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8.4 Level decomposition of E10

As we have seen, the Kac–Moody algebra E10 is one of the four hyperbolic algebras of maximal rank, the others being BE10, DE10 and CE10. It can be constructed as an overextension of E8 and is therefore often denoted by E++8. Similarly to E8 in the rank 8 case, E10 is the unique indefinite rank 10 algebra with an even self-dual root lattice, namely the Lorentzian lattice Π1,9.

Our first encounter with E10 in a physical application was in Section 5 where we have showed that the Weyl group of E10 describes the chaos that emerges when studying eleven-dimensional supergravity close to a spacelike singularity [45Jump To The Next Citation Point].

In Section 9.3, we will discuss how to construct a Lagrangian manifestly invariant under global E10-transformations and compare its dynamics to that of eleven-dimensional supergravity. The level decomposition associated with the removal of the “exceptional node” labelled “10” in Figure 49View Image will be central to the analysis. It turns out that the low-level structure in this decomposition precisely reproduces the bosonic field content of eleven-dimensional supergravity [47Jump To The Next Citation Point].

Moreover, decomposing E10 with respect to different regular subalgebras reproduces also the bosonic field contents of the Type IIA and Type IIB supergravities. The fields of the IIA theory are obtained by decomposition in terms of representations of the D9 = š”°š”¬(9,9,ā„ ) subalgebra obtained by removing the first simple root α1 [125Jump To The Next Citation Point]. Similarly the IIB-fields appear at low levels for a decomposition with respect to the A ⊕ A = š”°š”©(9,ā„) ⊕ š”°š”©(2,ā„) 9 1 subalgebra found upon removal of the second simple root α 2 [126Jump To The Next Citation Point]. The extra A1-factor in this decomposition ensures that the SL (2,ā„)-symmetry of IIB supergravity is recovered.

For these reasons, we investigate now these various level decompositions.

8.4.1 Decomposition with respect to š–˜š–‘(10, ā„)

Let α ,⋅⋅⋅ ,α 1 10 denote the simple roots of E 10 and α∨,⋅ ⋅⋅ ,α∨ 1 10 the Cartan generators. These span the root space ā‹† š”„ and the Cartan subalgebra š”„, respectively. Since E10 is simply laced the Cartan matrix is given by the scalar products between the simple roots:

( ) 2 − 1 0 0 0 0 0 0 0 0 | − 1 2 − 1 0 0 0 0 0 0 0| || || | 0 − 1 2 − 1 0 0 0 0 0 − 1| || 0 0 − 1 2 − 1 0 0 0 0 0|| || 0 0 0 − 1 2 − 1 0 0 0 0|| Aij[E10 ] = (αi|αj) = | 0 0 0 0 − 1 2 − 1 0 0 0| . (8.69 ) || 0 0 0 0 0 − 1 2 − 1 0 0|| || || | 0 0 0 0 0 0 − 1 2 − 1 0| ( 0 0 0 0 0 0 0 − 1 2 0) 0 0 − 1 0 0 0 0 0 0 2
The associated Dynkin diagram is displayed in Figure 49View Image. We will perform the decomposition with respect to the š”°š”©(10, ā„) subalgebra represented by the horizontal line in the Dynkin diagram so the level ā„“ of an arbitrary root α ∈ š”„ā‹† is given by the coefficient in front of the exceptional simple root, i.e.,
∑ 9 γ = mi αi + ā„“α10. (8.70 ) i=1

As before, the weight that is easiest to identify for each representation ā„› (Λ (ā„“)) at positive level ā„“ is the lowest weight Λ(ā„“) lw. We denote by ¯Λ (ā„“) lw the projection onto the spacelike slice of the root lattice defined by the level ā„“. The (conjugate) Dynkin labels p ,⋅⋅⋅ ,p 1 9 characterizing the representation (ā„“) ā„› (Λ ) are defined as before as minus the coefficients in the expansion of (ā„“) ¯Λ lw in terms of the fundamental weights λi of š”°š”©(10,ā„ ):

∑9 − ¯Λ(ā„“) = piλi. (8.71 ) lw i=1
View Image

Figure 49: The Dynkin diagram of E10. Labels i = 1,⋅⋅⋅ ,9 enumerate the nodes corresponding to simple roots α i of the š”°š”©(10,ā„ ) subalgebra and “10” labels the exceptional node.

