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5.3 Understanding the emerging Kac–Moody algebra

We shall now relate the Kac–Moody algebra whose fundamental Weyl chamber emerges in the BKL-limit to the U-duality group that appears upon toroidal dimensional reduction to three spacetime dimensions. We shall do this first in the case when 𝔲3 is a split real form. By this we mean that the real algebra 𝔲3 possesses the same Chevalley–Serre presentation as β„‚ 𝔲 3, but with coefficients restricted to be real numbers. This restriction is mathematically consistent because the coefficients appearing in the Chevalley–Serre presentation are all reals (in fact, integers).

The fact that the billiard structure is preserved under reduction turns out to be very useful for understanding the emergence of “overextended” algebras in the BKL-limit. By computing the billiard in three spacetime dimensions instead of in maximal dimension, the relation to U-duality groups becomes particularly transparent and the computation of the billiard follows a similar pattern for all cases. We will see that if 𝔲3 is the algebra representing the internal symmetry of the non-gravitational degrees of freedom in three dimensions, then the billiard is controlled by the Weyl group of the overextended algebra ++ 𝔲3. The analysis is somewhat more involved when 𝔲3 is non-split, and we postpone a discussion of this until Section 7.

5.3.1 Invariance under toroidal dimensional reduction

It was shown in [41Jump To The Next Citation Point] that the structure of the billiard for any given theory is completely unaffected by dimensional reduction on a torus. In this section we illustrate this by an explicit example rather than in full generality. We consider the case of reduction of eleven-dimensional supergravity on a circle.

The compactification ansatz in the conventions of [35Jump To The Next Citation Point41Jump To The Next Citation Point] is

( ) e− 2(3 4√2ˆΟ•) e−2(34√2ˆΟ•) ˆπ’œ gMN = −2(-4√-ˆΟ•) −2( −√1Ο•ˆ) −2(ν4√-ˆΟ•) , (5.33 ) e 3 2 ˆπ’œμ e 62 ˆgμν + e 3 2 π’œˆμπ’œˆν
where μ,ν = 0, 2,⋅⋅⋅ ,10, i.e., the compactification is performed along the first spatial direction16. We will refer to the new lower-dimensional fields Ο•ˆ and ˆ π’œ μ as the dilaton and the Kaluza–Klein (KK) vector, respectively. Quite generally, hatted fields are low-dimensional fields. The ten-dimensional Lagrangian becomes
SUGRA11 1- −2(2√32-ˆΟ•) ˆ(2) ˆ(2) β„’(10) = R(10) ⋆ 1 − ⋆d ˆΟ• ∧ dˆΟ• − 2 e ⋆ℱ ∧ β„± 1 −2(-1√-ˆΟ•) (4) (4) 1 − 2(−√1Ο•ˆ) (3) (3) − -e 2 2 ⋆ ˆF ∧ Fˆ − -e 2 ⋆Fˆ ∧ ˆF , (5.34 ) 2 2
where (2) (1) ˆβ„± = d ˆπ’œ and (4) (3) ˆF , ˆF are the field strengths in ten dimensions originating from the eleven-dimensional 3-form field strength F (4) = dA (3).

Examining the new form of the metric reveals that the role of the scale factor β1, associated to the compactified dimension, is now instead played by the ten-dimensional dilaton, Ο•ˆ. Explicitly we have

