon},\frac{\gamma^{5}-2\left(\gamma^{2}+2\epsilon\gamma-2\epsilon^{2}\right)\eta^{2}\gamma+(\gamma+4\epsilon)\eta^{4}}{16\gamma\epsilon\left(\gamma^{2}-\eta^{2}\right)}\right\\},\\\
&\left\\{\cdot,\cdot,\cdot,\cdot,\cdot,\cdot,\cdot,\frac{\gamma^{5}-2\left(\gamma^{2}+2\epsilon\gamma-2\epsilon^{2}\right)\eta^{2}\gamma+(\gamma+4\epsilon)\eta^{4}}{16\gamma\epsilon\left(\gamma^{2}-\eta^{2}\right)}\right\\}\end{split}$

The scalar curvature, for this family, was given by (12).

$K={\mathrm{U}}(1)_{I}$. Special case: Lorentzian Einstein metric

$\begin{array}[]{cccccccc}.&0.156539&0.0589315&0.0207884&0.0207884&0.0207884&0.0207884&0\\\
.&.&0.0589315&0.0207884&0.0207884&0.0207884&0.0207884&0\\\
.&.&.&0.000619797&0.000619797&0.000619797&0.000619797&0\\\
.&.&.&.&0.108778&-0.0335267&0.0410926&-0.054309\\\
.&.&.&.&.&0.0410926&-0.0335267&-0.054309\\\ .&.&.&.&.&.&0.108778&-0.054309\\\
.&.&.&.&.&.&.&-0.054309\\\ \end{array}$

Other features of this metric have been discussed in sect.3.

###  4.6 Ricci decomposition (examples)

The Ricci decomposition of the Riemann tensor associated with the Levi-Civita
connection defined by the metric $h$, namely
$R=C+\frac{1}{d-2}(\rho-\frac{\tau}{d}h)\mathbin{\mathchoice{\ooalign{$\displaystyle\bigcirc$\crcr$\displaystyle\land$\crcr}}{\ooalign{$\textstyle\bigcirc$\crcr$\textstyle\land$\crcr}}{\ooalign{$\scriptstyle\bigcirc$\crcr$\scriptstyle\land$\crcr}}{\ooalign{$\scriptscriptstyle\bigcirc$\crcr$\scriptscriptstyle\land$\crcr}}}h+\frac{\tau}{2d(d-1)}h\mathbin{\mathchoice{\ooalign{$\displaystyle\bigcirc$\crcr$\displaystyle\land$\crcr}}{\ooalign{$\textstyle\bigcirc$\crcr$\textstyle\land$\crcr}}{\ooalign{$\scriptstyle\bigcirc$\crcr$\scriptstyle\land$\crcr}}{\ooalign{$\scriptscriptstyle\bigcirc$\crcr$\scriptscriptstyle\land$\crcr}}}h$,
where $d$ is the dimension (here $d=8$), $C$ is the Weyl tensor, and
$\mathbin{\mathchoice{\ooalign{$\displaystyle\bigcirc$\crcr$\displaystyle\land$\crcr}}{\ooalign{$\textstyle\bigcirc$\crcr$\textstyle\land$\crcr}}{\ooalign{$\scriptstyle\bigcirc$\crcr$\scriptstyle\land$\crcr}}{\ooalign{$\scriptscriptstyle\bigcirc$\crcr$\scriptscriptstyle\land$\crcr}}}$
denotes the Kulkarni-Nomizu product of two $(0,2)$ tensors, expresses the
Riemann tensor (here thought of as a $(0,4)$ tensor), as an orthogonal direct
sum. Such a decomposition can be considered for an arbitrary metric, in
particular for homogeneous metrics. As a verification of our calculations
involving curvatures, we have checked this identity for all the familes of
metrics considered in this paper. It would be of course paper-consuming to
list all the non-zero entries of the relevant tensors, nevertheless, in a few
cases, it may be useful to mention a numerical consequence of this identity,
namely the following norm decomposition:

