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
,
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 ,
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 roots
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
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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