1 -4--- β = 3√2-ˆΟ•. (5.35 )
The nine remaining eleven-dimensional scale factors, β2, ⋅⋅⋅ ,β10, may in turn be written in terms of the new scale factors, ˆβa, associated to the ten-dimensional metric, ˆgμν, and the dilaton in the following way:
a ˆa -1--- β = β − 6√2-ˆΟ• (a = 2,⋅⋅⋅ ,10 ). (5.36 )
We are interested in finding the dominant wall forms in terms of the new scale factors ˆβ2,⋅⋅⋅ , ˆβ10 and Ο•ˆ. It is clear that we will have eight ten-dimensional symmetry walls,
ˆsmˆ(ˆβ) = ˆβmˆ+1 − ˆβmˆ (mˆ = 2,⋅⋅⋅ ,9), (5.37 )
which correspond to the eight simple roots of 𝔰𝔩(9,ℝ). Using Equation (5.35View Equation) and Equation (5.36View Equation) one may also check that the symmetry wall β2 − β1, that was associated with the compactified direction, gives rise to an electric wall of the Kaluza–Klein vector,
ˆπ’œ 2 3 ˆe2 (ˆβ ) = βˆ −-√--ˆΟ•. (5.38 ) 2 2
The metric in the dual space gets modified in a natural way,
( ) ( ) ∑10 1 ∑10 ∑10 (ˆαk|ˆαl) = ˆαkiˆαli − -- ˆαki ˆαlj + ˆαkˆΟ•αˆlΟ•ˆ, (5.39 ) i=2 8 i=2 j=2
i.e., the dilaton contributes with a flat spatial direction. Using this metric it is clear that π’œˆ ˆe2 has non-vanishing scalar product only with the second symmetry wall ˆs2 = ˆβ3 − βˆ2, (ˆeπ’œ2ˆ |ˆs2) = − 1, and it follows that the electric wall of the Kaluza–Klein vector plays the role of the first simple root of 𝔰𝔩(10,ℝ ), ˆπ’œ ˆα1 ≡ ˆe2. The final wall form that completes the set will correspond to the exceptional node labeled “10” in Figure 25View Image and is now given by one of the electric walls of the NS-NS 2-form ˆA(2), namely
Aˆ(2) ˆ ˆ2 ˆ3 -1-- ˆα10 ≡ ˆe23 (β ) = β + β + √2-ˆΟ•. (5.40 )
It is then easy to verify that this wall form has non-vanishing scalar product only with the third simple root ˆα3 = ˆs3, (ˆeˆA2(32)|ˆs3) = − 1, as desired.

We have thus shown that the E10 structure is sufficiently rigid to withstand compactification on a circle with the new simple roots explicitly given by

π’œˆ ˆA(2) {αˆ1, ˆα2,⋅⋅⋅ ,αˆ9, ˆα10} = {ˆe2 ,ˆs2,⋅⋅⋅ ,ˆs9,ˆe23 }. (5.41 )
This result is in fact true also for the general case of compactification on tori, Tn. When reaching the limiting case of three dimensions, all the non-gravity wall forms correspond to the electric and magnetic walls of the axionic scalars. We will discuss this case in detail below.

For non-toroidal reductions the above analysis is drastically modified [166165]. The topology of the internal manifold affects the dominant wall system, and hence the algebraic structure in the lower-dimensional theory is modified. In many cases, the billiard of the effective compactified theory is described by a (non-hyperbolic) regular Lorentzian subalgebra of the original hyperbolic Kac–Moody algebra [98].

The walls are also invariant under dualization of a p-form into a (D − p − 2)-form; this simply exchanges magnetic and electric walls.

5.3.2 Iwasawa decomposition for split real forms

We will now exploit the invariance of the billiard under dimensional reduction, by considering theories that – when compactified on a torus to three dimensions – exhibit “hidden” internal global symmetries 𝒰3. By this we mean that the three-dimensional reduced theory is described, after dualization of all vectors to scalars, by the sum of the Einstein–Hilbert Lagrangian coupled to the Lagrangian for the nonlinear sigma model 𝒰3βˆ• 𝒦(𝒰3 ). Here, 𝒦 (𝒰3) is the maximal compact subgroup defining the “local symmetries”. In order to understand the connection between the U-duality group 𝒰3 and the Kac–Moody algebras appearing in the BKL-limit, we must first discuss some important features of the Lie algebra 𝔲 3.

Let 𝔲3 be a split real form, meaning that it can be defined in terms of the Chevalley–Serre presentation of the complexified Lie algebra β„‚ 𝔲3 by simply restricting all linear combinations of generators to the real numbers ℝ. Let π”₯3 be the Cartan subalgebra of 𝔲3 appearing in the Chevalley–Serre presentation, spanned by the generators α ∨1,⋅⋅⋅ ,α ∨n. It is maximally noncompact (see Section 6). An Iwasawa decomposition of 𝔲 3 is a direct sum of vector spaces of the following form,

𝔲3 = 𝔨3 ⊕ π”₯3 ⊕ 𝔫3, (5.42 )
where 𝔨 3 is the “maximal compact subalgebra” of 𝔲 3, and 𝔫 3 is the nilpotent subalgebra spanned by the positive root generators E α, ∀α ∈ Δ+.

