The massless irreducible representation in E theory and how bosons can appear
as spinors.

Keith Glennon and Peter West

Department of Mathematics

King’s College, London WC2R 2LS, UK

Abstract

We study in detail the irreducible representation of E theory that corresponds
to massless particles. This has little algebra $I_{c}(E_{9})$ and contains 128
physical states that belong to the spinor representation of SO(16). These are
the degrees of freedom of maximal supergravity in eleven dimensions. This
smaller number of the degrees of freedom, compared to what might be expected,
is due to an infinite number of duality relations which in turn can be traced
to the existence of a subaglebra of $I_{c}(E_{9})$ which forms an ideal and
annihilates the representation. We explain how these features are inherited
into the covariant theory. We also comment on the remarkable similarity
between how the bosons and fermions arise in E theory.

1\. Introduction The symmetries of the non-linear realisation of
$E_{11}\otimes_{s}l_{1}$ with respect to the Cartan involution invariant
subalgebra of $E_{11}$, denoted by $I_{c}(E_{11})$, lead to equations of
motion at low levels that are precisely those of maximal supergravity provided
one discards the dependence on the coordinates of the spacetime that are
beyond those we usually consider [1,2,3,4]. In particular the degrees of
freedom they contain are those of the familiar graviton and the three form.
While there is no complete understanding of the role of all the higher level
fields a large class of them are known to provide different field descriptions
of the graviton and the three form degrees and freedom to which they are
related by invariant duality relations. The higher level fields fields also
account for the gauging of the maximal supergravity theories. Indeed the non-
linear realisation contains all the maximal supergravity theories in their
different dimensions including those that are gauged. For a review see
references [5,6].

In a different approach the irreducible representation of the semi-direct
product of $I_{c}(E_{11})$ with its vector representation $l_{1}$, denoted
$I_{c}(E_{11})\otimes_{s}l_{1}$ were formulated [7]. This latter algebra is
the analogue of the Poincare group which can be written in the form
$SO(1,D-1)\otimes_{s}T^{D}$ where $T^{D}$ are the translations in $D$
dimensions. The irreducible representations of the Poincare group were found
in 1939 by Wigner [8] and one can use a similar method in E theory. One
important difference is that while the construction of the irreducible
representations of the Poincare algebra begin by considering the possible
values of the momentum, those in E theory involve the vector, or first
fundamental representation of $E_{11}$ and these include all brane charges
[2,9,10,11,12]. Thus while the irreducible representations of the Poincare
algebra lead to all possible particles those in E theory lead to all possible
point particle and branes, that is, all extended objects in E theory.

The irreducible representation that arises when all the members of the vector
representation vanish except for the usual momentum, which was taken to be
massless, were studied briefly in reference [7]. It was found that the
corresponding little group was $I_{c}(E_{9})$ and it was argued that the
representation only contains a finite number of states which are those of the
graviton and three form in eleven dimensions and so the degrees of freedom of
eleven dimensional supergravity. In this paper we will study this irreducible
representation in detail. We will show that one can impose an infinite number
of duality equations on the representation that are invariant under
$I_{c}(E_{9})$ and reduce the number of independent fields to be 128. These
are the fields $h_{i_{1}i_{2}}=h_{(i_{1}i_{2})}$, $h^{i}{}_{i}=0$ and
$A_{i_{1}i_{2}i_{3}}=A_{[i_{1}i_{2}i_{3}]}$, $i_{1},i_{2},\ldots=2,\ldots,10$
and they belong to the 128 dimensional spinor representation of
$I_{c}(E_{8})={\rm SO}(16)$. Corresponding to the duality relations we find
that there exist an infinite number of generators of $I_{c}(E_{9})$ which
annihilate the representation and these form a subalgebra $I$ that is an
ideal. Indeed the Lie algebra ${I_{c}(E_{9})\over I}={\rm SO}(16)$.

To understand how it is that the bosonic states, and in particular the
graviton, can belong to the spinor representation of an algebra that involves
spacetime symmetries we will decompose the physical states into
representations of the subalgebra ${\rm SO(8)}\otimes{\rm SO(8)}$ of ${\rm
SO}(16)$. We find that they belong to the representations
$128=(8_{v},8_{v})\oplus(8_{c},8_{s})$. The graviton belongs to the first
representation.

Given an irreducible representation of the Poincare algebra one can embed it
in a larger representation to find a Lorentz covariant formulation of the
representation. The price is that the Lorentz covariant fields obey conditions
which are the physical sate conditions and are subject to gauge
transformations. We carry out the same procedure for the massless irreducible
representation of $I_{c}(E_{11})\otimes_{s}l_{1}$ and make contact with the
non-linear realisation of $E_{11}\otimes_{s}l_{1}$ with respect to the Cartan
involution invariant subalgebra of $E_{11}$, denoted by $I_{c}(E_{11})$.
Indeed the degrees of freedom of the massless irreducible representation are
the ones of the latter theory. This supports the asscertion that these are the
only degrees of freedom that arise in the non-linear realisation and it
explains the origin and structure of the duality relations that arise in the
non-linear realisation.

2\. The irreducible representation for a massless particle in E theory The
irreducible representations in E theory were discussed in reference [7]. To be
more precise this paper studied the irreducible representations of the semi-
direct product of $I_{c}(E_{11})$ and it’s vector ($l_{1}$) representation,
denoted $I_{c}(E_{11})\otimes_{s}l_{1}$. This is a natural extension of the
method used to find the irreducible representations of the Poincare group [8].
The similarity comes from the fact that the Poincare group is a semi-direct
product of the Lorentz group $SO(1,3)$ and the group of translations $T_{4}$,
denoted $SO(1,3)\otimes_{s}T_{4}$ Indeed at the lowest level
$I_{c}(E_{11})\otimes_{s}l_{1}$ is just the Poincare group in eleven
dimensions. For the case of the Poincare group one selects a value for the
momentum corresponding to if it is massive or massless and then one computes
the little group that preserves this choice. One then takes a representation
of this little group and finds a representation of the full Poincare group by
boosting, or said more technically, by taking an induced representation. In
this way one can find all irreducible representations of the Poincare group.

In E theory the vector ($l_{1}$) representation contains all the brane charges
and so the first step is to select a preferred value of the vector
representation and compute the little algebra that preserves this value. One
then takes an irreducible representation of this little algebra and boosts it
to the full $I_{c}(E_{11})\otimes_{s}l_{1}$ algebra. At lowest levels the
vector representation contains $P_{a},Z^{a_{1}a_{2}},Z^{a_{1}\ldots
a_{5}},\ldots$ with $a_{1},a_{2},...=0,1,...,10$ where $P_{a}$ are just the
usual momenta and $Z^{a_{1}a_{2}}$ and $Z^{a_{1}\ldots a_{5}}$ the well known
two form and five form charges that first appeared in the supersymmetry
algebra. In reference [7] the irreducible representation that arises when one
takes the momentum to take a value corresponding to it being massless with the
other components of the vector representation being zero was discussed. We
will refer to this irreducible representation as the massless irreducible
representation. Taking some of the higher charges to be non-zero one finds the
irreducible representations corresponding to branes.

The purpose of this section is to fully elucidate the properties of the
massless irreducible representation. To begin the construction we will take
the momentum to have the values $p_{0}=-m,p_{10}=m$ with all other components
being zero. It will be more convenient to use light-cone notation, for which
the components of a vector $V^{a}$ are defined as
$V^{\pm}={1\over\sqrt{2}}(V^{10}\pm V^{0}),V^{i},1,...,9$, and the Minkowski
metric becomes $\eta_{+-}=1,\eta_{ij}=\delta_{ij}$ so that
$V_{\pm}={1\over\sqrt{2}}(V_{10}\pm V_{0})$. In light-cone notation the
massless irreducible representation begins by taking the brane charges to be
$p^{+}=p_{-}=\sqrt{2}m$, with all other momenta and other brane charges being
set equal to zero. We next seek the subalgebra ${\cal H}$ of $I_{c}(E_{11})$
which preserves this choice of brane charges. The brane charges $l_{A}$
transform under $I_{c}(E_{11})$ and so the subalgebra ${\cal H}$ is determined
by the requirement that

$\delta l_{A}=[\Lambda^{\underline{\alpha}}S_{\underline{\alpha}},l_{A}]=0$
$None$

where the charges $l_{A}$ take the above values

It is straightforward to show that for a massless particle the parameters
$\Lambda_{\underline{\alpha}}$ must satisfy
$\Lambda_{+a}=0=\Lambda_{+-}=\Lambda_{+ab}=\ldots$, that is, any parameter
$\Lambda_{\underline{\alpha}}$ with a lowered $+$ index is zero. The resulting
algebra which leaves this choice invariant is [7]

${\cal H}=\\{J_{+i}\ ,\ J_{ij}\ ,\ S_{+ij}\ ,\ S_{i_{1}i_{2}i_{3}}\ ,\
S_{+i_{1}\ldots i_{5}}\ ,\ \ldots\\}\ \ ,\ \ i,j,\ldots=2,\ldots 10.$ $None$

The generators possessing a lowered $+$ index satisfy the commutation
relations

$[J_{+i},J_{+j}]=0\ ,\ [S_{+i_{1}i_{2}},S_{+j_{1}j_{2}}]=0\ ,\
[J_{+i},S_{+j_{1}j_{2}}]=0\ ,\ \ldots$ $None$

In order to obtain a finite-dimensional unitary representation, these
commutation relations imply that all generators with a lowered $+$ index be
trivially realized in the representation. The remaining generators obey the
$I_{c}(E_{11})$ commutation relations with indices restricted to
$i,j=2,\ldots,10$. The result is that the subalgebra which preserves the above
choice of charges is the Cartan involution invariant subalgebra of $E_{9}$,
denoted ${\cal H}=I_{c}(E_{9})$, given by the generators [7]

$J_{ij}\ ,\ S_{i_{1}i_{2}i_{3}}\ ,\ S_{i_{1}\ldots i_{6}}\ ,\ S_{i_{1}\ldots
i_{8},j}\ ,\ S_{i_{1}\ldots i_{9},j_{1}j_{2}j_{3}}\ ,\ \ldots$ $None$

These steps are similar to those of the irreducible massless representations
of the Poincare algebra. In this case of finds that the generators $J_{+i}$
commute and in order to obtain a finite dimensional irreducible representation
one takes a representation in which these act to give zero leaving us to take
an irreducible representation of the algebra SO(D-2).

The Dynkin diagram of $E_{9}$ is given by deleting the first two nodes, which
correspond to the $+$ and $-$ directions, from the Dynkin diagram of $E_{11}$;

$\matrix{&&&&&&&&&&&&&&\bullet&11&&&\cr&&&&&&&&&&&&&&|&&&&\cr\oplus&-&\oplus&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet\cr
1&&2&&3&&4&&5&&6&&7&&8&&9&&10\cr}$

Here $\oplus$ indicates that the node has been removed from the Dynkin diagram
of $E_{11}$.

In general the theory in $D$ dimensions can then be obtained from the $E_{11}$
Dynkin diagram by deleting node $D$ and analysing the theory with respect to
the algebra corresponding to the remaining nodes. Thus to find the theory in
eleven dimensions we delete node eleven, that is, the top node and analysis
the theory when decomposed into the algebra GL(11), while in $D$ dimensions we
decompose into representations of $A_{D-1}\otimes E_{9-D}$ algebra. To find
the irreducible representations in $D$ dimensions one carries out the same
steps and so for the massless irreducible representation we end up deleting
nodes one and two, as explained above, as well as node $D$. For the case of
eleven dimensions we should therefore delete nodes one and two to arrive at
$E_{9}$ and decompose this algebra in terms of representations of the $A_{8}$
subgroup of $E_{9}$ as shown in the Dynkin diagram below

$\matrix{&&&&&&&&&&&&&&\otimes&11&&&\cr&&&&&&&&&&&&&&|&&&&\cr\oplus&-&\oplus&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet\cr
1&&2&&3&&4&&5&&6&&7&&8&&9&&10\cr}$

Since the irreducible representations are representations of $I_{c}(E_{11})$,
rather than $E_{11}$ itself, we actually decompose the $I_{c}(E_{9})$ subgroup
of $E_{9}$ into representations of $I_{c}(A_{8})={\rm SO}(9)$. Indeed the
generators of equation (2.4) are listed in this decomposition as from the
beginning of our discussion we had in effect deleted node eleven as we are
concentrating on the eleven dimensional case.

The next step is to choose an irreducible representation of $I_{c}(E_{9})$.
Such an reducible representation is provided by the Cartan involution odd
generators of $E_{9}$ which are given by [7]

$T_{ij}=\eta_{ik}K^{k}{}_{k}+\eta_{jk}K^{k}{}_{i}\ \ ,\ \
T_{i_{1}i_{2}i_{3}}=R^{j_{1}j_{2}j_{3}}\eta_{j_{1}i_{1}}\eta_{j_{2}i_{2}}\eta_{j_{3}i_{3}}+R_{i_{1}i_{2}i_{3}}\
,$ $None$
$T_{i_{1}..i_{6}}=R^{j_{1}..j_{6}}\eta_{j_{1}i_{1}}..\eta_{j_{6}i_{6}}-R_{i_{1}..i_{6}}\
,$ $None$
$T_{i_{1}..i_{8},k}=R^{j_{1}..j_{8},l}\eta_{j_{1}i_{1}}..\eta_{j_{8}i_{8}}\eta_{lk}+R_{i_{1}..i_{8},k}\
,$ $None$
$T_{i_{1}..i_{9},j_{1}j_{2}j_{3}}=R^{m_{1}..m_{9},l_{1}l_{2}l_{3}}\eta_{m_{1}i_{1}}..\eta_{m_{9}i_{9}}\eta_{l_{1}j_{1}}\eta_{l_{2}j_{2}}\eta_{l_{3}j_{3}}-R_{i_{1}..i_{9},j_{1}j_{2}j_{3}}\
,\ \ldots$ $None$

These generators provide a linear representation of $I_{c}(E_{9})$ because the
involution operator $I_{c}$ is defined to act on the generators $A,B,...$ of a
Lie algebra as $I_{c}(AB)=I_{c}(A)I_{c}(B)$ and so $I_{c}([{\rm even},{\rm
odd}])=-[{\rm even},{\rm odd}]$ which guarantees that the commutator will
always be a Cartan involution odd generator. The generators of equation (2.8)
are the Cartan involution odd generators of $I_{c}(E_{11})$ when restricted to
their indices taking only the values $i,j=2,...,10$. Clearly one could for the
massless case take a different irreducible representation but in this paper we
will only consider this case.

As a result we take our representation to consist of fields corresponding to
the Cartan involution odd generators

$\\{h_{ij}(0)\ ,A_{i_{1}i_{2}i_{3}}(0)\ ,\ A_{i_{1}\ldots i_{6}}(0)\ ,\
h_{i_{1}\ldots i_{8},j}(0)\ ,\ \ldots\\},\ \ {\rm with}\ \ \ i,j,...=2,...,10\
.$ $None$

The value $(0)$ indicates that these fields are before the boost which takes
them to be a representation of $I_{c}(E_{11})\otimes_{s}l_{1}$.

We recognise $h_{ij}(0)$ and $A_{i_{1}i_{2}i_{3}}(0)$ as the degree of freedom
of the graviton and three form respectively. There are however, an infinite
number of higher level fields. It was proposed in [7] that these fields are
connected to the gravition and three form by certain duality relations. The
purpose of this section is to investigate these relations in more detail.

