We perform the level decomposition of with respect to the -subalgebra obtained by removing the exceptional node in the Dynkin diagram in Figure 49. This procedure was described in Section 8. When using this decomposition, a sum over (positive) roots becomes a sum over all -indices in each (positive) representation appearing in the decomposition.

We recall that up to level three the following representations appear

where all indices are -indices and so run from 1 to 10. The level zero generators correspond to the adjoint representation of and the higher level generators correspond to an infinite tower of raising operators of . As indicated by the square brackets, the level one and two representations are completely antisymmetric in all indices, while the level three representation has a mixed Young tableau symmetry: It is antisymmetric in the eight indices and is also subject to the constraint In the scale factor space (-basis), the roots of corresponding to the above generators act as follows on : We can use the scalar product in root space, , to compute the norms of these roots. Recall from Section 5 that the metric on is the inverse of the metric in Equation (9.88), and for it takes the form The level zero, one and two generators correspond to real roots of , We have split the roots corresponding to the level three generators into two parts, depending on whether or not the special index takes the same value as one of the other indices. The resulting two types of roots correspond to real and null roots, respectively, Thus, the first time that generators corresponding to imaginary roots appear in the level decomposition is at level three. This will prove to be important later on in our analysis.

Because of the importance and geometric significance of level zero, we shall first develop the formalism for the -sigma model. A general group element in the subgroup reads

where is a matrix (with being the row index and the column index). Although the ’s are generators of and can, within this framework, at best be viewed as infinite matrices, it will prove convenient – for streamlining the calculations – to view them in the present section also as matrices, since we confine our attention to the finite-dimensional subgroup . Namely, is treated as a matrix with 0’s everywhere except 1 in position (see Equation (6.83)). The final formulation in terms of the variables – which are matrices irrespectively as to whether one deals with per se or as a subgroup of – does not depend on this interpretation.It is also useful to describe as the set of linear combinations where the matrix is invertible. The product of the ’s is given by

One easily verifies that if and belong to , then where is the standard product of the matrices and . Furthermore, where is the standard matrix exponential.Under a general transformation, the representative is multiplied from the left by a time-dependent group element and from the right by a constant linear -group element . Explicitly, the transformation takes the form (suppressing the time-dependence for notational convenience)

In terms of components, with , , and , one finds where we have set . The indices on the coset representative have different covariance properties. To emphasize this fact, we shall write a bar over the first index, . Thus, barred indices transform under the local gauge group and are called “local”, or also “flat”, indices, while unbarred indices transform under the global and are called “world”, or also “curved”, indices. The gauge invariant matrix product is equal to with and The do not transform under local -transformations and transform as a (symmetric) contravariant tensor under rigid -transformations, They are components of a nondegenerate symmetric matrix that can be identified with an inverse Euclidean metric.Indeed, the coset space can be identified with the space of symmetric tensors of Euclidean signature, i.e., the space of metrics. This is because two symmetric tensors of Euclidean signature are equivalent under a change of frame, and the isotropy subgroup, say at the identity, is evidently . In that view, the coset representative is the spatial vielbein.

The action for the coset space with the metric of Equation (8.82) is easily found to be

Note that the quadratic form multiplying the time derivatives is just the “De Witt supermetric” [66]. Note also for future reference that the invariant form reads explicitly where is the inverse vielbein.

We now turn to the full nonlinear sigma model for . Rather than exponentiating the Cartan subalgebra separately as in Equation (9.75), it will here prove convenient to instead single out the level zero subspace . This permits one to control easily -covariance. To make this level zero covariance manifest, we shall furthermore assume that the Borel gauge has been fixed only for the non-zero levels, and we keep all level zero fields present. The residual gauge freedom is then just multiplication by an rotation from the left.

Thus, we take a coset representative of the form

where the sum in the first exponent would be restricted to all if we had taken a full Borel gauge also at level zero. The parameters and are coordinates on the coset space and will eventually be interpreted as physical time-dependent fields of eleven-dimensional supergravity.How do the fields transform under ? Let , and decompose according to Equation (9.117) as the product

with One has Now, the first matrix clearly belongs to , since it is the product of a rotation matrix by two -matrices. It has exactly the same transformation as in Equation (9.110) above in the context of the nonlinear sigma model for . Hence, the geometric interpretation of as the vielbein remains.Similarly, the matrix has exactly the same form as ,

where the variables , , ..., are obtained from the variables , , ..., by computing , , ..., using the commutation relations with . Explicitly, one gets Hence, the fields , , ... do not transform under local transformations. However, they do transform under rigid -transformations as tensors of the type indicated by their indices. Their indices are world indices and not flat indices.

