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 [47] 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
[141]. 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 [157
], 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 [41]. 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 [124].
This analysis suggests an interesting possibility for evading the no-go theorem of [13] 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 [13] 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 [47]. This is
supported by the fact that the known quantum corrections of M-theory do correspond to roots of
[54
].
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 [16] and was extended to higher dimensions in [15, 63]. It was rederived
in [54] 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
The analysis of [54] was completed in [42] 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 [11].
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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