The fields of equation (2.9) live in the representation associated to the
Cartan involution odd generators and so we consider the quantity

$\overline{{\cal
V}}=h_{ij}T^{ij}+A_{i_{1}i_{2}i_{3}}T^{i_{1}i_{2}i_{3}}+A_{i_{1}..i_{6}}T^{i_{1}..i_{6}}+h_{i_{1}..i_{8},j}T^{i_{1}..i_{8},j}+A_{i_{1}..i_{9},j_{1}j_{2}j_{3}}T^{i_{1}..i_{9},j_{1}j_{2}j_{3}}+\ldots$
$None$

The transformations of the fields follow from that of the Cartan involution
odd generators under $I_{c}(E_{9})$. In particular under the level one
generator $S_{i_{1}i_{2}i_{3}}$ of $I_{c}(E_{9})$ we take

$\delta\overline{{\cal
V}}=[\Lambda^{i_{1}i_{2}i_{3}}S_{i_{1}i_{2}i_{3}},\overline{{\cal
V}}]+[\Lambda^{i_{1}\ldots i_{6}}S_{i_{1}\ldots i_{6}},\overline{{\cal
V}}]+[\Lambda^{i_{1}\ldots i_{6}}S_{i_{1}\ldots i_{6}},\overline{{\cal
V}}]+[\Lambda^{i_{1}\ldots i_{8},j}S_{i_{1}\ldots i_{8},j},\overline{{\cal
V}}]$ $None$

Using the $E_{11}$ algebra to evaluate equation (2.11) we find that

$\delta
h_{ij}=18\,\Lambda_{(i|k_{1}k_{2}|}A_{j)}{}^{k_{1}k_{2}}-2\,\eta_{ij}\,\Lambda^{k_{1}k_{2}k_{3}}A_{k_{1}k_{2}k_{3}}-5!\,\eta_{ij}\,\Lambda^{k_{1}\ldots
k_{6}}A_{k_{1}\ldots k_{6}}-9\cdot 5!\,\Lambda^{k_{1}\ldots
k_{5}}{}_{(i}A_{j)k_{1}\ldots k_{5}}$ $+7\cdot 9\cdot 10\cdot
16\,(h_{k_{1}\ldots k_{8},(i|}\Lambda^{k_{1}\ldots
k_{8}}{}_{,|j)}+8h_{(i|k_{1}\ldots k_{7},l}\Lambda_{|j)}{}^{k_{1}\ldots
k_{7},l}-\eta_{ij}\Lambda^{k_{1}\ldots k_{8},l}h_{k_{1}\ldots k_{8},l})$
$None$ $\delta A_{i_{1}i_{2}i_{3}}=-3\cdot
2\,h_{[i_{1}}{}^{k}\Lambda_{i_{2}i_{3}]k}+{5!\over
2}\,A_{i_{1}i_{2}i_{3}}{}^{k_{1}k_{2}k_{3}}\Lambda_{k_{1}k_{2}k_{3}}+{5!\over
2}\,\Lambda_{i_{1}i_{2}i_{3}k_{1}k_{2}k_{3}}A^{k_{1}k_{2}k_{3}}$ $+{7!\cdot
2\over 3}\,\Lambda^{k_{1}\ldots k_{6}}(h_{i_{1}i_{2}i_{3}[k_{1}\ldots
k_{5},k_{6}]}-h_{k_{1}\ldots k_{6}[i_{1}i_{2},i_{3}]})$ $None$ $\delta
A_{i_{1}..i_{6}}=2\,\Lambda_{[i_{1}i_{2}i_{3}}A_{i_{4}i_{5}i_{6}]}+112\,h_{i_{1}..i_{6}k_{1}k_{2},k_{3}}\Lambda^{k_{1}k_{2}k_{3}}+112\,h_{[i_{1}..i_{5}|k_{1}k_{2}k_{3}|,i_{6}]}\Lambda^{k_{1}k_{2}k_{3}}$
$+12\,\Lambda_{k[i_{1}\ldots i_{5}}h_{i_{6}]}{}^{k}$ $None$ $\delta
h_{i_{1}..i_{8},j}=2\,(\Lambda_{[i_{1}i_{2}i_{3}}A_{i_{4}..i_{8}]j}-\Lambda_{j[i_{1}i_{2}}A_{i_{3}..i_{8}]})-12\cdot
8\cdot
3\,\Lambda^{k_{1}k_{2}k_{3}}(A_{i_{1}..i_{8}k_{1},k_{2}k_{3}j}+A_{[i_{1}..i_{7}|jk_{1},k_{2}k_{3}|i_{8}]})$
$+3\,\Lambda_{[i_{1}\ldots i_{6}}A_{i_{7}i_{8}]j}-3\,\Lambda_{[i_{1}\ldots
i_{6}}A_{i_{7}i_{8}j]}.$ $None$

In this section we will only use the $\Lambda_{i_{1}i_{2}i_{3}}$ variations.

We will now going to postulate duality relations and show they are preserved
under the $I_{c}(E_{9})$ symmetry and as a result the number of fields in the
representation is radically reduced, indeed, one can take the representation
to contain only the graviton and the three form. We first propose a duality
relation between the three-form and six-form which is given by

$E_{i_{1}i_{2}i_{3}}=A_{i_{1}i_{2}i_{3}}+c\,\varepsilon_{i_{1}i_{2}i_{3}}{}^{j_{1}..j_{6}}A_{j_{1}..j_{6}}=0$
$None$

where $c$ is a constant which we will fix by requiring that this relation is
part of an infinite set of relations that are, as a collection, left invariant
by $I_{c}(E_{9})$. Varying equation (2.16) under $I_{c}(E_{9})$ we find that

$\delta
E_{i_{1}i_{2}i_{3}}=2c\Lambda_{j_{1}j_{2}j_{3}}\varepsilon_{i_{1}i_{2}i_{3}}{}^{j_{1}..j_{6}}(A_{j_{4}j_{5}j_{6}}+{1\over
c}{1\over
3!4!}\varepsilon_{j_{4}j_{5}j_{6}}{}^{i_{1}i_{2}i_{3}j_{1}j_{2}j_{3}}A_{i_{1}i_{2}i_{3}j_{1}j_{2}j_{3}})+\ldots$
$None$

where $+\ldots$ denotes the gravity terms. Clearly we will only recover our
original relation if $c=\pm{1\over 12}$ and we choose $c=-{1\over 12}$. For
this choice the variation takes the form

$\delta E_{i_{1}i_{2}i_{3}}={1\over
6}\Lambda_{j_{1}j_{2}j_{3}}\varepsilon_{i_{1}i_{2}i_{3}}{}^{j_{1}..j_{6}}E_{j_{4}j_{5}j_{6}}-6E_{k[i_{1}}\Lambda^{k}{}_{i_{2}i_{3}]}$
$None$

where

$E_{i_{1}i_{2}i_{3}}=A_{i_{1}i_{2}i_{3}}+{1\over
12}\varepsilon_{i_{1}i_{2}i_{3}}{}^{j_{1}..j_{6}}A_{j_{1}..j_{6}}=0$ $None$
$E_{ij}=h_{ij}-{1\over
4}\varepsilon_{i}{}^{r_{1}..r_{8}}h_{r_{1}..r_{8},j}=0.$ $None$

Thus we not only recover our original duality relation but find the new
duality relation of equation (2.20) which relates the graviton and dual
graviton. The irreducibility condition $h_{[i_{1}..i_{8},j]}=0$ implies that
for this relation to be consistent, the field $h_{ij}$ must be traceless,
$h^{i}{}_{i}=0$. With this condition $h_{ij}$ has ${10.9\over 2}-1=44$ degrees
of freedom as it should.

We now vary the duality relation of equation (2.20) to find that

$\delta E_{ij}=-{1\over 2}{1\over 3\cdot
5!}\,\varepsilon_{i}{}^{r_{1}..r_{8}}(\Lambda_{r_{1}r_{2}r_{3}}\varepsilon_{r_{4}..r_{8}j}{}^{k_{1}k_{2}k_{3}}E_{k_{1}k_{2}k_{3}}-\Lambda_{jr_{1}r_{2}}\varepsilon_{r_{3}..r_{8}}{}^{k_{1}k_{2}k_{3}}E_{k_{1}k_{2}k_{3}})$
$+12\cdot 2\cdot
3\,\Lambda^{k_{1}k_{2}k_{3}}\varepsilon_{i}{}^{r_{1}..r_{8}}(E_{r_{1}..r_{8}k_{1},k_{2}k_{3}j}-E_{jr_{1}..r_{7}k_{1},k_{2}k_{3}r_{8}})=0$
$None$

where

$E_{i_{1}..i_{9},j_{1}j_{2}j_{3}}=A_{i_{1}..i_{9},j_{1}j_{2}j_{3}}+{1\over
9!}\ \varepsilon_{i_{1}..i_{9}}A_{j_{1}j_{2}j_{3}}=0.$ $None$

Thus the variation contains the previously-derived duality relation of
equation (2.13), and a new duality relation relating the three-form
$A_{i_{1}i_{2}i_{3}}$ and nine-three form $A_{i_{1}\ldots
i_{9},j_{1}j_{2}j_{3}}$.

These duality relations $E_{ij}=0$, $E_{i_{1}i_{2}i_{3}}=0$, $E_{i_{1}\ldots
i_{9},j_{1}j_{2}j_{3}}=0$ are the first of an infinite tower of duality
relations showing that the fields at levels, two, three, four, etc… can be
expressed in terms of the fields

$h_{ij},\ \ \ A_{i_{1}i_{2}i_{3}}.$ $None$

The six form is related to the three form by equation (2.19) and the dual
graviton field $h_{i_{1}\ldots i_{9},j}$ is related to the graviton by
equation (2.20). All the higher level fields carry the indices of these fields
as well a multiple sets of blocks of nine indices and we can expect that in
all the higher level duality relations these indices are carried by
$\varepsilon_{i_{1}..i_{9}}$ ’s in a way that is similar to how they appear in
equation (2.22) . These indices correspond to the action of the affine
generator of $I_{c}(E_{9})$. As a result it should be possible of show that
the complete infinite set of duality relations are invariant under the
$I_{c}(E_{9})$ symmetry.

Hence we have found that the massless irreducible representation contains the
fields of equation (2.23) and as a result it contains the $44+84=128$ bosonic
degrees of freedom of eleven-dimensional supergravity. Rather than write them
as in equation (2.23) we can write the degrees of freedom in terms of an
$E_{8}$ multiplet, or in our case here, an $I_{C}(E_{8})=SO(16)$ multiplet,
namely

$h_{i^{\prime}j^{\prime}}\ ,A_{i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}}\
,A_{i_{1}^{\prime}\ldots i_{6}^{\prime}}\ ,\ h_{j^{\prime}}\equiv{1\over
8!}\epsilon^{i_{1}^{\prime}\ldots i^{\prime}_{8}}h_{i_{1}^{\prime}\ldots
i^{\prime}_{8},j^{\prime}},\ i^{\prime},j^{\prime},\ldots=3,\ldots,10$ $None$

In this paper we will take un-primed indices to range over
$i,j,\ldots=2,\ldots,10$ and primed indices to range over
$i^{\prime},j^{\prime},\ldots=3,\ldots,10$. The fields of equation (2.24) have
been obtained from those of equation (2.23) by expressing
$A_{i_{1}^{\prime}i_{2}^{\prime}2}$ in terms of $A_{i_{1}^{\prime}\ldots
i_{6}^{\prime}}$ using the duality relation of equation (2.16) and also by
expressing $h_{2j^{\prime}}$ in terms of $h_{i_{1}^{\prime}\ldots
i_{8}^{\prime},j^{\prime}}$ using the duality relation of equation (2.20). The
fields of equation (2.24) also give $36+56+28+8=128$ degress of freedom. We
note that $h_{i^{\prime}}{}^{i^{\prime}}+h_{9}{}^{9}=0$ and as we have not
included $h_{9}{}^{9}$ we do not take $h_{i^{\prime}}{}^{i^{\prime}}$ equal to
zero. 3\. Reduction of degrees of freedom from the algebra viewpoint In the
previous section we have seen that the higher level fields can be expressed in
terms of the fields $h_{ij}$ and $A_{i_{1}i_{2}i_{3}}$ at levels zero and one
through the existence of duality relations such as those of equations (2.19),
(2.20) and (2.22). In this section we will show that the massless irreducible
representation is annihilated by an infinite set of generators and this
corresponds to the existence of the infinite set of duality relations. We
begin by considering the generator of the form

$N_{k_{1}k_{2}k_{3}}=S_{k_{1}k_{2}k_{3}}+c_{1}\
\varepsilon_{k_{1}k_{2}k_{3}}{}^{r_{1}\ldots r_{6}}S_{r_{1}\ldots r_{6}}$
$None$

for a constant $c_{1}$. Under the action of this generator the fields
transform under ${\Lambda}^{k_{1}k_{2}k_{3}}N_{k_{1}k_{2}k_{3}}$ as

$\delta
h_{ij}=18{\Lambda}_{(i|k_{1}k_{2}|}A_{j)}{}^{k_{1}k_{2}}-2\eta_{ij}{\Lambda}^{k_{1}k_{2}k_{3}}A_{k_{1}k_{2}k_{3}}$
$-5!c_{1}(\cdot 3\cdot 3\
{\Lambda}^{k_{1}k_{2}k_{3}}\varepsilon_{k_{1}k_{2}k_{3}}{}^{r_{1}\ldots
r_{5}}{}_{(i}A_{j)}{}_{r_{1}..r_{5}}+\eta_{ij}{\Lambda}^{k_{1}k_{2}k_{3}}\varepsilon_{k_{1}k_{2}k_{3}}{}^{r_{1}\ldots
r_{6}}A_{r_{1}..r_{6}})$ $None$ $\delta A_{i_{1}i_{2}i_{3}}=-3\cdot
2h_{[i_{1}}{}^{j}{\Lambda}_{i_{2}i_{3}]j}+60A_{i_{1}i_{2}i_{3}}{}^{k_{1}k_{2}k_{3}}{\Lambda}_{k_{1}k_{2}k_{3}}-{5!\over
2}c_{1}\
{\Lambda}^{k_{1}k_{2}k_{3}}A^{r_{1}r_{2}r_{3}}\varepsilon_{k_{1}k_{2}k_{3}r_{1}r_{2}r_{3}}{}^{i_{1}i_{2}i_{3}}$
$-{7!\cdot 2\over 3}c_{1}\
{\Lambda}^{k_{1}k_{2}k_{3}}(\varepsilon_{k_{1}k_{2}k_{3}}{}_{r_{1}\ldots
r_{6}}A^{r_{1}..r_{6}}{}_{[i_{1}i_{2},i_{3}]}-\varepsilon_{k_{1}k_{2}k_{3}}{}_{r_{1}\ldots
r_{5}j}A_{i_{1}i_{2}i_{3}}{}^{r_{1}..r_{5},j})$ $None$ $\delta A^{i_{1}\ldots
i_{6}}=2{\Lambda}^{[i_{1}i_{2}i_{3}}A^{i_{4}i_{5}i_{6}]}+112h^{i_{1}..i_{6}k_{1}k_{2},k_{3}}{\Lambda}_{k_{1}k_{2}k_{3}}+112h^{[i_{1}..i_{5}|k_{1}k_{2}k_{3}|,i_{6}]}\tilde{\Lambda}_{k_{1}k_{2}k_{3}}$
$+12\ c_{1}\
{\Lambda}^{k_{1}k_{2}k_{3}}\varepsilon_{k_{1}k_{2}k_{3}j}{}^{[i_{1}\ldots
i_{5}}h^{i_{6}]j}$ $-3\cdot
7!c_{1}\,{\Lambda}^{k_{1}k_{2}k_{3}}\varepsilon_{k_{1}k_{2}k_{3}}{}_{r_{1}\ldots
r_{6}}A^{[i_{1}..i_{4}|[r_{1}..r_{5},r_{6}]|i_{5}i_{6}]}$ $-15\cdot
7!c_{1}\,{\Lambda}^{k_{1}k_{2}k_{3}}\varepsilon_{k_{1}k_{2}k_{3}}{}_{r_{1}\ldots
r_{4}r_{5}r_{6}}A^{[r_{1}..r_{4}|[i_{1}..i_{5},i_{6}]|r_{5}r_{6}]}$ $-4\cdot
7!c_{1}{\Lambda}^{k_{1}k_{2}k_{3}}\varepsilon_{k_{1}k_{2}k_{3}}{}_{r_{1}\ldots
r_{6}}A^{r_{1}..r_{6}[i_{1}i_{2}i_{3},i_{4}i_{5}i_{6}]}$ $None$ $\delta
h_{i_{1}\ldots
i_{8},j}=2({\Lambda}_{[i_{1}i_{2}i_{3}}A_{i_{4}..i_{8}]j}-\tilde{\Lambda}_{j[i_{1}i_{2}}A_{i_{3}..i_{8}]})-12\cdot
8\cdot
3{\Lambda}^{k_{1}k_{2}k_{3}}(A_{i_{1}..i_{8}k_{1},k_{2}k_{3}j}+A_{[i_{1}..i_{7}|jk_{1},k_{2}k_{3}|i_{8}]})$
$+3c_{1}\
{\Lambda}^{k_{1}k_{2}k_{3}}\varepsilon_{k_{1}k_{2}k_{3}}{}^{[i_{1}\ldots
i_{6}}A^{i_{7}i_{8}]j}-3c_{1}\
{\Lambda}^{k_{1}k_{2}k_{3}}\varepsilon_{k_{1}k_{2}k_{3}}{}^{[i_{1}\ldots
i_{6}}A^{i_{7}i_{8}j]}$ $None$

