Spacelike Singularities and Hidden Symmetries of Gravity

Cosmological billiards and hidden symmetries of gravity

This important work has opened the way to many further fruitful investigations in theoretical cosmology. Recently, a new — and somewhat unanticipated — development has occurred in the field, with the realisation that for the gravitational theories that have been studied most (pure gravity and supergravities in various spacetime dimensions) the dynamics of the gravitational field exhibits strong connections with Lorentzian Kac-Moody algebras, as discovered by Damour and Henneaux [45], suggesting that these might be “hidden” symmetries of the theory.

These connections appear for the cases at hand because in the BKL-limit, not only can the equations of motion be reformulated as dynamical equations for billiard motion in a region of hyperbolic space, but also this region possesses unique features: It is the fundamental Weyl chamber of some Kac-Moody algebra. The dynamical motion in the BKL-limit is then a succession of reflections in the walls bounding the fundamental Weyl chamber and defines “words” in the Weyl group of the Kac-Moody algebra.

Which billiard region of hyperbolic space actually emerges — and hence which Kac-Moody algebra is relevant — depends on the theory at hand, i.e., on the spacetime dimension, the menu of matter fields, and the dilaton couplings. The most celebrated case is eleven-dimensional super-gravity, for which the billiard region is the fundamental region of , one of the four hyperbolic Kac-Moody algebras of highest rank 10. The root lattice of E₁₀ is furthermore one of the few even, Lorentzian, self-dual lattices — actually the only one in 10 dimensions — a fact that could play a key role in our ultimate understanding of M-theory.

Other gravitational theories lead to other billiards characterized by different algebras. These algebras are closely connected to the hidden duality groups that emerge upon dimensional reduction to three dimensions [41, 95].

That one can associate a regular billiard and an infinite discrete reflection group (Coxeter group) to spacelike singularities of a given gravitational theory in the BKL-limit is a robust fact (even though the BKL-limit itself is yet to be fully understood), which, in our opinion, will survive future developments. The mathematics necessary to appreciate the billiard structure and its connection to the duality groups in three dimensions involve hyperbolic Coxeter groups, Kac-Moody algebras and real forms of Lie algebras.

The appearance of infinite Coxeter groups related to Lorentzian Kac-Moody algebras has triggered fascinating conjectures on the existence of huge symmetry structures underlying gravity [47]. Similar conjectures based on different considerations had been made earlier in the pioneering works [113, 167]. The status of these conjectures, however, is still somewhat unclear since, in particular, it is not known how exactly the symmetry would act.

The main purpose of this article is to explain the emergence of infinite discrete reflection groups in gravity in a self-contained manner, including giving the detailed mathematical background needed to follow the discussion. We shall avoid, however, duplicating already existing reviews on BKL billiards.

Contrary to the main core of the review, devoted to an explanation of the billiard Weyl groups, which is indeed rather complete, we shall also discuss some paths that have been taken towards revealing the conjectured infinite-dimensional Kac-Moody symmetry. Our goal here will only be to give a flavor of some of the work that has been done along these lines, emphasizing its dynamical relevance. Because we feel that it would be premature to fully review this second subject, which is still in its infancy, we shall neither try to be exhaustive nor give detailed treatments.

Outline of the paper

Our article is organized as follows. In Section 2, we outline the key features of the BKL phenomenon, valid in any number of dimensions, and describe the billiard formulation which clearly displays these features. Since the derivation of these aspects have been already reviewed in [48], we give here only the results without proof. Next, for completeness, we briefly discuss the status of the BKL conjecture — assumed to be valid throughout our review.

In Sections 3 and 4, we develop the mathematical tools necessary for apprehending those aspects of Coxeter groups and Kac-Moody algebras that are needed in the BKL analysis. First, in Section 3, we provide a primer on Coxeter groups (which are the mathematical structures that make direct contact with the BKL billiards). We then move on to Kac-Moody algebras in Section 4, and we discuss, in particular, some prominent features of hyperbolic Kac-Moody algebras.

In Section 5 we then make use of these mathematical concepts to relate the BKL billiards to Lorentzian Kac-Moody algebras. We show that there is a simple connection between the relevant Kac-Moody algebra and the U-duality algebras that appear upon toroidal dimensional reduction to three dimensions, when these U-duality algebras are split real forms. The Kac-Moody algebra is then just the standard overextension of the U-duality algebra in question.

To understand the non-split case requires an understanding of real forms of finite-dimensional semi-simple Lie algebras. This mathematical material is developed in Section 6. Here, again, we have tried to be both rather complete and explicit through the use of many examples. We have followed a pedagogical approach privileging illustrative examples over complete proofs (these can be found in any case in the references given in the text). We explain the complementary Vogan and Tits-Satake approaches, where maximal compact and maximal noncompact Cartan subalgebras play the central roles, respectively. The concepts of restricted root systems and of the Iwasawa decomposition, central for understanding the emergence of the billiard, have been given particular attention. For completeness we provide tables listing all real forms of finite Lie algebras, both in terms of Vogan diagrams and in terms of Tits-Satake diagrams. In Section 7 we use these mathematical developments to relate the Kac-Moody billiards in the non-split case to the U-duality algebras appearing in three dimensions.

Up to (and including) Section 7, the developments present well-established results. With Section 8 we initiate a journey into more speculative territory. The presence of hyperbolic Weyl groups suggests that the corresponding infinite-dimensional Kac-Moody algebras might, in fact, be true underlying symmetries of the theory. How this conjectured symmetry should actually act on the physical fields is still unclear, however. We explore one approach in which the symmetry is realized nonlinearly on a (1 + 0)-dimensional sigma model based on , which is the case relevant to eleven-dimensional supergravity. To this end, in Section 8 we introduce the concept of a level decomposition of some of the relevant Kac-Moody algebras in terms of finite regular subalgebras. This is necessary for studying the sigma model approach to the conjectured infinite-dimensional symmetries, a task undertaken in Section 9. We show that the sigma model for spectacularly reproduces important features of eleven-dimensional supergravity. However, we also point out important limitations of the approach, which probably does not constitute the final word on the subject.

In Section 10 we show that the interpretation of eleven-dimensional supergravity in terms of a manifestly -invariant sigma model sheds interesting and useful light on certain cosmological solutions of the theory. These solutions were derived previously but without the Kac-Moody algebraic understanding. The sigma model approach also suggests a new method of uncovering novel solutions. Finally, in Section 11 we present a concluding discussion and some suggestions for future research.

The general action

We are interested in general theories describing Einstein gravity coupled to bosonic “matter” fields. The only known bosonic matter fields that consistently couple to gravity are p-form fields, so our collection of fields will contain, besides the metric, p-form fields, including scalar fields (p = 0). The action reads

2.1

where we have chosen units such that 16πG = 1. The spacetime dimension is left unspecified. The Einstein metric g_μν has Lorentzian signature (−, +, …, +) and is used to lower or raise the indices. Its determinant is^(D)g, where the index D is used to avoid any confusion with the determinant of the spatial metric introduced below. We assume that among the scalars, there is only one dilaton1, denoted ϕ, whose kinetic term is normalized with weight 1 with respect to the Ricci scalar. The real parameter λ^(p) measures the strength of the coupling to the dilaton. The other scalar fields, sometimes called axions, are denoted A⁽⁰⁾ and have dilaton coupling λ⁽⁰⁾ ≠ 0. The integer p ≥ 0 labels the various p-forms A^(p) present in the theory, with field strengths F^(p) = dA^(p),

2.2

We assume the form degree p to be strictly smaller than D − 1, since a (D − 1)-form in D dimensions carries no local degree of freedom. Furthermore, if p = D − 2 the p-form is dual to a scalar and we impose also λ^(D−2) ≠ 0.

The field strength, Equation (2.2), could be modified by additional coupling terms of Yang-Mills or Chapline-Manton type [20, 29] (e.g., F_C = dC⁽²⁾ − C⁽⁰⁾dB⁽²⁾ for two 2-forms C⁽²⁾ and B⁽²⁾ and a 0-form C⁽⁰⁾, as it occurs in ten-dimensional type IIB supergravity), but we include these additional contributions to the action in “more”. Similarly, “more” might contain Chern-Simons terms, as in the action for eleven-dimensional supergravity [38].

We shall at this stage consider arbitrary dilaton couplings and menus of p-forms. The billiard derivation given below remains valid no matter what these are; all theories described by the general action Equation (2.1) lead to the billiard picture. However, it is only for particular p-form menus, spacetime dimensions and dilaton couplings that the billiard region is regular and associated with a Kac-Moody algebra. This will be discussed in Section 5. Note that the action, Equation (2.1), contains as particular cases the bosonic sectors of all known supergravity theories.

Hamiltonian description

We assume that there is a spacelike singularity at a finite distance in proper time. We adopt a spacetime slicing adapted to the singularity, which “occurs” on a slice of constant time. We build the slicing from the singularity by taking pseudo-Gaussian coordinates defined by and Nⁱ = 0, where N is the lapse and Nⁱ is the shift [48]. Here, g = det(g_ij). Thus, in some spacetime patch, the metric reads2

2.3

where the local volume g collapses at each spatial point as x⁰ → +∞, in such a way that the proper time remains finite (and tends conventionally to 0+). Here we have assumed the singularity to occur in the past, as in the original BKL analysis, but a similar discussion holds for future spacelike singularities.

Action in canonical form

In the Hamiltonian description of the dynamics, the canonical variables are the spatial metric components g_ij, the dilaton ϕ, the spatial p-form components and their respective conjugate momenta π^ij, πϕ and. The Hamiltonian action in the pseudo-Gaussian gauge is given by

2.4

where the Hamiltonian is

2.5

In addition to imposing the coordinate conditions and Nⁱ = 0, we have also set the temporal components of the p-forms equal to zero (“temporal gauge”).

The dynamical equations of motion are obtained by varying the above action w.r.t. the canonical variables. Moreover, there are constraints on the dynamical variables, which are

2.6

Here we have set

2.7

where the subscript |m_p denotes the spatially covariant derivative. These constraints are preserved by the dynamical evolution and need to be imposed only at one “initial” time, say at x⁰ = 0.

Iwasawa change of variables

In order to study the dynamical behavior of the fields as x⁰ → ∞ (g → 0) and to exhibit the billiard picture, it is particularly convenient to perform the Iwasawa decomposition of the spatial metric. Let g(x⁰,xⁱ) be the matrix with entries g_ij(x⁰,xⁱ). We set

2.8

where is an upper triangular matrix with 1’s on the diagonal () and A is a diagonal matrix with positive elements, which we parametrize as

2.9

Both and depend on the spacetime coordinates. The spatial metric dσ² becomes

2.10

with

2.11

The variables βⁱ of the Iwasawa decomposition give the (logarithmic) scale factors in the new, orthogonal, basis. The variables characterize the change of basis that diagonalizes the metric and hence they parametrize the off-diagonal components of the original g_ij.

We extend the transformation Equation (2.8) in configuration space to a canonical transformation in phase space through the formula

2.12

Since the scale factors and the off-diagonal variables play very distinct roles in the asymptotic behavior, we split off the Hamiltonian as a sum of a kinetic term for the scale factors (including the dilaton),

2.13

plus the rest, denoted by V, which will act as a potential for the scale factors. The Hamiltonian then becomes

2.14

The kinetic term K is quadratic in the momenta conjugate to the scale factors and defines the inverse of a metric in the space of the scale factors. Explicitly, this metric reads

2.15

Since the metric coefficients do not depend on the scale factors, that metric in the space of scale factors is flat, and, moreover, it is of Lorentzian signature. A conformal transformation where all scale factors are scaled by the same number (βⁱ → βⁱ + ϵ) defines a timelike direction. It will be convenient in the following to collectively denote all the scale factors (the βⁱ’s and the dilaton ϕ) as β^μ, i.e., (β^μ) = (βⁱϕ).

The analysis is further simplified if we take for new p-form variables the components of the p-forms in the Iwasawa basis of the ω_k,

2.16

and again extend this configuration space transformation to a point canonical transformation in phase space,

2.17

using the formula ∑ p dq = ∑p′ dq′, which reads

2.18

Note that the scale factor variables are unaffected, while the momenta P_ij conjugate to get redefined by terms involving , and since the components of the p-forms in the Iwasawa basis involve the ’s. On the other hand, the new p-form momenta, i.e., the components of the electric field in the basis {ω^k} are simply given by

2.19

In terms of the new variables, the electromagnetic potentials become

2.20

Here, are the electric linear forms

2.21

(the indices i_j are all distinct because is completely antisymmetric) while are the components of the magnetic field in the basis {ω^k},

2.22

and are the magnetic linear forms

2.23

One sometimes rewrites as , where {i_p+2, i_p+3, …, i_d} is the set complementary to {i₁, i₂, … i_p+1}, e.g.,

2.24

The exterior derivative of in the non-holonomic frame {ω^k} involves of course the structure coefficients in that frame, i.e.,

2.25

where

2.26

is here the frame derivative. Similarly, the potential V_ϕ reads

2.27

where is

2.28

and

2.29

Decoupling of spatial points close to a spacelike singularity

So far we have only redefined the variables without making any approximation. We now start the discussion of the BKL-limit, which investigates the leading behavior of the fields as x⁰ → ∞ (g → 0). Although the more recent “derivations” of the BKL-limit treat both elements at once [43, 44, 45, 48], it appears useful — especially for rigorous justifications — to separate two aspects of the BKL conjecture3.

The first aspect is that the spatial points decouple in the limit x⁰ → ∞, in the sense that one can replace the Hamiltonian by an effective “ultralocal” Hamiltonian H^UL involving no spatial gradients and hence leading at each point to a set of dynamical equations that are ordinary differential equations with respect to time. The ultralocal effective Hamiltonian has a form similar to that of the Hamiltonian governing certain spatially homogeneous cosmological models, as we shall explain in this section.

The second aspect of the BKL-limit is to take the sharp wall limit of the ultralocal Hamiltonian. This leads directly to the billiard description, as will be discussed in Section 2.4.

Spatially homogeneous models

In spatially homogeneous models, the fields depend only on time in invariant frames, e.g., for the metric

2.30

where the invariant forms fulfill

Here, the are the structure constants of the spatial homogeneity group. Similarly, for a 1-form and a 2-form,

2.31

The Hamiltonian constraint yielding the field equations in the spatially homogeneous context4 is obtained by substituting the form of the fields in the general Hamiltonian constraint and contains, of course, no explicit spatial gradients since the fields are homogeneous. Note, however, that the structure constants contain implicit spatial gradients. The Hamiltonian can now be decomposed as before and reads

2.32

where K, V_S and which do not involve spatial gradients, are unchanged and where V_ϕ disappears since . The potential V_G is given by [61]

2.33

where the linear forms α_ijk(β) (with i, j, k distinct) read

2.34

and where “more” stands for the terms in the first sum that arise upon taking i = j or i = k. The structure constants in the Iwasawa frame (with respect to the coframe in Equation (2.30)) are related to the structure constants through

2.35

and depend therefore on the dynamical variables. Similarly, the potential becomes

2.36

where the field strengths reduce to the “AC” terms in dA and depend on the potentials and the off-diagonal Iwasawa variables.