The invariant form reads

The first term is the invariant form encountered above in the discussion of the level zero nonlinear sigma model for . So let us focus on the second term. It is clear that will contain only positive generators at level . So we set, in a manner similar to Equation (9.64), where the sum is over positive roots at levels one and higher and takes into account multiplicities (through the extra index ). The expressions are linear in the time derivatives . As before, we call them “covariant derivatives”. They are computed by making use of the Baker–Hausdorff formula, as in Section 9.1.6. Explicitly, up to level 3, one finds with as computed in [47]. The notation denotes projection onto the Young tableaux symmetry carried by the field upon which the covariant derivative acts

The action can now be computed using the bilinear form on ,

where is obtained by projecting orthogonally onto the subalgebra by using the generalized transpose, where as above (with being the Chevalley involution). We shall compute the action up to, and including, level 3,From Equation (9.123) and the fact that generators at level zero are orthogonal to generators at levels , we see that will be constructed from the level zero part and will coincide with the Lagrangian (9.115) for the nonlinear sigma model ,

To compute the other terms, we use the following trick. The Lagrangian must be a scalar. One can easily compute it in the frame where , i.e., where the metric is equal to . One can then covariantize the resulting expression by replacing everywhere by . To illustrate the procedure consider the level term. One has, for and at level , and thus, with the same gauge conditions, (where we have raised the indices of with , etc). Using (3! terms; see Section 8.4), one then gets in the frame where . This yields the level 1 Lagrangian in a general frame,

By a similar analysis, the level 2 and 3 contributions are Collecting all terms, the final form of the action for up to and including level is which agrees with the action found in [47].

We shall now relate the equations of motion for the sigma model to the equations of motion of eleven-dimensional supergravity. As the precise correspondence is not yet known, we shall here only sketch the main ideas. These work remarkably well at low levels but need unknown ingredients at higher levels.

We have seen that the sigma model for can be consistently truncated level by level. More precisely, one can consistently set equal to zero all covariant derivatives of the fields above a given level and get a reduced system whose solutions are solutions of the full system. We shall show here that the consistent truncations of at levels 0, 1 and 2 yields equations of motion that coincide with the equations of motion of appropriate consistent truncations of eleven-dimensional supergravity, using a prescribed dictionary presented below. We will also show that the correspondence extends up to parts of level 3.

We recall that in the gauge (vanishing shift) and (temporal gauge), the bosonic fields of eleven-dimensional supergravity are the spatial metric , the lapse and the spatial components of the vector potential 3-form. The physical field is and its electric and magnetic components are, respectively, denoted and . The electric field involves only time derivatives of , while the magnetic field involves spatial gradients.

If one keeps only levels zero and one, the sigma model action (9.133) reduces to

Consider now the consistent homogeneous truncation of eleven-dimensional supergravity in which the spatial metric, the lapse and the vector potential depend only on time (no spatial gradient). Then the reduced action for this truncation is precisely Equation (9.134) provided one makes the identification and

(see, for instance, [61]). Also the Hamiltonian constraints (the only one left) coincide. Thus, there is a perfect match between the sigma model truncated at level one and supergravity “reduced on a 10-torus”. If one were to drop level one, one would find perfect agreement with pure gravity. In the following, we shall make the gauge choice , equivalent to .

At levels 0 and 1, the supergravity fields and depend only on time. When going beyond this truncation, one needs to introduce some spatial gradients. Level 2 introduces spatial gradients of a very special type, namely allows for a homogeneous magnetic field. This means that acquires a space dependence, more precisely, a linear one (so that its gradient does not depend on ). However, because there is no room for -dependence on the sigma model side, where the only independent variable is , we shall use the trick to describe the magnetic field in terms of a dual potential . Thus, there is a close interplay between duality, the sigma model formulation, and the introduction of spatial gradients.