On setting $c_{1}=-{1\over 3\cdot 5!}$ in the generator of equation (3.1)
becomes

$N_{k_{1}k_{2}k_{3}}=S_{k_{1}k_{2}k_{3}}-{1\over 3\cdot 5!}\
\varepsilon_{k_{1}k_{2}k_{3}}{}^{r_{1}\ldots r_{6}}S_{r_{1}\ldots r_{6}}\ \ ,$
$None$

With this choice the variations of equations (3.2) to (3.5) can be expressed
in terms of the duality relations defined in equations (2.19) (2.20) and
(2.22) and become equal to zero

$\delta
h_{ij}=18{\Lambda}_{(i}{}^{k_{1}k_{2}}E_{j)k_{1}k_{2}}-2\eta_{ij}{\Lambda}^{k_{1}k_{2}k_{3}}E_{k_{1}k_{2}k_{3}}=0$
$None$ $\delta A_{i_{1}i_{2}i_{3}}={1\over
3!}\varepsilon_{i_{1}i_{2}i_{3}k_{1}k_{2}k_{3}}{}_{r_{1}r_{2}r_{3}}{\Lambda}^{k_{1}k_{2}k_{3}}E^{r_{1}r_{2}r_{3}}-6{\Lambda}_{[i_{1}i_{2}}{}^{k}E_{i_{3}]k}=0$
$None$ $\delta A_{i_{1}\ldots i_{6}}={1\over
30}{\Lambda}^{k_{1}k_{2}k_{3}}\varepsilon_{k_{1}k_{2}k_{3}[i_{1}\ldots
i_{5}|}{}^{j}E_{|i_{6}]}{}_{j}+28{\Lambda}_{k_{1}k_{2}k_{3}}\varepsilon^{k_{1}k_{2}k_{3}}{}^{r_{1}\ldots
r_{6}}E_{r_{1}..r_{6}[i_{1}i_{2}i_{3},i_{4}i_{5}i_{6}]}$ $None$
$-140{\Lambda}_{k_{1}k_{2}k_{3}}\varepsilon^{k_{1}k_{2}k_{3}r_{1}\ldots
r_{4}r_{5}r_{6}}E_{i_{1}..i_{6}[r_{1}r_{2}r_{3},r_{4}r_{5}r_{6}]}=0$ $None$

In a similar fashion one can identify other generators that annihilate the
massless irreducible representation. Indeed one finds that the generator

$N_{ij}=J_{ij}+{2\over 8!}\varepsilon_{[i}{}^{r_{1}\ldots
r_{8}}S_{|r_{1}\ldots r_{8}|,j]}$ $None$

The variation of the graviton under a transformation of this generator with
parameter ${\Lambda}^{ij}$ gives

$\delta h_{ij}=2{\Lambda}_{i}{}^{k}E_{kj}+2{\Lambda}_{j}{}^{k}E_{ki}=0,$
$None$

It would seem inevitable that one has an infinite number of generators that
annihilate the entire representation and we will take this to be the case. At
the next level we would expect to find the generator

$\tilde{N}_{i_{1}i_{2}i_{3}}\equiv S_{i_{1}i_{2}i_{3}}+{1\over
9!}\epsilon^{j_{1}\ldots j_{9}}S_{j_{1}\ldots j_{9},}{}^{i_{1}i_{2}i_{3}}$
$None$

where we have used the value of the coefficient that we will find below. The
pattern of the higher level generators that annihilate the representation is
apparent. To a given generator one adds with a suitable coefficient another
generator which possess an additional block of nine anti-symmetric indices
which are saturate with the $\epsilon$ symbol.

The set of all generators that annihilate the representation must form a
subalgebra, denoted $I$, of $I_{c}(E_{9})$ and we will now show that this is
true for such lowest level generators. Indeed we find that

$[N^{i_{1}i_{2}i_{3}},N_{j_{1}j_{2}j_{3}}]=2N^{i_{1}i_{2}i_{3}}{}_{j_{1}j_{2}j_{3}}-9\cdot
4\delta^{[i_{1}i_{2}}_{[j_{1}j_{2}}N^{i_{3}]}{}_{j_{3}]}-{1\over
2}\varepsilon^{i_{1}i_{2}i_{3}}{}_{j_{1}j_{2}j_{3}k_{1}k_{2}k_{3}}\tilde{N}^{k_{1}k_{2}k_{3}}$
$None$

where

$N_{i_{1}...i_{6}}=S_{i_{1}..i_{6}}+{1\over
12}\varepsilon_{i_{1}..i_{6}}{}^{k_{1}k_{2}k_{3}}S_{k_{1}k_{2}k_{3}}={1\over
12}\varepsilon_{i_{1}..i_{6}}{}^{k_{1}k_{2}k_{3}}N_{k_{1}k_{2}k_{3}}$ $None$

The generators that annihilate the irreducible representation also form an
ideal. We recall that an ideal $I$ of a Lie algebra $G$ is a subalgebra
consisting of elements $X\in I$ such that such that $[X,Y]\in I$ for all $Y\in
G$. One finds that

$[S^{i_{1}i_{2}i_{3}},N_{j_{1}j_{2}j_{3}}]=-{1\over
6}\varepsilon^{i_{1}i_{2}i_{3}}{}_{j_{1}j_{2}j_{3}k_{1}k_{2}k_{3}}N^{k_{1}k_{2}k_{3}}-18\delta^{[i_{1}i_{2}}_{[j_{1}j_{2}}N^{i_{3}]}{}_{j_{3}]}$
$None$

and that

$[S_{i_{1}i_{2}i_{3}},N_{jk}]=3\eta_{[i_{1}|j}\tilde{N}_{k|i_{2}i_{3}]}+3\eta_{[i_{1}|j}N_{k|i_{2}i_{3}]}$
$None$

Since the commutators of the generator $S^{i_{1}i_{2}i_{3}}$ lead to the whole
of $I_{c}(E_{9})$ it follows that the equations (3.16) and (3.17) together
with their higher level analogues, which we have not shown, imply that the
generators in the subalgebra $I$ are an ideal of $I_{c}(E_{9})$. Starting from
the level zero field $h_{ij}(0)$ of the massless irreducible representation we
can find all fields in the representation by the action of
$S^{i_{1}i_{2}i_{3}}$. Hence if the generators which annihilate the massless
irreducible representation form an ideal in $I_{c}(E_{9})$ it follows that if
all the elements of $I$ annihilate the lowest field, the graviton, then they
will annihilate all fields in the representation.

Given a Lie algebra $G$ (group) which contains an ideal subalgebra (subgroup)
$I$ then the coset $G\over I$ is also a Lie algebra (group). If $A_{1},A_{2}$
belong to $G$ the corresponding equivalence relation is $A_{1}\sim A_{2}$
means $A_{1}=A_{2}+i$ for some $i\in I$. Indeed $I_{c}(E_{9})\over I$ is a Lie
algebra. It is clear from the form of the generators in the ideal that all the
generators in $I_{c}(E_{9})$ are related to the generators $J_{i_{1}i_{2}}$
and $S_{i_{1}i_{2}i_{3}}$ by the equivalence relations. Hence
$I_{c}(E_{9})\over I$ contains just these two generators. These generators
contain $36+84=120$ generators and, as we will show in the next next section,
they generate S0(16). As a result

${I_{c}(E_{9})\over I}=SO(16)$ $None$

In this section we have seen that there is a radical reduction in the number
of states of the massless irreducible representation can be traced to the
existence of a ideal in $I_{c}(E_{9})$. It contains only the graviton and
three form. Indeed due to equation (3.18) these states belong to an
irreducible representation of the much smaller algebra SO(16). The situation
is a bit similar to the existence of highest weight states in representations
of the Virasoro algebra and the corresponding reduction in the number of
states in the representation. 4\. The bosonic states viewed as a spinor In the
section two we found that although the massless irreducible representation of
the little group $I_{c}(E_{9})$ contained, at first sight, an infinite number
of states it was subject to an infinite set of duality relations that reduced
the number of degrees of freedom it contained to 128. These can be listed as
in equation (2.24), or as an $I_{c}(E_{8})={\rm SO}(16)$ multiplet, in
equation (2.24). The SO(16) algebra has two 128 dimensional irreducible
representations both of which are spinors. The number of components of a
spinor in sixteen dimensions is $2^{{16\over 2}}=256$. However, in sixteen,
effectively Euclidean, dimensions we can have Majorana Weyl spinors which have
just 128 component. Hence it must be that our 128 bosonic states belong to a
spinor representation of SO(16). As $E_{8}$ contains the gravity line
consisting of the nodes three to ten it contains the SL(9) algebra and as a
result $I_{c}(E_{8})={\rm SO}(16)$ contains the SO(9) which acts on the
spacetime time coordinates. The purpose of this section is to understand how
the bosonic states can be assembled into this spinor representation.

We found that for the massless representation the little algebra is
$I_{c}(E_{9})$ which arises when we delete nodes one and two in the $E_{11}$
Dynkin diagram. Further decomposing the $I_{c}(E_{9})$ representations into
those of $I_{c}(E_{8})$ corresponds to also deleting node three in addition to
nodes one and two in the $E_{11}$ Dynkin diagram as shown below

$\matrix{&&&&&&&&&&&&&&\bullet&11&&&\cr&&&&&&&&&&&&&&|&&&&\cr\oplus&-&\oplus&-&\otimes&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet&-&\bullet\cr
1&&2&&3&&4&&5&&6&&7&&8&&9&&10\cr}$

The eleven dimensional theory appears when we deleted node eleven and
decompose with respect to the remaining $A_{10}$ algebra. In the context of
the massless irreducible representation, which has little group
$I_{c}(E_{9})$, this implies that we should decompose with respect to
$I_{c}(A_{8})={\rm SO}(9)$. Indeed the fields of equation (2.9), and so
equation (2.23), appear as representations of $I_{c}(A_{8})={\rm SO}(9)$.
However, we wish to study the massless irreducible representation from the
viewpoint of $I_{c}(E_{8})={\rm SO}(16)$ which is the algebra that appears in
the above $E_{11}$ Dynkin diagram. As such we have in effect to undo the
deletion of node eleven. To better understand the manner in which bosonic
fields belong to the 128 dimensional spinor representation we consider the
fields from the viewpoint of the ${\rm SO}(8)\times{\rm SO}(8)$ subalgebra of
${\rm SO}(16)$.

The first step is to identify the ${\rm SO}(8)\times{\rm SO}(8)$ generators
amongst those of $I_{c}(E_{8})$. The latter generators arise in $E_{11}$ as

$\\{J_{i^{\prime}j^{\prime}}\ ,\
S_{i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}}\ ,\ S_{i_{1}^{\prime}\ldots
i_{6}^{\prime}}\ ,\ S_{i_{1}^{\prime}\ldots i_{8}^{\prime},j^{\prime}}\\},\ \
{\rm with}\ \ i^{\prime}.j^{\prime},\ldots 3,\ldots,10$ $None$

These Cartan involution invariant generators appear in equations (2.5-2.8) if
we change the signs between the two generators from plus to minus and visa-
versa. In equation (4.1) one finds $28+56+28+8=120$ generators. The
$J_{i^{\prime}j^{\prime}}$ generate an ${\rm SO}(8)$ This subalgebra is the
$I_{c}(A_{7})$ that appears as the gravity line consisting of nodes four to
ten of the $E_{11}$ Dynkin diagram. As such this SO(8) is just part of the
familiar gravity line symmetries that are foremost in E theory discussions and
act on the spacetime coordinates. The ${\rm SO}(8)\times{\rm SO}(8)$
subalgebra consists of the generators

$J^{\pm}_{i^{\prime}j^{\prime}}=J_{i^{\prime}j^{\prime}}\pm
2\hat{J}_{i^{\prime}j^{\prime}},\quad\quad{\rm
where}\quad\quad\hat{J}_{i^{\prime}j^{\prime}}={1\over
6!}\varepsilon_{i^{\prime}j^{\prime}}{}^{k_{1}^{\prime}\ldots
k_{6}^{\prime}}S_{k_{1}^{\prime}\ldots k_{6}^{\prime}}.$ $None$

Using the well known commutators of the $E_{11}$ algebra one finds that they
obey the commutation relations

$[J^{+}{}^{i^{\prime}j^{\prime}},J^{-}_{k^{\prime}l^{\prime}}]=0\ ,$ $None$
$[J^{+}{}^{i^{\prime}j^{\prime}},J^{+}_{k^{\prime}l^{\prime}}]=-8\delta^{[i^{\prime}}{}_{[k^{\prime}}J^{+}{}^{j^{\prime}]}{}_{l^{\prime}]}\
,\
[J^{-}{}^{i^{\prime}j^{\prime}},J^{-}_{k^{\prime}l^{\prime}}]=-8\delta^{[i^{\prime}}{}_{[k^{\prime}}J^{-}{}^{j^{\prime}]}{}_{l^{\prime}]}.$
$None$

which are indeed those of ${\rm SO}(8)\times{\rm SO}(8)$. We note that the
gravity line SO(8) discussed above arises as the algebra which is the diagonal
subalgebra of ${\rm SO}(8)\times{\rm SO}(8)$. We note that we have three
different SO(8) algebras.