The ultralocal Hamiltonian

Let us now come back to the general, inhomogeneous case and express the dynamics in the frame {dx⁰, ψⁱ} where the ψⁱ’s form a “generic” non-holonomic frame in space,

2.37

Here the ’s are in general space-dependent. In the non-holonomic frame, the exact Hamiltonian takes the form

2.38

where the ultralocal part is given by Equations (2.32) and (2.33) with the relevant ’s, and where involves the spatial gradients of and .

The first part of the BKL conjecture states that one can drop asymptotically; namely, the dynamics of a generic solution of the Einstein-p-form-dilaton equations (not necessarily spatially homogeneous) is asymptotically determined, as one goes to the spatial singularity, by the ultralocal Hamiltonian

2.39

provided that the phase space constants are such that all exponentials in the above potentials do appear. In other words, the f’s must be chosen such that none of the coefficients of the exponentials, which involve f and the fields, identically vanishes — as would be the case, for example, if since then the potentials V_G and are equal to zero. This is always possible because the , even though independent of the dynamical variables, may in fact depend on x and so are not required to fulfill relations “f f = 0” analogous to the Bianchi identity since one has instead “∂f + f f = 0”.

Comments

1. As we shall see, the conditions on the f’s (that all exponentials in the potential should be present) can be considerably weakened. It is necessary that only the relevant exponentials (in the sense defined in Section 2.4) be present. Thus, one can correctly capture the asymptotic BKL behavior of a generic solution with fewer exponentials. In the case of eleven-dimensional supergravity the spatial curvature is asymptotically negligible with respect to the electromagnetic terms and one can in fact take a holonomic frame for which (and hence also .

2. The actual values of the (provided they fulfill the criterion given above or rather its weaker form just mentioned) turn out to be irrelevant in the BKL-limit because they can be absorbed through redefinitions. This is for instance why the Bianchi VIII and IX models, even though they correspond to different groups, can both be used to describe the BKL behavior in four spacetime dimensions.

Dynamics as a billiard in hyperbolic space

The second step in the BKL-limit is to take the sharp wall limit of the potentials.5 This leads to the billiard picture. It is crucial here that the coefficients in front of the dominant walls are all positive. Again, just as for the first step, this limit has not been fully justified. Only heuristic, albeit convincing, arguments have been put forward.

The idea is that as one goes to the singularity, the exponential potentials get sharper and sharper and can be replaced in the limit by the corresponding Θ∞-function, denoted for short Θ and defined by Θ(x) = 0 for x < 0 and Θ(x) = +∞ for x > 0. Taking into account the facts that aΘ(x) = Θ(x) for all a > 0, as well as that some walls can be neglected, one finds that the Hamiltonian becomes in the sharp wall limit

2.40

with

2.41

where s_ji(β) = β^j − βⁱ. See [48] for more information.

The description of the motion of the scale factors (at each spatial point) is easy to give in that limit. Because the potential walls are infinite (and positive), the motion is constrained to the region where the arguments of all Θ-functions are negative, i.e., to

2.42

In that region, the motion is governed by the kinetic term K, i.e., is a geodesic for the metric in the space of the scale factors. Since that metric is flat, this is a straight line. In addition, the constraint , which reduces to K=0 away from the potential walls, forces the straight line to be null. We shall assume that the time orientation in the space of the scale factors is such that the straight line is future-oriented (g → 0 in the future).

It is easy to check that all the walls appearing in Equation (2.41), collectively denoted , are timelike hyperplanes. This is because the squared norms of all the F_a’s are positive,

2.43

Explicitly, one finds

2.44

Because the potential walls are timelike, they have a non-empty intersection with the forward light cone in the space of the scale factors. When the null straight line representing the evolution of the scale factors hits one of the walls, it gets reflected according to the rule [43]

2.45

where v is the velocity vector (tangent to the straight line). This reflection preserves the time orientation since the hyperplanes are timelike and hence belong to the orthochronous Lorentz group O^↑ (k, 1) where k = d − 1 or d according to whether there is no or one dilaton. The conditions s_ji = 0 define the “symmetry” or “centrifugal” walls, the conditions β_ijk = 0 define the “curvature” or “gravitational” walls, the conditions define the “electric” walls, while the conditions define the “magnetic” walls.

The motion is thus a succession of future-oriented null straight line segments interrupted by reflections against the walls, where the motion undergoes a reflection belonging to O^↑(k, 1). Whether the collisions eventually stop or continue forever is better visualized by projecting the motion radially on the positive sheet of the unit hyperboloid, as was done first in the pioneering work of Chitre and Misner [31, 138] for pure gravity in four spacetime dimensions. We recall that the positive sheet of the unit hyperboloid ∑ (βⁱ)² − βⁱ)² + ϕ² = −1, ∑βⁱ > 0, provides a model of hyperbolic space (see, e.g., [146]).

The intersection of a timelike hyperplane with the unit hyperboloid defines a hyperplane in hyperbolic space. The region in hyperbolic space on the positive side of all hyperplanes is the allowed dynamical region and is called the “billiard table”. It is never compact in the cases relevant to gravity, but it may or may not have finite volume. The projection of the motion of the scale factors on the unit hyperboloid is the same as the motion of a billiard ball in a hyperbolic billiard: geodesic arcs in hyperbolic space within the billiard region, interrupted by collisions against the bounding walls where the motion undergoes a specular reflection.

When the volume of the billiard table is finite, the collisions with the potential walls never end (for generic initial data) and the motion is chaotic. When, on the other hand, the volume is infinite, generic initial data lead to a motion that ultimately freely runs away to infinity. This is non-chaotic. For more information, see [135, 170]. An interesting criterion for chaos (equivalent to finite volume of hyperbolic billiard region) has been given in [111] in terms of illuminations of spheres by point sources.

Rules for deriving the wall forms from the Lagrangian — Summary

We have recalled above that the generic behavior near a spacelike singularity of the system with action (2.1) can be described at each spatial point in terms of a billiard in hyperbolic space. The action for the billiard ball reads, in the gauge ,

2.46

where we recall that x⁰ → ∞ in the BKL-limit (proper time T → 0⁺), and G_μν is the metric in the space of the scale factors,

2.47

introduced in Equation (2.15) above. As stressed there, this metric is flat and of Lorentzian signature. Between two collisions, the motion is a free, geodesic motion. The collisions with the walls are controlled by the potential V(β^μ), which is a sum of sharp wall potentials. The walls are hyperplanes and can be inferred from the Lagrangian. They are as follows:

1. Gravity brings in the symmetry walls

   2.48

   with i = 1, 2, …,d − 1, and the curvature wall

   2.49

2. Each p-form brings in an electric wall

   2.50

   and a magnetic wall

   2.51

We have written here only the (potentially) relevant walls. There are other walls present in the potential, but because these are behind the relevant walls, which are infinitely steep in the BKL-limit, they are irrelevant. They are relevant, however, when trying to exhibit the symmetry in a complete treatment where the BKL-limit is the zeroth order term in a gradient expansion yet to be understood [47].

The scalar product dual to the scalar product in the space of the scale factors is

2.52

for two linear forms .

These recipes are all that we shall need for investigating the regularity properties of the billiards associated with the class of actions Equation (2.1).

More on the free motion: The Kasner solution

The free motion between two bounces is a straight line in the space of the scale factors. In terms of the original metric components, it takes the form of the Kasner solution with dilaton. Indeed, the free motion is given by

where the “velocities” q^μ are subject to

since the motion is lightlike by the Hamiltonian constraint. The proper time is then T = B exp(−Kx⁰), with K = ∑_iqⁱ and for some constant B (we assume, as before, that the singularity is at T = 0⁺). Redefining then

yields the celebrated Kasner solution

2.53

2.54

subject to the constraints

2.55

where A is a constant of integration and where the coordinates xⁱ have been suitably rescaled (if necessary).

Chaos and billiard volume

With our rules for writing down the billiard region, one can determine in which case the volume of the billiard is finite and in which case it is infinite. The finite-volume, chaotic case is also called “mixmaster case”, a terminology introduced in four dimensions in [137].

The following results have been obtained:

• Pure gravity in D ≤ 10 dimensions is chaotic, but ceases to be so for D ≥ 11 [63, 62].

• The introduction of a dilaton removes chaos [15, 3]. The gravitational four-derivative action in four dimensions, based on R², is dynamically equivalent to Einstein gravity coupled to a dilaton [160]. Hence, chaos is removed also for this case.

• p-form gauge fields (0 < p < d − 1) without scalar fields lead to a finite-volume billiard [44].

• When both p-forms and dilatons are included, the situation is more subtle as there is a competition between two opposing effects. One can show that if the dilaton couplings are in a “subcritical” open region that contains the origin — i.e., “not too big” — the billiard volume is infinite and the system is non chaotic. If the dilaton couplings are outside of that region, the billiard volume is finite and the system is chaotic [49].

A note on the constraints

We have focused in the above presentation on the dynamical equations of motion. The constraints were only briefly mentioned, with no discussion, except for the Hamiltonian constraint. This is legitimate because the constraints are first class and hence preserved by the Hamiltonian evolution. Thus, they need only be imposed at some “initial” time. Once this is done, one does not need to worry about them any more. Furthermore the momentum constraints and Gauss’ law constraints are differential equations relating the initial data at different spatial points. This means that they do not constrain the dynamical variables at a given point but involve also their gradients — contrary to the Hamiltonian constraint which becomes ultralocal. Consequently, at any given point, one can freely choose the initial data on the undifferentiated dynamical variables and then use these data as (part of) the appropriate boundary data necessary to integrate the constraints throughout space. This is why one can assert that all the walls described above are generically present even when the constraints are satisfied.

The situation is different in homogeneous cosmologies where the symmetry relates the values of the fields at all spatial points. The momentum and Gauss’ law constraints become then algebraic equations and might remove some relevant walls. But this feature (removal of walls by the momentum and Gauss’ law constraints) is specific to some homogeneous cosmologies and does not hold in the generic case where spatial gradients are non-zero.

A final comment: How the spatial diffeomorphism constraints and Gauss’ law fit in the conjectured infinite-dimensional symmetry is a point that is still poorly understood. See, however, [52] for recent progress in this direction.

On the validity of the BKL conjecture — A status report

Providing a complete rigorous justification of the above description of the behavior of the gravitational field in the vicinity of a spacelike singularity is a formidable task that has not been pushed to completion yet. The task is formidable because the Einstein equations form a complicated nonlinear system of partial differential equations. We shall assume throughout our review that the BKL description is correct, based on the original convincing arguments put forward by BKL themselves [16] and the subsequent fruitful investigations that have shed further important light on the validity of the conjecture. The billiard description will thus be taken for granted.

For completeness, we provide in this section a short guide to the work that has been accumulated since the late 1960’s to consolidate the BKL phenomenon.

As we have indicated, there are two aspects to the BKL conjecture:

1. The first part of the conjecture states that spatial points decouple as one goes to a spacelike singularity in the sense that the evolution can be described by a collection of systems of ordinary differential equations with respect to time, one such system at each spatial point. (“A spacelike singularity is local.”)

2. The second part of the conjecture states that the system of ordinary differential equations with respect to time describing the asymptotic dynamics at any given spatial point can be asymptotically replaced by the billiard equations. If the matter content is such that the billiard table has infinite volume, the asymptotic behavior at each point is given by a (generalized) Kasner solution (“Kasner-like spacelike singularities”). If, on the other hand, the matter content is such that the billiard table has finite volume, the asymptotic behavior at each point is a chaotic, infinite, oscillatory succession of Kasner epochs. (“Oscillatory, or mixmaster, spacelike singularities.”)

A third element of the original conjecture was that the matter could be neglected asymptotically. While generically true in four spacetime dimensions (the exception being a massless scalar field, equivalent to a fluid with the stiff equation of state p = ρ), this aspect of the conjecture does not remain valid in higher dimensions where the p-form fields might add relevant walls that could change the qualitative asymptotic behavior. We shall thus focus here only on Aspects 1 and 2.

• In the Kasner-like case, the mathematical situation is easier to handle since the conjectured asymptotic behavior of the fields is then monotone and known in closed form. There exist theorems validating (generically) this conjectured asymptotic behavior, starting from the pioneering work of [3] (where the singularities with this behavior are called “quiescent”), which was extended later in [49] to cover more general matter contents. See also [18, 108] for related work.

• The situation is much more complicated in the oscillatory case, where only partial results exist. However, even though as yet incomplete, the mathematical and numerical studies of the BKL analysis has provided overwhelming support for its validity. Most work has been done in four dimensions.

  The first attempts to demonstrate that spacelike singularities are local were done in the simpler context of solutions with isometries. It is only recently that general solutions without symmetries have been treated, but this has been found to be possible only numerically so far [87]. The literature on this subject is vast and we refer to [2, 87, 147] for points of entry into it. Let us note that an important element in the analysis has been a more precise reformulation of what is meant by “local”. This has been achieved in [163], where a precise definition involving a judicious choice of scale invariant variables has been proposed and given the illustrative name of “asymptotic silence” — the singularities being called “silent singularities” since propagation of information is asymptotically eliminated.

  If one accepts that generic spacelike singularities are silent, one can investigate the system of ordinary differential equations that arise in the local limit. In four dimensions, this system is the same as the system of ordinary differential equations describing the dynamics of spatially homogeneous cosmologies of Bianchi type IX. It has been effectively shown analytically in [151] that the Bianchi IX evolution equations can indeed be replaced, in the generic case, by the billiard equations (with only the dominant, sharp walls) that produce the mixmaster behavior. This validates the second element in the BKL conjecture in four dimensions.

The connection between the billiard variables and the scale invariant variables has been investigated recently in the interesting works [92, 162].

Finally, taking for granted the BKL conjecture, one might analyze the chaotic properties of the billiard map (when the volume is finite). Papers exploring this issue are [30, 32, 121, 132] (four dimensions) and [68] (five dimensions).

Let us finally mention the interesting recent paper [40], in which a more precise formulation of the BKL conjecture, aimed towards the chaotic case, is presented. In particular, the main result of this work is an extension of the Fuchsian techniques, employed, e.g., in [49], which are applicable also for systems exhibiting chaotic dynamics. Furthermore, [40] examines the geometric structure which is preserved close to the singularity, and it is shown that this structure has a mathematical description in terms of a so called “partially framed flag”.

Preliminary example: The BKL billiard (vacuum D = 4 gravity)

To illustrate the regularity of the gravitational billiards and motivate the mathematical developments through an explicit example, we first compute in detail the billiard characterizing vacuum, D = 4 gravity. Since this corresponds to the case originally considered by BKL, we call it the “BKL billiard”. We show in detail that the billiard reflections in this case are governed by the “extended modular group” PGL(2. ℤ), which, as we shall see, is isomorphic to the hyperbolic Coxeter group .

Billiard reflections

There are three scale factors so that after radial projection on the unit hyperboloid, we get a billiard in two-dimensional hyperbolic space. The billiard region is defined by the following relevant wall inequalities,

3.1

(symmetry walls) and

3.2

(curvature wall). The remarkable properties of this region from our point of view are:

• It is a triangle (i.e., a simplex in two dimensions) because even though we had to begin with 6 walls (3 symmetry walls and 3 curvature walls), only 3 of them are relevant.

• The walls intersect at angles that are integer submultiples of π, i.e., of the form

  3.3

  where n is an integer. The symmetry walls intersect indeed at sixty degrees (n = 3) since the scalar product of the corresponding linear forms (of norm squared equal to 2) is −1, while the gravitational wall makes angles of zero (n = ∞, scalar product = − 2) and ninety (n = 2, scalar product = 0) degrees with the symmetry walls.