The Killing form on each such slice is positive definite so the projected weight (ā„“) ¯Λhw is of course real. The fundamental weights of š”°š”©(10,ā„ ) can be computed explicitly from their definition as the duals of the simple roots:

9 i ∑ ij λ = B αj, (8.72 ) j=1
where Bij is the inverse of the Cartan matrix of A9,
( ) 9 8 7 6 5 4 3 2 1 || 8 16 14 12 10 8 6 4 2|| | 7 14 21 18 15 12 9 6 3| || 6 12 18 24 20 16 12 8 4|| −1 -1-|| || (Bij[A9]) = 10 | 5 10 15 20 25 20 15 10 5| . (8.73 ) || 4 8 12 16 20 24 18 12 6|| || 3 6 9 12 15 18 21 14 7|| ( 2 4 6 8 10 12 14 16 8) 1 2 3 4 5 6 7 8 9
Note that all the entries of Bij are positive which will prove to be important later on. As we saw for the AE3 case we want to find the possible allowed values for (m1, ⋅⋅⋅ ,m9 ), or, equivalently, the possible Dynkin labels [p1,⋅⋅⋅ ,p9] for each level ā„“.

The corresponding diophantine equation, Equation (8.50View Equation), for E10 was found in [47Jump To The Next Citation Point] and reads

9 i i3 ∑ ij m = B ā„“ − B pj ≥ 0. (8.74 ) j=1
Since the two sets {pi} and {mi } both consist of non-negative integers and all entries of Bij are positive, these equations put strong constraints on the possible representations that can occur at each level. Moreover, each lowest weight vector (ā„“) Λ = γ must be a root of E10, so we have the additional requirement
∑9 1 (Λ (ā„“)|Λ (ā„“)) = Bijpipj − --ā„“2 ≤ 2. (8.75 ) i,j=1 10

The representation content at each level is represented by š”°š”©(10,ā„ )-tensors whose index structure are encoded in the Dynkin labels [p1,⋅⋅⋅ ,p9]. At level ā„“ = 0 we have the adjoint representation of š”°š”©(10,ā„ ) represented by the generators Kab obeying the same commutation relations as in Equation (8.32View Equation) but now with š”°š”©(10,ā„ )-indices.

All higher (lower) level representations will then be tensors transforming contravariantly (covariantly) under the level ā„“ = 0 generators. The resulting representations are displayed up to level 3 in Table 39. We see that the level 1 and 2 representations have the index structures of a 3-form and a 6-form respectively. In the E10-invariant sigma model, to be constructed in Section 9, these generators will become associated with the time-dependent physical “fields” Aabc(t) and Aa1⋅⋅⋅a6(t) which are related to the electric and magnetic component of the 3-form in eleven-dimensional supergravity. Similarly, the level 3 generator Ea |b1⋅⋅⋅b9 with mixed Young symmetry will be associated to the dual of the spatial part of the eleven-dimensional vielbein. This field is therefore sometimes referred to as the “dual graviton”.


Table 39: The low-level representations in a decomposition of the adjoint representation of E10 into representations of its A 9 subalgebra obtained by removing the exceptional node in the Dynkin diagram in Figure 49View Image.
ā„“ (ā„“) Λ = [p1,⋅⋅⋅ ,p9] (ā„“) Λ = (m1, ⋅⋅⋅ ,m10 ) A9-representation E10-generator
1 [0,0,1,0,0, 0,0,0,0] (0,0, 0,0,0,0,0,0,0,1 ) 1201 abc E
2 [0,0,0,0,0, 1,0,0,0] (1,2, 3,2,1,0,0,0,0,2 ) 2102 Ea1⋅⋅⋅a6
3 [1,0,0,0,0, 0,0,1,0] (1,3, 5,4,3,2,1,0,0,3 ) 440 3 Ea |b1⋅⋅⋅b8

Algebraic structure at low levels

Let us now describe in a little more detail the commutation relations between the low-level generators in the level decomposition of E10 (see Table 39). We recover the Chevalley generators of A9 through the following realisation:

ei = Ki+1i, fi = Kii+1, hi = Ki+1i+1 − Kii (i = 1,⋅⋅⋅ ,9), (8.76 )
where, as before, the i K j’s obey the commutation relations
i k k i i k [K j,K l] = δjK l − δlK j. (8.77 )
At levels ±1 we have the positive root generators Eabc and their negative counterparts abc Fabc = − τ(E ), where τ denotes the Chevalley involution as defined in Section 4. Their transformation properties under the š”°š”©(10,ā„)-generators a K b follow from the index structure and reads explicitly
a cde [c de]a [K b,E ] = 3δb E , [Kab, Fcde] = − 3δa[cFde]b, 10 (8.78 ) abc [ab c] abc ∑ a [E ,Fdef] = 18δ[deK f] − 2δdef K a, a=1
where we defined
δab = 1(δaδb− δbδa) cd 2 c d c d (8.79 ) abc 1- a b c δdef = 3!(δdδeδf ± 5 permutations ).
The “exceptional” generators e10 and f10 are fixed by Equation (8.76View Equation) to have the following realisation:
123 e10 = E , f10 = F123. (8.80 )
The corresponding Cartan generator is obtained by requiring [e10,f10] = h10 and upon inspection of the last equation in Equation (8.78View Equation) we find
1 ∑ a 2 1 2 3 h10 = − -- K a + -(K 1 + K 2 + K 3), (8.81 ) 3iā„=1,2,3 3
enlarging š”°š”©(10, ā„) to š”¤š”©(10,ā„ ).

