The motivation is that the construction of a geodesic sigma model that exhibits this Kac–Moody symmetry in a manifest way, would provide a link to understanding the role of the full algebra beyond the BKL-limit.
For definiteness, we consider only the case when the Lorentzian algebra is a split real form, although this is not really necessary as the Iwasawa decomposition holds also in the non-split case.
A very important difference from the finite-dimensional case is that we now have nontrivial multiplicities of the imaginary roots (see Section 4). Recall that if a root has multiplicity , then the associated root space is -dimensional. Thus, it is spanned by generators ,
Our main object of study is the coset representative , which must now be seen as “formal” exponentiation of the infinite number of generators in . We can then proceed as before and choose to be in the Borel gauge, i.e., of the form
A -valued “one-form” can be constructed analogously to the finite-dimensional case,
The action for a particle moving on the infinite-dimensional coset space can now be constructed using the invariant bilinear form on ,null geodesic on ,
Defining, as before, a covariant derivative with respect to the local symmetry as
Although is infinite-dimensional we still have the notion of “formal integrability”, owing to the existence of an infinite number of conserved charges, defined by the equations of motion in Equation (9.86). We can define the generalized metric for any as
Although the explicit form of contains infinitely many terms, we have seen that each coefficient can, in principle, be computed exactly for each root . This, however, is not the case for the current . To find the form of one must conjugate with the coset representative and this requires an infinite number of commutators to get the correct coefficient in front of any generator in .
One method for dealing with infinite expressions like Equation (9.80) consists in considering successive finite expansions allowing more and more terms, while still respecting the dynamics of the sigma model.
This leads us to the concept of a consistent truncation of the sigma model for . We take as definition of such a truncation any sub-model of whose solutions are also solutions of the original model.
There are two main approaches to finding suitable truncations that fulfill this latter criterion. These are the so-called subgroup truncations and the level truncations, which will both prove to be useful for our purposes, and we consider them in turn below.
The first consistent truncation we shall treat is the case when the dynamics of a sigma model for some global group is restricted to that of an appropriately chosen subgroup . We consider here only subgroups which are obtained by exponentiation of regular subalgebras of . The concept of regular embeddings of Lorentzian Kac–Moody algebras was discussed in detail in Section 4.
To restrict the dynamics to that of a sigma model based on the coset space , we first assume that the initial conditions and are such that the following two conditions are satisfied:
When these conditions hold, then belongs to for all . Moreover, there always exists such that
Now recall that from Equation (9.93), we know that is a solution to the equations of motion for the sigma model on . But since we have found that preserves the Borel gauge for , it follows that is a solution to the equations of motion for the full sigma model. Thus, the dynamical evolution of the subsystem preserves the Borel gauge of . These arguments show that initial conditions in remain in , and hence the dynamics of a sigma model on can be consistently truncated to a sigma model on .
Finally, we recall that because the embedding is regular, the restriction of the bilinear form on coincides with the bilinear form on . This implies that the Hamiltonian constraints for the two models, arising from time reparametrization invariance of the action, also coincide.
We shall make use of subgroup truncations in Section 10.
Alternative ways of consistently truncating the infinite-dimensional sigma model rest on the use of gradations of ,level decomposition of the adjoint representation of into representations of a finite regular subalgebra . We will now use this construction to truncate the sigma model based of , by “terminating” the gradation of at some finite level . More specifically, the truncation will involve setting to zero all coefficients , in the expansion of , corresponding to roots whose generators belong to subspaces with . Part of this section draws inspiration from the treatment in [47, 48, 124].
The level might be the height, or it might count the number of times a specified single simple root appears. In that latter case, the actual form of the level decomposition must of course be worked out separately for each choice of algebra and each choice of decomposition. We will do this in detail in Section 9.3 for a specific level decomposition of the hyperbolic algebra . Here, we shall display the general construction in the case of the height truncation, which exists for any algebra.
Let be a positive root, . It has the following expansion in terms of the simple rootsheight of is defined as (see Section 4)
To achieve the height truncation, we assume that the sum over all roots in the expansion of , Equation (9.80), is ordered in terms of increasing height. Then we can consistently set to zero all coefficients corresponding to roots with greater height than some, suitably chosen, finite height . We thus find that the finitely truncated coset element is
For further use, we note here some properties of the coefficients . By examining the structure of Equation (9.81), we see that takes the form of a temporal derivative acting on , followed by a sequence of terms whose individual components, for example , are all associated with roots of lower height than , . It will prove useful to think of as representing a kind of “generalized” derivative operator acting on the field . Thus we define the operator by we shall call the operators “covariant derivatives”.
It is clear from the equations of motion , that if all covariant derivatives above a given height are set to zero, this choice is preserved by the dynamical evolution. Hence, the height (and any level) truncation is indeed a consistent truncation. Let us here emphasize that it is not consistent by itself to merely put all fields above a certain level to zero, but one must take into account the fact that combinations of lower level fields may parametrize a higher level generator in the expansion of , and therefore it is crucial to define the truncation using the derivative operator .
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