We will now decompose the spinor representation of SO(16), which contains the
${\rm 128}$ degrees of freedom, into representations of ${\rm SO}(8)\times{\rm
SO}(8)$. To begin with rather than consider the fields we will consider the
corresponding Cartan involution odd generators of equation (2.5-2.8). While
these transform into each other in the usual way under the SO(8) rotations
generated by $J_{i^{\prime}j^{\prime}}$, under the $S_{i_{1}^{\prime}\ldots
i_{6}^{\prime}}$ generator we find that

$[S_{i_{1}^{\prime}\ldots
i_{6}^{\prime}},T^{j^{\prime}k^{\prime}}]=-6T_{[i_{1}^{\prime}\ldots
i_{5}^{\prime}}{}^{j^{\prime}}\delta^{k^{\prime}}{}_{i_{6}^{\prime}]}-6T_{[i_{1}^{\prime}\ldots
i_{5}^{\prime}}{}^{k^{\prime}}\delta^{j^{\prime}}{}_{i_{6}^{\prime}]},$ $None$
$[S_{i_{1}^{\prime}\ldots
i_{6}^{\prime}},T^{j_{1}^{\prime}j_{2}^{\prime}j_{3}^{\prime}}]=60T_{[i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}}\delta^{j_{1}^{\prime}j_{2}^{\prime}j_{3}^{\prime}}_{i_{4}^{\prime}i_{5}^{\prime}i_{6}^{\prime}]}+3T_{i_{1}^{\prime}\ldots
i_{6}^{\prime}}{}^{[j_{1}^{\prime}j_{2}^{\prime},j_{3}^{\prime}]},$ $None$
$[S_{i_{1}^{\prime}\ldots i_{6}^{\prime}},T^{j_{1}^{\prime}\ldots
j_{6}^{\prime}}]=-120T^{k^{\prime}}{}_{k^{\prime}}\delta^{j_{1}^{\prime}\ldots
j_{6}^{\prime}}_{i_{1}^{\prime}\ldots
i_{6}^{\prime}}+1080T_{[i_{1}^{\prime}}{}^{[j_{1}^{\prime}}\delta^{j_{2}^{\prime}\ldots
j_{6}^{\prime}]}_{i_{2}^{\prime}\ldots i_{6}^{\prime}]},$ $None$
$[S_{i_{1}^{\prime}\ldots i_{6}^{\prime}},T^{j_{1}^{\prime}\ldots
j_{8}^{\prime},k^{\prime}}]=3360(T^{[j_{1}^{\prime}j_{2}^{\prime}j_{3}^{\prime}}\delta^{j_{4}^{\prime}\ldots
j_{8}^{\prime}]k^{\prime}}_{i_{1}^{\prime}\ldots
i_{5}^{\prime}i_{6}^{\prime}}-T^{[j_{1}^{\prime}j_{2}^{\prime}|k^{\prime}|}\delta^{j_{3}^{\prime}\ldots
j_{8}^{\prime}]}_{i_{1}^{\prime}\ldots i_{6}^{\prime}}),$ $None$

Examining these equations we notice that $T^{j^{\prime}k^{\prime}}$ and
$\tilde{T}_{i^{\prime}j^{\prime}}\equiv{1\over
6!}\varepsilon_{i^{\prime}j^{\prime}}{}^{k_{1}^{\prime}\ldots
k_{6}^{\prime}}T_{k_{1}^{\prime}\ldots k_{6}^{\prime}}$ transform into each
other as do $T^{j_{1}^{\prime}j_{2}^{\prime}j_{3}^{\prime}}$ and
$T^{i^{\prime}}\equiv{1\over 8!}\varepsilon_{k_{1}^{\prime}\ldots
k_{8}^{\prime}}T^{k_{1}^{\prime}\ldots k_{8}^{\prime},i^{\prime}}$.

Motivated by the result just above we define the combination

$\hat{T}{}_{i^{\prime}}{}^{j^{\prime}}=T_{i^{\prime}}{}^{j^{\prime}}+2\tilde{T}_{i^{\prime}}{}^{j^{\prime}}-{1\over
6}\delta_{i^{\prime}}^{j^{\prime}}T_{k^{\prime}}{}^{k^{\prime}}$ $None$

which contains $8.8=64$ degrees of freedom as $T_{i^{\prime}j^{\prime}}$ and
$\tilde{T}_{i^{\prime}j^{\prime}}$ are symmetric and antisymmetric
respectively. Under ${\rm SO}(8)\times{\rm SO}(8)$ the objects
$\hat{T}_{i^{\prime}j^{\prime}}$ transforms as

$[J^{+i^{\prime}j^{\prime}},\hat{T}{}_{k^{\prime}}{}_{l^{\prime}}]=-4\delta^{[i^{\prime}}{}_{k^{\prime}}\hat{T}{}^{j^{\prime}]}{}_{l^{\prime}},\
\
[J^{-i^{\prime}j^{\prime}},\hat{T}{}_{k^{\prime}}{}_{l^{\prime}}]=-4\delta^{[i^{\prime}}{}_{l^{\prime}}\hat{T}{}_{k^{\prime}}{}^{j^{\prime}]}$
$None$

We recognise that the first ${\rm SO}(8)$ transforms the first index on
$\hat{T}{}_{k^{\prime}}{}_{l^{\prime}}$ as a vector and the second ${\rm
SO}(8)$ transforms the second index on
$,\hat{T}{}_{k^{\prime}}{}_{l^{\prime}}$ as a vector. Thus we recognise that
$\hat{T}{}_{k^{\prime}}{}_{l^{\prime}}$ transforms as the $(8_{v},8_{v})$
representation of ${\rm SO}(8)\times{\rm SO}(8)$ where $8_{v}$ is the vector
representation of SO(8).

We now turn our attention to the objects
$T^{j_{1}^{\prime}j_{2}^{\prime}j_{3}^{\prime}}$ and $T^{i^{\prime}}$ which
have $56$ and $8$ degrees of freedom respectively. Under ${\rm
SO}(8)\times{\rm SO}(8)$ they transform as

$[J_{i^{\prime}j^{\prime}}^{\pm},T_{k^{\prime}}]=2T_{[i^{\prime}}\delta_{j^{\prime}]k^{\prime}}\mp{1\over
2}T_{i^{\prime}j^{\prime}k^{\prime}}$ $None$

and

$[J^{\pm}{}_{i^{\prime}j^{\prime}},T_{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}}]=6T_{[k^{\prime}_{1}k^{\prime}_{2}|[i^{\prime}}\delta_{j^{\prime}]|k^{\prime}_{3}]}\mp{1\over
6}\varepsilon_{i^{\prime}j^{\prime}k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}l^{\prime}_{1}l^{\prime}_{2}l^{\prime}_{3}}T^{l^{\prime}_{1}l^{\prime}_{2}l^{\prime}_{3}}\pm
12\delta_{i^{\prime}j^{\prime}}^{[k_{1}^{\prime}k_{2}^{\prime}}T^{k_{3}^{\prime}]}$
$None$

Hence $T_{k^{\prime}}$ and $T_{i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}}$
and form the $56+8=64$ dimensional representation of ${\rm SO}(8)\times{\rm
SO}(8)$.

We will now identify what representation this is. We observe that the eight
dimensional gamma matrices in Euclidean space obey the equations

$\gamma_{i^{\prime}j^{\prime}}\gamma_{k^{\prime}}=\gamma_{i^{\prime}j^{\prime}k^{\prime}}+2\delta_{k^{\prime}[j^{\prime}}\gamma_{i^{\prime}]}$
$None$

and

$\ \ \
\gamma_{i^{\prime}j^{\prime}}\gamma^{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}}=6\delta_{[j^{\prime}}{}^{[k^{\prime}_{1}}\gamma_{i^{\prime}]}{}^{k^{\prime}_{2}k^{\prime}_{3}]}+\gamma_{i^{\prime}j^{\prime}}{}^{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}}-6\delta_{i^{\prime}j^{\prime}}^{[k_{1}^{\prime}k_{2}^{\prime}}\gamma^{k_{3}^{\prime}]}$
$=6\delta_{[j^{\prime}}{}^{[k^{\prime}_{1}}\gamma_{i^{\prime}]}{}^{k^{\prime}_{2}k^{\prime}_{3}]}+{1\over
3!}\varepsilon_{i^{\prime}j^{\prime}}{}^{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}l^{\prime}_{1}l^{\prime}_{2}l^{\prime}_{3}}\gamma_{l^{\prime}_{1}l^{\prime}_{2}l^{\prime}_{3}}\gamma_{9}-6\delta_{i^{\prime}j^{\prime}}^{[k_{1}^{\prime}k_{2}^{\prime}}\gamma^{k_{3}^{\prime}]}$
$None$

where

$\gamma_{9}\equiv\gamma^{1\ldots 8}$ $None$

Hence if we were to make the identifications

$T_{k^{\prime}}=\gamma_{k^{\prime}}\ ,\
T_{i^{\prime}j^{\prime}k^{\prime}}=-{2}\gamma_{i^{\prime}j^{\prime}k^{\prime}}$
$None$

then multiplication on the left by $\gamma^{i^{\prime}j^{\prime}}$ has the
same result as $J_{i^{\prime}j^{\prime}}^{+}$ in the commutators of equations
(4.11) and (4.12) provided $\gamma_{9}$ takes the value $-1$. Similarly one
can verify that right multiplication by $-\gamma^{i^{\prime}j^{\prime}}$ has
the same result as $J_{i^{\prime}j^{\prime}}^{-}$ in the commutators of
equations (4.11) and (4.12).

As a result we can think of $T^{j_{1}^{\prime}j_{2}^{\prime}j_{3}^{\prime}}$
and $T^{i^{\prime}}$ as belonging to a bi-spinor which is Weyl projected. A
bi-spinor takes the form

$\Gamma=c_{0}I+c_{i^{\prime}}\gamma^{i^{\prime}}+\ldots+c_{i^{\prime}_{1}\ldots
i^{\prime}_{8}}\gamma^{i^{\prime}_{1}\ldots i^{\prime}_{8}}$ $None$

However, if we require $\gamma_{9}\Gamma=\Gamma$ then we find that
$c_{0}={1\over 8!}\epsilon^{i^{\prime}_{1}\ldots
i^{\prime}_{8}}c_{i^{\prime}_{1}\ldots i^{\prime}_{8}}$ etc and we can take
the bi-spinor to be of the form

$\Gamma=(1+\gamma_{9})(c_{0}I+c_{i^{\prime}}\gamma^{i^{\prime}}+\ldots+c_{i^{\prime}_{1}\ldots
i^{\prime}_{4}}\gamma^{i^{\prime}_{1}\ldots i^{\prime}_{4}})$ $None$

However, we can also demand that $\Gamma\gamma_{9}=-\Gamma$ then find that
$\Gamma$ takes the form

$\Gamma=(1+\gamma_{9})(c_{i^{\prime}}\gamma^{i^{\prime}}+c_{i^{\prime}_{1}\ldots
i^{\prime}_{3}}\gamma^{i^{\prime}_{1}\ldots i^{\prime}_{3}})$ $None$

Thus we find the representation carried by
$T^{j^{\prime}_{1}j^{\prime}_{2}j^{\prime}_{3}}$ and $T^{i^{\prime}}$ is a
bispinor with the above values of Weyl projection. The coefficients
$c_{i^{\prime}}$ and $c_{i^{\prime}_{1}i^{\prime}_{2}i^{\prime}_{3}}$
correspond to the fields $h_{i^{\prime}}\equiv{1\over 8!}\epsilon^{j_{1}\ldots
j_{8}}h_{j_{1}\ldots j_{8},i}$ and $A_{i^{\prime}_{1}\ldots i^{\prime}_{3}}$
of equation (2.24). It we denote the eight dimensional Majorana Weyl spinor
representation of SO(8) with Weyl projection $+1$, that is
$\gamma_{9}\epsilon=\epsilon$, by $8_{c}$ and the one with Weyl projection
$-1$ by $8_{s}$ then the generators
$T^{j_{1}^{\prime}j_{2}^{\prime}j_{3}^{\prime}}$ and $T^{i^{\prime}}$ belong
to the $(8_{c},8_{s})$ representation of ${\rm SO}(8)\times{\rm SO}(8)$.

Hence we find that the the spinor representation of $SO(16)$ which contains
the 128 bosonic degrees of freedom of the massless irreducible representation
decomposes into the

$128=(8_{v},8_{v})\oplus(8_{c},8_{s})$ $None$

representations of ${\rm SO}(8)\times{\rm SO}(8)$. The $(8_{v},8_{v})$
contains the fields $h_{i^{\prime}j^{\prime}}$ and $A_{i^{\prime}_{1}\ldots
i^{\prime}_{6}}$ of equation (2.24) while the $(8_{c},8_{s})$ contains the
$A_{i^{\prime}_{1}i^{\prime}_{2},i^{\prime}_{3}}$ and $h_{i^{\prime}_{1}\ldots
i^{\prime}_{8},k^{\prime}}$ fields. The duality equations relate these two
representations. The higher level fields are related by duality equations to
the fields of these two representations and so these duality relations can be
thought of arising from the action of the affine operator that takes
$I_{c}(E_{8})$ to $I_{c}(E_{9})$.

Rather than discuss the transformations of the Cartan involution odd
generators we can consider the transformations of the corresponding fields
which we will now derive. The Cartan form containing these fields can be
written as

$\tilde{{\cal
V}}=\hat{A}{}_{i_{1}^{\prime}i_{2}^{\prime}}\hat{T}{}^{i_{1}^{\prime}i_{2}^{\prime}}+A_{i^{\prime}}T^{i^{\prime}}+A_{i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}}T^{i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}}$
$None$

In terms of our original generators this takes the form

$\tilde{{\cal V}}=[(\hat{A}{}_{i_{1}^{\prime}i_{2}^{\prime}}-{1\over
6}\hat{A}{}^{k^{\prime}}{}_{k^{\prime}}\delta_{i_{1}^{\prime}i_{2}^{\prime}})]T^{i_{1}^{\prime}i_{2}^{\prime}}+A_{i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}}T^{i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}}+{2\over
6!}\varepsilon_{i_{1}^{\prime}\ldots
i_{6}^{\prime}}{}^{k_{1}^{\prime}k_{2}^{\prime}}\hat{A}{}_{k_{1}^{\prime}k_{2}^{\prime}}T^{i_{1}^{\prime}\ldots
i_{6}^{\prime}}$ $+{1\over 8!}\varepsilon_{i_{1}^{\prime}\ldots
i_{8}^{\prime}}A_{k^{\prime}}T^{i_{1}^{\prime}\ldots
i_{8}^{\prime},k^{\prime}}$ $None$

Using the commutation relations (4.10), (4.11), (4.12) that

$\delta\tilde{{\cal
V}}=[\Lambda^{\pm}{}_{i_{1}^{\prime}i_{2}^{\prime}}J^{\pm}{}^{i_{1}^{\prime}i_{2}^{\prime}},\tilde{{\cal
V}}]$ $None$

$\delta_{\Lambda^{+}}\hat{A}{}_{i_{1}^{\prime}i_{2}^{\prime}}=4\Lambda^{+}{}_{i_{1}^{\prime}k^{\prime}}\hat{A}{}^{k^{\prime}}{}_{i_{2}^{\prime}}\
\ ,\ \
\delta_{\Lambda^{-}}\hat{A}{}_{i_{1}^{\prime}i_{2}^{\prime}}=-4\hat{A}{}_{i_{1}^{\prime}k^{\prime}}\Lambda^{-}{}^{k^{\prime}}{}_{i_{2}^{\prime}}\
\ ,$
$\delta_{\Lambda^{\pm}}A_{i^{\prime}}=2\Lambda^{\pm}{}_{i^{\prime}k^{\prime}}\hat{A}^{k^{\prime}}\pm
12\Lambda^{+}{}^{k_{1}^{\prime}k_{2}^{\prime}}A_{k_{1}^{\prime}k_{2}^{\prime}i^{\prime}}$
$None$

$\delta_{\Lambda^{\pm}}A_{i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}}=\mp{1\over
2}\Lambda^{\pm}{}_{[i_{1}^{\prime}i_{2}^{\prime}}A_{i_{3}^{\prime}]}+6\Lambda^{\pm}{}_{[i_{1}^{\prime}|k^{\prime}}A^{k^{\prime}}{}_{|i_{2}^{\prime}i_{3}^{\prime}]}\pm{1\over
6}\varepsilon_{i_{1}^{\prime}i_{2}^{\prime}i_{3}^{\prime}j_{1}^{\prime}j_{2}^{\prime}k_{1}^{\prime}k_{2}^{\prime}k_{3}^{\prime}}\Lambda^{\pm}{}^{j_{1}^{\prime}j_{2}^{\prime}}A^{k_{1}^{\prime}k_{2}^{\prime}k_{3}^{\prime}}$
$None$