There is no tractable, fully satisfactory variational formulation of eleven-dimensional supergravity where both the 3-form potential and its dual appear as independent variables in the action, with a quadratic dependence on the time derivatives (this would be double-counting, unless an appropriate self-duality condition is imposed [35, 36]). This means that from now on, we shall not compare the actions of the sigma model and of supergravity but, rather, only their respective equations of motion. As these involve the electromagnetic field and not the potential, we rewrite the correspondence found above at levels 0 and 1 in terms of the metric and the electromagnetic field as

The equations of motion for the nonlinear sigma model, obtained from the variation of the Lagrangian Equation (9.133), truncated at level two, read explicitlyIn addition, we have the constraint obtained by varying ,

On the supergravity side, we truncate the equations to metrics and electromagnetic fields , that depend only on time. We take, as in Section 2, the spacetime metric to be of the form

but now with . In the following we use Greek letters to denote eleven-dimensional spacetime indices, and Latin letters to denote ten-dimensional spatial indices.The equations of motion and the Hamiltonian constraint for eleven-dimensional supergravity have been explicitly written in [61], so they can be expediently compared with the equations of motion of the sigma model. The dynamical equations for the metric read

and for the electric and magnetic fields we have, respectively, the equations of motion and the Bianchi identity, Furthermore we have the Hamiltonian constraint (We shall not consider the other constraints here; see remarks as at the end of this section.)One finds again perfect agreement between the sigma model equations, Equation (9.139) and (9.140), and the equations of eleven-dimensional supergravity, Equation (9.142) and (9.144), provided one extends the above dictionary through [47]

This result appears to be quite remarkable, because the Chern–Simons term in Equation (9.143) is in particular reproduced with the correct coefficient, which in eleven-dimensional supergravity is fixed by invoking supersymmetry.

Level 3 should correspond to the introduction of further controlled spatial gradients, this time for the metric. Because there is no room for spatial derivatives as such on the sigma model side, the trick is again to introduce a dual graviton field. When this dual graviton field is non-zero, the metric does depend on the spatial coordinates.

Satisfactory dual formulations of non-linearized gravity do not exist. At the linearized level, however, the problem is well understood since the pioneering work by Curtright [39] (see also [167, 14, 21]). In eleven spacetime dimensions, the dual graviton field is described precisely by a tensor with the mixed symmetry of the Young tableau appearing at level 3 in the sigma model description. Exciting this field, i.e., assuming amounts to introducing spatial gradients for the metric – and, for that matter, for the other fields as well – as follows. Instead of considering fields that are homogeneous on a torus, one considers fields that are homogeneous on non-Abelian group manifolds. This introduces spatial gradients (in coordinate frames) in a well controlled manner.

Let be the group invariant one-forms, with structure equations

We shall assume that (“Bianchi class A”). Truncation at level 3 assumes that the metric and the electric and magnetic fields depend only on time in this frame and that the are constant (corresponding to a group). The supergravity equations have been written in that case in [61] and can be compared with the sigma model equations. There is almost a complete match between both sets of equations provided one extends the dictionary at level 3 through (with the equations of motion of the sigma model implying indeed that does not depend on time). Note that to define an invertible mapping between the level three fields and the , it is important that be traceless; there is no “room” on level three on the sigma model side to incorporate the trace of .With this correspondence, the match works perfectly for real roots up to, and including, level three. However, it fails for fields associated with imaginary roots (level 3 is the first time imaginary roots appear, at height 30) [47]. In fact, the terms that match correspond to “-covariantized ”, i.e., to fields associated with roots of and their images under the Weyl group of .

Since the match between the sigma model equations and supergravity fails at level 3 under the present line of investigation, we shall not provide the details but refer instead to [47] for more information. The correspondence up to level 3 was also checked in [53] through a slightly different approach, making use of a formulation with local frames, i.e., using local flat indices rather than global indices as in the present treatment.

Let us note here that higher level fields of , corresponding to imaginary roots, have been considered from a different point of view in [24], where they were associated with certain brane configurations (see also [23, 9]).

One may view the above failure at level 3 as a serious flaw to the sigma model approach to exhibiting the
symmetry^{36}.
Let us, however, be optimistic for a moment and assume that these problems will somehow get
resolved, perhaps by changing the dictionary or by including higher order terms. So, let us
proceed.

What would be the meaning of the higher level fields? As discussed in Section 9.3.7, there are indications that fields at higher levels contain higher order spatial gradients and therefore enable us to reconstruct completely, through something similar to a Taylor expansion, the most general field configuration from the fields at a given spatial point.

From this point of view, the relation between the supergravity degrees of freedom and would be given, at a specific spatial point and in a suitable spatial frame (that would also depend on ), by the following “dictionary”:

which reproduces in the homogeneous case what we have seen up to level 3.This correspondence goes far beyond that of the algebraic description of the BKL-limit in terms of Weyl reflections in the simple roots of a Kac–Moody algebra. Indeed, the dynamics of the billiard is controlled entirely by the walls associated with simple roots and thus does not transcend height one. Here, we go to a much higher height and successfully extend (unfortunately incompletely) the intriguing connection between eleven-dimensional supergravity and .