$|R|^{2}=|C|^{2}+|\frac{1}{d-2}(\rho-\frac{\tau}{d}h)\mathbin{\mathchoice{\ooalign{$\displaystyle\bigcirc$\crcr$\displaystyle\land$\crcr}}{\ooalign{$\textstyle\bigcirc$\crcr$\textstyle\land$\crcr}}{\ooalign{$\scriptstyle\bigcirc$\crcr$\scriptstyle\land$\crcr}}{\ooalign{$\scriptscriptstyle\bigcirc$\crcr$\scriptscriptstyle\land$\crcr}}}h|^{2}+|\frac{\tau}{2d(d-1)}h\mathbin{\mathchoice{\ooalign{$\displaystyle\bigcirc$\crcr$\displaystyle\land$\crcr}}{\ooalign{$\textstyle\bigcirc$\crcr$\textstyle\land$\crcr}}{\ooalign{$\scriptstyle\bigcirc$\crcr$\scriptstyle\land$\crcr}}{\ooalign{$\scriptscriptstyle\bigcirc$\crcr$\scriptscriptstyle\land$\crcr}}}h|^{2}$
(26)

In those cases where the results are reasonably short we give $|R|^{2}$
followed by a triple containing the three contributions, in the same order as
in the previous equation.

$K={\rm SU}(3)$, (bi-invariant metrics):
$\left\\{\frac{\alpha^{2}}{2},\\{\frac{5\alpha^{2}}{14},0,\frac{\alpha^{2}}{7}\\}\right\\}$

$K={\mathrm{U}}(2)$:

$\begin{split}&\\{\frac{8\alpha^{4}\gamma^{2}+\alpha^{2}\beta^{2}\left(51\beta^{2}-144\beta\gamma+176\gamma^{2}\right)+6\alpha\beta^{3}\gamma(5\beta-16\gamma)+23\beta^{4}\gamma^{2}}{96\alpha^{2}\gamma^{2}},\\\
&\\{\frac{80\alpha^{4}\gamma^{2}-24\alpha^{3}\beta\gamma(\beta-8\gamma)+\alpha^{2}\beta^{2}\left(909\beta^{2}-2448\beta\gamma+2600\gamma^{2}\right)+6\alpha\beta^{3}\gamma(79\beta-240\gamma)+377\beta^{4}\gamma^{2}}{2016\alpha^{2}\gamma^{2}},\\\
&\frac{20\alpha^{4}\gamma^{2}+12\alpha^{3}\beta\gamma(\beta-8\gamma)+\alpha^{2}\beta^{2}\left(45\beta^{2}-144\beta\gamma+236\gamma^{2}\right)+6\alpha\beta^{3}\gamma(7\beta-24\gamma)+29\beta^{4}\gamma^{2}}{576\alpha^{2}\gamma^{2}},\frac{1}{448}\left(-\frac{\beta^{2}}{\alpha}+2\alpha+\beta\left(8-\frac{\beta}{\gamma}\right)\right)^{2}\\}\\}\end{split}$

$K={\mathrm{S}O}(3)$:
$\left\\{\frac{295\alpha^{4}-660\alpha^{3}\beta+460\alpha^{2}\beta^{2}+\beta^{4}}{192\beta^{2}},\\{\frac{5\left(508\alpha^{4}-1110\alpha^{3}\beta+733\alpha^{2}\beta^{2}+12\alpha\beta^{3}+\beta^{4}\right)}{2016\beta^{2}},\frac{5\left(11\alpha^{2}-12\alpha\beta+\beta^{2}\right)^{2}}{2304\beta^{2}},\frac{\left(-5\alpha^{2}+20\alpha\beta+\beta^{2}\right)^{2}}{1792\beta^{2}}\\}\right\\}$

In particular, for the Einstein solution (Jensen case: $\beta=11\alpha$), we
have
$\left\\{\frac{2639\alpha^{2}}{968},\\{\frac{2135\alpha^{2}}{968},0,\frac{63\alpha^{2}}{121}\\}\right\\}$.