It will also be instructive to decompose the 120 dimensional adjoint
representation of $I_{c}(E_{8})={\rm SO}(16)$, whose generators are given in
equation equation (4.1), into representations of terms of ${\rm
SO}(8)\times{\rm SO}(8)$. The generators not included in ${\rm
SO}(8)\times{\rm SO}(8)$ are $S_{k^{\prime}}\equiv{1\over
8!}\epsilon^{j^{\prime}_{1}\ldots j^{\prime}_{8}}S_{j^{\prime}_{1}\ldots
j^{\prime}_{8},k^{\prime}}$ and
$S_{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}}$ and their commutation
relations with the ${\rm SO}(8)\times{\rm SO}(8)$ generators are given by

$[J_{i^{\prime}j^{\prime}}^{\pm},S_{k^{\prime}}]=2S_{[i^{\prime}}\delta_{j^{\prime}]k^{\prime}}\mp{1\over
2}S_{i^{\prime}j^{\prime}k^{\prime}}$ $None$

and

$[J^{\pm}{}^{i^{\prime}j^{\prime}},S_{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}}]=6S_{[k^{\prime}_{1}k^{\prime}_{2}}{}^{[i^{\prime}}\delta_{k^{\prime}_{3}]}{}^{j^{\prime}]}\pm{1\over
6}\varepsilon^{i^{\prime}j^{\prime}}{}_{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}l^{\prime}_{1}l^{\prime}_{2}l^{\prime}_{3}}S^{l^{\prime}_{1}l^{\prime}_{2}l^{\prime}_{3}}\pm
12\delta^{i^{\prime}j^{\prime}}_{[k_{1}^{\prime}k_{2}^{\prime}}S_{k_{3}^{\prime}]}$
$None$

Comparing with the commutators of equations (4.11) and (4.12) we see that
$S_{k^{\prime}}$ and $S_{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}}$ have the
same commutators with ${\rm SO}(8)\times{\rm SO}(8)$ as $T_{k^{\prime}}$ and
$T_{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}}$ except for an opposite sign
in the second term on the right-hand side of equation (4.27). Examining the
gamma matrix algebra of equations (4.13) and (4.14) we see that we can
identify $S_{k^{\prime}}$ and $S_{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}}$
with $\gamma_{k^{\prime}}$ and
$-2\gamma_{k^{\prime}_{1}k^{\prime}_{2}k^{\prime}_{3}}$ provided we take
$\gamma_{9}$ in equation (4.14) to take the value $+1$. Hence these generators
belong to the $(8_{s},8_{c})$ representation of ${\rm SO}(8)\times{\rm
SO}(8)$. Hence the 120 dimensional adjoint representation of SO(16) consists
of the

$120=(28,1)\oplus(1,28)\oplus(8_{s},8_{c})$ $None$

representation of ${\rm SO}(8)\times{\rm SO}(8)$. The $(28,1)\oplus(1,28)$
contain the adjoint representation of ${\rm SO}(8)\times{\rm SO}(8)$. We note
that the decompositions we have found are not quite the same as those one
finds in certain books.

5\. Spinors of $I_{c}(E_{9})$ decomposed into representations of ${\rm
SO}(8)\times{\rm SO}(8)$ Having decomposed the 128 dimensional SO(16) spinor
representation that contains the bosonic degrees of freedom in terms of
representation of ${\rm SO}(8)\times{\rm SO}(8)$ it will be educational to
also analyse the 128 dimensional spinor representation to which the fermionic
degrees of freedom belong. Long ago it was shown that the fermionic degrees of
freedom appear in maximal supergravity in $D$ dimensions as a linear
representation of the Cartan involution invariant subgroup of the duality
group $E_{11-D}$, for example in four dimensions this group is
$I_{c}(E_{7})=SU(8)$ [16]. It was therefore natural to take the fermions in
$E$ theory to be a linear representation of the Cartan involution invariant
subalgebra of $I_{c}(E_{11})$. In fact fermions were first introduced [17-20]
in this way in the context of the $E_{10}$ theory and subsequently [21] in the
$E_{11}$ theory. The key to these constructions was the realisation that
$I_{c}(E_{10})$ and $I_{c}(E_{11})$ admit highly unfaithful representations.
In particular it was shown that $I_{c}(E_{11})$ has a representation based on
a spinor of SO(1,10), $\epsilon_{\alpha}$ in which the low lowest level
generators take the form [2]

$J_{a_{1}a_{2}}\epsilon_{\alpha}=-{1\over
2}(\gamma_{a_{1}a_{2}})_{\alpha}{}^{\beta}\epsilon_{\beta},\ \
S_{a_{1}a_{2}a_{3}}\epsilon_{\alpha}={1\over
2}(\gamma_{a_{1}a_{2}a_{3}})_{\alpha}{}^{\beta}\epsilon_{\beta},\ \ $
$S_{a_{1}\ldots a_{6}}\epsilon_{\alpha}=-{1\over 4}(\gamma_{a_{1}\ldots
a_{6}})_{\alpha}{}^{\beta}\epsilon_{\beta},\ \ S_{a_{1}\ldots
a_{8},}{}^{b}=\delta^{b}_{[a_{1}}\epsilon_{a_{2}\ldots
a_{8}]c_{1}c_{2}}\gamma^{c_{1}c_{2}},\ldots$ $None$

The generators at higher levels can be found by substituting the above actions
into the $I_{c}(E_{11})$ commutation relations. Clearly, some parts of the
higher level generators are trivially realised and so the representation is
unfaithful.

We will be interested in the fermionic degrees of freedom from the viewpoint
of the irreducible representations of $I_{c}(E_{11})\otimes_{s}l_{1}$ and in
particular as a representation of $I_{c}(E_{9})$. The $I_{c}(E_{9})$
transformations of the spinor $\epsilon_{\alpha}$ are given by equation (5.1)
with the indices restricted to take the values $2,\ldots,10$. It is straight
forward to show that the representation, $\epsilon$ is annihilated by
precisely the same generators of equation (3.6) and (3.11), that is,

$N_{i_{1}i_{2}i_{3}}\epsilon=0=N_{i_{1}i_{2}}\epsilon$ $None$

It must be true that there are an infinite number of such generators that also
annihilate $\epsilon$. Thus the spinor $\epsilon$ is annihilated by the same
generators as the representation that contains the bosonic degrees of freedom
and so they form the same ideal $I$. As such the spinor is really a
representation of ${K(E_{9})\over I}=SO(16)$.

The generalisation of the unfaithful representation of equation (5.1) to the
gravitino which was first done in the context of $E_{10}$ [17-20] and the
result for $E_{11}$ [21] is

$J_{a_{1}a_{2}}\psi_{b}=-{1\over
2}\gamma_{a_{1}a_{2}}\psi_{b}-2\eta_{b[a_{1}}\psi_{a_{2}]}$
$S_{a_{1}a_{2}a_{3}}\psi_{b}=\lambda\\{{1\over
2}\gamma_{a_{1}a_{2}a_{3}}\psi_{b}-\gamma_{b[a_{1}a_{2}}\psi_{a_{3}]}+4\eta_{b[a_{1}}\gamma_{a_{2}}\psi_{a_{3}]}\\}$
$S_{a_{1}\ldots a_{6}}\psi_{b}=-{1\over 4}\gamma_{a_{1}\ldots
a_{6}}\psi_{b}-2\gamma_{b[a_{1}\ldots
a_{5}}\psi_{a_{6}]}+5\eta_{b[a_{1}}\gamma_{a_{2}\ldots a_{5}}\psi_{a_{6}]}$
$S_{a_{1}\ldots a_{8},c}\psi_{b}=\lambda\\{\gamma_{b[a_{1}\ldots
a_{8}}\psi_{c]}-\gamma_{ba_{1}\ldots
a_{8}}\psi_{c}+8(\eta_{b[c}\gamma_{a_{1}\ldots
a_{7}}\psi_{a_{8}]}-\eta_{bc}\gamma_{[a_{1}\ldots a_{7}}\psi_{a_{8}]})$
$-2\eta_{c[a_{1}}\gamma_{a_{2}\ldots a_{8}]}\psi_{b}-28\gamma_{b[a_{1}\ldots
a_{6}}\psi_{a_{7}}\eta_{a_{8}]c}\\}$ $None$

We can take the constant $\lambda$ to take the values $\lambda=-1$ or
$\lambda=1$ and still have a representation of $I_{c}(E_{11})$. This reflects
the way the generator $S_{a_{1}a_{2}a_{3}}$ appears in the algebra resulting
from its level one character. In reference [21] we took $\lambda=1$ but in
this paper we will find that the other sign is better.

When the indices in equation (5.2) take the range $3$ to $10$ we can interpret
this equation as containing the representation that contains the fermionic
degrees of freedom as they occur in the little algebra $I_{c}(E_{9})$.

In this section we will decompose the spinor representation of equation (5.2)
into those of ${\rm SO}(8)\times{\rm SO}(8)$ which is generated by
$J^{\pm}_{i^{\prime}j^{\prime}}$ defined in equation (4.2), but we will begin
with the representation of equation (5.1). We find that

$J^{\pm}_{i^{\prime}j^{\prime}}\epsilon=-{1\over
2}\gamma_{i^{\prime}j^{\prime}}(1\mp\gamma_{9})\epsilon$ $None$

where $\gamma_{9}=\gamma_{1}\ldots\gamma_{8}$. Defining
$\epsilon_{\pm}={1\over 2}(I\pm\gamma_{9})\epsilon$ we can rewrite this
equation as

$J^{+}_{i^{\prime}j^{\prime}}\epsilon_{+}=0,\ \
J^{-}_{i^{\prime}j^{\prime}}\epsilon_{+}=-{1\over
2}\gamma_{i^{\prime}j^{\prime}}\epsilon_{+};\ \
J^{+}_{i^{\prime}j^{\prime}}\epsilon_{-}=-{1\over
2}\gamma_{i^{\prime}j^{\prime}},\ \ \epsilon_{-}\ \
J^{+}_{i^{\prime}j^{\prime}}\epsilon_{-}=0$ $None$

If we take $\epsilon_{+}$ and $\epsilon_{-}$ to be the $8_{c}$ and $8_{s}$
representations of ${\rm SO}(8)$ respectively then the spinor
$\epsilon=\epsilon_{+}\oplus\epsilon_{-}$ is in the $(I\otimes
8_{c})\oplus(8_{s}\otimes I)$ of ${\rm SO}(8)\times{\rm SO}(8)$.

We now wish to decompose the gravitino into representations of ${\rm
SO}(8)\times{\rm SO}(8)$. The action of the SO(16) generators can be read off
from equation (5.3) by taking the indices to take the values
$i^{\prime}.j^{\prime}=3,\ldots 10$. The gravitino obeys the condition
$\gamma^{i}\psi_{i}=\gamma\cdot\psi+\gamma_{9}\psi_{9}=0$ and so we can
eliminate the $\psi_{9}$ component in terms of
$\gamma\cdot\psi\equiv\gamma^{i^{\prime}}\psi_{i^{\prime}}$. We find that the
${\rm SO}(8)\times{\rm SO}(8)$ generators
$J_{i_{1}^{\prime}i_{2}^{\prime}}^{\pm}$ act on the gravitino
$\psi_{k^{\prime}}$ as

$J^{\pm}_{i_{1}^{\prime}i_{2}^{\prime}}\psi_{j^{\prime}}=-{1\over
2}\gamma_{i_{1}^{\prime}i_{2}^{\prime}}(1\mp\gamma_{9})\psi_{j^{\prime}}-2\eta_{j^{\prime}[i_{1}^{\prime}}\psi_{i_{2}^{\prime}]}\pm{1\over
3}\gamma_{9}\gamma_{i_{1}^{\prime}i_{2}^{\prime}}\gamma_{j^{\prime}}\gamma\cdot\psi-\gamma_{9}\gamma_{i_{1}^{\prime}i_{2}^{\prime}}\psi_{j^{\prime}}$
$\mp{2\over
3}\gamma_{9}\eta_{j^{\prime}[i_{1}^{\prime}}\gamma_{i_{2}^{\prime}]}\gamma\cdot\psi\mp{2\over
3}\gamma_{9}\gamma_{j^{\prime}[i_{1}^{\prime}}\psi_{i_{2}^{\prime}]}\pm{4\over
3}\gamma_{9}\eta_{j^{\prime}[i_{1}^{\prime}}\psi_{i_{2}^{\prime}]}$ $None$

In deriving this equation we have used the identity

$\gamma_{k_{1}\ldots k_{n}}={(-1)^{{n(n+1)\over 2}}\over
m!}\epsilon_{k_{1}\ldots k_{n}}{}^{j_{1}\ldots
j_{m}}\gamma_{9}\gamma_{j_{1}\ldots j_{m}}$ $None$

where $n+m=8$. While the action of $J^{\pm}_{i^{\prime}j^{\prime}}$ on
$\gamma\cdot\psi$ follows from equation (5.6) and it is given by

$J_{i_{1}^{\prime}i_{2}^{\prime}}^{\pm}\gamma\cdot\psi=-{1\over
2}\gamma_{i_{1}^{\prime}i_{2}^{\prime}}\gamma\cdot\psi\mp{1\over
6}\gamma_{9}\gamma_{i_{1}^{\prime}i_{2}^{\prime}}\gamma\cdot\psi\pm{4\over
3}\gamma_{9}\gamma_{[i_{1}^{\prime}}\psi_{i_{2}^{\prime}]}.$ $None$

We now take the combination

$\Psi_{i}=\psi_{i}-{1\over 2}\gamma_{i}\gamma\cdot\psi$ $None$

which transforms as

$J_{i_{1}^{\prime}i_{2}^{\prime}}^{\pm}\Psi_{k}=-{1\over
2}\gamma_{i_{1}^{\prime}i_{2}^{\prime}}(1\pm\gamma_{9})\Psi_{k^{\prime}}-2\eta_{k^{\prime}[i_{1}^{\prime}}(1\mp\gamma_{9})\Psi_{i_{2}^{\prime}]}$
$None$

If we define $\Psi_{k}^{\pm}={1\over 2}(1\pm\gamma_{9})\Psi_{k}$ then equation
(5.10) can be rewritten as

$J_{i_{1}^{\prime}i_{2}^{\prime}}^{+}\Psi^{+}_{k}=-{1\over
2}\gamma_{i_{1}^{\prime}i_{2}^{\prime}}\Psi^{+}_{k},\ \
J_{i_{1}^{\prime}i_{2}^{\prime}}^{-}\Psi^{+}_{k}=-2\eta_{k[i_{1}}\Psi^{+}_{i_{2}]},\
\ $
$J_{i_{1}^{\prime}i_{2}^{\prime}}^{+}\Psi^{-}_{k}=-2\eta_{k[i_{1}}\Psi^{-}_{i_{2}]},\
\ J_{i_{1}^{\prime}i_{2}^{\prime}}^{-}\Psi^{-}_{k}=-{1\over
2}\gamma_{i_{1}^{\prime}i_{2}^{\prime}}\Psi^{-}_{k}$ $None$

As a result we find that $\Psi_{k}=\Psi_{k}^{+}+\Psi_{k}^{-}$ belongs to the
$(8_{c},8_{v})\oplus(8_{v},8_{s})$ representation of ${\rm SO}(8)\times{\rm
SO}(8)$. Thus both the fermionic and bosonic degrees of freedom belong to
spinor representations of SO(16) and they have a rather similar decompositions
into representations of ${\rm SO}(8)\times{\rm SO}(8)$. Indeed the bosonic
degrees of freedom belong to the $(8_{v},8_{v})\oplus(8_{c},8_{s})$ and one
can interchange the bosons and fermions by interchanging $8_{v}$ with $8_{c}$
for the first ${\rm SO}(8)$ factor.