The corresponding Iwasawa decomposition at the group level enables one to write uniquely any group element as a product of an element of the maximally compact subgroup times an element in the subgroup whose Lie algebra is π”₯3 times an element in the subgroup whose Lie algebra is 𝔫3. An arbitrary element 𝒱(x) of the coset 𝒰3 βˆ•π’¦ (𝒰3) is defined as the set of equivalence classes of elements of the group modulo elements in the maximally compact subgroup. Using the Iwasawa decomposition, one can go to the “Borel gauge”, where the elements in the coset are obtained by exponentiating only generators belonging to the Borel subalgebra,

π”Ÿ3 = π”₯3 ⊕ 𝔫3 ⊂ 𝔲3. (5.43 )
In that gauge we have
𝒱 (x) = Exp [φ(x) ⋅ π”₯3] Exp [χ (x) ⋅ 𝔫3 ], (5.44 )
where φ and χ are (sets of) coordinates on the coset space 𝒰3 βˆ•π’¦ (𝒰3). A Lagrangian based on this coset will then take the generic form (see Section 9)
dim π”₯3 β„’ = ∑ ∂ φ(i)(x)∂ φ (i)(x) + ∑ e2α(φ)[∂ χ(α)(x ) + ⋅⋅⋅] [∂ χ(α)(x ) + ⋅⋅⋅] , (5.45 ) 𝒰3βˆ•π’¦(𝒰3) x x x x i=1 α∈Δ+
where x denotes coordinates in spacetime and the “ellipses” denote correction terms that are of no relevance for our present purposes. We refer to the fields {φ} collectively as dilatons and the fields {χ } as axions. There is one axion field (α) χ for each positive root α ∈ Δ+ and one dilaton field (i) φ for each Cartan generator ∨ αi ∈ π”₯3.

The Lagrangian (5.45View Equation) coupled to the pure three-dimensional Einstein–Hilbert term is the key to understanding the appearance of the Lorentzian Coxeter group 𝔲++ 3 in the BKL-limit.

5.3.3 Starting at the bottom – Overextensions of finite-dimensional Lie algebras

To make the point explicit, we will again limit our analysis to the example of eleven-dimensional supergravity. Our starting point is then the Lagrangian for this theory compactified on an 8-torus, T 8, to D = 2 + 1 spacetime dimensions (after all form fields have been dualized into scalars),

8 120 β„’SUGRA11 = R ⋆ 1 − ∑ ⋆d ˆΟ•(i) ∧ dˆΟ•(i) − 1-∑ e2αq(ˆΟ•) ⋆ (dˆχ(q) + ⋅⋅⋅) ∧ (d ˆχ(q) + ⋅⋅⋅). (5.46 ) (3) (3) 2 i=1 q=1
The second two terms in this Lagrangian correspond to the coset model β„°8(8)βˆ•(Spin (16 )βˆ•β„€2), where β„°8(8) denotes the group obtained by exponentiation of the split form E8(8) of the complex Lie algebra E8 and Spin(16)βˆ•β„€2 is the maximal compact subgroup of β„°8(8) [33Jump To The Next Citation Point134Jump To The Next Citation Point35Jump To The Next Citation Point]. The 8 dilatons ˆΟ• and the 120 axions χ(q) are coordinates on the coset space17. Furthermore, the αq(Ο•ˆ) are linear forms on the elements of the Cartan subalgebra h = ˆΟ•iα∨i and they correspond to the positive roots of E8(8)18. As before, we do not write explicitly the corrections to the curvatures dˆχ that appear in the compactification process. The entire set of positive roots can be obtained by taking linear combinations of the seven simple roots of 𝔰𝔩(8,ℝ) (we omit the “hatted” notation on the roots since there is no longer any risk of confusion),
( √-- ) ( √ -- ) α1(Ο•ˆ) = √1-- -7-ˆΟ•2 − 3-ˆΟ•1 , α2(Ο•ˆ) = √1-- 2√--3ˆΟ•3 − √4--ˆΟ•2 , 2 2 2 2 7 7 ( √-- √ -- ) ( √ -- √ -- ) 1 5 7 1 2 2 2 3 α3(Ο•ˆ) = √--- √--ˆΟ•4 − √--Ο•ˆ3 , α4(Ο•ˆ) = √--- √---ˆΟ•5 − -√---ˆΟ•4 , 2 3 3 2 5 5 (5.47 ) ( √-- √ -- ) ( √ -- ) α (Ο•ˆ) = √1-- √3-ˆΟ• − √-5Ο•ˆ , α (Ο•ˆ) = √1-- √2-ˆΟ• − 2√-2-ˆΟ• , 5 2 2 6 2 5 6 2 3 6 3 5 ( ) √1-- √ -- α7(Ο•ˆ) = 2 ˆΟ•8 − 3ˆΟ•7 ,
and the exceptional root
( √ -- ) -1-- -3-- 2--3- α10(ˆΟ•) = √2-- ˆΟ•1 + √7-Ο•ˆ2 + √7--ˆΟ•3 . (5.48 )
These correspond exactly to the root vectors βƒ—bi,i+1 and βƒ—a123 as they appear in the analysis of [35Jump To The Next Citation Point], except for the additional factor of √1- 2 needed to compensate for the fact that the aforementioned reference has an additional factor of 2 in the Killing form. Hence, using the Euclidean metric δij (i,j = 1,⋅⋅⋅ ,8) one may check that the roots defined above indeed reproduce the Cartan matrix of E8.

