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9.2 Geodesic sigma models on infinite-dimensional coset spaces

In the following we shall both “generalize and specialize” the construction from Section 9.1. The generalization amounts to relaxing the restriction that the algebra 𝔤 be finite-dimensional. Although in principle we could consider 𝔤 to be any indefinite Kac–Moody algebra, we shall be focusing on the case where it is of Lorentzian type. The analysis will also be a specialization, in the sense that we consider only geodesic sigma models, meaning that the Riemannian space X is the one-dimensional worldline of a particle, parametrized by one variable t. This restriction is of course motivated by the billiard description of gravity close to a spacelike singularity, where the dynamics at each spatial point is effectively described by a particle geodesic in the fundamental Weyl chamber of a Lorentzian Kac–Moody algebra.

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 𝔤 beyond the BKL-limit.

9.2.1 Formal construction

For definiteness, we consider only the case when the Lorentzian algebra 𝔤 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 α ∈ Δ has multiplicity m α, then the associated root space 𝔤α is m α-dimensional. Thus, it is spanned by m α generators [s] E α (s = 1,⋅⋅⋅ ,m α),

𝔤α = ℝE [1α]+ ⋅⋅⋅ + ℝE [αmα]. (9.74 )
The root multiplicities are not known in closed form for any indefinite Kac–Moody algebra, but must be computed recursively as described in Section 8.

Our main object of study is the coset representative 𝒱 (t) ∈ 𝒢 ∕𝒦 (𝒢 ), which must now be seen as “formal” exponentiation of the infinite number of generators in 𝔭. We can then proceed as before and choose 𝒱 (t) to be in the Borel gauge, i.e., of the form

[dim𝔥 ] ⌊ ⌋ ∑ μ ∨ ∑ ∑mα [s] [s] 𝒱(t) = Exp β (t)αμ Exp ⌈ ξα (t)E α ⌉ ∈ 𝒢 ∕𝒦 (𝒢 ). (9.75 ) μ=1 α∈Δ+ s=1
Here, the index μ does not correspond to “spacetime” but instead is an index in the Cartan subalgebra 𝔥, or, equivalently, in “scale-factor space” (see Section 2). In the following we shall dispose of writing the sum over μ explicitly. The second exponent in Equation (9.75View Equation) contains a formal infinite sum over all positive roots Δ+. We will describe in detail in subsequent sections how it can be suitably truncated. The coset representative 𝒱 (t) corresponds to a nonlinear realisation of 𝒢 and transforms as
𝒢 : 𝒱 (t) ↦− → k(𝒱 (t),g) 𝒱(t)g, k (𝒱(t),g) ∈ 𝒦 (𝒢), g ∈ 𝒢. (9.76 )

A 𝔤-valued “one-form” can be constructed analogously to the finite-dimensional case,

−1 ∂𝒱 (t)𝒱 (t) = 𝒬 (t) + 𝒫 (t), (9.77 )
where ∂ ≡ ∂t. The first term on the right hand side represents a 𝔨-connection that is fixed under the Chevalley involution,
τ(𝒬 ) = 𝒬, (9.78 )
while 𝒫 (t) lies in the orthogonal complement 𝔭 and so is anti-invariant,
τ(𝒫) = − 𝒫 (9.79 )
(for the split form, the Cartan involution coincides with the Chevalley involution). Using the projections onto the coset 𝔭 and the compact subalgebra 𝔨, as defined in Equation (9.20View Equation), we can compute the forms of 𝒫(t) and 𝒬 (t) in the Borel gauge, and we find
1 ∑ m∑α ( ) 𝒫(t) = ∂β μ(t)α ∨μ + -- eα(β)ℜ [sα] (t) E[αs]+ E−[s]α , 2α ∈Δ+ s=1 ∑ m∑ α ( ) (9.80 ) 𝒬(t) = 1- eα(β)ℜ [sα](t) E [sα]− E [s−]α , 2 α∈Δ+ s=1
where [s] ℜ α (t) is the analogue of ℛ α(x) in the finite-dimensional case, i.e., it takes the form
ℜ [s](t) = ∂ ξ[s](t) + 1-ξ[s](t)∂ ξ[s](t)+ ⋅⋅⋅ , (9.81 ) α α 2 ◟ζ---◝◜ζ′--◞ ζ+ζ′= α
which still contains a finite number of terms for each positive root α. The value of the root ⋆ α ∈ 𝔥 acting on μ ∨ β = β (t)α μ ∈ 𝔥 is
α (β) = αμ βμ. (9.82 )
Note that here the notation αμ does not correspond to a simple root, but denotes the components of an arbitrary root vector α ∈ 𝔥 ⋆.