We have seen that the correspondence between the -invariant sigma model and eleven-dimensional supergravity fails when we include spatial gradients beyond first order. It is nevertheless believed that the information about spatial gradients is somehow encoded within the algebraic description: One idea is that space is “smeared out” among the infinite number of fields contained in and it is for this reason that a direct dictionary for the inclusion of spatial gradients is difficult to find. If true, this would imply that we can view the level expansion on the algebraic side as reflecting a kind of “Taylor expansion” in spatial gradients on the supergravity side. Below we discuss some speculative ideas about how such a correspondence could be realized in practice.

One intriguing suggestion put forward in [47] was that fields associated to certain “affine representations” of could be interpreted as spatial derivatives acting on the level one, two and three fields, thus providing a direct conjecture for how space “emerges” through the level decomposition of . The representations in question are those for which the Dynkin label associated with the overextended root of vanishes, and hence these representations are realized also in a level decomposition of the regular -subalgebra obtained by removing the overextended node in the Dynkin diagram of .

The affine representations were discussed in Section 8 and we recall that they are given in terms of three infinite towers of generators, with the following -tensor structures,

where the upper indices have the same Young tableau symmetries as the and representations, while the lower indices are all completely symmetric. In the sigma model these generators of are parametrized by fields exhibiting the same index structure, i.e., , and .The idea is now that the three towers of fields have precisely the right index structure to be interpreted as spatial gradients of the low level fields

Although appealing and intuitive as it is, this conjecture is difficult to prove or to check explicitly, and not much progress in this direction has been made since the original proposal. However, recently [73] this problem was attacked from a rather different point of view with some very interesting results, indicating that the gradient conjecture may need to be substantially modified. For completeness, we briefly review here some of the main features of [73].

Recall from Section 4 that the infinite-dimensional Kac–Moody algebras and can be obtained from through prescribed extensions of the Dynkin diagram: is obtained by extending with one extra node, and by extending with two extra nodes. This procedure can be continued and after extending three times, one finds the Lorentzian Kac–Moody algebra , which is also believed to be relevant as a possible underlying symmetry of M-theory [167, 74].

These algebras are part of the chain of exceptional regular embeddings,

which was used in [69] to show that a sigma model for the coset space can be consistently truncated to a sigma model for the coset space , which coincides with Equation (9.133). This result builds upon previous work devoted to general constructions of sigma models invariant under Lorentzian Kac–Moody algebras of -type [74, 71, 70, 72].It was furthermore shown in [69] that by performing a suitable Weyl reflection before truncation, yet another sigma model based on could be obtained. It differs from Equation (9.133) because the parameter along the geodesic is a spacelike, not timelike, variable in spacetime. This follows from the fact that the sigma model is constructed from the coset space , where coincides with the noncompact group at level zero, and not as is the case for Equation (9.133). The two sigma model actions were referred to in [69] as and , since solutions to the first model translate to time-dependent (cosmological) solutions of eleven-dimensional supergravity, while the second model gives rise to stationary (brane) solutions, which are smeared in all but one spacelike direction. In particular, the and fields correspond to potentials for the - and -branes, respectively.

In [73], solutions associated to the infinite tower of affine representations for the brane sigma model based on were investigated. The idea was that by restricting the indices to be -indices, any such representation coincides with a generator of , and so different fields in these affine towers must be related by Weyl reflections in .

The Weyl group is a subgroup of the U-duality group of M-theory compactified on to two spacetime dimensions. Moreover, the continuous group is the M-theory analogue of the Geroch group, i.e., it is a symmetry of the space of solutions of supergravity in two dimensions [140]. Under these considerations it is natural to expect that the fields associated with the affine representations should somehow be related to the infinite number of “dual potentials” appearing in connection with the Geroch group in two dimensions. Indeed, the authors of [73] were able to show, using the embedding , that given, e.g., a representation in the affine tower, there exists a -transformation that relates the associated field to the lowest generator . The resulting solution, however, is different from the standard brane solution obtained from the -field because the new solution is smeared in all directions except two spacelike directions, i.e., the solution is an -brane solution which depends on two spacelike variables.

Thus, by taking advantage of the embedding , it was shown that the three towers of “gradient representations” encode a kind of “de-compactification” of one spacelike variable. In a way this therefore indicates that part of the gradient conjecture must be correct, in the sense that the towers of affine representations indeed contain information about the emergence of spacelike directions. On the other hand, it also seems that the correspondence is more complicated than was initially believed, perhaps deeply connected to U-duality in some, as of yet, unknown way.

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