It was observed in reference [18] that the fermionic degrees of freedom
encoded in the gravitino carried a highly unfaithful representation of the
Cartan involution algebra of the relevant algebra and as a result the
representation should be annihilated certain generators. This paper also
pointed out that this set of generators would form a subalgebra that was an
ideal. We will now compute the lowest level generators in the context of
irreducible representation, that is $I_{c}(E_{9})$. Using equation (5.3) with
the indices taking the values $3,\ldots,,10$, we find, taking $\lambda=-1$
that

$(S_{i_{1}i_{2}i_{3}}-{2\over 6!}\epsilon^{i_{1}i_{2}i_{3}}{}^{j_{1}\ldots
j_{6}}S_{j_{1}\ldots j_{6}})\psi_{k}=0$ $None$

and that

$(J_{ij}+{2\over 8!}\varepsilon_{[i|}{}^{r_{1}\ldots r_{8}}S_{r_{1}\ldots
r_{8},|j]})\psi_{k}=0$ $None$

These are precisely the same generators that annihilated the bosonic states.
There are no doubt an infinite similar equations at higher levels. One could
also take the choice $\lambda=1$ and find the same ideal if one systematically
changed our definition of $\gamma_{9}$ by introducing a minus sign. Thus seen
from the perspective of the irreducible representations of $I_{c}(E_{9})$, the
bosonic and fermionic states have a remarkably similar structure. They both
belong to 128-dimensional spinor representations of SO(16), they have the same
ideal which annihilates the two representations which carry a representation
of the coset of $I_{c}(E_{9})$ with the ideal $I$ whose coset
${I_{c}(E_{9})\over I}$ which is just SO(16).

However, there is an important difference in the way the bosons and fermions
appear. The original irreducible representation for the bosons contains an
infinite number of fields whose number is reduced by duality relations, while
for the fermions we just have the gravitino. This latter representation was
not deduced from some general theory but by hand following the example of the
maximal supergravity theories in lower dimensions. It would seem natural to
introduce higher level spinors and have these linked to the gravitino by
duality relations. One possibility is to introduce the fields

$\psi_{\alpha},\ \psi_{i_{1}\ldots i_{8}\alpha},\ \psi_{i_{1}\ldots
i_{9}\alpha},\ \psi_{i_{1}\ldots i_{9},j_{1}\ldots j_{8}\alpha},\ \ldots$
$None$

and corresponding duality relations

$\psi_{i_{1}\ldots i_{8}}=\gamma_{i_{1}\ldots i_{8}j}\psi_{j},\ldots$ $None$

It might be straightforward to extend this representation to $I_{c}(E_{11})$
and even $E_{11}$. This latter step could be possible if one exploited the
fact that the algebras SL(D) do admit spinor representation. These have not
been popular as they infinite dimensional, but this is what we need in this
context. [25]. 6\. The covariant formulation In the above section we examined
how the bosonic degrees of freedom of supergravity arose as the massless
irreducible representation of $I_{c}(E_{11})\otimes_{s}l_{1}$. In particular
we saw that they occurred as an irreducible representation of the little group
$I_{c}(E_{9})$ as carried by the fields of equation (2.9). To find the full
representation of $I_{c}(E_{11})\otimes_{s}l_{1}$ one has to carry out a
boost. This procedure was discussed in reference [7]. In carrying out this
last step one does not introduce any additional fields and so even though all
the symmetries are present they are not manifest. However, in field theory one
usually requires representations that have some of the symmetries realised in
a covariant manner. For example, for the irreducible representations of the
Poincare algebra one usually requires the Lorentz algebra to be manifest. In
our case we want the $I_{c}(E_{11})$ symmetry to be essentially manifest. How
to achieve this was very briefly outlined in reference [7] and in this section
we will discuss this procedure in detail. We will also examine how the duality
relations and annihilation generators which are present in the irreducible
representation are inherited into the covariant formulation.

Before we begin we will briefly recall the general theory for obtaining a
covariant formulation of an irreducible representation of
$I_{c}(E_{11})\otimes_{s}l_{1}$. We will use the same notation as in section
four of reference [7]. Let the little algebra be denoted by ${\cal H}$ and
then any group element $g$ of $I_{c}(E_{11})\otimes_{s}l_{1}$ can be written
as $g=e^{\varphi\cdot S}h$ for $h\in{\cal H}$. Before the boost we choose a
linear representation $u_{\alpha}(0)$ of ${\cal H}$ which we can take to
transform as

$U(g)u_{\alpha}(0)=D(g^{-1})_{\alpha}{}^{\beta}u_{\beta}(0),\ \
L_{A}u_{\alpha}(0)=l^{(0)}_{A}u_{\alpha}(0)$ $None$

This representation will contain the irreducible representation of the little
algebra we are considering. To be more precise, our irreducible representation
of the little algebra is embedded in the representation $u_{\alpha}(0)$ which,
as a result, must be a reducible representation of ${\cal H}$.

The next step is to boost the representation $u_{\alpha}(0)$ to find a
representation of $I_{c}(E_{11})\otimes_{s}l_{1}$ in much the same way as we
boosted the irreducible representation of ${\cal H}$ (see section four of
reference [7]). In particular, we take $u_{\alpha}(\varphi)\equiv
U(e^{\varphi\cdot S})u_{\alpha}(0)$ where the symbol $U$ denotes the action of
the group element. However, $u_{\alpha}(\varphi)$ does not transform
covariantly and so instead we consider the objects

${\cal A}_{\alpha}(\varphi)=D(e^{\varphi\cdot
S})_{\alpha}{}^{\beta}u_{\beta}(\varphi)=D(e^{\varphi\cdot
S})_{\alpha}{}^{\beta}U(e^{\varphi\cdot S})u_{\beta}(0)$ $None$

It is straight forward to show that $U_{\alpha}(\varphi)$ transforms
covariantly, and in particular

$U(g){\cal A}_{\alpha}(\varphi)=D(g^{-1})_{\alpha}{}^{\beta}{\cal
A}_{\beta}(\varphi^{\prime})$ $None$

where $g\in I_{c}(E_{11})$ and we have used the relation $ge^{\varphi\cdot
S}=e^{\varphi^{\prime}\cdot S}h(g,\varphi)$ with $h(g,\varphi)\in{\cal H}$.
Although we have embedded the original irreducible representation into a
bigger representation of ${\cal H}$ we still have to ensure that we still have
the irreducible representation and as a result we have to impose projection
conditions and, for the massless case, equivalence relations. How to do this
in general has yet to be understood but we will do it for the massless
irreducible representation.

For comparison we very briefly give an account of the spin one massless
irreducible representation of the Poincare algebra. We can choose all momenta
to vanish except for $p^{+}=p_{-}=\sqrt{2}m$. The little algebra is SO(D-2)
since we have to take the generator $J_{+i}$ to vanish in order to have a
unitary representation. The representation of SO(D-2) is carried by the fields
$A_{i}(0)$, $i=1,\ldots,D-2$. We embed this irreducible representation into
the SO(D-2) reducible representation $A_{\mu}(0),\ \mu=0,1\ldots,D-1$. To get
the same fields as in the original irreducible representation we demand that
$A_{+}(0)=0$ and require the equivalence relation $A_{-}(0)\sim
A_{-}(0)+p_{-}\Lambda(0)$. The latter allows us to set $A_{-}(0)=0$. To find
the effect of these conditions in the covariant formulation in terms of
$A_{\mu}$ we can impose these conditions in the rest frame fields, as given in
equation (6.2), and we recover, in $x$-space the familiar gauge fixing
condition $\partial^{a}A_{a}(x)=0$ and the well known gauge transformation
$A_{a}(x)\sim A_{a}(x)+\partial_{a}\Lambda(x)$.

We will now carry out the above discussion in the context of the massless
irreducible representation we studied in the earlier sections. The first step
is to embed the fields of the irreducible of of $I_{c}(E_{9})\otimes_{s}l_{1}$
contained in equation (2.9) into a larger representation. There is an obvious
candidate, namely, we take the fields corresponding to the Cartan involution
odd generators of $E_{11}$ rather than just those of $E_{9}$. As such we
consider the fields

${\cal A}_{\alpha}(0)=\\{h_{a_{1}a_{2}}(0)\ ,\ A_{a_{1}a_{2}a_{3}}(0)\ ,\
A_{a_{1}\ldots a_{6}}(0)\ ,\ h_{a_{1}\ldots a_{8},b}(0)\ ,\ \ldots\\}\ \
a,b,\ldots=0,\ldots,10.$ $None$

which carry a linear reducible representation of $I_{c}(E_{11})$. These
correspond to the fields ${\cal A}_{\alpha}(0)$ of equation (6.2).

We will now address the issue of the embedding condition that will ensure that
we are really still dealing with the same massless irreducible representation.
We will first do this before the fields are boosted as in equation (6.2). We
take the fields ${\cal A}_{\alpha}(0)$ of equation (6.4) to be subject to two
conditions. The first of these is

$K^{AB}G_{A,B}{}^{C}=K^{AB}(D^{\underline{\alpha}})_{B}{}^{C}\partial_{A}{\cal
A}_{\underline{\alpha}}=0$ $None$

where $K^{AB}$ is the metric on a tangent space which transforms under
$I_{c}(E_{11})$ as its indices suggest [13], and the second is the equivalence
relation

${\cal A}_{\underline{\alpha}}\simeq{\cal
A}_{\underline{\alpha}}+(D_{\underline{\alpha}}+D_{-\underline{\alpha}})_{A}{}^{B}\partial_{B}\Lambda^{A}.$
$None$

The matrices $(D^{\underline{\alpha}})_{B}{}^{C}$ are those of the first
fundamental representation and are defined in the equation
$[R^{\underline{\alpha}},l_{B}]=-(D^{\underline{\alpha}})_{B}{}^{C}l_{C}$. As
we will see equation (6.5) eliminates all fields with a lower $+$ index, while
the second equation (6.6) eliminating fields which take on a lower $-$ index,
leaving us with the fields of equation (2.9), that is, those of the original
massless irreducible representation.

When acting on the fields of equation (6.4), the only component of
$\partial_{A}$ which is non-zero is $\partial_{-}={\partial\over\partial
x^{-}}$. Thus the only non-zero component of $K^{AB}$ in equation (6.5) is
$K^{-+}=\eta^{-+}$, and so equation (6.5) reduces to
$(D^{{\alpha}})_{+}{}^{C}\partial_{-}{\cal A}_{{\alpha}}=0$. Since
$p_{-}=\sqrt{2}m$ is a constant we have the condition
$(D^{{\alpha}})_{+}{}^{C}{\cal A}_{{\alpha}}=0$. The above matrix occurs in
the equation $[R^{{\alpha}},P_{+}]=-(D^{{\alpha}})_{+}{}^{B}l_{B}$ and so our
condition can be expressed as $[{\cal A}_{\alpha}R^{\alpha},P_{+}]=0$ which
consists of transformations ${\cal A}_{\alpha}\ R^{\alpha}$ that preserve the
$p_{+}=0$ with the only non-zero momentum being $p_{-}$. By definition this
includes the little algebra transformations $E_{9}$ and so the condition of
equation (6.5) places no constraints on the fields in equation (2.9). At
levels zero, one and two the condition of equation (6.5) lead to

$h_{++}=0=h_{+i}=h^{i}{}_{i};\ \ A_{+a_{1}a_{2}}=0=A_{+a_{1}\ldots a_{5}}$
$None$

At higher levels we find that the commutator $[A_{\alpha}R^{\alpha},P_{+}]=0$
contains a factor $\delta^{a}_{+}$ times ${\cal A}_{\alpha}$ times the $l_{1}$
generator at the corresponding level. Hence we find that the condition of
equation (6.5) implies that any field ${\cal A}_{\alpha}$ with a lower $+$
index is set to zero.

We can recover the same results by directly examining the component equations
that following from equation (6.5). At levels zero, one and three, these are
given by [13]

$\partial^{e}h_{e}{}^{a}-{1\over 2}\partial^{a}h_{e}{}^{e}=0\ \ \ ,\ \ \
\partial^{e}A_{ea_{1}a_{2}}=0\ \ \ ,\ \ \ \partial^{e}A_{ea_{1}\ldots
a_{5}}=0\ \ ,\ \ \ \ldots$ $None$

where we have thrown away derivatives with respect to the higher level
coordinates. By taking all the momentum except $p_{-}$ non-zero in equation
(6.8) we recover the above result.

We now consider the equivalence relation (6.6). In the rest frame, that is,
for the fields of equation (6.4), this reduces to
$(D_{{\alpha}}+D_{-{\alpha}})_{A}{}^{-}\partial_{-}\Lambda^{A}$ which is
proportional to $(D_{{\alpha}}+D_{-{\alpha}})_{A}{}^{-}\Lambda^{A}$. This
matrix occurs in the commutator
$[R^{{\alpha}}+R_{{\alpha}},l_{A}]=-(D^{{\alpha}}+D_{{\alpha}})_{A}{}^{-}P_{-}$.
At level one we have the commutator
$[R_{a_{1}a_{2}a_{3}},Z^{b_{1}b_{2}}]=6\delta^{b_{1}b_{2}}_{[a_{1}a_{2}}P_{a_{3}]}$
and so we require $R_{a_{1}a_{2}-}$ in order to get $P_{-}$. At higher levels
one also finds that this expression will only be non-zero if the index
${{\alpha}}$ contains a lower $-$ index. To see this we note that the level
zero $P_{-}$ on the right-hand side arises from the commutator of a level $n$
$l_{1}$ generator and a level $-n$ level generator of $E_{11}$ with a
resulting structure constant that contains a $\delta_{-}^{a}$ factor. Hence
all the fields ${\cal A}_{\alpha}$ with a lower $-$ index are subject to an
equivalence relation and for a suitable choice of $\Lambda^{A}$ they can be
set to zero.

The equivalence relation (6.6) have been worked out at low levels [14] and
they lead to the expected gauge transformations;

$\delta h_{(ab)}=\partial_{a}\xi_{b}+\partial_{b}\xi_{a}\ \ ,\ \ \delta
A_{a_{1}a_{2}a_{3}}=\partial_{[a_{1}}\Lambda_{a_{2}a_{3}]}\ \ ,\ \ \delta
A_{a_{1}\ldots a_{6}}=2\partial_{[a_{1}}\Lambda_{a_{2}\ldots a_{6}]}\ \ ,\ \
\ldots$ $None$

As the only non-zero momenta is $p_{-}$ we find that at levels zero, one and
two we can set to zero $h_{-+}$, $A_{-a_{1}a_{2}}$ and $A_{-a_{1}\ldots
a_{5}},\ldots$, in agreement with the above discussion.

So far we only considered the effect of the conditions of equation (6.5) and
(6.6) in the rest frame, that is, $p_{-}=\sqrt{2}m$ all other members of the
vector representation being zero. To find the analogue of these conditions for
the covariant theory we take the conditions to act on $u_{\alpha}(0)$ in
equation (6.2) and then apply the boost and matrix multiplication that this
equation contains. Since equation (6.5) contains the $I_{c}(E_{11})$ covariant
expression $K^{AB}(D^{{\alpha}})_{B}{}^{C}\partial_{A}$ it retains the same
form under the boost and so we can take this equation to hold for the theory
after the boost. Taking equation (6.6) to hold in the rest frame and carrying
out the boost and matrix multiplication,as in equation (6.2), we find that it
takes the form as in equation (6.6) but with a parameter
$\Lambda^{A}(\varphi)=D(e^{-\varphi\cdot S})_{C}{}^{A}U(e^{\varphi\cdot
S})\Lambda^{C}(0)$. We recognise this as a gauge transformation of reference
[14].

