We have just seen that some of the higher level fields might have an interpretation in terms of spatial gradients. This would account for a subclass of representations at higher levels. The existence of other representations at each level besides the “gradient representations” shows that the sigma model contains further degrees of freedom besides the supergravity fields, conjectured in  to correspond to M-theoretic degrees of freedom and (quantum) corrections.
The gradient representations have the interesting properties that their highest -weight is a real root. There are other representations with the same properties. An interesting interpretation of some of those has been put forward recently using dimensional reduction, as corresponding to the -forms that generate the cosmological constant for maximal gauged supergravities in spacetime dimensions [19, 150, 73]. (A cosmological constant that appears as a constant of integration can be described by a -form [7, 99].) For definiteness, we shall consider here only the representations at level 4, related to the mass term of type IIA theory.
There are two representations at level 4, both of them with a highest weight which is a real root of , namely and . The lowest weight of the first one is, in terms of the scale factors, . The lowest weight of the second one is . Both weights are easily verified to have squared length equal to 2 and, since they are on the root lattice, they are indeed roots by the criterion for roots of hyperbolic algebras. The first representation is described by a tensor with mixed symmetry corresponding, as we have seen, to the conjectured gradient representation (with one derivative) of the level 1 field . We shall thus focus on the second representation, described by a tensor .
By dimensional reduction along the first direction, the representation splits into various representations, one of which is described by the completely antisymmetric field , i.e., a 9-form (in ten spacetime dimensions). It is obtained by taking in and corresponds precisely to the lowest weight given above. If one rewrites the corresponding term in the Lagrangian in terms of ten-dimensional scale factors and dilatons, one reproduces, using the field equations for , the mass term of massive Type IIA supergravity.
The fact that contains information about the massive Type IIA theory is in our opinion quite profound because, contrary to the low level successes which are essentially a covariantization of known results, this is a true test. The understanding of the massive Type IIA theory in the light of infinite Kac–Moody algebras was studied first in , where the embedding of the mass term in a nonlinear realisation of was constructed. The precise connection between the mass term and an positive real root was first explicitly made in Section of . It is interesting to note that even though the corresponding representation does not appear in , it is present in without having to go to . The mass term of Type IIA was also studied from the point of view of the coset model in .
This analysis suggests an interesting possibility for evading the no-go theorem of  on the impossibility to generate a cosmological constant in eleven-dimensional supergravity. This should be tried by introducing new degrees of freedom described by a mixed symmetry tensor . If this tensor can be consistently coupled to gravity (a challenge in the context of field theory with a finite number of fields!), it would provide the eleven-dimensional origin of the cosmological constant in massive Type IIA. There would be no contradiction with  since in eleven dimensions, the new term would not be a standard cosmological constant, but would involve dynamical degrees of freedom. This is, of course, quite speculative.
Finally, there are extra fields at higher levels besides spatial gradients and the massive Type IIA term. These might correspond to higher spin degrees of freedom [47, 21, 25, 169].
Another attractive aspect of the -sigma model formulation is that it can easily account for the fermions of supergravity up to the levels that work in the bosonic sector. The fermions transform in representations of the compact subalgebra . An interesting feature of the analysis is that -covariance leads to -covariant derivatives that coincides with the covariant derivatives dictated by supersymmetry. This has been investigated in detail in [56, 50, 57, 51, 128], to which we refer the interested reader.
If the gradient conjecture is correct (perhaps with a more sophisticated dictionary), then one sees that the sigma model action would contain spatial derivatives of higher order. It has been conjectured that these could perhaps correspond to higher quantum corrections . This is supported by the fact that the known quantum corrections of M-theory do correspond to roots of .
The idea is that with each correction curvature term of the form , where is a generic monomial of order in the Riemann tensor, one can associate a linear form in the scale factors ’s in the BKL-limit. This linear form will be a root of only for certain values of . Hence compatibility of the corresponding quantum correction with the structure constrains the power .
The evaluation of the curvature components in the BKL-limit goes back to the paper by BKL themselves in four dimensions  and was extended to higher dimensions in [15, 63]. It was rederived in  for the purpose of evaluating quantum corrections. It is shown in these references that the leading terms in the curvature expressed in an orthonormal frame adapted to the slicing are, in the BKL-limit, and () which behave as
Now, is not on the root lattice. It is not an integer combination of the simple roots and it has length squared equal to . Integer combinations of the simple roots contains ’s, where is the level. Since 10 and 3 are relatively primes, the only multiples of that are on the root lattice are of the form , . These are negative, imaginary roots. The smallest value is , corresponding to the imaginary root, since it it only in this case that has on the root lattice. The first corrections are thus of the form , , etc. This in in remarkable agreement with the quantum computations of  (see also ).
The analysis of  was completed in  where it was observed that the imaginary root (9.155) was actually one of the fundamental weights of , namely, the fundamental weight conjugate to the exceptional root that defines the level. In the case of , the root lattice and the weight lattice coincides, but this observation was useful in the analysis of the quantum corrections for other theories where the weight lattice is strictly larger than the root lattice. The compatibility conditions seem in those cases to be that quantum corrections should be associated with vectors on the weight lattice. (See also [130, 131, 12, 136].)
Finally, we note that recent work devoted to investigations of U-duality symmetries of compactified higher curvature corrections indicates that the results reported here in the context of might require reconsideration .
The previous analysis has revealed that the hyperbolic Kac–Moody algebra contains a large amount of information about the structure and the properties of M-theory. How this should ultimately be incorporated in the final formulation of the theory is, however, not clear.
The sigma model approach exhibits some important drawbacks and therefore it does not appear to be the ultimate formulation of the theory. In addition to the absence of a complete dictionary enabling one to go satisfactorily beyond level 3 (the level where the first imaginary root appears), more basic difficulties already appear at low levels. These are:
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