$K={\mathrm{U}}(1)_{I}$: The results giving the norms are too large to be
displayed. We only consider the Lorentzian Einstein solution. In that case,
one can obtain these four norms as roots of appropriate 15th degree
polynomials with (very) large integer coefficients. We shall not print them.
Numerically, one finds $\\{0.115257,\\{0.0813543,0,0.0339023\\}\\}$.

The expressions of the four norms, for
$K={\mathrm{U}}(1)\times{\mathrm{U}}(1)$ and for $K={\mathrm{U}}(1)_{Y}$, are
also very large, and we shall not display them.

Notice that for all Einstein spaces the square norm
$|(\rho-\frac{\tau}{d}h)\mathbin{\mathchoice{\ooalign{$\displaystyle\bigcirc$\crcr$\displaystyle\land$\crcr}}{\ooalign{$\textstyle\bigcirc$\crcr$\textstyle\land$\crcr}}{\ooalign{$\scriptstyle\bigcirc$\crcr$\scriptstyle\land$\crcr}}{\ooalign{$\scriptscriptstyle\bigcirc$\crcr$\scriptscriptstyle\land$\crcr}}}h|^{2}$
vanishes, as it should.

##  5 Physical applications

###  5.1 Particle physics and the GMO formula

##### Physical considerations.

In the standard model of elementary particles, more precisely in the quark
model based on the Lie group $G={\rm SU}(N)$, the particles called mesons are
associated with the space of intertwiners ($G$-equivariant morphisms) from
$V_{f}\otimes{\overline{V}_{f}}$, where $V_{f}$ is the defining representation
and ${\overline{V}_{f}}$ its conjugate, to the irreducible representations
that appear in the product, namely the adjoint representation, and the trivial
one. This decomposition, for ${\rm SU}(3)$, in terms of dimensions, read
$[3]\otimes[\overline{3}]=[8]\oplus[1]$. Mesons are described as basis vectors
of the tensor product, specified by appropriate $G$-Clebsch-Gordan
coefficients (or by the $3J$ Wigner symbols of the group $G$) associated with
the chosen spaces of intertwiners, but this is not our concern here.

Quarks (resp. anti-quarks) are basis vectors in the space of the defining
representation of $G$ (resp. its conjugate), and mesons are called “bound
states ” of a quark and an anti-quark. In particle physics parlance $G$ is the
“flavor group”, which, in the presently accepted model, means ${\rm SU}(N)$,
and where $N$ can only be $2,3,4,5$ or $6$. When $N=2$, $G$ is called the
isospin group and the basis vectors of the defining representation (the
quarks) are nicknamed ‘up’ and ‘down’. When $N=3$ (in this paper we restrict
our attention to $G={\rm SU}(3)$) they are nicknamed ‘up’, ‘down’, ‘strange’,
and are respectively denoted by $u,d,s$.

The classical (i.e., not quantum field theoretical) description of mesons, as
sections of appropriate vector bundles over the space-time manifold, also
specifies their behavior with respect to space-time symmetries (i.e., under
action of the Lorentz group or of the Poincaré group), but we don’t have to be
more precise here, it is enough to say that there are several families of
mesons differing by their space-time properties, the two most important
families being the so-called pseudo-scalar mesons and the vector mesons.

Quarks are not observable but mesons are, and they have masses. Experimentally
the pseudo-scalar mesons have masses that are close (same remark for the
vector mesons), and this is precisely the reason why, historically, they were
described as members of the same Lie group multiplet (basis vectors of some
irrep). Calculating meson masses in terms of more fundamental parameters is a
task that goes beyond the possibilities of (perturbative) quantum field
theory, in particular of quantum chromodynamics, but it remains that,
phenomenologically, one can assume that interactions of mesons, in particular
the quadratic operator responsible for their masses, or their mass splitting,
commutes with the generators of the chosen flavor group, or of a subgroup of
the latter. This hypothesis is at the origin of several mass relations.

