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10.2 Geometric configurations and regular subalgebras of E10

We prove here that the conditions on the electric field embodied in the geometric configurations (nm, g3) have a direct Kac–Moody algebraic interpretation. They simply correspond to a consistent truncation of the E10 nonlinear sigma model to a ¯ 𝔤 nonlinear sigma model, where ¯ 𝔤 is a rank g Kac–Moody subalgebra of E10 (or a quotient of such a Kac–Moody subalgebra by an appropriate ideal when the relevant Cartan matrix has vanishing determinant), with three crucial properties: (i) It is regularly embedded in E10 (see Section 4 for the definition of regular subalgebras), (ii) it is generated by electric roots only, and (iii) every node P in its Dynkin diagram 𝔻¯ 𝔤 is linked to a number k of nodes that is independent of P (but depend on the algebra). We find that the Dynkin diagram 𝔻¯𝔤 of ¯𝔤 is the line incidence diagram of the geometric configuration (nm, g3), in the sense that (i) each line of (nm, g3) defines a node of 𝔻 ¯𝔤, and (ii) two nodes of 𝔻¯𝔤 are connected by a single bond iff the corresponding lines of (nm, g3) have no point in common. This defines a geometric duality between a configuration (n ,g ) m 3 and its associated Dynkin diagram 𝔻 ¯𝔤. In the following we shall therefore refer to configurations and Dynkin diagrams related in this way as dual.

None of the algebras ¯𝔤 relevant to the truncated models turn out to be hyperbolic: They can be finite, affine, or Lorentzian with infinite-volume Weyl chamber. Because of this, the solutions are non-chaotic. After a finite number of collisions, they settle asymptotically into a definite Kasner regime (both in the future and in the past).

10.2.1 General considerations

In order to match diagonal Bianchi I cosmologies with the sigma model, one must truncate the ℰ ∕𝒦 (ℰ ) 10 10 action in such a way that the sigma model metric g ab is diagonal. This will be the case if the subalgebra ¯𝔤 to which one truncates has no generator i K j with i ⁄= j. Indeed, recall from Section 9 that the off-diagonal components of the metric are precisely the exponentials of the associated sigma model fields. The set of simple roots of ¯𝔤 should therefore not contain any root at level zero.

Consider “electric” regular subalgebras of E 10, for which the simple roots are all at level one, where the 3-form electric field variables live. These roots can be parametrized by three indices corresponding to the indices of the electric field, with i1 < i2 < i3. We denote them αi1i2i3. For instance, α123 ≡ α10. In terms of the β-parametrization of [45Jump To The Next Citation Point48Jump To The Next Citation Point], one has αi ii = βi1 + βi2 + βi3 1 23.

Now, for ¯ 𝔤 to be a regular subalgebra, it must fulfill, as we have seen, the condition that the difference between any two of its simple roots is not a root of E10: αii i − αi′i′i′∕∈ ΦE 123 12 3 10 for any pair αii i 123 and α ′ ′′ i1i2i3 of simple roots of ¯𝔤. But one sees by inspection of the commutator of Ei1i2i3 with F ′ ′′ i1i2i3 in Equation (8.78View Equation) that αi1i2i3 − αi′1i′2i′3 is a root of E10 if and only if the sets {i1,i2,i3} and {i′1,i′2,i′3} have exactly two points in common. For instance, if i1 = i′1, i2 = i′2 and i3 ⁄= i′3, the commutator of Ei1i2i3 with Fi′1i′2i′3 produces the off-diagonal generator Ki3i′3 corresponding to a level zero root of E10. In order to fulfill the required condition, one must avoid this case, i.e., one must choose the set of simple roots of the electric regular subalgebra ¯ 𝔤 in such a way that given a pair of indices (i1,i2), there is at most one i3 such that the root αijk is a simple root of ¯𝔤, with (i,j,k) being the re-ordering of (i1,i2,i3) such that i < j < k.

To each of the simple roots α i1i2i3 of ¯𝔤, one can associate the line (i ,i,i ) 1 2 3 connecting the three points i1, i2 and i3. If one does this, one sees that the above condition is equivalent to the following statement: The set of points and lines associated with the simple roots of ¯𝔤 must fulfill the third rule defining a geometric configuration, namely, that two points determine at most one line. Thus, this geometric condition has a nice algebraic interpretation in terms of regular subalgebras of E 10.