Hence we have shown that the conditions of equations (6.5) and (6.6) do not
affect the fields of equation (2.9). However for the fields of equation (6.4)
we find that they set all fields with a lower $+$ to zero and we can, using
the equivalence relation remove all fields with a lower $-$ index. As a result
equations (6.5) and (6.6) ensure that $A_{\alpha}$, given in equation (6.4),
contains the same fields as occurred in the original massless irreducible
representation, given in equation (2.9). Hence it is just the same massless
irreducible representation of $I_{c}(E_{11})\otimes_{s}l_{1}$.

In section two we found that the fields of the massless irreducible
representation obeyed the duality relations of equation (2.19), (2.20) and
(2.22) as well as similar higher level equations. While in section three we
found that as a result this representation was annihilated by the generators
which belonged to an ideal, for example those in equations (3.6) and (3.11).
As the covariant formulation of the irreducible representation is essentially
the same as the original irreducible representation we can expect that the
generators in equations (3.6) and (3.11), as well as those at higher level,
imply the presence of analogous equations which we will not find. Indeed,
starting from the equations that the generators in the ideal $I$ annihilate
the fields in the irreducible representation in the rest frame, we can carry
out a boost, as in equation (6.2), to find an infinite number of constraints
in the covariant theory. However, as the boost contains generators which
levels greater than zero it will transform the elements in the ideal with
different levels into each other.

Rather than carry out the boost, we will find some covariant operators that
agree with elements in the ideal in the rest frame. We begin by considering
the operator

$\hat{N}_{a_{1}\ldots a_{4}}=P_{[a_{1}}S_{a_{2}a_{3}a_{4}]}+{1\over
2.6!}\varepsilon_{a_{1}\ldots a_{4}}{}^{c_{1}\ldots
c_{7}}P_{c_{1}}S_{c_{2}\ldots c_{7}}+{1\over
4}Z^{e_{1}e_{2}}S_{e_{1}e_{2}a_{1}\ldots a_{4}}$ $+{1\over
7.2!}\epsilon_{a_{1}\ldots a_{4}}{}^{c_{1}\ldots
c_{7}}Z^{e_{1}e_{2}}S_{e_{1}c_{1}\ldots c_{7},e_{2}}+{1\over
12}Z_{[a_{1}a_{2}}J_{a_{3}a_{4}]}$ $None$

Acting on the fields of the massless irreducible representation of equation
(2.9), which are in the rest frame, it obviously vanishes except for
$\hat{N}_{-i_{1}i_{2}i_{3}}$ which is equal to $p_{-}N_{i_{1}i_{2}i_{3}}$,
where $N_{i_{1}i_{2}i_{3}}$ is defined in equation (3.6). However, as we have
shown in section three $N_{i_{1}i_{2}i_{3}}$ vanishes on the massless
irreducible representation of equation (2.9) and as a result
$\hat{N}_{a_{1}\ldots a_{4}}$ also vanishes on this representation on the
fields the rest frame. The terms in $\hat{N}_{a_{1}\ldots a_{4}}$ that contain
higher level elements of $l_{1}$ vanish in the rest frame and the precise
coefficients that we have given will be derived below. Clearly,
$\hat{N}_{a_{1}\ldots a_{4}}$ is one of the covariant operators resulting from
the action of the boost on the annihilation operators in the rest frame as
discussed just above.

We now consider if $\hat{N}_{a_{1}\ldots a_{4}}$ really does annihilate the
covariant states of equation (6.4). Since the covariant fields are only
defined up to an equivalence, or equivalently gauge symmetry, it simplifies
the calculation considerably if we act on objects that are gauge invariant. At
the linearised level such objects are

$\omega_{a,b_{1}b_{2}}=-\partial_{b_{1}}h_{(b_{2}a)}+\partial_{b_{2}}h_{(b_{1}a)}+\partial_{a}h_{[b_{1}b_{2}]},\
G_{a_{1}\ldots a_{4}}=\partial_{[a_{1}}A_{a_{2}a_{3}a_{4}]},\ G_{a_{1}\ldots
a_{7}}=\partial_{[a_{1}}A_{a_{2}\ldots a_{7}]},\ \ldots$ $None$

We find that

$\hat{N}^{b_{1}b_{2}b_{3}b_{4}}G_{a_{1}a_{2}a_{3}a_{4}}=-{1\over
4!}P_{a_{1}}\varepsilon_{a_{2}a_{3}a_{4}}{}^{b_{1}..b_{4}}{}^{c_{1}c_{2}c_{3}c_{4}}E_{c_{1}c_{2}c_{3}c_{4}}+3\delta^{[b_{1}b_{2}}_{a_{1}a_{2}}P_{a_{3}}E_{a_{4}],}{}^{b_{3}b_{4}]}$
$None$

and

$\hat{N}^{b_{1}b_{2}b_{3}b_{4}}\omega_{a_{1},a_{2}a_{3}}=-4\eta_{a_{1}a_{3}}P_{a_{2}}E^{b_{1}\ldots
b_{4}}+4\eta_{a_{1}a_{2}}P_{a_{3}}E^{b_{1}\ldots
b_{4}}-90P_{c_{1}}\eta_{c_{2}a_{3}}\delta^{[c_{1}c_{2}}_{a_{1}a_{2}}E^{b_{1}\ldots
b_{4}]}$
$-90P_{c_{1}}\eta_{c_{2}a_{2}}\delta^{[c_{1}c_{2}}_{a_{3}a_{1}}E^{b_{1}\ldots
b_{4}]}-90P_{c_{1}}\eta_{c_{2}a_{1}}\delta^{[c_{1}c_{2}}_{a_{3}a_{2}}E^{b_{1}\ldots
b_{4}]}$ $+\partial_{a_{1}}\tilde{\Lambda}_{a_{2}a_{3}}{}^{b_{1}\ldots b_{4}}$
$None$

where

$E_{a_{1}\ldots a_{4}}=G_{a_{1}\ldots a_{4}}-{1\over 2\cdot
4!}\varepsilon_{a_{1}\ldots a_{4}}{}^{b_{1}\ldots b_{7}}G_{b_{1}\ldots
b_{7}}=0,$ $None$ $E_{a,b_{1}b_{2}}=\omega_{a,b_{1}b_{2}}-{1\over
4}\varepsilon_{b_{1}b_{2}}{}^{c_{1}\ldots c_{9}}G_{c_{1}\ldots c_{9},a}=0,$
$None$

and

$\partial_{a_{1}}\tilde{\Lambda}_{a_{2}a_{3}}{}^{b_{1}\ldots
b_{4}}=P_{a_{1}}[12\delta_{a_{3}}{}^{[b_{1}}G_{a_{2}}{}^{b_{2}b_{3}b_{4}]}-12\delta_{a_{2}}{}^{[b_{1}}G_{a_{3}}{}^{b_{2}b_{3}b_{4}]}+3\delta_{a_{3}}{}^{[b_{1}}P_{a_{2}}A^{b_{2}b_{3}b_{4}]}$
$-3\delta_{a_{2}}{}^{[b_{1}}P_{a_{3}}A^{b_{2}b_{3}b_{4}]}+{3\over
8}\varepsilon^{b_{1}\ldots b_{4}}{}^{c_{1}\ldots
c_{7}}P_{c_{1}}\eta_{c_{2}a_{2}}A_{a_{3}c_{3}\ldots c_{7}}-{3\over
8}\varepsilon^{b_{1}\ldots b_{4}}{}^{c_{1}\ldots
c_{7}}P_{c_{1}}\eta_{c_{2}a_{3}}A_{a_{2}c_{3}\ldots c_{7}}]$ $None$

We recognise equations (6.14) and (6.15) as the duality relations that occur
in the covariant theory arsing from the non-linear realisation of
$E_{11}\otimes_{s}l_{1}$ with local subalgebra $I_{c}(E_{11})$ [ 1,2,3,4].
Thus we find that $N_{a_{1}\ldots a_{4}}$ will indeed vanish on the covariant
fields provided the duality relations hold. The last term in equation (6.13)
is defined in equation (6.16) and it just corresponds to the fact that
$\omega_{a,b_{1}b_{2}}$ is subject to local Lorentz transformations. We note
that $\tilde{\Lambda}_{a_{2}a_{3}}{}^{b_{1}\ldots b_{4}}$ is indeed of the
form of a local Lorentz transformation since the $b_{1}\ldots b_{4}$ indices
can be contracted with a parameter with these indices.

We will now consider the operator

$\hat{N}_{b_{1}b_{2}b_{3}}=P_{[b_{1}}J_{b_{2}b_{3}]}-{1\over
8!}\varepsilon_{[b_{1}b_{2}|}{}^{e_{1}\ldots e_{9}}P_{e_{1}}S_{e_{2}\ldots
e_{9},|b_{3}]}+\ldots$ $None$

where $+\ldots$ means terms which contain higher order $l_{1}$ generators.
Acting on the fields of equation (2.9) in the rest frame this operator we find
that it obviously vanishes except for
$\hat{N}_{-i_{2}i_{3}}=p_{-}N_{i_{2}i_{3}}$ where $N_{i_{2}i_{3}}$ is defined
in equation (3.11). However, this generator also vanishes on the rest frame
states and so $\hat{N}_{b_{1}b_{2}b_{3}}$ vanishes on the rest frame states.
As such we would expect it to vanish on the covariant fields of equation (6.4)
One finds that

$\hat{N}^{b_{1}b_{2}b_{3}}\omega_{a_{1},a_{2}a_{3}}=\eta_{a_{1}a_{3}}P_{a_{2}}E^{[b_{1},b_{2}b_{3}]}-P_{a_{2}}\delta^{[b_{1}}{}_{(a_{3}}E_{a_{1}),}{}^{b_{2}b_{3}]}-\eta_{a_{1}a_{2}}P_{a_{3}}E^{[b_{1},b_{2}b_{3}]}$
$+P_{a_{3}}\delta^{[b_{1}}{}_{(a_{2}}E_{a_{1}),}{}^{b_{2}b_{3}]}$
$-6P_{r_{1}}(\eta_{r_{2}a_{3}}\delta^{[b_{1}b_{2}}_{a_{2}a_{1}}E^{|b_{3}|,r_{1}r_{2}}+\eta_{r_{2}a_{1}}\delta^{[b_{1}b_{2}}_{a_{2}a_{3}}E^{|b_{3}|,r_{1}r_{2}]}-\eta_{r_{2}a_{2}}\delta^{[b_{1}b_{2}}_{a_{3}a_{1}}E^{|b_{3}|,r_{1}r_{2}]})$
$+\partial_{a_{1}}\tilde{\Lambda}_{a_{2}a_{3}}{}^{b_{1}b_{2}b_{3}}=0,$ $None$

where

$\partial_{a_{1}}\tilde{\Lambda}_{a_{2}a_{3}}{}^{b_{1}b_{2}b_{3}}=P_{a_{1}}\\{\varepsilon^{[b_{1}b_{2}|r_{1}\ldots
r_{9}}P_{r_{1}}(\eta_{r_{2}a_{3}}h_{a_{2}r_{3}\ldots
r_{9}}{}^{,b_{3}]}-\eta_{r_{2}a_{2}}h_{a_{3}r_{3}\ldots r_{9}}{}^{,b_{3}]})$
$+P^{[b_{2}}(\delta^{b_{1}}{}_{a_{2}}h_{a_{3}}{}^{b_{3}]}-\delta^{b_{1}}{}_{a_{3}}h_{a_{2}}{}^{b_{3}]})\\}$
$None$

is an expected Lorentz transformation.

Since the operators $\hat{N}_{a_{1}\ldots a_{4}}$ and
$\hat{N}_{a_{1}a_{2}a_{3}}$, and presumably their higher level analogues,
annihilate the covariant fields one would expect their commutator with the
generators of $I_{c}(E_{11})$ to give them back. At lowest order, and assuming
we can find a missing factor of two, we find that

$[S_{a_{1}a_{2}a_{3}},\hat{N}_{b_{1}b_{2}b_{3}b_{4}}]=+9.3\delta^{[a_{1}a_{2}}_{[b_{1}b_{2}}\hat{N}^{a_{3}]}{}_{b_{3}b_{4}]}+{1\over
24}\epsilon_{b_{1}b_{2}b_{3}b_{4}}{}^{a_{1}a_{2}a_{3}}{}^{c_{1}c_{2}c_{3}c_{4}}\hat{N}_{c_{1}c_{2}c_{3}c_{4}}$
$None$

Similar results can be expected for $\hat{N}_{a_{1}\ldots a_{3}}$ and the
higher level generators of $I_{c}(E_{11})$. It would be interesting to compute
the commutators of $\hat{N}_{a_{1}\ldots a_{4}}$ and $\hat{N}_{a_{1}\ldots
a_{3}},\ldots$ with themselves. Given the above result one would expect the
result to be proportional to $\hat{N}$’s times $l_{1}$ generators. A result
which would again annihilate the covariant fields.

The covariant fields are subject to the Casimir condition
$L^{2}\Psi=K^{AB}L_{A}L_{B}\Psi=0$ as well as higher level conditions [15]. We
would expect that the generators $\hat{N}_{a_{1}\ldots a_{4}}$ and
$\hat{N}_{a_{1}\ldots a_{3}},\ldots$ will commute with these up to terms
proportional to themselves. They should also be consistent with the gauge
transformation of equation (6.5).

In this section we have found that the generators in the ideal $I$ that
annihilate the massless irreducible states of equation (2.9) in the rest frame
lead to an infinite set of operators which annihilate the massless irreducible
representation when expressed in terms covariant fields provided the covariant
fields obey an infinite set of duality relations. These covariant duality
relations are precisely those that occur in the non-linear realisation of
$I_{c}(E_{11})\otimes_{s}l_{1}$ with local subalgebra $I_{c}(E_{11})$. This is
consistent with the observation that the duality conditions on the covariant
fields are just covariant versions of the duality identities that exist in the
original massless irreducible representation [7]. Thus the existence of the
duality relations which contain the dynamics arise from the irreducible
representation and in particular many of them correspond to the action of the
affine action of $I_{c}(E_{9})$ on the $I_{c}(E_{8})$ representation of
equation (2.24).

7\. An alternative formulation of fermions in E theory  The spinors were
introduced in E theory by hand in the sense that they do not follow in natural
way from the $E_{11}$ algebra but started from the familiar gravitino and
insisted that it carry a representation of $I_{c}(E_{11})$, which was
constructed by hand. In this section we will take a different approach which
generalises, in a natural way, the way spinors appear in the context of the
Poincare algebra. For the Poincare algebra, that is,
$SO(1,D-1)\otimes_{s}T^{D}$ the vector representation $T^{D}$ leads to a $D$
dimensional spacetime on which $SO(1,D-1)$ acts. For each coordinate of the
spacetime we introduce a $\gamma^{a}$ matrix which act on a spinor.