The action for a particle moving on the infinite-dimensional coset space 𝒢∕𝒦 (𝒢) can now be constructed using the invariant bilinear form (⋅|⋅) on 𝔤,

∫ S 𝒢∕𝒦(𝒢) = dtn (t)−1 (𝒫(t)|𝒫 (t)) , (9.83 )
where n(t) ensures invariance under reparametrizations of t. The variation of the action with respect to n (t) constrains the motion to be a null geodesic on 𝒢∕ 𝒦(𝒢 ),
(𝒫 (t)|𝒫(t)) = 0. (9.84 )

Defining, as before, a covariant derivative 𝔇 with respect to the local symmetry 𝒦 (𝒢) as

𝔇𝒫 (t) ≡ ∂𝒫(t) − [𝒬 (t),𝒫 (t)], (9.85 )
the equations of motion read simply
( ) 𝔇 n(t)−1𝒫(t) = 0. (9.86 )
The explicit form of the action in the parametrization of Equation (9.75View Equation) becomes
⌊ ⌋ ∫ 1 ∑ ∑mα S𝒢∕𝒦(𝒢) = dt n(t)−1⌈G μν ∂βμ(t)∂ βν(t) + -- e2α(β)ℜ [sα] (t)ℜ [αs](t)⌉ , (9.87 ) 2α∈Δ+ s=1
where Gμν is the flat Lorentzian metric, defined by the restriction of the bilinear form (⋅|⋅) to the Cartan subalgebra 𝔥 ⊂ 𝔤. The metric G μν is exactly the same as the metric in scale-factor space (see Section 2), and the kinetic term for the Cartan parameters βμ(t) reads explicitly
dim∑ 𝔥−1 (dim∑ 𝔥−1 ) (dim∑𝔥− 1 ) G μν ∂β μ(t)∂β ν(t) = ∂ βi(t) ∂βi(t) − ∂βi(t) ∂βj (t) + ∂φ(t)∂φ (t). (9.88 ) i=1 i=1 j=1

Although 𝔤 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 (9.86View Equation). We can define the generalized metric for any 𝔤 as

𝒯 ℳ (t) ≡ 𝒱(t) 𝒱 (t), (9.89 )
where the transpose ()𝒯 is defined as before in terms of the Chevalley involution,
()𝒯 = − τ( ). (9.90 )
Then the equations of motion 𝔇 𝒫(t) = 0 are equivalent to the conservation ∂𝔍 = 0 of the current
1- −1 𝔍 ≡ 2ℳ (t) ∂ℳ (t). (9.91 )
This can be formally solved in closed form
t𝔍𝒯 t𝔍 ℳ (t) = e ℳ (0)e , (9.92 )
and so an arbitrary group element g ∈ 𝒢 evolves according to
g(t) = k (t)g(0)et𝔍, k(t) ∈ 𝒦 (𝒢). (9.93 )

Although the explicit form of 𝒫(t) contains infinitely many terms, we have seen that each coefficient ℜ [αs](t) can, in principle, be computed exactly for each root α. This, however, is not the case for the current 𝔍. To find the form of 𝔍 one must conjugate 𝒫 (t) with the coset representative 𝒱(t) and this requires an infinite number of commutators to get the correct coefficient in front of any generator in 𝔍.

9.2.2 Consistent truncations

One method for dealing with infinite expressions like Equation (9.80View Equation) 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 𝒢 ∕𝒦 (𝒢). We take as definition of such a truncation any sub-model S ′ of S𝒢∕𝒦 (𝒢) 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.

Subgroup truncation

The first consistent truncation we shall treat is the case when the dynamics of a sigma model for some global group 𝒢 is restricted to that of an appropriately chosen subgroup ¯ 𝒢 ⊂ 𝒢. We consider here only subgroups ¯ð’¢ which are obtained by exponentiation of regular subalgebras ¯ð”¤ of 𝔤. 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 ¯ð’¢âˆ•ð’¦ (¯ð’¢), we first assume that the initial conditions || g(t)t=0 = g(0) and || ∂g(t)t=0 are such that the following two conditions are satisfied:

  1. The group element g(0) belongs to ¯ð’¢.
  2. The conserved current 𝔍 belongs to ¯ 𝔤.

When these conditions hold, then t𝔍 g(0)e belongs to ¯ 𝒢 for all t. Moreover, there always exists ¯ ¯ k(t) ∈ 𝒦 (𝒢) such that

t𝔍 ¯g(t) ≡ ¯k(t)g(0)e ∈ ¯ð’¢âˆ•ð’¦ (¯ð’¢), (9.94 )
i.e, ¯g(t) belongs to the Borel subgroup of 𝒢¯. Because the embedding is regular, ¯k(t) belongs to 𝒦 (𝒢) and we thus have that ¯g(t) also belongs to the Borel subgroup of the full group 𝒢.