These angles are captured in the matrix A = (A_ij)_i,j=1,2,3 of scalar products,

3.4

which reads explicitly

3.5

Recall from the previous section that the scalar product of two linear forms F = F_iβⁱ and is, in a three-dimensional scale factor space,

3.6

where we have taken α₁(β) = 2β¹, α₂(β) = β² − β¹ and α₃(β) = β³ − β². The corresponding billiard region is drawn in Figure 1.

Figure 1.

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Because the angles between the reflecting planes are integer submultiples of π, the reflections in the walls bounding the billiard region6,

3.7

obey the following relations,

3.8

The product s₁s₃ is a rotation by 2π/2 = π and hence squares to one; the product s₂s₃ is a rotation by 2π/3 and hence its cube is equal to one. There is no power of the product s₁s₂ that is equal to one, something that one conventionally writes as

3.9

The group generated by the reflections s₁, s₂ and s₃ is denoted , for reasons that will become clear in the following, and coincides with the arithmetic group PGL(2, ℤ), as we will now show (see also [75, 116, 107]).

On the group PGL(2, ℤ)

The group PGL(2. ℤ) is defined as the group of 2 × 2 matrices C with integer entries and determinant equal to ±1, with the identification of C and −C,

3.10

Note that although elements of the real general linear group GL(2, ℝ) have (non-vanishing) unrestricted determinants, the discrete subgroup GL(2. ℝ) ⊂ GL(2. ℝ) only allows for det C = ±1 in order for the inverse C⁻¹ to also be an element of GL(2. ℤ).

There are two interesting realisations of PGL(2. ℤ) in terms of transformations in two dimensions:

• One can view PGL(2. ℤ) as the group of fractional transformations of the complex plane

  3.11

  with

  3.12

  Note that one gets the same transformation if C is replaced by −C, as one should. It is an easy exercise to verify that the action of PGL(2. ℤ) when defined in this way maps the complex upper half-plane,

  3.13

  onto itself whenever the determinant ad − bc of C is equal to +1. This is not the case, however, when det C = −1.
• For this reason, it is convenient to consider alternatively the following action of PGL(2. ℤ),

  3.14

  (a. b. c. d ∈ ℤ), which does map the complex upper-half plane onto itself, i.e., which is such that whenever .

  The transformation (3.14) is the composition of the identity with the transformation (3.11) when det C =1, and of the complex conjugation transformation, with the transformation (3.11) when det C = −1. Because the coefficients a, b, c, and d are real, f commutes with C and furthermore the map (3.11) → (3.14) is a group isomorphism, so that we can indeed either view the group PGL(2, →) as the group of fractional transformations (3.11), or as the group of transformations (3.14).

An important subgroup of the group PGL(2, ℤ) is the group PSL(2, ℤ) for which ad − cb = 1, also called the “modular group”. The translation T: z → z +1 and the inversion S: z → −1/z are examples of modular transformations,

3.15

It is a classical result that any modular transformation can be written as the product

3.16

but the representation is not unique [4].

Let s₁, s₂ and s₃ be the PGL(2, ℤ)-transformations

3.17

to which there correspond the matrices

3.18

The s_i’s are reflections in the straight lines x = 0, x =1/2 and the unit circle |z| = 1, respectively. These are in fact just the transformations of hyperbolic space s₁, s₂ and s₃ described in Section 3.1.1, since the reflection lines intersect at 0, 90 and 60 degrees, respectively.

One easily verifies that T = s₂s₁ and that S = s₁s₃ = s₃s₁. Since any transformation of PGL(2, ℤ) not in PSL(2, ℤ) can be written as a transformation of PSL(2, ℤ) times, say, s₁ and since any transformation of PSL(2, ℤ) can be written as a product of S’s and T’s, it follows that the group generated by the 3 reflections s₁, s₂ and s₃ coincides with PGL(2, ℤ), as announced above. (Strictly speaking, PGL(2, ℤ) could be a quotient of that group by some invariant subgroup, but one may verify that the kernel of the homomorphism is trivial (see Section 3.2.5 below).) The fundamental domains for PGL(2, ℤ) and PSL(2, ℤ) are drawn in Figure 2. The equivalence between PGL(2, ℤ) and the Coxeter group has been discussed previously in [75, 116, 107].

Figure 2.

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Coxeter groups — The general theory

We have just shown that the billiard group in the case of pure gravity in four spacetime dimensions is the group PGL(2, ℤ). This group is generated by reflections and is a particular example of a Coxeter group. Furthermore, as we shall explain below, this Coxeter group turns out to be the Weyl group of the (hyperbolic) Kac-Moody algebra . Our first encounter with Lorentzian Kac-Moody algebras in more general gravitational theories will also be through their Weyl groups, which are, exactly as in the four-dimensional case just described, particular instances of (non-Euclidean) Coxeter groups, and which arise as the groups of billiard reflections.

For this reason, we start by developing here some aspects of the theory of Coxeter groups. An excellent reference on the subject is [107], to which we refer for more details and information. We consider Kac-Moody algebras in Section 4.

Examples

Coxeter groups generalize the familiar notion of reflection groups in Euclidean space. Before we present the basic definition, let us briefly discuss some more illuminating examples.

The dihedral group I₂(3) = A₂

Consider the dihedral group I₂(3) of order 6 of symmetries of the equilateral triangle in the Euclidean plane.

This group contains the identity, three reflections s₁, s₂ and s₃ about the three medians, the rotation R₁ of 2π/3 about the origin and the rotation R₂ of 4π/3 about the origin (see Figure 3),

3.19

The reflections act as follows7,

3.20

where (|) is here the Euclidean scalar product and where α_i is a vector orthogonal to the hyper-plane (here, line) of reflection.

Figure 3.

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Now, all elements of the dihedral group I₂(3) can be written as products of the two reflections s₁ and s₂:

3.21

Hence, the dihedral group I₂(3) is generated by s₁ and s₂. The writing Equation (3.21) is not unique because s₁ and s₂ are subject to the following relations,

3.22

The first two relations merely follow from the fact that s₁ and s₂ are reflections, while the third relation is a consequence of the property that the product s₁s₂ is a rotation by an angle of 2π/3. This follows, in turn, from the fact that the hyperplanes (lines) of reflection make an angle of π/3. There is no other relation between the generators s₁ and s₂ because any product of them can be reduced, using the relations Equation (3.22), to one of the 6 elements in Equation (3.21), and these are independent.

The dihedral group I₂(3) is also denoted A₂ because it is the Weyl group of the simple Lie algebra A2 (see Section 4). It is isomorphic to the permutation group S3 of three objects.

The infinite dihedral group

Consider now the group of isometries of the Euclidean line containing the symmetries about the points with integer or half-integer values of x (x is a coordinate along the line) as well as the translations by an integer. This is clearly an infinite group. It is generated by the two reflections s1 about the origin and s₂ about the point with coordinate 1/2,

3.23

The product s₂s₁ is a translation by +1 while the product s₁s₂ is a translation by −1, so no power of s₁s₂ or s₂s₁ gives the identity. All the powers (s₂s₁)^k and (s₁s₂)^j are distinct (translations by +k and −j, respectively). The only relations between the generators are

3.24

This infinite dihedral group I₂(∞) is also denoted by because it is the Weyl group of the affine Kac-Moody algebra .

Definition

A Coxeter group is a group generated by a finite number of elements s_i (i = 1. …,n) subject to relations that take the form

3.25

and

3.26

where the integers m_ij associated with the pairs (i, j) fulfill

3.27

Note that Equation (3.25) is a particular case of Equation (3.26) with m_ii = 1. If there is no power of s_is_j that gives the identity, as in our second example, we set, by convention, m_ij = ∞. The generators s_i are called “reflections” because of Equation (3.25), even though we have not developed yet a geometric realisation of the group. This will be done in Section 3.2.4 below.

The number n of generators is called the rank of the Coxeter group. The Coxeter group is completely specified by the integers m_ij. It is useful to draw the set {m_ij} pictorially in a diagram Γ, called a Coxeter graph. With each reflection s_i, one associates a node. Thus there are n nodes in the diagram. If m_ij > 2, one draws a line between the node i and the node j and writes m_ij over the line, except if m_ij is equal to 3, in which case one writes nothing. The default value is thus “3”. When there is no line between i and j (i ≠ j), the exponent m_ij is equal to 2. We have drawn the Coxeter graphs for the Coxeter groups I₂(3), I₂(m) and for the Coxeter group H₃ of symmetries of the icosahedron.

Note that if m_ij = 2, the generators s_i and s_j commute, s_is_j = s_is_j. Thus, a Coxeter group is the direct product of the Coxeter subgroups associated with the connected components of its Coxeter graph. For that reason, we can restrict the analysis to Coxeter groups associated with connected (also called irreducible) Coxeter graphs.

The Coxeter group may be finite or infinite as the previous examples show.

Another example:

It should be stressed that the Coxeter group can be infinite even if none of the Coxeter exponent is infinite. Consider for instance the group of isometries of the Euclidean plane generated by reflections in the following three straight lines: (i) the x-axis (s₁), (ii) the straight line joining the points (1,0) and (0,1) (s₂), and (iii) the y-axis (s₃). The Coxeter exponents are finite and equal to 4 (m₁₂ = m₂₁ = m₂₃ = m₃₂ = 4) and 2 (m₁₃ = m₃₁ = 2). The Coxeter graph is given in Figure 7. The Coxeter group is the symmetry group of the regular paving of the plane by squares and contains translations. Indeed, the product s₂s₁s₂ is a reflection in the line parallel to the y-axis going through (1, 0) and thus the product t = s₂ s₁s₂s₃ is a translation by +2 in the x-direction. All powers of t are distinct; the group is infinite. This Coxeter group is of affine type and is called (which coincides with )

Figure 7.

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The isomorphism problem

The Coxeter presentation of a given Coxeter group may not be unique. Consider for instance the group I₂(6) of order 12 of symmetries of the regular hexagon, generated by two reflections s₁ and s₂ with

This group is isomorphic with the rank 3 (reducible) Coxeter group I₂ (3) × ℤ₂, with presentation

the isomorphism being given by f(r₁) = s₁, f(r₂) = s₁s₂s₁s₂s₁, f(r₃) = (s₁s₂)³. The question of determining all such isomorphisms between Coxeter groups is known as the “isomorphism problem of Coxeter groups”. This is a difficult problem whose general solution is not yet known [10].

Geometric realization

We now construct a geometric realisation for any given Coxeter group. This enables one to view the Coxeter group as a group of linear transformations acting in a vector space of dimension n, equipped with a scalar product preserved by the group.

To each generator s_i, associate a vector α_i of a basis {α₁, …, α_n} of an n-dimensional vector space V. Introduce a scalar product defined as follows,

3.28

on the basis vectors and extend it to V by linearity. Note that for i = j, m_ii = 1 implies B(α_i,α_i) = 1 for all i. In the case of the dihedral group A₂, this scalar product is just the Euclidean scalar product in the two-dimensional plane where the equilateral triangle lies, as can be seen by taking the two vectors α₁ and α₂ respectively orthogonal to the first and second lines of reflection in Figure 3 and oriented as indicated. But in general, the scalar product (3.28) might not be of Euclidean signature and might even be degenerate. This is the case for the infinite dihedral group I₂(∞), for which the matrix B reads

3.29

and has zero determinant. We shall occasionally use matrix notations for the scalar product, B(α,γ) = α^T Bγ.

However, the basis vectors are always all spacelike since they have norm squared equal to 1. For each i, the vector space V splits then as a direct sum

3.30

where H_i is the hyperplane orthogonal to α_i (δ ∈ H_i iff B(γ, α_i) =0). One defines the geometric reflection σ_i as

3.31

It is clear that σ_i fixes H_i pointwise and reverses α_i. It is also clear that and that σ_i preserves B,

3.32

Note that in the particular case of A₂, we recover in this way the reflections s₁ and s₂.

We now verify that the σ_i’s also fulfill the relations . To that end we consider the plane Π spanned by α_i and α_j. This plane is left invariant under σ_i and σ_j. Two possibilities may occur:

1. The induced scalar product on Π is nondegenerate and in fact positive definite, or

2. the induced scalar product is positive semi-definite, i.e., there is a null direction orthogonal to any other direction.

The second case occurs only when m_ij = ∞. The null direction is given by γ = α_i + α_j.

• In Case 1, V splits as is clearly the identity on Π^⊥ since both σ_i and σ_j leave Π^⊥ pointwise invariant. One needs only to investigate on Π, where the metric is positive definite. To that end we note that the reflections σ_i and σ_j are, on Π, standard Euclidean reflections in the lines orthogonal to α_i and α_j, respectively. These lines make an angle of π/m_ij and hence the product σ_iσ_j is a rotation by an angle of 2π/m_ij. It follows that also on Π.

• In Case 2, is infinite and we must show that no power of the product σ_iσ_j gives the identity. This is done by exhibiting a vector γ for which (σ_iσ_j)^k(γ) ≠ γ for all integers k different from zero. Take for instance α_i. Since one has (σ_iσ_j)(α_i) = α_i + 2λ and (σ_iσ_j)(λ) = λ, it follows that (σ_iσ_j)^k(α_i) = α_i + 2kλ ≠ α_i unless k = 0.

As the defining relations are preserved, we can conclude that the map f from the Coxeter group generated by the s_i’s to the geometric group generated by the σ_i’s defined on the generators by f (s_i) = σ_i is a group homomorphism. We will show below that its kernel is the identity so that it is in fact an isomorphism.

Finally, we note that if the Coxeter graph is irreducible, as we assume, then the matrix B_ij is indecomposable. A matrix A_ij is called decomposable if after reordering of its indices, it decomposes as a non-trivial direct sum, i.e., if one can slit the indices i, j in two sets J and Λ such that A_ij = 0 whenever i ∈ J, j ∈ Λ or i ∈ Λ, j ∈ J. The indecomposability of B follows from the fact that if it were decomposable, the corresponding Coxeter graph would be disconnected as no line would join a point in the set Λ to a point in the set J.

Positive and negative roots

A root is any vector in the space V of the geometric realisation that can be obtained from one of the basis vectors σ_i by acting with an element w of the Coxeter group (more precisely, with its image f(w) under the above homomorphism, but we shall drop “f” for notational simplicity). Any root α can be expanded in terms of the α_i’s,

3.33

If the coefficients c_i are all non-negative, we say that the root a is positive and we write α > 0. If the coefficients c_i are all non-positive, we say that the root a is negative and we write α < 0. Note that we use strict inequalities here because if c_i = 0 for all i, then a is not a root. In particular, the α_i’s themselves are positive roots, called also “simple” roots. (Note that the simple roots considered here differ by normalization factors from the simple roots of Kac-Moody algebras, as we shall discuss below.) We claim that roots are either positive or negative (there is no root with some c_i’s in Equation (3.33) > 0 and some other c_i’s < 0). The claim follows from the fact that the image of a simple root by an arbitrary element w of the Coxeter group is necessarily either positive or negative.

This, in turn, is the result of the following theorem, which provides a useful criterion to tell whether the length l(ws_i) of ws_i is equal to l(w) + 1 or l(w) − 1.

Theorem: l(ws_i) = l(w) + 1 if and only if w(α_i) > 0.

