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 .
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 .
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 . Similarly the IIB-fields appear at low levels for a decomposition with respect to the subalgebra found upon removal of the second simple root . 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  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”.
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
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. The right hand side of Equation (8.89) therefore only contains the index structure compatible with the generators ,
For later reference, we list here some additional commutators that are useful :
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 . 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. ). 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:
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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