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9.4 Further comments

9.4.1 Massive type IIA supergravity

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 [47Jump To The Next Citation Point] to correspond to M-theoretic degrees of freedom and (quantum) corrections.

The gradient representations have the interesting properties that their highest 𝔰𝔩(10,ℝ )-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 (D − 1)-forms that generate the cosmological constant for maximal gauged supergravities in D spacetime dimensions [1915073]. (A cosmological constant that appears as a constant of integration can be described by a (D − 1)-form [799].) 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 E10, namely [0,0,1,0,0,0,0,0, 1] and [2,0,0,0, 0,0,0,0,0] [141]. The lowest weight of the first one is, in terms of the scale factors, 2(β1 + β2 + β3) + β4 + β5 + β6 + β7 + β8 + β9. The lowest weight of the second one is 1 2 3 4 5 6 7 8 9 10 3β + β + β + β + β + β + β + β + β + β. 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 𝒜a1a2a3|b1b2⋅⋅⋅b9 corresponding, as we have seen, to the conjectured gradient representation (with one derivative) of the level 1 field 𝒜 a1a2a3. We shall thus focus on the second representation, described by a tensor 𝒜a1|b1|c1c2⋅⋅⋅c10.

By dimensional reduction along the first direction, the representation [2,0,0,0, 0,0,0,0,0] splits into various 𝔰𝔩(9,ℝ ) representations, one of which is described by the completely antisymmetric field 𝒜c ⋅⋅⋅c 2 10, i.e., a 9-form (in ten spacetime dimensions). It is obtained by taking a = b = c = 1 1 1 1 in 𝒜 a1|b1|c1c2⋅⋅⋅c10 and corresponds precisely to the lowest weight 1 2 3 4 5 6 7 8 9 10 γ = 3β + β + β + β + β + β + β + β + β + β given above. If one rewrites the corresponding term ∼ 𝒟 𝒜21|1|1c2⋅⋅⋅c10e2γ in the Lagrangian in terms of ten-dimensional scale factors and dilatons, one reproduces, using the field equations for 𝒜1|1|1c2⋅⋅⋅c10, the mass term of massive Type IIA supergravity.

The fact that E10 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 E8 results, this is a true E 10 test. The understanding of the massive Type IIA theory in the light of infinite Kac–Moody algebras was studied first in [157Jump To The Next Citation Point], where the embedding of the mass term in a nonlinear realisation of E11 was constructed. The precise connection between the mass term and an E10 positive real root was first explicitly made in Section 6.5 of [41]. It is interesting to note that even though the corresponding representation does not appear in E9, it is present in E10 without having to go to E11. The mass term of Type IIA was also studied from the point of view of the E10 coset model in [124].

This analysis suggests an interesting possibility for evading the no-go theorem of [13Jump To The Next Citation Point] 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 𝒜 a1|b1|c1c2⋅⋅⋅c10. 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 [47Jump To The Next Citation Point21Jump To The Next Citation Point25Jump To The Next Citation Point169].

9.4.2 Including fermions

Another attractive aspect of the E 10-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 𝔚E10 ⊂ E10. An interesting feature of the analysis is that E10-covariance leads to 𝔚E10-covariant derivatives that coincides with the covariant derivatives dictated by supersymmetry. This has been investigated in detail in [56505751128], to which we refer the interested reader.

9.4.3 Quantum corrections

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 [47Jump To The Next Citation Point]. This is supported by the fact that the known quantum corrections of M-theory do correspond to roots of E10 [54Jump To The Next Citation Point].

The idea is that with each correction curvature term of the form ∘ ------ RN − (11)g, where RN is a generic monomial of order N in the Riemann tensor, one can associate a linear form in the scale factors μ β’s in the BKL-limit. This linear form will be a root of E10 only for certain values of N. Hence compatibility of the corresponding quantum correction with the E 10 structure constrains the power N.