In E theory we start from the algebra $I_{c}(E_{11})\otimes_{s}l_{1}$ which is
also a semi-direct product. The vector representation $l_{1}$ leads in the
nonlinear realisation to the spacetime and we now introduce matrices
$\Gamma^{A}$’s which are in one to one correspondence with the vector
representation, hence their label $A$. As a result they are also carry the
indices of the spacetime coordinates in E theory. For example, in eleven
dimensions we introduce the matrices

$\Gamma^{A}=\\{\Gamma^{a},\ \Gamma^{a_{1}a_{2}},\ \Gamma^{a_{1}\ldots a_{5}},\
\Gamma^{a_{1}\ldots a_{7},b},\ \Gamma^{a_{1}\ldots a_{8}},\ldots\\}$ $None$

These matrices are to obey the equation

$\Gamma^{A}\Gamma^{B}+\Gamma^{B}\Gamma^{A}=2K^{AB}$ $None$

where $K^{AB}$ is the $I_{c}(E_{11})$ invariant tangent space metric. We now
introduce a spinor $\Psi$ which carries a representation of the $\Gamma$
matrices and transforms under $I_{c}(E_{11})$ as

$\delta\Psi=U(S^{\alpha})\Psi=-{\cal S}^{\alpha}\Psi$ $None$

where $U(S^{\alpha})$ is the action of the generator $S^{\alpha}$ and the
matrix ${\cal S}^{\alpha}$ its effect. We require that

$[{\cal S}^{\alpha},\Gamma^{A}]=\Gamma^{B}(\tilde{D}^{\alpha})_{B}{}^{A}{}$
$None$

where $\tilde{D}^{\alpha}\equiv D^{\alpha}-D_{\alpha}$. By its definition the
matrix of the vector representation appears in the commutator
$[R^{\alpha},l_{A}]=-(D^{\alpha})_{A}{}^{B}l_{B}$. We recall that
$S^{\alpha}=R^{\alpha}-R_{\alpha}$ and that
$(\tilde{D}^{\alpha})_{A}{}^{C}K_{CB}$ is an antisymmetric matrix. As such the
commutator of ${\cal S}^{\alpha}$ with the $\Gamma^{A}$’s leads to a transform
under $I_{c}(E_{11})$ which is that of the vector representation.

A generalised Dirac equation is given by

$\Gamma^{A}\partial_{A}\Psi=0$ $None$

It is invariant under $I_{c}(E_{11})$ transformations as the derivatives
$\partial_{A}\equiv{\partial\over\partial x^{A}}$ transforms as
$\delta(\partial_{A})=-(\tilde{D}^{\alpha})_{A}{}^{B}\partial_{B}$.

At level zero $I_{c}(E_{11})$ is just SO(1,10) and so at this level the above
discussion just reduces to the standard discussion of the Dirac equation. The
Gamma matrices corresponding to the higher level coordinates occur with
derivatives with respect to these coordinates. Thus if we neglect the higher
level derivatives this is just the familiar Dirac equation. We note that we
can not take $\Gamma^{a_{1}a_{2}},\ldots$ to be proportional to the standard
gamma matrix $\gamma^{a_{1}a_{2}},\ldots$ as this does not satisfy equation
(7.2)

The above applies if we replace $E_{11}$ by any Kac-Moody algebra and the
vector representation by anyone of its representations. To give a simple
example we consider the non-linear realisation of $A_{1}^{+++}\otimes l_{1}$
which corresponds to gravity in four dimensions. At levels zero and one the
generators of the vector representation are $P_{a}$ and $Z^{a}$ and so we
introduce the gamma matrices $\Gamma^{A}=\Gamma^{a},\tilde{\Gamma}^{a},\ldots$
with $a=0,1,\ldots,3$. The algebra $I_{c}(A_{1}^{+++})$ contains the
generators $J_{ab}$ and $S_{ab}$ at levels zero and one respectively. For an
account of this theory see reference [26] and the earlier references it
contains. We can take the gamma matrices to be

$\Gamma^{a}=\gamma^{a}\otimes
I,\quad\tilde{\Gamma}^{a}=2i\gamma^{5}\otimes\gamma^{a}$ $None$

where $\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\eta^{ab}$ are the usual
gamma matrices in four dimensions, $(i\gamma^{5})^{2}=I$ and
$\gamma^{5}\gamma^{b}+\gamma^{b}\gamma^{5}=0$. Thus this spinor has eight
components.

The vector representation of the algebra $I_{c}(E_{11})$ contains at level one
the coordinates $x^{a_{1}a_{2}}$ and so the corresponding spinor will carry a
representation of the matrices $\Gamma^{a_{1}a_{2}}$ which obey

$\Gamma^{a_{1}a_{2}}\Gamma_{b_{1}b_{2}}+\Gamma_{b_{1}b_{2}}\Gamma^{a_{1}a_{2}}=2\delta_{b_{1}b_{2}}^{a_{1}a_{2}}$
$None$

Such matrices are often required if we consider other very extended Kac-Moody
algebras. If we have the index range in three dimensions then we can define
$\gamma^{a}={1\over 2}\epsilon^{ab_{1}b_{2}}\Gamma_{b_{1}b_{2}}$ and we can
use our usual representation of gamma matrices in three dimensions. So we
require the spinor in this sector to have two components. If the index range
is in four, for example, Euclidean dimensions then we can define

$\Gamma^{+}{}^{a_{1}a_{2}}={1\over 2}(\Gamma^{+}{}^{a_{1}a_{2}}\pm{1\over
2}\epsilon^{a_{1}a_{2}b_{1}b_{2}}\Gamma_{b_{1}b_{2}})$ $None$

which obey the relations

${1\over
2}\epsilon^{a_{1}a_{2}b_{1}b_{2}}\Gamma^{\pm}_{b_{1}b_{2}}=\pm\Gamma^{\pm}{}^{a_{1}a_{2}},\
\ {\rm and}\ \
\\{\Gamma^{+}{}^{a_{1}a_{2}},\Gamma^{-}{}_{b_{1}b_{2}}\\}=\delta_{b_{1}b_{2}}^{a_{1}a_{2}}$
$None$

As such we can take the independent matrices to be $\Gamma^{\pm}{}^{12}$,
$\Gamma^{\pm}{}^{13}$ and $\Gamma^{\pm}{}^{14}$ and a representation is formed
by defining the vacuum $|>$ to satisfy

$\Gamma^{-}{}_{12}|>=0=\Gamma^{-}{}_{13}|>=\Gamma^{-}{}_{14}|>$ $None$

We can regard the matrices $\Gamma^{+}{}^{12}$, $\Gamma^{+}{}^{13}$ and
$\Gamma^{+}{}^{14}$ as creation operators acting on this vacuum, that is,

$|>,\ \Gamma^{+}{}_{12}|>,\ldots,\Gamma^{+}{}_{12}\Gamma^{+}{}_{13}|>,\
\ldots$ $None$

Thus we have $(1+1)^{3}=8$ states, or spinor components, due to this sector of
the spinor.

Let us also consider the algebra $I_{c}(E_{8})\otimes_{s}l_{1}$. The vector
representation of $E_{8}$ has dimension 248 and as explained in section four
this decomposes into the 120 plus 128 dimensional representations of
$I_{c}(E_{8})={\rm SO}(16)$. We can take Gamma matrices corresponding to both
of these representations or just one of them. If we take just the spinor 128
dimensional representation the we should take the Gamma matrices

$\Gamma^{i},\ \Gamma^{i_{1}i_{2}},\ \Gamma^{i_{1}i_{2}i_{3}},\ \Gamma^{(ij)},\
\ {\rm where}\ \ i,j=1,2,\ldots,8$ $None$

where we have given them in the ${\rm SO}(8)\otimes{\rm SO}(8)$ decomposition
of section four. The generalised Dirac equation would have the form

$(\Gamma^{i}\partial_{i}+\Gamma^{i_{1}i_{2}}\partial_{i_{1}i_{2}}+\ldots)\Psi=0$
$None$

The Gamma matrices can be written in block diagonal form using the analogue of
the $\gamma_{5}$ matrices as above. The $\Gamma^{i}$ can be taken to be the
usually gamma matrices in the first block and as their are eight of them this
part of the spinor will have $2^{4}=16$ components. More generally the spinor
which carries a representation can be found by defining annihilation and
creation operators. We might expect 64 creation operators and 64 destruction
operators and so the corresponding part of the spinor should have $2^{64}$
components.

The above considerations were at the linearised order but it is
straightforward to generalise it to the full symmetries of the
$E_{11)}\otimes_{s}l_{1}$ by taking

$\Gamma^{A}E_{A}{}^{M}(\partial_{M}+Q_{M})\Psi=0$ $None$

where $E_{M}{}^{A}$ is the vielbein and $Q_{M}$ the connection found in the
non-linear realisation.

We end this section with some speculative remarks. To account for the
gravitino we can introduce the object $\Psi_{A}$ which is a spinor with the
vector index $A$. We could take this to obey the on-shell conditions

$\Gamma^{B}\partial_{B}\Psi_{A}=0=\Gamma^{A}\Psi_{A}=\partial^{A}\Psi_{A}$
$None$

which does indeed contain the correct on-shell states if we restrict to the
first part of the spinor $\Psi_{a}$. We leave it to the future to examine how
this fits into the full theory.

We could also introduce a generalised supersymmetry generator $Q^{A}$ which is
a generalised spinor and so with $I_{c}(E_{11})$ generators it has the
relation

$[S^{\alpha},Q]=-{\cal S}^{\alpha}Q$ $None$

One might speculate that they obey an anti-commutator of the generic form

$\\{Q,Q\\}=\Gamma^{A}l_{A}$ $None$

How the generalised spinor introduced in this section fits in detail into E
theory and what are its mathematical properties for future study. 8 Discussion
In this paper we have analysed in detail the irreducible representation of
$I_{c}(E_{11})\otimes_{s}l_{1}$ corresponding to a massless point particle.
The corresponding little algebra, $I_{c}(E_{9})\otimes_{s}l_{1}$, is an
infinite dimensional Lie algebra and as a result one may expect it to contain
an infinite number of degrees of freedom. However, the fields are subject to
an infinite number of duality relations which are preserved by $I_{c}(E_{9})$
and these reduce the number of independent degrees of freedom to be just 128
which belong to the spinor representation of $I_{c}(E_{8})={\rm SO}(16)$.
These can be taken to be $h_{ij}\ (h^{i}{}_{i}=0$) and $A_{i_{1}i_{2}i_{3}}$
where $i,j=2,\ldots,10$. They are indeed the bosonic degrees of freedom of
eleven dimensional supergravity. The infinite number of duality equations
relate all the other fields in the representation to these fields. A
consequence of these duality relations is that the irreducible representation
is annihilated by an infinite number of generators of $I_{c}(E_{9})$ which
form a subalgebra $I$ which is an ideal. The Lie algebra $I_{c}(E_{9})\over I$
is SO(16).

The 128 bosonic independent degrees of freedom belong to the
$(8_{v},8_{v})\oplus(8_{c},8_{s})$ representations when decomposed into the
subalgebra ${\rm SO}(8)\times{\rm SO}(8)$ subalgebra of SO(16). These two
representations contain the fields ($h_{i^{\prime}j^{\prime}}$,
$A_{i^{\prime}_{1}\ldots i^{\prime}_{6}}$) and
($A_{i^{\prime}_{1}i^{\prime}_{2}i^{\prime}_{3}}$, $h_{i^{\prime}_{1}\ldots
i^{\prime}_{8},k^{\prime}}$) respectively where
$i^{\prime},j^{\prime}=3,\ldots,10$. The SO(16) transformations not in the
${\rm SO}(8)\times{\rm SO}(8)$ subalgebra change these representations into
each other. The remaining fields which appear in the irreducible
representation of $I_{c}(E_{9})$ are affine copies of these 128 fields and are
related by duality relations to the fields in the
$(8_{v},8_{v})\oplus(8_{c},8_{s})$ representations. Like all such irreducible
representations we can formulate the massless irreducible representation in a
covariant manner. The infinite number of duality relations become covariant
duality equations which contain the dynamics and one finds a corresponding set
of operators that annihilate the representation.

The dynamics that results from the non-linear realisation of
$E_{11}\otimes_{s}l_{1}$ with local subalgebra $I_{c}(E_{11})$ has been found
at low levels and the resulting equations of motion agree precisely with those
of maximal supergravity if one discards the dependence on the spacetime
coordinates beyond those usually considered. The dynamics appears through an
infinite set of duality equations that relate an infinite number of the higher
level fields to the graviton and three form and it is by taking space time
derivatives of these that one can find the standard equations of motion of
maximal supergravity. The duality relations that arise in the covariant
formulation of the massless irreducible of $I_{c}(E_{11})\otimes_{s}l_{1}$ in
the covariant formulation are essentially those that arise in the non-linear
realisation of $E_{11}\otimes_{s}l_{1}$ with local subalgebra $I_{c}(E_{11})$
at the linearised level. This provides strong support for the strongly
suspected fact that the only degrees of freedom contained in the non-linear
realisation are the 128 bosonic degrees of freedom of supergravity. We also
see that the infinite number of duality relations that appear in the non-
linear realisation are a consequence of the duality relations that occur in
the irreducible representation and this allows us to predict the structure of
the duality relations in the former theory. In the non-linear realisation
there are also higher level fields which have blocks of ten, or eleven,
indices and so these fields do not appear in the irreducible representation.
These fields obey equations predicted by the non-linear realisation and they
lead to the gauged supergravities.

In the irreducible representations in the E theory approach the bosonic and
fermionic degrees of freedom appear in a unified way. They belong to the two
different spinor representations of $I_{c}(E_{8})={\rm SO}(16)$ which are
associated with the two nodes at the end of the SO(16) Dynkin diagram. Thus
swopping thee two nodes results in swopping bosons and fermions. One puzzle
with E theory is the way it leads to predictions that were usually seen as a
result of supersymmetry. Examples are the appearance of the two and five brane
charges and the BPS conditions [15]. Supersymmetry was discovered, at least
from the Russian viewpoint, by demanding that internal and spacetime
symmetries were contained in the same symmetry algebra. This required the
introduction of the supersymmetry generators. However, $E_{11}$ achieves the
same objective as it does contain the symmetries of spacetime, such as Lorentz
symmetry, and also internal symmetries such as the internal exceptional
symmetries of the maximal supergravities.

In this paper we have concentrated on the massless irreducible representation
of $E_{11}\otimes_{s}l_{1}$ which corresponds to the maximal supergravity
theories. The vector representation of $E_{11}$ contains the brane charges and
by taking these to be non-zero we will find other irreducible representations
of $E_{11}\otimes_{s}l_{1}$ corresponding to branes. In a future paper we hope
to examine some of these representations in detail and make the connection to
the the brane dynamics as also appears in the context of E theory, see for
example references [27].

The interesting paper [28] also remarks on the similar properties of bosons
and fermions although from a different viewpoint. It noticed that the SO(9)
representations to which bosonic ($44\oplus 84$)and fermionic (128) degrees of
freedom emerge in a very natural way as a solution of the equation $K\Psi=0$.
In this equation $K$ is the Kostant operator which is of the form
$K=\sum_{a}T_{a}\gamma^{a}$ where $\gamma^{a}$ are gamma matrices and $T^{a}$
are generators of $F_{4}$ that belong to the coset ${F_{4}\over{\rm SO}(9)}$.
The algebra $F_{4}$, dimension 52, can be found from the algebra SO(9),
dimension 36, by adding to the latter the 16 generators that belong to the
spinor representation of SO(16).

One can speculate that this picture can be generalised to incorporate the
features of the irreducible representations found in this paper. The bosonic
degrees of freedom belong to one spinor representation of SO(16), while the
fermionic degrees of freedom belong to the other spinor representation, both
of which have dimension 128. The algebra $E_{8}$ emerges from the SO(16)
algebra if we add to the 128 generators belonging to the spinor representation
of SO(16) to the 120 generators of SO(16). We can then consider the coset of
$E_{8}\over{\rm SO}(16)$. The corresponding Kostant operator would consist of
128 Gamma matrices multiplied by the generators in the coset. Could it be that
the solutions of the equation $K\Psi=0$ contain the two 128 dimensional spinor
representation of SO(16)? We can also wonder what are the higher spin
solutions? Speculating even further we can suppose that some of these 128
Gamma matrices can be taken to be the supercharges similar to what took place
in the discussion of the $N=2$ hypermultiplet given in reference [28]. We hope
to study these matter further and how they might fit into E theory.
Acknowledgements Peter West wishes to thank the SFTC for support from
Consolidated grants number ST/J002798/1 and ST/P000258/1, while Keith Glennon
would like to thank Kings College for support during his PhD studies.
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