Experimentally, particles of the same ${\rm SU}(3)$ multiplet have
approximately the same masses, but this is even more so when they are members
of the same irreducible representation of the ${\mathrm{U}}(2)$ subgroup
(locally ${\rm SU}(2)\times{\mathrm{U}}(1)$) defined in table (2). For this
reason, it is natural to describe (or approximate) the unknown mass operator
as the Laplacian on ${\rm SU}(3)$ associated with an appropriate left-
invariant metric for which the right isometry group is $K={\mathrm{U}}(2)$,
and to look at the consequences of this ansatz. We rescale the dual metric by
$\mu^{2}$ to fix the dimensions ($\mu$ will have the dimensions of a mass).
The eigenvalues of the Laplacian are given by (20), we write them again below
(the whole expression is now multiplied by $\mu^{2}$). To an irrep
$(o_{1},o_{2})$ of ${\rm SU}(3)$ branching to irreps of ${\mathrm{U}}(2)$
labelled by the isospin value $I$ and hypercharge $Y$ we associate the square
mass $m^{2}=\mu^{2}\,C(o_{1},o_{2})$ given by

$m^{2}={{C_{2}}(o_{1},o_{2})}\,\beta\,\mu^{2}+\left(\frac{1}{3}I(I+1)(\alpha-\beta)\mu^{2}+\frac{1}{4}Y^{2}(\gamma-\beta)\mu^{2}\right)$
(27)

##### Pseudo-scalar mesons and the Gell-Mann-Okubo formula.

The branching of the adjoint representation (octet) of ${\rm SU}(3)$, when
restricted to the previously defined ${\mathrm{U}}(2)$ subgroup, in terms of
${\rm SU}(2)$ highest weights, reads292929the components are written in the
basis of fundamental weights.: $(1,1)\rightarrow(2)+(1)+(1)+(0)$; this is
often written in terms of dimensions of irreps
($[8]\rightarrow[3]_{0}+[2]_{3}+[2]_{-3}+[1]_{0}$), with the same notations
$[2I+1]_{3Y}$ as in sect. 4.2, since there is no possible confusion in the
present case. The corresponding mesons are the three pions
$\\{\pi^{+},\pi^{0},\pi^{-}\\}$, for which $I=1$, $Y=0$, the four kaons
$\\{K^{+},K^{0}\\}$ for which $I=1$, $Y=1$, and $\\{\overline{K^{0}},K^{-}\\}$
for which $I=1$, $Y=-1$, and the eta particle, for which $I=0$, $Y=0$.

In the adjoint representation of ${\rm SU}(3)$, ${C_{2}}=1$, so that one
obtains immediately:

$\left\\{m^{2}{}_{\pi},\,m^{2}{}_{K},\,m^{2}{}_{\eta}\right\\}=\left\\{\frac{1}{3}(2\alpha+\beta)\mu^{2},\frac{1}{4}(\alpha+2\beta+\gamma)\mu^{2},\beta\mu^{2}\right\\}$

Experimentally: $m_{\pi^{+}}=m_{\pi^{-}}=139.57\,\text{MeV}$,
$m_{\pi^{0}}=134.976\,\text{MeV}$, $m_{K^{+}}=m_{K^{-}}=493.677\,\text{MeV}$,
$m_{K^{0}}=m_{\overline{K^{0}}}=497.64\,\text{MeV}$, and
$m_{\eta}=549\,\text{MeV}$. For pions and kaons we use averaged masses
$m_{\pi}\simeq 137\,\text{MeV}$, $m_{K}\simeq 496\,\text{MeV}$,
$m_{\eta}=549\,\text{MeV}$. The corresponding values303030These values where
already obtained in [4], up to a scaling factor equal to $12$ coming from the
fact that the bi-invariant metric used in that reference for normalizing
purposes was a multiple of the Killing metric. of parameters are then
$\mu^{2}\alpha_{exp}\simeq-(350\,\text{MeV})^{2},\,\mu^{2}\beta_{exp}\simeq(549\,\text{MeV})^{2},\,\mu^{2}\gamma_{exp}\simeq(710\,\text{MeV})^{2}$.
Their ratios, normalized by $\beta$, are
$(\alpha_{exp},\beta_{exp},\gamma_{exp})/\beta_{exp}=(-0.406591,1,1.67156)$.

