Our first encounter with in a physical application was in Section 5 where we have showed that the
Weyl group of
describes the chaos that emerges when studying eleven-dimensional supergravity close
to a spacelike singularity [45
].
In Section 9.3, we will discuss how to construct a Lagrangian manifestly invariant under global
-transformations and compare its dynamics to that of eleven-dimensional supergravity. The level
decomposition associated with the removal of the “exceptional node” labelled “10” in Figure 49
will be
central to the analysis. It turns out that the low-level structure in this decomposition precisely reproduces
the bosonic field content of eleven-dimensional supergravity [47
].
Moreover, decomposing with respect to different regular subalgebras reproduces also the bosonic
field contents of the Type IIA and Type IIB supergravities. The fields of the IIA theory are obtained by
decomposition in terms of representations of the
subalgebra obtained by removing the
first simple root
[125
]. Similarly the IIB-fields appear at low levels for a decomposition with respect to
the
subalgebra found upon removal of the second simple root
[126
].
The extra
-factor in this decomposition ensures that the
-symmetry of IIB supergravity is
recovered.
For these reasons, we investigate now these various level decompositions.
Let denote the simple roots of
and
the Cartan generators. These span
the root space
and the Cartan subalgebra
, respectively. Since
is simply laced the Cartan
matrix is given by the scalar products between the simple roots:
As before, the weight that is easiest to identify for each representation at positive level
is
the lowest weight
. We denote by
the projection onto the spacelike slice of the root lattice
defined by the level
. The (conjugate) Dynkin labels
characterizing the representation
are defined as before as minus the coefficients in the expansion of
in terms of the
fundamental weights
of
:
The Killing form on each such slice is positive definite so the projected weight is of course real.
The fundamental weights of
can be computed explicitly from their definition as the duals of the
simple roots:
The corresponding diophantine equation, Equation (8.50), for
was found in [47
] and reads
The representation content at each level is represented by -tensors whose index structure are
encoded in the Dynkin labels
. At level
we have the adjoint representation of
represented by the generators
obeying the same commutation relations as in
Equation (8.32
) but now with
-indices.
All higher (lower) level representations will then be tensors transforming contravariantly (covariantly)
under the level generators. The resulting representations are displayed up to level 3 in Table 39.
We see that the level 1 and 2 representations have the index structures of a 3-form and a 6-form
respectively. In the
-invariant sigma model, to be constructed in Section 9, these generators will
become associated with the time-dependent physical “fields”
and
which are related to
the electric and magnetic component of the 3-form in eleven-dimensional supergravity. Similarly, the level 3
generator
with mixed Young symmetry will be associated to the dual of the spatial part
of the eleven-dimensional vielbein. This field is therefore sometimes referred to as the “dual
graviton”.
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Let us now describe in a little more detail the commutation relations between the low-level generators in the
level decomposition of (see Table 39). We recover the Chevalley generators of
through the
following realisation:
The bilinear form at level zero is
and can be extended level by level to the full algebra by using its invariance,Now, by using the graded structure of the level decomposition we can infer that the level 2 generators can be obtained by commuting the level 1 generators
Concretely, this means that the level 2 content should be found from the commutator We already know that the only representation at this level is For later reference, we list here some additional commutators that are useful [53]:
So far, we have only discussed the representations occurring at the first four levels in the
decomposition. This is due to the fact that a physical interpretation of higher level fields is yet to be found.
There are, however, among the infinite number of representations, a subset of three (infinite) towers of
representations with certain appealing properties. These are the “gradient representations”,
so named due to their conjectured relation to the emergence of space, through a Taylor-like
expansion in spatial gradients [47
]. We explain here how these representations arise and we
emphasize some of their important properties, leaving a discussion of the physical interpretation to
Section 9.
The gradient representations are obtained by searching for “affine representations”, for which the
coefficient in front of the overextended simple root of
vanishes, i.e., the lowest weights of the
representations correspond to the following subset of
roots,
The Dynkin labels allowed by this restricting are parametrized by an integer which is related to the
level at which a specific representation occurs in the following way:
Thus, in this way we obtain the three infinite towers of generators
The existence of these towers of representations is not special for among the exceptional algebras,
although the symmetric Young structure of the lower indices is actually a very special and
important feature of
. In Section 9 we will discuss the tantalizing possibility that these
representations encode an infinite set of spatial gradients that describe the emergence, or “unfolding”, of
space.
To illustrate the difference from other exceptional algebras, we consider, for instance, a similar search for
affine representations within (see, e.g. [141
]). The same sets of 9-tuples appear, but now these should
be dualized with the rank 11 epsilon tensor of
, leaving us with three towers of generators that
have
pairs of antisymmetric indices, i.e.,
Finally, we note that because all these representations were found by setting , we are really
dealing with representations that also exist within
, in the sense that when restricting all indices to
-indices, these generators can be found in a level decomposition of
with respect to its
-subalgebra. However, it is important to note that in
and
the affine representations
constitute merely a small subset of all representations occurring in the level decomposition, while in
they are actually the only ones and so they provide (together with their transposed partners) the full
structure of the algebra. Moreover, in
the epsilon tensor is of rank 9 so all the 9-tuples of
antisymmetric indices are “swallowed” by the epsilon tensor. This reflects the fact that for
affine algebras the level decomposition corresponds to an infinite repetition of the low-level
representations.
A level decomposition can be performed with respect to any of the regular subalgebras encoded in the Dynkin diagram. We mention here two additional cases which are specifically interesting for our purposes, since they give rise to low-level field contents that coincide with the bosonic spectrum of Type IIA and IIB supergravity. The relevant decompositions are the following:
The corresponding levels are defined as It turns out that in the On the contrary, in the decomposition the additional factor of
causes
mixing between the R-R and NS-NS fields at each level. This is to be expected since we know that for
example the fundamental string (F1) and the D1-brane couples to the NS-NS 2-form
and the R-R
2-form
, respectively, which transform as a doublet under the
-symmetry of Type IIB
supergravity.
In the the level
content is of course just the adjoint representation in
the same way as in the
decomposition considered above. In the other case instead
we find the adjoint representation
of
with commutation relations
The procedure follows a similar structure as for the previous cases so we will not give the details here. We refer the interested reader to [125, 126] for a detailed account. The result of the decompositions up to level 3 for the two cases discussed here is displayed in Tables 40 and 41.
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