The proof is given in [107], page 111.

It easily follows from this theorem that l(ws_i) = l(w) − 1 if and only if w(α_i) < 0. Indeed, l(ws_i) = l(w) − 1 is equivalent to l(w) = l(ws_i) + 1, i.e., l((ws_i)s_i) = l(ws_i) + 1 and thus, by the theorem, ws_i(α_i) > 0. But since s_i(α_i) = −α_i, this is equivalent to w(α_i) < 0.

We have seen in Section 3.2.3 that there are only two possibilities for the length l(ws_i). It is either equal to l(w) + 1 or to l(w) − 1. From the theorem just seen, the root w(α_i) is positive in the first case and negative in the second. Since any root is the Coxeter image of one of the simple roots α_i, i.e., can be written as w(α_i) for some w and α_i, we can conclude that the roots are either positive or negative; there is no alternative.

The theorem can be used to provide a geometric interpretation of the length function. One can show [107] that l(w) is equal to the number of positive roots sent by w to negative roots. In particular, the fundamental reflection s associated with the simple root α_s maps α_s to its negative and permutes the remaining positive roots.

Note that the theorem implies also that the kernel of the homomorphism that appears in the geometric realisation of the Coxeter group is trivial. Indeed, assume f(w) = 1 where w is an element of the Coxeter group that is not the identity. It is clear that there exists one group generator s_i such that l(ws_i) = l(w) − 1. Take for instance the last generator occurring in a reduced expression of w. For this generator, one has w(α_i) < 0, which is in contradiction with the assumption f(w) = 1.

Because f is an isomorphism, we shall from now on identify the Coxeter group with its geometric realisation and make no distinction between s_i and σ_i.

Fundamental domain

In order to describe the action of the Coxeter group, it is useful to introduce the concept of fundamental domain. Consider first the case of the symmetry group A₂ of the equilateral triangle. The shaded region in Figure 4 contains the vectors γ such that B(α₁, γ) ≥ 0 and B(α₂, γ) ≤ 0. It has the following important property: Any orbit of the group A₂ intersects once and only once. It is called for this reason a “fundamental domain”. We shall extend this concept to all Coxeter groups. However, when the scalar product B is not positive definite, there are inequivalent types of vectors and the concept of fundamental domain can be generalized a priori in different ways, depending on which region one wants to cover. (The entire space? Only the timelike vectors? Another region?) The useful generalization turns out not to lead to a fundamental domain of the action of the Coxeter group on the entire vector space V, but rather to a fundamental domain of the action of the Coxeter group on the so-called Tits cone , which is such that the inequalities B(α_i, γ) ≥ 0 continue to play the central role.

Figure 5.

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Figure 6.

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Figure 4.

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We assume that the scalar product is nondegenerate. Define for each simple root α_i the open half-space

3.34

We define to be the intersection of all A_i,

3.35

This is a convex open cone, which is non-empty because the metric is nondegenerate. Indeed, as B is nondegenerate, one can, by a change of basis, assume for simplicity that the bounding hyperplanes B(α_i, γ) = 0 are the coordinate hyperplanes x_i = 0. is then the region x_i > 0 (with appropriate orientation of the coordinates) and is x_i ≥ 0. The closure

3.36

is then a closed convex cone8.

We next consider the union of the images of under the Coxeter group,

3.37

One can show [107] that this is also a convex cone, called the Tits cone. Furthermore, is a fundamental domain for the action of the Coxeter group on the Tits cone; the orbit of any point in intersects once and only once [107]. The Tits cone does not coincide in general with the full space V and is discussed below in particular cases.

Finite Coxeter groups

An important class of Coxeter groups are the finite ones, like I₂(3) above. One can show that a Coxeter group is finite if and only if the scalar product defined by Equation (3.28) on V is Euclidean [107]. Finite Coxeter groups coincide with finite reflection groups in Euclidean space (through hyperplanes that all contain the origin) and are discrete subgroups of O(n). The classification of finite Coxeter groups is known and is given in Table 1 for completeness. For finite Coxeter groups, one has the important result that the Tits cone coincides with the entire space V [107].

Table 1.

Finite Coxeter groups.

Name	Coxeter graph
A n
Bn ≡ Cn
D n
I 2(m)
F 2
E 6
E 7
E 8
H 3
H 4

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Affine Coxeter groups

Affine Coxeter groups are by definition such that the bilinear form B is positive semi-definite but not positive definite. The radical V^⊥ (defined as the subspace of vectors x for which B(x,y) = x^T By = 0 for all y) is then one-dimensional (in the irreducible case). Indeed, since B is positive semi-definite, its radical coincides with the set N of vectors such that λ^TBλ = 0 as can easily be seen by going to a basis in which B is diagonal (the eigenvalues of B are non-negative). Furthermore, N is at least one-dimensional since B is not positive definite (one of the eigenvalues is zero). Let μ be a vector in Vα = N. Let ν be the vector whose components are the absolute values of those of μ, ν_i = |μ_i|Because B_ij ≤ 0 for i ≠ j (see definition of B in Equation (3.28)), one has

and thus the vector ν belongs also to V^⊥. All the components of ν are strictly positive, ν_i > 0. Indeed, let J be the set of indices for which ν_j > 0 and I the set of indices for which ν_i = 0. From ∑_jB_kjν_j = 0 (ν ∈ V^⊥) one gets, by taking k in I, that B_ij = 0 for all i ∈ I, j ∈ J, contrary to the assumption that the Coxeter system is irreducible (B is indecomposable). Hence, none of the components of any zero eigenvector μ can be zero. If V^⊥ were more than one-dimensional, one could easily construct a zero eigenvector of B with at least one component equal to zero. Hence, the eigenspace V^⊥ of zero eigenvectors is one-dimensional.

Affine Coxeter groups can be identified with the groups generated by affine reflections in Euclidean space (i.e., reflections through hyperplanes that may not contain the origin, so that the group contains translations) and have also been completely classified [107]. The translation subgroup of an affine Coxeter group is an invariant subgroup and the quotient is finite; the affine Coxeter group is equal to the semi-direct product of its translation subgroup by ₀. We list all the affine Coxeter groups in Table 2.

Table 2.

Affine Coxeter groups.

Name	Coxeter graph

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Lorentzian and hyperbolic Coxeter groups

Coxeter groups that are neither of finite nor of affine type are said to be of indefinite type. An important property of Coxeter groups of indefinite type is the following. There exists a positive vector (c_i) such that ∑_jB_ijc_j is negative [116]. A vector is said to be positive (respectively, negative) if all its components are strictly positive (respectively, strictly negative). This is denoted c_i > 0 (respectively, c_i < 0). Note that a vector may be neither positive nor negative, if some of its components are positive while some others are negative. Note also that these concepts refer to a specific basis. This property is demonstrated in Appendix A.

We assume, as already stated, that the scalar product B is nondegenerate. Let {ω_i} be the basis dual to the basis {α_i} in the scalar product B,

3.38

The ω_i’s are called “fundamental weights”. (The fundamental weights are really defined by Equation (3.38) up to normalization, as we will see in Section 3.6 on crystallographic Coxeter groups. They thus differ from the solutions of Equation (3.38) only by a positive multiplicative factor, irrelevant for the present discussion.)

Consider the vector υ = ∑_ic_iα_i, where the vector c_i is such that c_i > 0 and ∑_jB_ijc_j < 0. This vector exists since we assume the Coxeter group to be of indefinite type. Let Σ be the hyperplane orthogonal to υ. Because c_i > 0, the vectors ω_i’s all lie on the positive side of Σ, B(υ, ω_i) = c_i > 0. By contrast, the vectors α_i’s all lie on the negative side of Σ since B(α_i, υ) = ∑_jB_ijc_j < 0. Furthermore, υ has negative norm squared, B(υ, υ) =∑_jc_j(∑_jB_ijc_j) < 0. Thus, in the case of Coxeter groups of indefinite type (with a nondegenerate metric), one can choose a hyperplane such that the positive roots lie on one side of it and the fundamental weights on the other side. The converse is true for Coxeter group of finite type: In that case, there exists c_i > 0 such that ∑_jB_ijc_j is positive, implying that the positive roots and the fundamental weights are on the same side of the hyperplane Σ.

We now consider a particular subclass of Coxeter groups of indefinite type, called Lorentzian Coxeter groups. These are Coxeter groups such that the scalar product B is of Lorentzian signature (n − 1,1). They are discrete subgroups of the orthochronous Lorentz group O⁺(n − 1, 1) preserving the time orientation. Since the α_i are spacelike, the reflection hyperplanes are timelike and thus the generating reflections s_i preserve the time orientation. The hyperplane Σ from the previous paragraph is spacelike. In this section, we shall adopt Lorentzian coordinates so that Σ has equation x⁰ = 0 and we shall choose the time orientation so that the positive roots have a negative time component. The fundamental weights have then a positive time component. This choice is purely conventional and is made here for convenience. Depending on the circumstances, the other time orientation might be more useful and will sometimes be adopted later (see for instance Section 4.8).

Turn now to the cone defined by Equation (3.35). This cone is clearly given by

3.39

Similarly, its closure is given by

3.40

The cone is thus the convex hull of the vectors ω_i, which are on the boundary of .

By definition, a hyperbolic Coxeter group is a Lorentzian Coxeter group such that the vectors in are all timelike, B(λ, λ) < 0 for all . Hyperbolic Coxeter groups are precisely the groups that emerge in the gravitational billiards of physical interest. The hyperbolicity condition forces B(λ, λ) < 0 for all , and in particular, B(ω_i, ω_i) ≤ 0: The fundamental weights are timelike or null. The cone then lies within the light cone. This does not occur for generic (non-hyperbolic) Lorentzian algebras.

The following theorem enables one to decide whether a Coxeter group is hyperbolic by mere inspection of its Coxeter graph.

Theorem: Let be a Coxeter group with irreducible Coxeter graph Γ. The Coxeter group is hyperbolic if and only if the following two conditions hold:

• The bilinear form B is nondegenerate but not positive definite.

• For each i, the Coxeter graph obtained by removing the node i from Γ is of finite or affine type.

(Note: By removing a node, one might get a non-irreducible diagram even if the original diagram is connected. A reducible diagram defines a Coxeter group of finite type if and only if each irreducible component is of finite type, and a Coxeter group of affine type if and only if each irreducible component is of finite or affine type with at least one component of affine type.)

Proof:

• It is clear that if a Coxeter group is hyperbolic, then its bilinear form fulfills the first condition. Let ω_i be one of the vectors of the dual basis. The vectors α_j with j ≠ i form a basis of the hyperplane Π_i orthogonal to ω_i. Because ω_i is non-spacelike (the group is hyperbolic), the hyperplane Π_i is spacelike or null. The Coxeter graph defined by the α_j with j ≠ i (i.e., by removing the node α_i) is thus of finite or affine type.

• Conversely, assume that the two conditions of the theorem hold. From the first condition, it follows that the set N = {λ ∈ V | B(λ, λ) < 0} is non-empty. Let Π_i be the hyperplane spanned by the α_j with j ≠ i, i.e., orthogonal to ω_i. From the second condition, it follows that the intersection of N with each Π_i is empty. Accordingly, each connected component of N lies in one of the connected components of the complement of Π_i, namely, is on a definite (positive or negative) side of each of the hyperplanes Π_i. These sets are of the form ∈_ic_iα_i with c_i > 0 for some i’s (fixed throughout the set) and c_i < 0 for the others. This forces the signature of B to be Lorentzian since otherwise there would be at least a two-dimensional subspace Z of V such that Z {0} ⊂ N. Because Z {0} is connected, it must lie in one of the subsets just described. But this is impossible since if λ ∈ Z {0}, then −λ ∈ Z {0}.

We now show that . Because the signature of B is Lorentzian, N is the inside of the standard light cone and has two components, the “future” component and the “past” component. From the second condition of the theorem, each ω_i lies on or inside the light cone since the orthogonal hyperplane is non-timelike. Furthermore, all the ω_i’s are future pointing, which implies that the cone lies in N, as had to be shown (a positive sum of future pointing non spacelike vectors is non-spacelike). This concludes the proof of the theorem.

In particular, this theorem is useful for determining all hyperbolic Coxeter groups once one knows the list of all finite and affine ones. To illustrate its power, consider the Coxeter diagram of Figure 8, with 8 nodes on the loop and one extra node attached to it (we shall see later that it is called ).

The bilinear form is given by

3.41

and is of Lorentzian signature. If one removes the node labelled 9, one gets the affine diagram (see Figure 9). If one removes the node labelled 8, one gets the finite diagram of the direct product group A₁ × A₇ (see Figure 10). Deleting the nodes labelled 1 or 7 yields the finite diagram of A₈(see Figure 11). Removing the nodes labelled 2 or 6 gives the finite diagram of D₈ (see Figure 12). If one removes the nodes labelled 3 or 5, one obtains the finite diagram of E₈ (see Figure 13). Finally, deleting the node labelled 4 yields the affine diagram of (see Figure 14). Hence, the Coxeter group is hyperbolic.

Consider now the same diagram, with one more node in the loop . In that case, if one removes one of the middle nodes 4 or 5, one gets the Coxeter group , which is neither finite nor affine. Hence, is not hyperbolic.

Using the two conditions in the theorem, one can in fact provide the list of all irreducible hyperbolic Coxeter groups. The striking fact about this classification is that hyperbolic Coxeter groups exist only in ranks 3 ≤ n ≤ 10, and, moreover, for 4 ≤ n ≤ 10 there is only a finite number. In the n =3 case, on the other hand, there exists an infinite class of hyperbolic Coxeter groups. In Figure 15 we give a general form of the Coxeter graphs corresponding to all rank 3 hyperbolic Coxeter groups, and in Tables 3–9 we give the complete classification for 4 ≤ n ≤ 10.

Table 4.

Hyperbolic Coxeter groups of rank 5.

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Table 5.

Hyperbolic Coxeter groups of rank 6.

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Table 6.

Hyperbolic Coxeter groups of rank 7.

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Table 7.

Hyperbolic Coxeter groups of rank 8.

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Table 8.

Hyperbolic Coxeter groups of rank 9.

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Table 3.

Hyperbolic Coxeter groups of rank 4.

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Table 9.

Hyperbolic Coxeter groups of rank 10.

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Note that the inverse metric (B⁻¹)_ij, which gives the scalar products of the fundamental weights, has only negative entries in the hyperbolic case since the scalar product of two future-pointing non-spacelike vectors is strictly negative (it is zero only when the vectors are both null and parallel, which does not occur here).

One can also show [116, 107] that in the hyperbolic case, the Tits cone coincides with the future light cone. (In fact, it coincides with either the future light cone or the past light cone. We assume that the time orientation in V has been chosen as in the proof of the theorem, so that the Tits cone coincides with the future light cone.) This is at the origin of an interesting connection with discrete reflection groups in hyperbolic space (which justifies the terminology). One may realize hyperbolic space as the upper sheet of the hyperboloid B(λ, λ) = −1 in V. Since the Coxeter group is a subgroup of O⁺(n − 1,1), it leaves this sheet invariant and defines a group of reflections in . The fundamental reflections are reflections through the hyperplanes in hyperbolic space obtained by taking the intersection of the Minkowskian hyperplanes B(α_i, λ) = 0 with hyperbolic space. These hyperplanes bound the fundamental region, which is the domain to the positive side of each of these hyperplanes. The fundamental region is a simplex with vertices , where are the intersection points of the lines ℝω_i with hyperbolic space. This intersection is at infinity in hyperbolic space if ω_i is lightlike. The fundamental region has finite volume but is compact only if the ω_i are timelike.