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 [1563]. It was rederived in [54Jump To The Next Citation Point] 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, R ⊥a⊥b and Rabab (a ⁄= b) which behave as

2σ 2σ R ⊥a⊥b ∼ e , Rabab ∼ e , (9.152 )
where σ is the sum of all the scale factors
σ = β1 + β2 + ⋅ ⋅⋅ + β10, (9.153 )
and where we have set R ⊥a⊥b = N −2R0a0b. This implies that
------ R ∼ e2σ, RN ∼ e2N σ, RN ∘ − (11)g ∼ e2(N −1)σ. (9.154 )

Now, σ is not on the root lattice. It is not an integer combination of the simple roots and it has length squared equal to − 10∕9. Integer combinations of the simple roots contains 3ℓ βμ’s, where ℓ is the level. Since 10 and 3 are relatively primes, the only multiples of σ that are on the root lattice are of the form 3kσ, k = 1,2,3, ⋅⋅⋅. These are negative, imaginary roots. The smallest value is k = 1, corresponding to the imaginary root

ω(β ) = 3σ (9.155 )
at level − 10, with squared length − 10. It follows that the only quantum corrections compatible with the E10 structure must have N − 1 = 3k, i.e., N = 3k + 1 [54Jump To The Next Citation Point], since it it only in this case that N ∘ --(11)- −2γ R − g ∼ e has γ = − (N − 1)σ on the root lattice. The first corrections are thus of the form 4 R, 7 R, 10 R etc. This in in remarkable agreement with the quantum computations of [88] (see also [152]).

The analysis of [54] was completed in [42] where it was observed that the imaginary root (9.155View Equation) was actually one of the fundamental weights of E 10, namely, the fundamental weight conjugate to the exceptional root that defines the level. In the case of E10, 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 [13013112136].)

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 E10 might require reconsideration [11].

9.4.4 Understanding duality

The previous analysis has revealed that the hyperbolic Kac–Moody algebra E10 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:

The Hamiltonian constraint
There is an obvious discrepancy between the Hamiltonian constraint of the sigma model and the Hamiltonan constraint of supergravity. In the sigma model case, all terms are positive, except for the kinetic term of the scale factors, which contain a negative sign related to the conformal factor. On the supergravity side, the kinetic term of the scale factors matches correctly, but there are extra negative contributions coming from level 3 (something perhaps not too surprising if level 3 is to be thought as a dual formulation of gravity and hence contains in particular dual scale factors). How this problem can be cured by tractable redefinitions is far from obvious.
Gauge invariance
The sigma model formulation corresponds to a partially gauge-fixed formulation since there are no arbitrary functions of time in the solutions of the equations of motion (except for the lapse function n(t)). The only gauge freedom left corresponds to time-independent gauge transformation (this is the equivalent of the “temporal gauge” of electromagnetism). The constraints associated with the spatial diffeomorphisms and with the 3-form gauge symmetry have not been eliminated. How they are expressed in terms of the sigma model variables and how they fit with the E10-symmetry is a question that should be answered. Progress along these lines may be found in recent work [52].
Electric-magnetic duality
The sigma model approach contains both the graviton and its dual, as well as both the 3-form and its dual 6-form. Since these obey second-order equations of motion, there is a double-counting of degrees of freedom. For instance, the magnetic field of the 3-form would also appear as a spatial gradient of the 3-form at level 4, but nothing in the formalism tells that this is the same magnetic field as the time derivative of the 6-form at level 2. A generalized self-duality condition should be imposed [3536], not just in the 3-form sector but also for the graviton. Better yet, one might search for a duality-invariant action without double-counting. Such actions have been studied both for p-forms [6564] and for gravity [21Jump To The Next Citation Point101114] and are not manifestly spacetime covariant (this is not an issue here since manifest spacetime covariance has been given up anyway in the (1 + 0)-dimensional ℰ10-sigma model). One must pick a spacetime coordinate, which might be time, or one spatial direction [100159]. We feel that a better understanding of duality might yield an important clue [212526148].

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