Next, we want to determine the billiard structure for this Lagrangian. As was briefly mentioned before, in the reduction from eleven to three dimensions all the non-gravity walls associated to the eleven-dimensional 3-form (3) A have been transformed, in the same spirit as for the example given above, into electric and magnetic walls of the axionic scalars ˆχ. Since the terms involving the electric fields ∂tˆχ (i) possess no spatial indices, the corresponding wall forms do not contain any of the remaining scale factors βˆ9, ˆβ10, and are simply linear forms on the dilatons only. In fact the dominant electric wall forms are just the simple roots of E 8,

ˆeˆχa(ˆΟ•) = αa (ˆΟ•) (a = 1, ⋅⋅⋅ ,7), χˆ (5.49 ) ˆe10(ˆΟ•) = α10(Ο•ˆ).
The magnetic wall forms naturally come with one factor of ˆβ since the magnetic field strength ∂iχˆ carries one spatial index. The dominant magnetic wall form is then given by
ˆχ 9 ˆm 9(ˆβ,Ο•ˆ) = ˆβ − θ(ˆΟ•), (5.50 )
where θ(ˆΟ•) denotes the highest root of E8 which takes the following form in terms of the simple roots,
√ -- θ = 2α1 + 4α2 + 6α3 + 5α4 + 4α5 + 3α6 + 2α7 + 3α10 = 2 ˆΟ•8. (5.51 )
Since we are in three dimensions there is no curvature wall and hence the only wall associated to the Einstein–Hilbert term is the symmetry wall
ˆs9 = ˆβ10 − ˆβ9, (5.52 )
coming from the three-dimensional metric ˆg μν (μ,ν = 0, 9,10). We have thus found all the dominant wall forms in terms of the lower-dimensional variables.

The structure of the corresponding Lorentzian Kac–Moody algebra is now easy to establish in view of our discussion of overextensions in Section 4.9. The relevant walls listed above are the simple roots of the (untwisted) overextension E++ 8. Indeed, the relevant electric roots are the simple roots of E8, the magnetic root of Equation (5.50View Equation) provides the affine extension, while the gravitational root of Equation (5.52View Equation) is the overextended root.

What we have found here in the case of eleven-dimensional supergravity also holds for the other theories with U-duality algebra 𝔲3 in 3 dimensions when 𝔲3 is a split real form. The Coxeter group and the corresponding Kac–Moody algebra are given by the untwisted overextension ++ 𝔲3. This overextension emerges as follows [41Jump To The Next Citation Point]:

Thus we see that the appearance of overextended algebras in the BKL-analysis of supergravity theories is a generic phenomenon closely linked to hidden symmetries.

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