Thus, we see that the hyperbolic Coxeter groups are the reflection groups in hyperbolic space with a fundamental domain which (i) is a simplex, and which (ii) has finite volume. The fact that the fundamental domain is a simplex (n vectors in ) follows from our geometric construction where it is assumed that the n vectors α_i form a basis of V.

The group PGL(2, ℤ) relevant to pure gravity in four dimensions is easily verified to be hyperbolic.

For general information, we point out the following facts:

• Compact hyperbolic Coxeter groups (i.e., hyperbolic Coxeter groups with a compact fundamental region) exist only for ranks 3, 4 and 5, i.e., in two, three and four-dimensional hyperbolic space. All hyperbolic Coxeter groups of rank > 5 have a fundamental region with at least one vertex at infinity. The hyperbolic Coxeter groups appearing in gravitational theories are always of the noncompact type.

• There exist reflection groups in hyperbolic space whose fundamental domains are not simplices. Amazingly enough, these exist only in hyperbolic spaces of dimension < 995. If one imposes that the fundamental domain be compact, these exist only in hyperbolic spaces of dimension < 29. The bound can probably be improved [164].

• Non-hyperbolic Lorentzian Coxeter groups are associated through the above construction with infinite-volume fundamental regions since some of the vectors ω_i are spacelike, which imply that the corresponding reflection hyperplanes intersect beyond hyperbolic infinity.

Crystallographic Coxeter groups

Among the Coxeter groups, only those that are crystallographic correspond to Weyl groups of Kac-Moody algebras. Therefore we now introduce this important concept. By definition, a Coxeter group is crystallographic if it stabilizes a lattice in V. This lattice need not be the lattice generated by the α_i’s. As discussed in [107], a Coxeter group is crystallographic if and only if two conditions are satisfied: (i) The integers m_ij (i ≠ j) are restricted to be in the set {2, 3, 4, 6, ∞}, and (ii) for any closed circuit in the Coxeter graph of , the number of edges labelled 4 or 6 is even.

Given a crystallographic Coxeter group, it is easy to exhibit a lattice L stabilized by it. We can construct a basis for that lattice as follows. The basis vectors μ_i of the lattice are multiples of the original simple roots, μ_i = c_iα_i for some scalars c_i which we determine by applying the following rules:

• .

• .

• .

• .

One easily verifies that σ_i(μ_j) = μ_j − d_ijμ_i for some integers d_ij. Hence L is indeed stabilized. The integers d_ij are equal to .

The rules are consistent as can be seen by starting from an arbitrary node, say α₁, for which one takes c₁ = 1. One then proceeds to the next nodes in the (connected) Coxeter graph by applying the above rules. If there is no closed circuit, there is no consistency problem since there is only one way to proceed from α₁ to any given node. If there are closed circuits, one must make sure that one comes back to the same vector after one turn around any circuit. This can be arranged if the number of steps where one multiplies or divides by (respectively, ) is even.

Our construction shows that the lattice L is not unique. If there are only two different lengths for the lattice vectors μ_i it is convenient to normalize the lengths so that the longest lattice vectors have length squared equal to two. This choice simplifies the factors .

The rank 10 hyperbolic Coxeter groups are all crystallographic. The lattices preserved by E₁₀ and DE₁₀ are unique up to an overall rescaling because the non-trivial m_ij (i ≠ j) are all equal to 3 and there is no choice in the ratios c_i/c_j, all equal to one (first rule above). The Coxeter group BE₁₀ preserves two (dual) lattices.

On the normalization of roots and weights in the crystallographic case

Since the vectors μ_i and α_i are proportional, they generate identical reflections. Even though they do not necessarily have length squared equal to unity, the vectors μ_i are more convenient to work with because the lattice preserved by the Coxeter group is simply the lattice ∈_iℤμ_i of points with integer coordinates in the basis {μ_i}. For this reason, we shall call from now on “simple roots” the vectors μ_i and, to follow common practice, will sometimes even rename them α_i. Thus, in the crystallographic case, the (redefined) simple roots are appropriately normalized to the lattice structure. It turns out that it is with this normalization that simple roots of Coxeter groups correspond to simple roots of Kac-Moody algebras defined in the Section 4.6.3. A root is any point on the root lattice that is in the Coxeter orbit of some (redefined) simple root. It is these roots that coincide with the (real) roots of Kac-Moody algebras.

It is also useful to rescale the fundamental weights. The rescaled fundamental weights, of course proportional to ω_i, are denoted Λ_i. The convenient normalization is such that

3.42

With this normalization, they coincide with the fundamental weights of Kac-Moody algebras, to be considered in Section 4.

Definitions

An n × n matrix A is called a “generalized Cartan matrix” (or just “Cartan matrix” for short) if it satisfies the following conditions9:

4.1

4.1

4.3

where ℤ_ denotes the non-positive integers. One can encode the Cartan matrix in terms of a Dynkin diagram, which is obtained as follows:

1. For each i = 1, …, n, one associates a node in the diagram.

2. One draws a line between the node i and the node j if A_ij ≠ 0; if A_ij = 0 (= A_ij), one draws no line between i and j.

3. One writes the pair (A_ij, A_ij) over the line joining i to j. When the products A_ij · A_ij are all ≤ 4 (which is the only situation we shall meet in practice), this third rule can be replaced by the following rules:

   1. one draws a number of lines between i and j equal to max(| A_ij|, | A_ij|);
   2. one draws an arrow from j to i if |A_ij | > |A_ij|.

So, for instance, the Dynkin diagrams in Figure 16 correspond to the Cartan matrices

4.4

4.5

4.6

4.7

4.8

respectively. If the Dynkin diagram is connected, the matrix A is indecomposable. This is what shall be assumed in the following.

Although this is not necessary for developing the general theory, we shall impose two restrictions on the Cartan matrix. The first one is that det A ≠ 0; the second one is that A is symmetrizable. The restriction det A ≠ 0 excludes the important class of affine algebras and will be lifted below. We impose it at first because the technical definition of the Kac-Moody algebra when det A = 0 is then slightly more involved.

The second restriction imposes that there exists an invertible diagonal matrix D with positive elements ϵ_i and a symmetric matrix S such that

4.9

The matrix S is called a symmetrization of A and is unique up to an overall positive factor because A is indecomposable. To prove this, choose the first (diagonal) element ϵ₁ > 0 of D arbitrarily. Since A is indecomposable, there exists a nonempty set J₁ of indices j such that A_1j ≠ 0. One has A_1j = ϵ₁S_1j and A_j1 = ϵ_jS_j1. This fixes the ϵ_j’s > 0 in terms of ϵ₁ since S_1j = S_j1. If not all the elements ϵ_j are determined at this first step, we pursue the same construction with the elements A_jk = ϵ_jS_jk and A_kj = ϵ_kS_kj = ϵ_kS_kj with j ϵ J₁ and, more generally, at step p, with j ∈ J₁ ∩ J₂ ⋯ ∩ J_p. As the matrix A is assumed to be indecomposable, all the elements ϵ_i of D and S_ij of S can be obtained, depending only on the choice of ϵ₁. One gets no contradicting values for the ϵ_j’s because the matrix A is assumed to be symmetrizable.

In the symmetrizable case, one can characterize the Cartan matrix according to the signature of (any of) its symmetrization(s). One says that A is of finite type if S is of Euclidean signature, and that it is of Lorentzian type if S is of Lorentzian signature.

Given a Cartan matrix A (with det A ≠ 0), one defines the corresponding Kac-Moody algebra as the algebra generated by 3n generators h_i, e_i, f_i subject to the following “Chevalley-Serre” relations (in addition to the Jacobi identity and anti-symmetry of the Lie bracket),

4.10

4.11

The relations (4.11), called Serre relations, read explicitly

4.12

(and likewise for the f_k’s).

Any multicommutator can be reduced, using the Jacobi identity and the above relations, to a multicommutator involving only the e_i’s, or only the f_i’s. Hence, the Kac-Moody algebra splits as a direct sum (“triangular decomposition”)

4.13

where is the subalgebra involving the multicommutators is the subalgebra involving the multicommutators and is the Abelian subalgebra containing the h_i’s. This is called the Cartan subalgebra and its dimension n is the rank of the Kac-Moody algebra . It should be stressed that the direct sum Equation (4.13) is a direct sum of , and as vector spaces, not as subalgebras (since these subalgebras do not commute).

A priori, the numbers of the multicommutators

are infinite, even after one has taken into account the Jacobi identity. However, the Serre relations impose non-trivial relations among them, which, in some cases, make the Kac-Moody algebra finite-dimensional. Three worked examples are given in Section 4.4 to illustrate the use of the Serre relations. In fact, one can show [116] that the Kac-Moody algebra is finite-dimensional if and only if the symmetrization S of A is positive definite. In that case, the algebra is one of the finite-dimensional simple Lie algebras given by the Cartan classification. The list is given in Table 10.

Table 10.

Finite Lie algebras.

Name	Dynkin diagram
A n
B n
C n
D n
G 2
F 4
E 6
E 7
E 8

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When the Cartan matrix A is of Lorentzian signature the Kac-Moody algebra , constructed from A using the Chevalley-Serre relations, is called a Lorentzian Kac-Moody algebra. This is the case of main interest for the remainder of this paper.

Roots

The adjoint action of the Cartan subalgebra on and is diagonal. Explicitly,

4.14

for any element , where α_i is the linear form on (i.e., the element of the dual ) defined by α_i(h_j) = A_ji. The α_i’s are called the simple roots. Similarly,

4.15

and, if is non-zero, one says that is a positive root. On the negative side, , one has

4.16

and is called a negative root when is non-zero. This occurs if and only if is non-zero: −α is a negative root if and only if α is a positive root.

We see from the construction that the roots (linear forms α such that [h, x] = α(h)x has nonzero solutions x) are either positive (linear combinations of the simple roots α_i with integer non-negative coefficients) or negative (linear combinations of the simple roots with integer non-positive coefficients). The set of positive roots is denoted by Δ₊; that of negative roots by Δ₋. The set of all roots is Δ, so we have Δ = Δ₊ ∪Δ₋. The simple roots are positive and form a basis of . One sometimes denotes the h_i by (and thus, etc). Similarly, one also uses the notation ⟨·,·⟩ for the standard pairing between and its dual , i.e., ⟨α, h⟩ = α(h). In this notation the entries of the Cartan matrix can be written as

4.17

Finally, the root lattice Q is the set of linear combinations with integer coefficients of the simple roots,

4.18

All roots belong to the root lattice, of course, but the converse is not true: There are elements of Q that are not roots.

The Chevalley involution

The symmetry between the positive and negative subalgebras and of the Kac-Moody algebra can be rephrased formally as follows: The Kac-Moody algebra is invariant under the Chevalley involution τ, defined on the generators as

4.19

The Chevalley involution is in fact an algebra automorphism that exchanges the positive and negative sides of the algebra.

Finally, we quote the following useful theorem.

Theorem: The Kac-Moody algebra defined by the relations (4.10, 4.11) is simple. The proof may be found in Kac’ book [116], page 12.

We note that invertibility and indecomposability of the Cartan matrix A are central ingredients in the proof. In particular, the theorem does not hold in the affine case, for which the Cartan matrix is degenerate and has non-trivial ideals10 (see [116] and Section 4.5).

Three examples

To get a feeling for how the Serre relations work, we treat in detail three examples.

• A₂: We start with A₂, the Cartan matrix of which is Equation (4.4). The defining relations are then:

  4.20

  The commutator [e₁, e₂] is not killed by the defining relations and hence is not equal to zero (the defining relations are all the relations). All the commutators with three (or more) e’s are however zero. A similar phenomenon occurs on the negative side. Hence, the algebra A₂ is eight-dimensional and one may take as basis {h₁, h₂, e₁, e₂, [e₁, e₂], f₁, f₂, [f₁, f₂]}. The vector [e₁, e₂] corresponds to the positive root α₁ + α₂.
• B₂: The algebra B₂, the Cartan matrix of which is Equation (4.5), is defined by the same set of generators, but the Serre relations are now [e₁, [e₁, [e₁, e₂]]] = 0 and [e₂, [e₂, e₁]] = 0 (and similar relations for the f’s). The algebra is still finite-dimensional and contains, besides the generators, the commutators [e₁, e₂], [e₁, [e₁, e₂]], their negative counterparts [f₁, f₂] and [f₁, [f₁, f₂]], and nothing else. The triple commutator [e₁, [e₁, [e₁, e₂]]] vanishes by the Serre relations. The other triple commutator [e₂, [e₁, [e₁, e₂]]] vanishes also by the Jacobi identity and the Serre relations, (Each term on the right-hand side is zero: The first by antisymmetry of the bracket and the second because [e₂, [e₁, e₂]] = −[e₂, [e₂, e₁]] = 0.) The algebra is 10-dimensional and is isomorphic to .
• : We now turn to , the Cartan matrix of which is Equation (4.8). This algebra is defined by the same set of generators as A₂, but with Serre relations given by

  4.21

  (and similar relations for the f’s). This innocent-looking change in the Serre relations has dramatic consequences because the corresponding algebra is infinite-dimensional. (We analyze here the algebra generated by the h’s, e’s and f’s, which is in fact the derived Kac-Moody algebra — see Section 4.5 on affine Kac-Moody algebras. The derived algebra is already infinite-dimensional.) To see this, consider the current algebra, defined by

  4.22

  where are the structure constants of and where k^ab is the invariant metric on which we normalize here so that k⁻⁺ = 1. The subalgebra with n = 0 is isomorphic to , The current algebra (4.22) is generated by , c, and since any element can be written as a multi-commutator involving them. The map

  4.23

  preserves the defining relations of the Kac-Moody algebra and defines an isomorphism of the (derived) Kac-Moody algebra with the current algebra. The Kac-Moody algebra is therefore infinite-dimensional. One can construct non-vanishing infinite multi-commutators, in which e₁ and e₂ alternate:

  4.24

  The Serre relations do not cut the chains of multi-commutators to a finite number.

We see from these examples that the exact consequences of the Serre relations might be intricate to derive explicitly. This is one of the difficulties of the theory.

The affine case

The affine case is characterized by the conditions that the Cartan matrix has vanishing determinant, is symmetrizable and is such that its symmetrization S is positive semi-definite (only one zero eigenvalue). As before, we also take the Cartan matrix to be indecomposable. By a reasoning analogous to what we did in Section 3.4 above, one can show that the radical of S is one-dimensional and that the ranks of S and A are equal to n − 1.

One defines the corresponding Kac-Moody algebras in terms of 3n + 1 generators, which are the same generators h_i, e_i, f_i subject to the same conditions (4.10, 4.11) as above, plus one extra generator η which can be taken to fulfill

4.25

The algebra admits the same triangular decomposition as above,

4.26

but now the Cartan subalgebra has dimension n + 1 (it contains the extra generator n).