Now recall that from Equation (9.93View Equation), we know that ¯g(t) = k¯(t)g(0)et𝔍 is a solution to the equations of motion for the sigma model on ¯ð’¢âˆ•ð’¦ (¯ð’¢). But since we have found that ¯g(t) preserves the Borel gauge for 𝒢∕𝒦 (𝒢), it follows that ¯ t𝔍 k(t)g(0)e is a solution to the equations of motion for the full sigma model. Thus, the dynamical evolution of the subsystem ′ S = S¯ð’¢âˆ•ð’¦(¯ð’¢) preserves the Borel gauge of 𝒢. These arguments show that initial conditions in ¯ð’¢ remain in 𝒢¯, and hence the dynamics of a sigma model on 𝒢∕𝒦 (𝒢 ) can be consistently truncated to a sigma model on ¯ð’¢ ∕𝒦 (¯ð’¢).

Finally, we recall that because the embedding ¯ð”¤ ⊂ 𝔤 is regular, the restriction of the bilinear form on 𝔤 coincides with the bilinear form on ¯ð”¤. 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.

Level truncation and height truncation

Alternative ways of consistently truncating the infinite-dimensional sigma model rest on the use of gradations of 𝔤,

𝔤 = ⋅⋅⋅ + 𝔤− 2 + 𝔤 −1 + 𝔤0 + 𝔤1 + 𝔤2 + ⋅⋅⋅ , (9.95 )
where the sum is infinite but each subspace is finite-dimensional. One also has
[𝔤 ′,𝔤 ′′] ⊆ 𝔤 ′ ′′. (9.96 ) ℓ ℓ ℓ+ℓ
Such a gradation was for instance constructed in Section 8 and was based on a so-called level decomposition of the adjoint representation of 𝔤 into representations of a finite regular subalgebra 𝔯 ⊂ 𝔤. We will now use this construction to truncate the sigma model based of 𝒢 ∕𝒦 (𝒢), by “terminating” the gradation of 𝔤 at some finite level ¯ ℓ. More specifically, the truncation will involve setting to zero all coefficients [s] ℜα (t), in the expansion of 𝒫(t), corresponding to roots α whose generators [s] E α belong to subspaces 𝔤 ℓ with ℓ > ℓ¯. Part of this section draws inspiration from the treatment in [47Jump To The Next Citation Point48Jump To The Next Citation Point124Jump To The Next Citation Point].

The level ℓ 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 𝔤 and each choice of decomposition. We will do this in detail in Section 9.3 for a specific level decomposition of the hyperbolic algebra E10. Here, we shall display the general construction in the case of the height truncation, which exists for any algebra.

Let α be a positive root, α ∈ Δ+. It has the following expansion in terms of the simple roots

∑ α = mi αi (mi ≥ 0). (9.97 ) i
Then the height of α is defined as (see Section 4)
ht(α ) = ∑ m . (9.98 ) i i
The height can thus be seen as a linear integral map ht : Δ → ℤ, and we shall sometimes use the notation ht(α ) = h α to denote the value of the map ht acting on a root α ∈ Δ.

To achieve the height truncation, we assume that the sum over all roots in the expansion of 𝒫 (t), Equation (9.80View Equation), is ordered in terms of increasing height. Then we can consistently set to zero all coefficients [s] ℜα (t) corresponding to roots with greater height than some, suitably chosen, finite height ¯h. We thus find that the finitely truncated coset element 𝒫0(t) is

| μ ∨ 1 ∑ ∑mα α(β) [s] ( [s] [s]) 𝒫0(t) ≡ 𝒫(t)|ht≤¯h = ∂ β (t)αμ + -- e ℜ α (t) E α + E− α , (9.99 ) 2 α∈Δ+ s=1 ht(α)≤¯h
which is equivalent to the statement
ℜ [s](t) = 0 ∀ γ ∈ Δ , ht(γ ) > h¯. (9.100 ) γ +

For further use, we note here some properties of the coefficients ℜ [αs](t). By examining the structure of Equation (9.81View Equation), we see that [s] ℜ α (t) takes the form of a temporal derivative acting on [s] ξα (t), followed by a sequence of terms whose individual components, for example ξ[ζs](t), are all associated with roots of lower height than α, ht(ζ) < ht(α ). It will prove useful to think of [s] ℜ α (t) as representing a kind of “generalized” derivative operator acting on the field ξ[αs]. Thus we define the operator 𝒟 by

[s] [s] [s]( 2 ) 𝒟 ξα (t) ≡ ∂ ξα (t) + ℱα ξ∂ξ,ξ ∂ξ,⋅⋅⋅ , (9.101 )
where ℱ[αs](t) is a polynomial function of the coordinates ξ(t), whose explicit structure follows from Equation (9.81View Equation). It is common in the literature to refer to the level truncation as “setting all higher level covariant derivatives to zero”, by which one simply means that all [s] 𝒟ξγ (t) corresponding to roots γ above a given finite level ¯â„“ should vanish. Following [47Jump To The Next Citation Point] we shall call the operators 𝒟 “covariant derivatives”.

It is clear from the equations of motion 𝔇 𝒫(t) = 0, that if all covariant derivatives 𝒟 ξ[sγ](t) 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 ξ[γs](t) 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 𝒫 (t), and therefore it is crucial to define the truncation using the derivative operator [s] 𝒟 ξγ (t).

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