The first rule, which states that each line contains 3 points, is a consequence of the fact that the E10-generators at level one are the components of a 3-index antisymmetric tensor. The second rule, that each point is on m lines, is less fundamental from the algebraic point of view since it is not required to hold for ¯𝔤 to be a regular subalgebra. It was imposed in [61Jump To The Next Citation Point] in order to allow for solutions isotropic in the directions that support the electric field. We keep it here as it yields interesting structure.

10.2.2 Incidence diagrams and Dynkin diagrams

We have just shown that each geometric configuration (nm, g3) with n ≤ 10 defines a regular subalgebra ¯𝔤 of E 10. In order to determine what this subalgebra ¯𝔤 is, one needs, according to the theorem recalled in Section 4, to compute the Cartan matrix

[( )] C = [Ci1i2i3,i′1i′2i′3] = αi1i2i3|αi′1i′2i′3 (10.10 )
(the real roots of E10 have length squared equal to 2). According to that same theorem, the algebra ¯𝔤 is then just the rank g Kac–Moody algebra with Cartan matrix C, unless C has zero determinant, in which case ¯𝔤 might be the quotient of that algebra by a nontrivial ideal.

Using for instance the root parametrization of [4548] and the expression of the scalar product in terms of this parametrization, one easily verifies that the scalar product (α |α ′′ ′) i1i2i3 i1i2i3 is equal to

( ||{ 2 if all three indices coincide, ( ′′′) 1 if two and only two indices coincide, αi1i2i3|αi1i2i3 = || 0 if one and only one index coincides, (10.11 ) ( − 1 if no indices coincide.
The second possibility does not arise in our case since we deal with geometric configurations. For completeness, we also list the scalar products of the electric roots αijk (i < j < k) with the symmetry roots α ℓm (ℓ < m) associated with the raising operators Km ℓ:
( − 1 if ℓ ∈ {i,j,k} and m ∕∈ {i,j,k}, { (αijk|α ℓm ) = ( 0 if { ℓ,m } ⊂ {i,j,k} or { ℓ,m } ∩ {i,j,k} = ∅, (10.12 ) 1 if ℓ ∕∈ {i,j,k} and m ∈ {i,j,k},
as well as the scalar products of the symmetry roots among themselves,
(| − 1 if j = ℓ or i = m, |{ (αij|αℓm) = 0 if {ℓ,m } ∩ {i,j} = ∅, (10.13 ) ||( 1 if i = ℓ or j ⁄= m, 2 if {ℓ,m } = {i,j}.
Given a geometric configuration (nm, g3), one can associate with it a “line incidence diagram” that encodes the incidence relations between its lines. To each line of (nm, g3) corresponds a node in the incidence diagram. Two nodes are connected by a single bond if and only if they correspond to lines with no common point (“parallel lines”). Otherwise, they are not connected37. By inspection of the above scalar products, we come to the important conclusion that the Dynkin diagram of the regular, rank g, Kac–Moody subalgebra ¯𝔤 associated with the geometric configuration (nm, g3) is just its line incidence diagram. We shall call the Kac–Moody algebra ¯𝔤 the algebra “dual” to the geometric configuration (nm, g3).

Because the geometric configurations have the property that the number of lines through any point is equal to a constant m, the number of lines parallel to any given line is equal to a number k that depends only on the configuration and not on the line. This is in fact true in general and not only for n ≤ 10 as can be seen from the following argument. For a configuration with n points, g lines and m lines through each point, any given line Δ admits 3(m − 1) true secants, namely, (m − 1) through each of its points38. By definition, these secants are all distinct since none of the lines that Δ intersects at one of its points, say P, can coincide with a line that it intersects at another of its points, say P ′, since the only line joining P to ′ P is Δ itself. It follows that the total number of lines that Δ intersects is the number of true secants plus Δ itself, i.e., 3 (m − 1) + 1. As a consequence, each line in the configuration admits k = g − [3(m − 1) + 1] parallel lines, which is then reflected by the fact that each node in the associated Dynkin diagram has the same number k of adjacent nodes.

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