Because the matrix A_ij has vanishing determinant, one can find a_i such that ∑_ia_iA_ij = 0. The element c = ∑_ia_ih_i is in the center of the algebra. In fact, the center of the Kac-Moody algebra is one-dimensional and coincides with ℂc [116]. The derived algebra is the subalgebra generated by h_i, e_i, f_i and has codimension one (it does not contain η). One has

4.27

(direct sum of vector spaces, not as algebras). The only proper ideals of the affine Kac-Moody algebra are and ℂc.

Affine Kac-Moody algebras appear in the BKL context as subalgebras of the relevant Lorentzian Kac-Moody algebras. Their complete list is known and is given in Tables 11 and 12.

Table 11.

Untwisted affine Kac-Moody algebras.

Name	Dynkin diagram

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Table 12.

Twisted affine Kac-Moody algebras. We use the notation of Kac [116].

Name	Dynkin diagram

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The invariant bilinear form

Definition

To proceed, we assume, as announced above, that the Cartan matrix is invertible and symmetrizable since these are the only cases encountered in the billiards. Under these assumptions, an invertible, invariant bilinear form is easily defined on the algebra. We denote by ϵ_i the diagonal elements of D,

4.28

First, one defines an invertible bilinear form in the dual of the Cartan subalgebra. This is done by simply setting

4.29

for the simple roots. It follows from A_ii = 2 that

4.30

and thus the Cartan matrix can be written as

4.31

It is customary to fix the normalization of S so that the longest roots have (α_i|α_i) = 2. As we shall now see, the definition (4.29) leads to an invariant bilinear form on the Kac-Moody algebra.

Since the bilinear form (·|·) is nondegenerate on , one has an isomorphism defined by

4.32

This isomorphism induces a bilinear form on the Cartan subalgebra, also denoted by (·|·). The inverse isomorphism is denoted by v and is such that

4.33

Since the Cartan elements obey

4.34

it is clear from the definitions that

4.35

and thus also

4.36

The bilinear form (·|·) can be uniquely extended from the Cartan subalgebra to the entire algebra by requiring that it is invariant, i.e., that it fulfills

4.37

for instance, for the e_i’s and f_i’s one finds

4.38

and similarly

4.39

In the same way we have

4.40

and thus

4.41

Quite generally, if e_α and e_γ are root vectors corresponding respectively to the roots α and γ,

then (e_α|e_γ) = 0 unless γ = −α. Indeed, one has

and thus

4.42

It is proven in [116] that the invariance condition on the bilinear form defines it indeed consistently and that it is nondegenerate. Furthermore, one finds the relations

4.43

Real and imaginary roots

Consider the restriction (·|·)_ℝ of the bilinear form to the real vector space obtained by taking the real span of the simple roots,

4.44

This defines a scalar product with a definite signature. As we have mentioned, the signature is Euclidean if and only if the algebra is finite-dimensional [116]. In that case, all roots — and not just the simple ones — are spacelike, i.e., such that (α|α) > 0.

When the algebra is infinite-dimensional, the invariant scalar product does not have Euclidean signature. The spacelike roots are called “real roots”, the non-spacelike ones are called “imaginary roots” [116]. While the real roots are nondegenerate (i.e., the corresponding eigenspaces, called “root spaces”, are one-dimensional), this is not so for imaginary roots. In fact, it is a challenge to understand the degeneracy of imaginary roots for general indefinite Kac-Moody algebras, and, in particular, for Lorentzian Kac-Moody algebras.

Another characteristic feature of real roots, familiar from standard finite-dimensional Lie algebra theory, is that if α is a (real) root, no multiple of α is a root except ±α. This is not so for imaginary roots, where 2α (or other non-trivial multiples of α) can be a root even if α is. We shall provide explicit examples below.

Finally, while there are at most two different root lengths in the finite-dimensional case, this is no longer true even for real roots in the case of infinite-dimensional Kac-Moody algebras11. When all the real roots have the same length, one says that the algebra is “simply-laced”. Note that the imaginary roots (if any) do not have the same length, except in the affine case where they all have length squared equal to zero.

Fundamental weights and the Weyl vector

The fundamental weights {Λ_i} of the Kac-Moody algebra are vectors in the dual space of the Cartan subalgebra defined by

4.45

This implies

4.46

The Weyl vector ρ ∈ is defined by

4.47

and is thus equal to

4.48

The generalized Casimir operator

From the invariant bilinear form, one can construct a generalized Casimir operator as follows.

We denote the eigenspace associated with α by . This is called the “root space” of α and is defined as

4.49

A representation of the Kac-Moody algebra is called restricted if for every vector v of the representation subspace V, one has for all but a finite number of positive roots α.

Let be a basis of and let be the basis of dual to in the B-metric,

4.50

Similarly, let {u_i} be a basis of and {uⁱ} the dual basis of with respect to the bilinear form

4.51

We set

4.52

where ρ is the Weyl vector. Recall from Section 4.6.1 that μ is an isomorphism from to , so, since ρ ∈ , the expression μ(ρ) belongs to as required. When acting on any vector of a restricted representation, Ω is well-defined since only a finite number of terms are different from zero.

It is proven in [116] that Ω commutes with all the operators of any restricted representation. For that reason, it is known as the (generalized) Casimir operator. It is quadratic in the generators12.

Note

This definition — and, in particular, the presence of the linear term μ(ρ) — might seem a bit strange at first sight. To appreciate it, turn to a finite-dimensional simple Lie algebra. In the above notations, the usual expression for the quadratic Casimir operator reads

4.53

(without degeneracy index K since the roots are nondegenerate in the finite-dimensional case). Here, κ^AB is the Killing metric and {T_A} a basis of the Lie algebra. This expression is not “normal-ordered” because there are, in the last term, lowering operators standing on the right. We thus replace the last term by

4.54

Using the fact that in a finite-dimensional Lie algebra, , (see, e.g., [85]) one sees that the Casimir operator can be rewritten in “normal ordered” form as in Equation (4.52). The advantage of the normal-ordered form is that it makes sense also for infinite-dimensional Kac-Moody algebras in the case of restricted representations.

The Weyl group

The Weyl group of a Kac-Moody algebra is a discrete group of transformations acting on . It is defined as follows. One associates a “fundamental Weyl reflection” to each simple root through the formula

4.55

The Weyl group is just the group generated by the fundamental Weyl reflections. In particular,

4.56

The Weyl group enjoys a number of interesting properties [116]:

• It preserves the scalar product on .

• It preserves the root lattice and hence is crystallographic.

• Two roots that are in the same orbit have identical multiplicities.

• Any real root has in its orbit (at least) one simple root and hence, is nondegenerate.

• The Weyl group is a Coxeter group. The connection between the Coxeter exponents and the Cartan integers A_ij is given in Table 13 (i ≠ j).

This close relationship between Coxeter groups and Kac-Moody algebras is the reason for denoting both with the same notation (for instance, A_n denotes at the same time the Coxeter group with Coxeter graph of type A_n and the Kac-Moody algebra with Dynkin diagram A_n).

Note that different Kac-Moody algebras may have the same Weyl group. This is in fact already true for finite-dimensional Lie algebras, where dual algebras (obtained by reversing the arrows in the Dynkin diagram) have the same Weyl group. This property can be seen from the fact that the Coxeter exponents are related to the duality-invariant product A_ijA_ji. But, on top of this, one sees that whenever the product A_ijA_ji exceeds four, which occurs only in the infinite-dimensional case, the Coxeter exponent m_ij is equal to infinity, independently of the exact value of A_ijA_ji. Information is thus clearly lost. For example, the Cartan matrices

4.57

lead to the same Weyl group, even though the corresponding Kac-Moody algebras are not isomorphic or even dual to each other.

Because the Weyl groups are (crystallographic) Coxeter groups, we can use the theory of Coxeter groups to analyze them. In the Kac-Moody context, the fundamental region is called “the fundamental Weyl chamber”.

We also note that by (standard vector space) duality, one can define the action of the Weyl group in the Cartan subalgebra , such that

4.58

One has using Equations (4.30, 4.32, 4.33, 4.35),

4.59

Finally, we leave it to the reader to verify that when the products A_ijA_ji are all ≤ 4, then the geometric action of the Coxeter group considered in Section 3.2.4 and the geometric action of the Weyl group considered here coincide. The (real) roots and the fundamental weights differ only in the normalization and, once this is taken into account, the metrics coincide. This is not the case when some products A_ijA_ji exceed 4. It should be also pointed out that the imaginary roots of the Kac-Moody algebras do not have immediate analogs on the Coxeter side.

Hyperbolic Kac-Moody algebras

Hyperbolic Kac-Moody algebras are by definition Lorentzian Kac-Moody algebras with the property that removing any node from their Dynkin diagram leaves one with a Dynkin diagram of affine or finite type. The Weyl group of hyperbolic Kac-Moody algebras is a crystallographic hyperbolic Coxeter group (as defined in Section 3.5). Conversely, any crystallographic hyperbolic Coxeter group is the Weyl group of at least one hyperbolic Kac-Moody algebra. Indeed, consider one of the lattices preserved by the Coxeter group as constructed in Section 3.6. The matrix with entries equal to the d_ij of that section is the Cartan matrix of a Kac-Moody algebra that has this given Coxeter group as Weyl group.

The hyperbolic Kac-Moody algebras have been classified in [154] and exist only up to rank 10 (see also [59]). In rank 10, there are four possibilities, known as and , BE₁₀ and CE₁₀ being dual to each other and possessing the same Weyl group (the notation will be explained below).

Examples

We discuss here briefly two examples, namely , for which all simple roots have equal length, and , with respective Dynkin diagrams shown in Figures 17 and 18.

Figure 17.

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Figure 18.

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The Kac-Moody Algebra

This is the algebra associated with vacuum four-dimensional Einstein gravity and the BKL billiard. We encountered its Weyl group PGL(2, ℤ) already in Section 3.1.1. The algebra is also denoted AE₃, or H₃. The Cartan matrix is

4.63

As it follows from our analysis in Section 3.1.1, the simple roots may be identified with the following linear forms α_i(β) in the three-dimensional space of the βⁱ’s,

4.64

with scalar product

4.65

for two linear forms F = F_iβⁱ and G = G_iβⁱ. It is sometimes convenient to analyze the root system in terms of an “affine” level ℓ that counts the number of times the root α₃ occurs: The root kα₁ + mα₂ + ℓα₃ has by definition level ℓ13. We shall consider here only positive roots for which k, m, ℓ ≥ 0.

Applying the first theorem, one easily verifies that the only positive roots at level zero are the roots kα₁ + mα₂, |k − m| ≤ 1 (k, m ≥ 0) of the affine subalgebra . When k = m, the root is imaginary and has length squared equal to zero. When |k − m| = 1, the root is real and has length squared equal to two.

Similarly, the only roots at level one are (m + a)α₁ + mα₂ + α₃ with a² ≤ m, i.e., . Whenever is an integer, the roots have squared length equal to two and are real. The roots (m + a)α₁ + mα₂ + α₃ with a² < m are imaginary and have squared length equal to 2(a² + 1 − m) ≤ 0. In particular, the root m(α₁ + α₂) + α₃ has length squared equal to 2(1 − m). Of all the roots at level one with m > 1, these are the only ones that are in the fundamental domain (i.e., that fulfill (β|α_i) ≤ 0). When m = 1, none of the level-1 roots is in and is either in the Weyl orbit of α₁ + α₂, or in the Weyl orbit of α₃.

We leave it to the reader to verify that the roots at level two that are in the fundamental domain take the form (m − 1)α₁ + mα₂ + 2α₃ and m(α₁ + α₂) + 2α₃ with m ≥ 4. Further information on the roots of may be found in [116], Chapter 11, page 215.

The Kac-Moody Algebra

This is the algebra associated with the Einstein-Maxwell theory (see Section 7). The notation will be explained in Section 4.9. The Cartan matrix is

4.66

and there are now two lengths for the simple roots. The scalar products are

4.67

One may realize the simple roots as the linear forms

4.68

in the three-dimensional space of the βⁱ’s with scalar product Equation (4.65).

The real roots, which are Weyl conjugate to one of the simple roots α₁ or α₂ (α₃ is in the same Weyl orbit as α₂), divide into long and short real roots. The long real roots are the vectors on the root lattice with squared length equal to two that fulfill the extra condition in the theorem. This condition expresses here that the coefficient of α₁ should be a multiple of 4. The short real roots are the vectors on the root lattice with length squared equal to one-half. The imaginary roots are all the vectors on the root lattice with length squared ≤ 0.

We define again the level ℓ as counting the number of times the root α₃ occurs. The positive roots at level zero are the positive roots of the twisted affine algebra , namely, α₁ and (2m + a)α₁+mα₂, m = 1, 2, 3, ⋯, with a = −2, −1, 0, 1, 2 for m odd and a = −1, 0, 1 for m odd. Although belonging to the root lattice and of length squared equal to two, the vectors (2m ± 2)α₁ + mα₂ are not long real roots when m is even because they fail to satisfy the condition that the coefficient (2m ± 2) of α₁ is a multiple of 4. The roots at level zero are all real, except when a = 0, in which case the roots m(2α₁ + α₂) have zero norm.

To get the long real roots at level one, we first determine the vectors α = α₃ + kα₁ + mα₂ of squared length equal to two. The condition (α|α) = 2 easily leads to m = p² for some integer p ≥ 0 and k = 2p² ± 2p = 2p(p ± 1). Since k is automatically a multiple of 4 for all p = 0, 1, 2, 3, ⋯, the corresponding vectors are all long real roots. Similarly, the short real roots at level one are found to be (2p² + 1)α₁ + (p² + p + 1)α₂ + α₃ and (2p² + 4p + 3)α₁ + (p² + p + 1)α₂ + α₃ for p a non-negative integer.

Finally, the imaginary roots at level one in the fundamental domain read (2m − 1)α₁ + mα₂ + α₃ and 2mα₁ + mα₂ + α₃ where m is an integer greater than or equal to 2. The first roots have length squared equal to , the second have length squared equal to −2m + 2.

Overextensions of finite-dimensional Lie algebras

An interesting class of Lorentzian Kac-Moody algebras can be constructed by adding simple roots to finite-dimensional simple Lie algebras in a particular way which will be described below. These are called “overextensions”.

In this section, we let be a complex, finite-dimensional, simple Lie algebra of rank r, with simple roots α₁, ⋯, α_r. As stated above, normalize the roots so that the long roots have length squared equal to 2 (the short roots, if any, have then length squared equal to 1 (or 2/3 for G₂)). The roots of simply-laced algebras are regarded as long roots.

Let α = ∑_in_iα_i, n_i ≥ 0 be a positive root. One defines the height of α as

4.69

Among the roots of , there is a unique one that has highest height, called the highest root. We denote it by θ. It is long and it fulfills the property that (θ|α_i) ≥ 0 for all simple roots α_i, and

4.70

(see, e.g., [85]). We denote by V the r-dimensional Euclidean vector space spanned by α_i (i = 1, ⋯, r). Let M₂ be the two-dimensional Minkowski space with basis vectors u and v so that (u|u) = (v|v) = 0 and (u|v) = 1. The metric in the space V ⊕ M₂ has clearly Minkowskian signature (−, +, +, ⋯, +) so that any Kac-Moody algebra whose simple roots span V ⊕ M₂ is necessarily Lorentzian.

Root systems in Euclidean space

In order to describe the “twisted” overextensions, we need to introduce the concept of a “root system”.