We stress the fact that the above is nothing else than an educated fit: it is
neither a prediction nor a “post-diction” since the number of unknown
parameters is the same as the number of values coming from experiment. In
order to get a prediction, one needs at least one more relation between the
parameters $\alpha,\beta,\gamma$; such a relation (expressed in a rather
different way) was postulated in the sixties, by making the hypothesis that
the ${\rm SU}(3)$ the mass operator could be well approximated by keeping only
its singlet and octet components, therefore neglecting the contribution from
the representation of dimension $27$ (see for instance [20]). In our language,
this amounts to neglect the ${}_{27}h^{-1}$ component of the (dual)
pseudo313131The signature of the bilinear form, with the previous values of
$\alpha_{exp},\beta_{exp},\gamma_{exp}$, is $(5,3)$.-Riemannian metric in the
decomposition $h^{-1}={{}_{1}h^{-1}}+{{}_{8}h^{-1}}+{{}_{27}h^{-1}}$ discussed
in sect. 2.3. In other words the coefficient
$C=\frac{1}{40}(\alpha-4\beta+3\gamma)$ given in (4) is set to $0$. One can
then eliminate the parameter $\gamma$, for example, and obtain

$\left\\{m^{2}{}_{\pi},\,m^{2}{}_{K},\,m^{2}{}_{\eta}\right\\}\simeq\left(\frac{1}{3}(2\alpha+\beta)\mu^{2},\,\frac{1}{6}(\alpha+5\beta)\mu^{2},\,\beta\mu^{2}\right),$
$\text{which implies the relation:}{\hskip
28.45274pt}m_{\eta}^{2}\simeq\frac{1}{3}\left(4m_{K}^{2}-m_{\pi}^{2}\right).$

This is the celebrated Gell-Mann-Okubo formula for pseudo-scalar mesons (the
formula using square masses, see for instance the article on GMO formulae in
Wikipedia). It holds reasonably well323232The first published mass relation of
this type (1961) was for baryons, and in particular for hyperons of the
decuplet. As it is well known this equation lead to the discovery of the
$\Omega^{-}$ particle (and to the Nobel Prize in Physics 1969). In our
language, this latter formula, which is linear in masses (not quadratic) could
be obtained from the eigenvalues of a Dirac operator on ${\rm SU}(3)$
associated with a left-invariant metric for which the right isometry group is
${\mathrm{U}}(2)$., although, using the experimental values of $m_{\pi}$ and
$m_{K}$, it leads to a value of $567\,\text{MeV}$ for the mass of the $\eta$,
which is slightly too big.

Remarks: Rather than using a (rough) Laplacian, for some appropriate ${\rm
SU}(3)\times{\mathrm{U}}(2)$ invariant metric, one could be tempted of using
the Yamabe (conformal) Laplacian that differs333333The conformal Laplacian is
$\Delta-\tfrac{d-2}{4(d-1)}\,\tau$, where $\tau$ is the scalar curvature. In
our case $d=8$. from the latter by a simple shift (equal to
$-\tfrac{3}{14}\,\frac{1}{4}\left(-\frac{\beta^{2}}{\alpha}+2\alpha+\beta\left(8-\frac{\beta}{\gamma}\right)\right)$).
but this does not seem to lead to anything physically particularly
interesting. One could also play with the idea of using analogous
considerations to study other particle multiplets, to generalize the previous
analysis to Lie groups $SU(N)$ for $N>3$, or even to consider other kinds of
“symmetry breaking” scenarios (selecting other right-invariant isometry
groups), but this would lie beyond the intended scope of these notes.