The bilinear form at level zero is

(Kij |Kkl) = δilδkj − δijδkl (8.82 )
and can be extended level by level to the full algebra by using its invariance, ([x, y]|z) = (x|[y,z]) for x, y,z ∈ E10 (see Section 4). For level 1 this yields
( abc ) abc E |Fdef = 3!δdef, (8.83 )
where the normalization was chosen such that
(e |f ) = (E123|F ) = 1. (8.84 ) 10 10 123

Now, by using the graded structure of the level decomposition we can infer that the level 2 generators can be obtained by commuting the level 1 generators

[š”¤1,š”¤1] ⊆ š”¤2. (8.85 )
Concretely, this means that the level 2 content should be found from the commutator
a1a2a3 a4a5a6 [E ,E ]. (8.86 )
We already know that the only representation at this level is 2102, realized by an antisymmetric 6-form. Since the normalization of this generator is arbitrary we can choose it to have weight one and hence we find
Ea1⋅⋅⋅a6 = [Ea1a2a3,Ea4a5a6]. (8.87 )
The bilinear form is lifted to level 2 in a similar way as before with the result
(Ea1⋅⋅⋅a6|Fb1⋅⋅⋅b6) = 6!δab11⋅⋅⋅⋅⋅⋅ba66 . (8.88 )
Continuing these arguments, the level 3-generators can be obtained from
[[š”¤1,š”¤1],š”¤1] ⊆ š”¤3. (8.89 )
From the index structure one would expect to find a 9-form generator a1⋅⋅⋅a9 E corresponding to the Dynkin labels [0,0,0,0, 0,0,0,0,1]. However, we see from Table 39 that only the representation [1,0,0,0,0,0, 0,1,0] appears at level 3. The reason for the disappearance of the representation [0,0,0,0,0,0, 0,0,1] is because the generator Ea1⋅⋅⋅a9 is not allowed by the Jacobi identity. A detailed explanation for this can be found in [77]. The right hand side of Equation (8.89View Equation) therefore only contains the index structure compatible with the generators a|b1⋅⋅⋅b8 E,
ab1b2 b3b4b5 b6b7b8 [a|b1b2]b3⋅⋅⋅b8 [[E ,E ],E ] = − E , (8.90 )
where the minus sign is purely conventional.

For later reference, we list here some additional commutators that are useful [53Jump To The Next Citation Point]:

[Ea1⋅⋅⋅a6,Fb b b] = − 5!δ[a1a2a3Ea1a2a3], 1 23 b1b2b3 [a⋅⋅⋅a 2 a⋅⋅⋅a ∑10 [Ea1⋅⋅⋅a6,Fb1⋅⋅⋅b6] = 6 ⋅ 6!δ[b11⋅⋅⋅b55Ka6]b6] −-⋅ 6!δb11⋅⋅⋅b66 Kaa, ( 3 a=1 ) (8.91 ) [Ea1 |a2⋅⋅⋅a9,F ] = − 7 ⋅ 48 δa1[a2a3Ea4⋅⋅⋅a9] − δ[a2a3a4Ea5 ⋅⋅⋅a9]a1 , b1b2b3 b1b2b3 b1b2b3 ( a1[a2⋅⋅⋅a6 aa a ] [a2⋅⋅⋅a7 a a]a ) [Ea1|a2⋅⋅⋅a9,Fb1⋅⋅⋅b6] = − 8! δb1⋅⋅⋅b6 E 78 9 − δb1⋅⋅⋅b6 E 8 9 1 .

8.4.2 “Gradient representations”

So far, we have only discussed the representations occurring at the first four levels in the E10 decomposition. This is due to the fact that a physical interpretation of higher level fields is yet to be found. There are, however, among the infinite number of representations, a subset of three (infinite) towers of representations with certain appealing properties. These are the “gradient representations”, so named due to their conjectured relation to the emergence of space, through a Taylor-like expansion in spatial gradients [47Jump To The Next Citation Point]. We explain here how these representations arise and we emphasize some of their important properties, leaving a discussion of the physical interpretation to Section 9.