A root system in a real Euclidean space V is by definition a finite subset Δ of nonzero elements of V obeying the following two conditions:

4.71

4.72

The elements of Δ are called the roots. From the definition one can prove the following properties [93]:

1. If α ∈ Δ, then −α ∈ Δ.

2. If α ∈ Δ, then the only elements of Δ proportional to α are ±½α, ±α, ±2α. If only ±α occurs (for all roots α), the root system is said to be reduced (proper in “Araki terminology” [5]).

3. If α, β ∈ Δ, then 0 ≤ A_α,βA_β,α ≤ 4, i.e., A_α,β = 0, ±1, ±2, ±3, ±4; the last occurrence appearing only for β = ±2α, i.e., for nonreduced systems. (The proof of this point requires the use of the Schwarz inequality.)

4. If α, β ∈ Δ are not proportional to each other and (α|α) ≤ (β|β) then A_α,β = 0, ±1. Moreover if (α|β) = 0, then (β|β) = (α|α), 2 (α|α), or 3 (α|α).

5. If If α, β ∈ Δ, but α — β ∉ Δ ∪ 0, then (α|β) ≤ 0 and, as a consequence, if α, β ∈ Δ but α ± β ∉ Δ ∪ 0 then (α|β) = 0. That (α|β) ≤ 0 can be seen as follows. Clearly, α and β can be assumed to be linearly independent14. Now, assume (α|β) > 0. By the previous point, A_α,β = 1 or A_β,α = 1. But then either α − A_α,ββ = α − β ∈ Δ or −(β − A_β,αα) = α − β ∈ Δ by (4.72), contrary to the assumption. This proves that (α|β) ≤ 0.

Since Δ spans the vector space V, one can chose a basis {α_i} of elements of V within Δ. This can furthermore be achieved in such a way the α_i enjoy the standard properties of simple roots of Lie algebras so that in particular the concepts of positive, negative and highest roots can be introduced [93].

All the abstract root systems in Euclidean space have been classified (see, e.g., [93]) with the following results:

• The most general root system is obtained by taking a union of irreducible root systems. An irreducible root system is one that cannot be decomposed into two disjoint nonempty orthogonal subsets.

• The irreducible reduced root systems are simply the root systems of finite-dimensional simple Lie algebras (A_n with n ≥ 1, B_n with n ≥ 3, C_n with n ≥ 2, D_n with n ≥ 4, G₂, F₄, E₆, E₇ and E₈).

• Irreducible nonreduced root systems are all given by the so-called (BC)_n-systems. A (BC)_n-system is obtained by combining the root system of the algebra B_n with the root system of the algebra C_n in such a way that the long roots of B_n are the short roots of C_n. There are in that case three different root lengths. Explicitly Δ is given by the n unit vectors and their opposite along the Cartesian axis of an n-dimensional Euclidean space, the 2n vectors obtained by multiplying the previous vectors by 2 and the 2n(n − 1) diagonal vectors , with k ≠ k′ and k, k′ = 1, …, n. The n = 3 case is pictured in Figure 19. The Dynkin diagram of (BC)_r is the Dynkin diagram of B_r with a double circle ⊚ over the simple short root, say α₁, to indicate that 2α₁ is also a root.

Figure 19.

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It is sometimes convenient to rescale the roots by the factor so that the highest root θ = 2(α₁ + α₂ + ⋯ + α_r) [93] of the (BC)-system has length 2 instead of 4.

Twisted overextensions

We follow closely [95]. Twisted affine algebras are related to either the (BC)-root systems or to extensions by the highest short root (see [116], Proposition 6.4).

Twisted overextensions associated with the (BC)-root systems

These are the overextensions relevant for some of the gravitational billiards. The construction proceeds as for the untwisted overextensions, but the starting point is now the (BC)_r root system with rescaled roots. The highest root has length squared equal to 2 and has non-vanishing scalar product only with α_r ((α_r|θ) = 1). The overextension procedure (defined by the same formulas as in the untwisted case) yields the algebra , also denoted .

There is an alternative overextension that can be defined by starting this time with the algebra C_r but taking one-half the highest root of C_r to make the extension (see [116], formula in Paragraph 6.4, bottom of page 84). The formulas for α₀ and α₋₁ are 2α₀ = u−θ and 2α₋₁ = −u−v (where θ is now the highest root of C_r). The Dynkin diagram of is dual to that of . (Duality amounts to reversing the arrows in the Dynkin diagram, i.e., replacing the (generalized) Cartan matrix by its transpose.)

The algebras and have rank r + 2 and are hyperbolic for r ≤ 4. The intermediate affine algebras are in all cases the twisted affine algebras . We shall see in Section 7 that by coupling to three-dimensional gravity a coset model , where the so-called restricted root system (see Section 6) of the (real) Lie algebra of the Lie group is of (BC)_r-type, one can realize all the algebras.

Twisted overextensions associated with the highest short root

We denote by θ_s the unique short root of heighest weight. It exists only for non-simply laced algebras and has length 1 (or 2/3 for G₂). The twisted overextensions are defined as the standard overextensions but one uses instead the highest short root θ_s. The formulas for the affine and overextended roots are

or

(We choose the overextended root to have the same length as the affine root and to be attached to it with a single link. This choice is motivated by considerations of simplicity and yields the fourth rank ten hyperbolic algebra when .)

The affine extensions generated by α₀, ⋯, α_r are respectively the twisted affine algebras and . These twisted affine algebras are related to external automorphisms of D_r+1, A_2r−1, E₆ and D₄, respectively (the same holds for above) [116]. The corresponding twisted overextensions have the following features.

• The overextensions have rank r + 2 and are hyperbolic for r ≤ 4.

• The overextensions have rank r + 2 and are hyperbolic for r ≤ 8. The last hyperbolic case, r = 8, yields the algebra , also denoted CE₁₀. It is the fourth rank-10 hyperbolic algebra, besides E₁₀, BE₁₀ and DE₁₀.

• The overextensions (rank 6) and (rank 4) are hyperbolic.

We list in Table 14 the Dynkin diagrams of all twisted overextensions.

Table 14.

Twisted overextended Kac-Moody algebras.

Name	Dynkin diagram

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A satisfactory feature of the class of overextensions (standard and twisted) is that it is closed under duality. For instance, is dual to . In fact, one could get the twisted overextensions associated with the highest short root from the standard overextensions precisely by requiring closure under duality. A similar feature already holds for the affine algebras.

Note also that while not all hyperbolic Kac-Moody algebras are symmetrizable, the ones that are obtained through the process of overextension are.

Regular subalgebras of Kac-Moody algebras

This section is based on [96].

Definitions

Let be a Kac-Moody algebra, and let be a subalgebra of with triangular decomposition . We assume that is canonically embedded in , i.e., that the Cartan subalgebra of is a subalgebra of the Cartan subalgebra of , , so that . We shall say that is regularly embedded in (and call it a “regular subalgebra”) if and only if two conditions are fulfilled: (i) The root generators of are root generators of , and (ii) the simple roots of are real roots of . It follows that the Weyl group of is a subgroup of the Weyl group of and that the root lattice of is a sublattice of the root lattice of .

The second condition is automatic in the finite-dimensional case where there are only real roots. It must be separately imposed in the general case. Consider for instance the rank 2 Kac-Moody algebra with Cartan matrix

Let

4.73

4.74

4.75

It is easy to verify that x, y, z define an A₁ subalgebra of since [z, x] = 2x, [z, y] = −2y and [x, y] = z. Moreover, the Cartan subalgebra of A₁ is a subalgebra of the Cartan subalgebra of g, and the step operators of A₁ are step operators of . However, the simple root α = α₁ + α₂ of A₁ (which is an A₁-real root since A₁ is finite-dimensional), is an imaginary root of has norm squared equal to −2. Even though the root lattice of A₁ (namely, {±α}) is a sublattice of the root lattice of , the reflection in α is not a Weyl reflection of . According to our definition, this embedding of A₁ in is not a regular embedding.

Examples — Regular subalgebras of E₁₀

We shall describe some regular subalgebras of E₁₀. The Dynkin diagram of E₁₀ is displayed in Figure 21.

Figure 21.

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A first, simple, example of a regular embedding is the embedding of A₉ in E₁₀ which will be used to define the level when trying to reformulate eleven-dimensional supergravity as a nonlinear sigma model. This is not a maximal embedding since one can find a proper subalgebra of E₁₀ that contains A₉. One may take for the Kac-Moody subalgebra of E₁₀ generated by the operators at levels 0 and ±2, which is a subalgebra of the algebra containing all operators of even level15. It is regularly embedded in E₁₀. Its Dynkin diagram is shown in Figure 22.

Figure 22.

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In terms of the simple roots of E₁₀, the simple roots of are α₁ through α₉ and ᾱ₁₀ = 2α₁₀ + α₁ + 2α₂ + 3α₃ + 2α₄ + α₅. The algebra is Lorentzian but not hyperbolic. It can be identified with the “very extended” algebra [86].

DE₁₀ ⊂ E₁₀

In [67], Dynkin has given a method for finding all maximal regular subalgebras of finite-dimensional simple Lie algebras. The method is based on using the highest root and is not generalizable as such to general Kac-Moody algebras for which there is no highest root. Nevertherless, it is useful for constructing regular embeddings of overextensions of finite-dimensional simple Lie algebras. We illustrate this point in the case of E₈ and its overextension . In the notation of Figure 21, the simple roots of E₈ (which is regularly embedded in E₁₀) are α₁, ⋯, α₇ and α₁₀.

Applying Dynkin’s procedure to E₈, one easily finds that D₈ can be regularly embedded in E₈. The simple roots of D₈ ⊂ E₈ are α₂, α₃, α₄, α₅, α₆, α₇, α₁₀ and , where

4.76

is the highest root of E₈. One can replace this embedding, in which a simple root of D₈, namely β, is a negative root of E₈ (and the corresponding raising operator of D₈ is a lowering operator for E₈), by an equivalent one in which all simple roots of D₈ are positive roots of E₈.

This is done as follows. It is reasonable to guess that the searched-for Weyl element that maps the “old” D₈ on the “new” D₈ is some product of the Weyl reflections in the four E₈-roots orthogonal to the simple roots α₃, α₄, α₅, α₆ and α₇, expected to be shared (as simple roots) by E₈, the old D₈ and the new D₈ — and therefore to be invariant under the searched-for Weyl element. This guess turns out to be correct: Under the action of the product of the commuting E₈-Weyl reflections in the E₈-roots μ₁ = 2α₁ + 3α₂ + 5α₃ + 4α₄ + 3α₅ + 2α₆ + α₇ + 3α₁₀ and μ₂ = 2α₁+4α₂ +5α₃+4α₄+3α₅ + 2α₆+α₇+2α₁₀, the set of D₈-roots {α₂, α₃, α₄, α₅, α₆, α₇, α₁₀, β} is mapped on the equivalent set of positive roots {α₁₀, α₃, α₄, α₅, α₆, α₇, α₂, β}, where

4.77

In this equivalent embedding, all raising operators of D₈ are also raising operators of E₈. What is more, the highest root of D₈,

4.78

is equal to the highest root of E₈. Because of this, the affine root α₈ of the untwisted affine extension can be identified with the affine root of , and the overextended root α₉ can also be taken to be the same. Hence, DE₁₀ can be regularly embedded in E₁₀ (see Figure 23).

Figure 23.

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The embedding just described is in fact relevant to string theory and has been discussed from various points of view in previous papers [125, 23]. By dimensional reduction of the bosonic sector of eleven-dimensional supergravity on a circle, one gets, after dropping the Kaluza-Klein vector and the 3-form, the bosonic sector of pure ten-dimensional supergravity. The simple roots of DE₁₀ are the symmetry walls and the electric and magnetic walls of the 2-form and coincide with the positive roots given above [45]. A similar construction shows that can be regularly embedded in E₁₀, and that DE₁₀ can be regularly embedded in . See [106] for a recent discussion of DE₁₀ in the context of Type I supergravity.

More on Coxeter billiards

The Coxeter billiard of pure gravity in D spacetime dimensions

We start by providing other examples of theories leading to regular billiards, focusing first on pure gravity in any number of D (> 3) spacetime dimensions. In this case, there are d = D − 1 scale factors βⁱ and the relevant walls are the symmetry walls, Equation (2.48),

5.1

and the curvature wall, Equation (2.49),

5.2

There are thus d relevant walls, which define a simplex in (d − 1)-dimensional hyperbolic space The scalar products of the linear forms defining these walls are easily computed. One finds as non-vanishing products

5.3

The matrix of the scalar products of the wall forms is thus the Cartan matrix of the (simply-laced) Lorentzian Kac-Moody algebra with Dynkin diagram as in Figure 24. The roots of the underlying finite-dimensional algebra A_d−2 are given by s_i (i = 1, ⋯, d − 3) and r. The affine root is s_d−2 and the overextended root is s_d−1.

Figure 24.

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Accordingly, in the case of pure gravity in any number of spacetime dimensions, one finds also that the billiard region is regular. This provides new examples of Coxeter billiards, with Coxeter groups , which are also Kac-Moody billiards since the Coxeter groups are the Weyl groups of the Kac-Moody algebras .

The Coxeter billiard for the coupled gravity-3-Form system Coxeter polyhedra

Let us review the conditions that must be fulfilled in order to get a Kac-Moody billiard and let us emphasize how restrictive these conditions are. The billiard region computed from any theory coupled to gravity with n dilatons in D = d + 1 dimensions always defines a convex polyhedron in a (d + n − 1)-dimensional hyperbolic space . In the general case, the dihedral angles between adjacent faces of can take arbitrary continuous values, which depend on the dilaton couplings, the spacetime dimensions and the ranks of the p-forms involved. However, only if the dihedral angles are integer submultiples of π (meaning of the form π/k for k ∈ ℤ_≥2) do the reflections in the faces of define a Coxeter group. In this special case the polyhedron is called a Coxeter polyhedron. This Coxeter group is then a (discrete) subgroup of the isometry group of .

In order for the billiard region to be identifiable with the fundamental Weyl chamber of a Kac-Moody algebra, the Coxeter polyhedron should be a simplex, i.e., bounded by d + n walls in a d + n − 1-dimensional space. In general, the Coxeter polyhedron need not be a simplex.

There is one additional condition. The angle ϑ between two adjacent faces i and j is given, in terms of the Coxeter exponents, by

5.4

Coxeter groups that correspond to Weyl groups of Kac-Moody algebras are the crystallographic Coxeter groups for which m_ij ∈ {2, 3, 4, 6, ∞}. So, the requirement for a gravitational theory to have a Kac-Moody algebraic description is not just that the billiard region is a Coxeter simplex but also that the angles between adjacent walls are such that the group of reflections in these walls is crystallographic.

These conditions are very restrictive and hence gravitational theories which can be mapped to a Kac-Moody algebra in the BKL-limit are rare.

The Coxeter billiard of eleven-dimensional supergravity

Consider for instance the action (2.1) for gravity coupled to a single three-form in D = d + 1 spacetime dimensions. We assume D ≥ 6 since in lower dimensions the 3-form is equivalent to a scalar (D = 5) or has no degree of freedom (D < 5).

Theorem: Whenever a p-form (p ≥ 1) is present, the curvature wall is subdominant as it can be expressed as a linear combination with positive coefficients of the electric and magnetic walls of the p-forms. (These walls are all listed in Section 2.5.)