##### Warning.

The physicist reader certainly knows, and the mathematician reader should be
warned, that the above way of obtaining mass relations for some elementary
particles, from considerations on Laplacians (or Dirac operators) associated
with left-invariant metrics (actually ${\rm SU}(3)\times{\mathrm{U}}(2)$
invariant metrics on the Lie group ${\rm SU}(3)$ is not standard, in the sense
that, although not a new observation (see [4]), this approach is not widely
known and it is not the way it is taught. It can be noticed that, whatever the
starting point one chooses (the elementary quark model, or more sophisticated
approaches like QCD or chiral perturbation theory), the elementary
mathematical considerations leading to these mass relations are similar: they
involve representation theory of ${\rm SU}(3)$, the branching to ${\rm
SU}(2)\times{\mathrm{U}}(1)$ (of isospin and hypercharge), the fact that
masses should be related to eigenvalues of linear or quadratic operators
(would-be Hamiltonian or mass operators) that are not explicitly known, and an
approximation of “octet dominance” (in the present case it is the hypothesis
that one can neglect, in the metric, a contribution associated with the $27$
dimensional representation of ${\rm SU}(3)$ —see sect 2.3 and the discussion
in [20]). However, one thing is to cook up a physico-mathematical formula that
works at least approximately in some particular cases, another is to derive it
in the framework of a physical theory. The contemporary attitude is to view
GMO mass formulae for hadrons as remote consequences of a fundamental theory
called “The Standard Model”; this theory, in its usual formulation, does not
explicitly involve considerations on the left-invariant geometry of Lie groups
${\rm SU}(N)$ —this $N$ standing for the number of “flavors”. Our observation
that the same mass formulae can be interpreted as expressions describing the
spectrum of appropriate differential operators associated with particular
left-invariant metrics on Lie groups (it is not difficult to extend the
previous results to $N>3$) may not be significant but, notwithstanding this
possibility, the result suggests that it could be, or should be, justified, as
a formal consequence of some currently accepted physical theory (a subject
that we don’t investigate in these notes) and maybe trigger some new
developments of the latter.

###  5.2 Other possibly physical considerations

The fact that some particular left-invariant metrics, which may be Riemannian
or pseudo-Riemannian, can sometimes be Einstein metrics played no role in the
previous discussion on particle masses. Now, for the last thirty years or so,
many theoretical physicists have been concerned with the construction of
classical field models, quantum field theories, and string theories,
incorporating, on top of space and time, several “extra-dimensions” that we do
not perceive (the prototype being the old Kaluza-Klein theory). Such models,
that are often speculative, are described by equations that sometimes require
the total space of the theory (or maybe a quotient of the latter) to be an
Einstein manifold. We have no wish to comment this endeavor and shall refrain
from suggesting anything in that direction but one may notice that, if needed,
the examples of eight dimensional Einstein manifolds described in sect 3 can
be used to construct higher dimensional pseudo-Riemannian spaces, of given
signature, that are also Einstein. For instance one can build a Lorentzian
homogeneous Einstein metric on the $11$-dimensional compact manifold ${\rm
SU}(3)\times{\rm SU}(2)$ where the first factor of the Cartesian product is
endowed with the Lorentzian Einstein structure described previously and where
the second factor (diffeomorphic with the sphere $S^{3}$) is endowed with its
standard ${\mathrm{S}O}(4)$ invariant Einstein metric.

## References

  * [1] S. Aloff, Nolan R. Wallach. An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Amer. Math. Soc. 81(1): 93-97 (January 1975).
  * [2] B.L. Beers and R.S. Millman. The Spectra of the Laplace-Beltrami Operator on Compact, Semisimple Lie Groups. American Journal of Mathematics 99, no. 4 (1977): 801-07. Accessed May 30, 2021. doi:10.2307/2373866.
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