Proof: The dominant electric wall is (assuming the presence of a dilaton)

5.5

while one of the magnetic wall reads

5.6

so that the dominant curvature wall is just the sum e_1⋯p(β) + m_{1,p+1, ⋯, d−2}(β).

It follows that in the case of gravity coupled to a single three-form in D = d + 1 spacetime dimensions, the relevant walls are the symmetry walls, Equation (2.48),

5.7

(as always) and the electric wall

5.8

(D ≥ 8) or the magnetic wall

5.9

(D ≤ 8). Indeed, one can express the magnetic walls as linear combinations with (in general non-integer) positive coefficients of the electric walls for D ≥ 8 and vice versa for D ≤ 8. Hence the billiard table is always a simplex (this would not be true had one a dilaton and various forms with different dilaton couplings).

However, it is only for D = 11 that the billiard is a Coxeter billiard. In all the other spacetime dimensions, the angle between the relevant p-form wall and the symmetry wall that does not intersect it orthogonally is not an integer submultiple of π. More precisely, the angle between

• the magnetic wall β¹ and the symmetry wall β² − β¹ (D = 6),

• the magnetic wall β¹ + β² and the symmetry wall β³ − β² (D = 7), and

• the electric wall β¹ + β² + β³ and the symmetry wall β⁴ − β³ (D ≥ 8),

is easily verified to be an integer submultiple of π only for D = 11, for which it is equal to π/3. From the point of view of the regularity of the billiard, the spacetime dimension D = 11 is thus privileged. This is of course also the dimension privileged by supersymmetry. It is quite intriguing that considerations a priori quite different (billiard regularity on the one hand, supersymmetry on the other hand) lead to the same conclusion that the gravity-3-form system is quite special in D = 11 spacetime dimensions.

For completeness, we here present the wall system relevant for the special case of D = 11. We obtain ten dominant wall forms, which we rename α₁, ⋯, α₁₀,

5.10

then, defining a new collective index i = (m, 10), we see that the scalar products between these wall forms can be organized into the matrix

5.11

which can be identified with the Cartan matrix of the hyperbolic Kac-Moody algebra E₁₀ that we have encountered in Section 4.10.2. We again display the corresponding Dynkin diagram in Figure 25, where we point out the explicit relation between the simple roots and the walls of the Einstein-3-form theory. It is clear that the nine dominant symmetry wall forms correspond to the simple roots α_m of the subalgebra . The enlargement to E₁₀ is due to the tenth exceptional root realized here through the dominant electric wall form e₁₂₃.

Figure 25.

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Dynamics in the Cartan subalgebra

We have just learned that, in some cases, the group of reflections that describe the (possibly chaotic) dynamics in the BKL-limit is a Lorentzian Coxeter group . In this section we fully exploit this algebraic fact and show that whenever is crystallographic, the dynamics takes place in the Cartan subalgebra of the Lorentzian Kac-Moody algebra , for which is the Weyl group. Moreover, we show that the “billiard table” can be identified with the fundamental Weyl chamber in .

Billiard dynamics in the Cartan subalgebra

Scale factor space and the wall system

Let us first briefly review some of the salient features encountered so far in the analysis. In the following we denote by the Lorentzian “scale factor”-space (or β-space) in which the billiard dynamics takes place. Recall that the metric in , induced by the Einstein-Hilbert action, is a flat Lorentzian metric, whose explicit form in terms of the (logarithmic) scale factors reads

5.12

where d counts the number of physical spatial dimensions (see Section 2.5). The role of all other “off-diagonal” variables in the theory is to interrupt the free-flight motion of the particle, by adding walls in that confine the motion to a limited region of scale factor space, namely a convex cone bounded by timelike hyperplanes. When projected onto the unit hyperboloid, this region defines a simplex in hyperbolic space which we refer to as the “billiard table”.

One has, in fact, more than just the walls. The theory provides these walls with a specific normalization through the Lagrangian, which is crucial for the connection to Kac-Moody algebras. Let us therefore discuss in somewhat more detail the geometric properties of the wall system. The metric, Equation (5.12), in scale factor space can be seen as an extension of a flat Euclidean metric in Cartesian coordinates, and reflects the Lorentzian nature of the vector space . In this space we may identify a pair of coordinates (βⁱ, φ) with the components of a vector , with respect to a basis {ū_μ} of , such that

5.13

The walls themselves are then defined by hyperplanes in this linear space, i.e., as linear forms ω = ω_μσ^μ, for which ω = 0, where {σ^μ} is the basis dual to {ū^μ}. The pairing ω(β) between a vector and a form is sometimes also denoted by ⟨ω, β⟩, and for the two dual bases we have, of course,

5.14

We therefore find that the walls can be written as linear forms in the scale factors:

5.15

We call ω(β) wall forms. With this interpretation they belong to the dual space , i.e.,

5.16

From Equation (5.16) we may conclude that the walls bounding the billiard are the hyperplanes ω = 0 through the origin in which are orthogonal to the vector with components ω^μ = G^μvω_v.

It is important to note that it is the wall forms that the theory provides, as arguments of the exponentials in the potential, and not just the hyperplanes on which these forms ω vanish. The scalar products between the wall forms are computed using the metric in the dual space , whose explicit form was given in Section 2.5,

5.17

Scale factor space and the Cartan subalgebra

The crucial additional observation is that (for the “interesting” theories) the matrix A associated with the relevant walls ω_A,

5.18

is a Cartan matrix, i.e., besides having 2’s on its diagonal, which is rather obvious, it has as off-diagonal entries non-positive integers (with the property A_AB ≠ 0 ⇒ A_BA ≠ 0). This Cartan matrix is of course symmetrizable since it derives from a scalar product.

For this reason, one can usefully identify the space of the scale factors with the Cartan subalgebra of the Kac-Moody algebra defined by A. In that identification, the wall forms become the simple roots, which span the vector space dual to the Cartan subalgebra. The rank r of the algebra is equal to the number of scale factors β^μ, including the dilaton(s) if any ((β^μ) ≡ (βⁱ, φ)). This number is also equal to the number of walls since we assume the billiard to be a simplex. So, both A and μ run from 1 to r. The metric in , Equation (5.12), can be identified with the invariant bilinear form of , restricted to the Cartan subalgebra . The scale factors β^μ of become then coordinates h^μ on the Cartan subalgebra .

The Weyl group of a Kac-Moody algebra has been defined first in the space as the group of reflections in the walls orthogonal to the simple roots. Since the metric is non degenerate, one can equivalently define by duality the Weyl group in the Cartan algebra itself (see Section 4.7). For each reflection r_i on we associate a dual reflection as follows,

5.19

which is the reflection relative to the hyperplane α_i(β) = ⟨α_i, β⟩ = 0. This expression can be rewritten (see Equation (4.59)),

5.20

or, in terms of the scale factor coordinates β^μ,

5.21

This is precisely the billiard reflection Equation (2.45) found in Section 2.4.

Thus, we have the following correspondence:

5.22

As we have also seen, the Kac-Moody algebra is Lorentzian since the signature of the metric Equation (5.12) is Lorentzian. This fact will be crucial in the analysis of subsequent sections and is due to the presence of gravity, where conformal rescalings of the metric define timelike directions in scale factor space.

We thereby arrive at the following important result [45, 46, 48]:

The dynamics of (a restricted set of) theories coupled to gravity can in the BKL-limit be mapped to a billiard motion in the Cartan subalgebra of a Lorentzian Kac-Moody algebra .

The fundamental Weyl chamber and the billiard table

Let denote the region in scale factor space to which the billiard motion is confined,

5.23

where the index A runs over all relevant walls. On the algebraic side, the fundamental Weyl chamber in is the closed convex (half) cone given by

5.24

We see that the conditions α_A(h) ≥ 0 defining are equivalent, upon examination of Equation (5.22), to the conditions ω_A(β) ≥ 0 defining the billiard table .

We may therefore make the crucial identification

5.25

which means that the particle geodesic is confined to move within the fundamental Weyl chamber of . From the billiard analysis in Section 2 we know that the piecewise motion in scale-factor space is controlled by geometric reflections with respect to the walls ω_A(β) = 0. By comparing with the dominant wall forms and using the correspondence in Equation (5.22) we may further conclude that the geometric reflections of the coordinates β^μ(τ) are controlled by the Weyl group in the Cartan subalgebra of .

Hyperbolicity implies chaos

We have learned that the BKL dynamics is chaotic if and only if the billiard table is of finite volume when projected onto the unit hyperboloid. From our discussion of hyperbolic Coxeter groups in Section 3.5, we see that this feature is equivalent to hyperbolicity of the corresponding Kac-Moody algebra. This leads to the crucial statement [45, 46, 48]:

If the billiard region of a gravitational system in the BKL-limit can be identified with the fundamental Weyl chamber of a hyperbolic Kac-Moody algebra, then the dynamics is chaotic.

As we have also discussed above, hyperbolicity can be rephrased in terms of the fundamental weights Λ_i defined as

5.26

Just as the fundamental Weyl chamber in can be expressed in terms of the fundamental weights (see Equation (3.40)), the fundamental Weyl chamber in can be expressed in a similar fashion in terms of the fundamental coweights:

5.27

As we have seen (Sections 3.5 and 4.8), hyperbolicity holds if and only if none of the fundamental weights are spacelike,

5.28

for all .

Example: Pure gravity in D = 3 + 1 and

Let us return once more to the example of pure four-dimensional gravity, i.e., the original “BKL billiard”. We have already found in Section 3 that the three dominant wall forms give rise to the Cartan matrix of the hyperbolic Kac-Moody algebra [46, 48]. Since the algebra is hyperbolic, this theory exhibits chaotic behavior. In this example, we verify that the Weyl chamber is indeed contained within the lightcone by computing explicitly the norms of the fundamental weights.

It is convenient to first write the simple roots in the β-basis as follows¿

5.29

Since the Cartan matrix of is symmetric, the relations defining the fundamental weights are

5.30

By solving these equations for Λ_i we deduce that the fundamental weights are

5.31

where in the last step we have written the fundamental weights in the β-basis. The norms may now be computed with the metric in root space and become

5.32

We see that Λ₁ and Λ₂ are timelike and that Λ₃ is lightlike. Thus, the Weyl chamber is indeed contained inside the lightcone, the algebra is hyperbolic and the billiard is of finite volume, in agreement with what we already found [46].

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 is a split real form. By this we mean that the real algebra possesses the same Chevalley-Serre presentation as , 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 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 . The analysis is somewhat more involved when is non-split, and we postpone a discussion of this until Section 7.

Invariance under toroidal dimensional reduction

It was shown in [41] 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 [35, 41] is

5.33

where μ, v = 0, 2, ⋯, 10, i.e., the compactification is performed along the first spatial direction16. We will refer to the new lower-dimensional fields and as the dilaton and the Kaluza-Klein (KK) vector, respectively. Quite generally, hatted fields are low-dimensional fields. The ten-dimensional Lagrangian becomes

5.34

where and 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, . Explicitly we have

5.35

The nine remaining eleven-dimensional scale factors, β², ⋯, β¹⁰, may in turn be written in terms of the new scale factors, , associated to the ten-dimensional metric, ĝ_μv, and the dilaton in the following way:

5.36

We are interested in finding the dominant wall forms in terms of the new scale factors and . It is clear that we will have eight ten-dimensional symmetry walls,

5.37

which correspond to the eight simple roots of . Using Equation (5.35) and Equation (5.36) 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,

5.38

The metric in the dual space gets modified in a natural way,

5.39

i.e., the dilaton contributes with a flat spatial direction. Using this metric it is clear that has non-vanishing scalar product only with the second symmetry wall , and it follows that the electric wall of the Kaluza-Klein vector plays the role of the first simple root of . The final wall form that completes the set will correspond to the exceptional node labeled “10” in Figure 25 and is now given by one of the electric walls of the NS-NS 2-form Â⁽²⁾, namely

5.40

It is then easy to verify that this wall form has non-vanishing scalar product only with the third simple root , 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

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 [166, 165]. 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.

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 . 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 . Here, is the maximal compact subgroup defining the “local symmetries”. In order to understand the connection between the U-duality group and the Kac-Moody algebras appearing in the BKL-limit, we must first discuss some important features of the Lie algebra .

Let be a split real form, meaning that it can be defined in terms of the Chevalley-Serre presentation of the complexified Lie algebra by simply restricting all linear combinations of generators to the real numbers ℝ. Let be the Cartan subalgebra of appearing in the Chevalley-Serre presentation, spanned by the generators . It is maximally noncompact (see Section 6). An Iwasawa decomposition of is a direct sum of vector spaces of the following form,

5.42

where is the “maximal compact subalgebra” of , and 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 times an element in the subgroup whose Lie algebra is . An arbitrary element of the coset 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,

5.43

In that gauge we have

5.44

where φ and χ are (sets of) coordinates on the coset space . A Lagrangian based on this coset will then take the generic form (see Section 9)

5.45

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 χ^(α) for each positive root α ∈ Δ₊ and one dilaton field φ⁽ⁱ⁾ for each Cartan generator .

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

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, T⁸, to D = 2 + 1 spacetime dimensions (after all form fields have been dualized into scalars),

5.46

The second two terms in this Lagrangian correspond to the coset model Ɛ₈₍₈₎/(Spin(16)/ℤ₂), where Ɛ₈₍₈₎ denotes the group obtained by exponentiation of the split form E₈₍₈₎ of the complex Lie algebra E₈ and Spin(16)/ℤ₂ is the maximal compact subgroup of Ɛ₈₍₈₎ [33, 134, 35]. The 8 dilatons and the 120 axions χ^(q) are coordinates on the coset space17. Furthermore, the are linear forms on the elements of the Cartan subalgebra and they correspond to the positive roots of E₈₍₈₎ 18. As before, we do not write explicitly the corrections to the curvatures 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 (we omit the “hatted” notation on the roots since there is no longer any risk of confusion),

5.47

and the exceptional root

5.48

These correspond exactly to the root vectors and as they appear in the analysis of [35], except for the additional factor of 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 δ_ij (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 A⁽³⁾ have been transformed, in the same spirit as for the example given above, into electric and magnetic walls of the axionic scalars . Since the terms involving the electric fields possess no spatial indices, the corresponding wall forms do not contain any of the remaining scale factors , and are simply linear forms on the dilatons only. In fact the dominant electric wall forms are just the simple roots of E₈,

5.49

The magnetic wall forms naturally come with one factor of since the magnetic field strength carries one spatial index. The dominant magnetic wall form is then given by

5.50

where denotes the highest root of E₈ which takes the following form in terms of the simple roots,

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

5.52

coming from the three-dimensional metric ĝ_μv (μ, v = 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 . Indeed, the relevant electric roots are the simple roots of E₈, the magnetic root of Equation (5.50) provides the affine extension, while the gravitational root of Equation (5.52) 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 in 3 dimensions when is a split real form. The Coxeter group and the corresponding Kac-Moody algebra are given by the untwisted overextension . This overextension emerges as follows [41]:

• The dominant electric wall forms for the supergravity theory in question are in one-to-one correspondence with the simple roots of the associated U-duality algebra .

• Adding the dominant magnetic wall form corresponds to an affine extension of .

• Finally, completing the set of dominant wall forms with the symmetry wall , which is the only gravitational wall form existing in three dimensions, is equivalent to an overextension of .

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.