The gradient representations are obtained by searching for “affine representations”, for which the coefficient 9 m in front of the overextended simple root of E10 vanishes, i.e., the lowest weights of the representations correspond to the following subset of E10 roots,

∑ 8 γ = mi αi + ā„“α10. (8.92 ) i=1

The Dynkin labels allowed by this restricting are parametrized by an integer k which is related to the level at which a specific representation occurs in the following way:

ā„“ = 3k + 1 [0,0, 1,0,0,0,0,0,k ], (8.93 ) ā„“ = 3k + 2 [0,0, 0,0,0,1,0,0,k ], (8.94 ) ā„“ = 3k + 3 [1,0, 0,0,0,0,0,1,k ]. (8.95 )
One easily verifies that these representations fulfill the diophantine constraints (8.74View Equation) and the lowest weight has length squared, Equation (8.75View Equation), equal to 2 and is thus indeed a real root of E10. For k = 0 these representations reduce to the ones for ā„“ = 1,2 and 3, and hence the gradient representations correspond to generalizations of these standard low-level structures. The corresponding generators have, respectively at levels ā„“ = 3k + 1,3k + 2 or 3k + 3, additional sets of k “9-tuples” of antisymmetric indices,
[0,0,1, 0,0,0,0,0,k] =⇒ Ea1 ⋅⋅⋅a9,b1⋅⋅⋅b9,⋅⋅⋅,c1c2c3, [0,0,0, 0,0,1,0,0,k] =⇒ Ea1 ⋅⋅⋅a9,b1⋅⋅⋅b9,⋅⋅⋅,c1⋅⋅⋅c6, (8.96 ) [1,0,0, 0,0,0,0,1,k] =⇒ Ea1 ⋅⋅⋅a9,b1⋅⋅⋅b9,⋅⋅⋅,|c|d1⋅⋅⋅d8
(with the irreducibility conditions expressing that antisymmetrizations involving one more index over the explicit antisymmetry are zero). Since these are š”°š”©(10,ā„ )-representations we can use the rank 10 antisymmetric epsilon tensor ε a1⋅⋅⋅a10 to dualize these representations, for instance for the ā„“ = 3k + 1 tower we get
a1a2a3 c1⋅⋅⋅c9,d1⋅⋅⋅d9,⋅⋅⋅,e1⋅⋅⋅e9,a1a2a3 E b1⋅⋅⋅bk = εb1c1⋅⋅⋅c9εb2d1⋅⋅⋅d9 ⋅⋅⋅εbke1⋅⋅⋅e9E , (8.97 )
where the lower indices b ⋅⋅⋅b 1 k are now completely symmetric and furthermore, obey appropriate tracelessness conditions when contracted with an upper index.

Thus, in this way we obtain the three infinite towers of E10 generators

Ea1a2a3b1⋅⋅⋅bk, Ea1 ⋅⋅⋅a6b1⋅⋅⋅bk, Ea1|a2⋅⋅⋅a9b1⋅⋅⋅bk. (8.98 )
The lowest weight vectors of these representations are all spacelike and so these representations always come with outer multiplicity one.

The existence of these towers of representations is not special for E10 among the exceptional algebras, although the symmetric Young structure of the lower indices is actually a very special and important feature of E10. In Section 9 we will discuss the tantalizing possibility that these representations encode an infinite set of spatial gradients that describe the emergence, or “unfolding”, of space.

To illustrate the difference from other exceptional algebras, we consider, for instance, a similar search for affine representations within E11 (see, e.g. [141Jump To The Next Citation Point]). The same sets of 9-tuples appear, but now these should be dualized with the rank 11 epsilon tensor of š”°š”©(11,ā„ ), leaving us with three towers of generators that have k pairs of antisymmetric indices, i.e.,

μ1μ2μ3 μ1⋅⋅⋅μ6 μ1|σ2⋅⋅⋅σ9 E [ν1ρ1]⋅⋅⋅[νkρk], E [ν1ρ1]⋅⋅⋅[νkρk], E [ν1ρ1]⋅⋅⋅[νkρk], (8.99 )
where all indices are š”°š”©(11, ā„)-indices and so run from 1 to 11. No interpretation in terms of spatial gradients exist for these generators. Note, however, that these representations have recently been interpreted as dual to scalars [149].

Finally, we note that because all these representations were found by setting 9 m = 0, we are really dealing with representations that also exist within E9, in the sense that when restricting all indices to š”°š”©(9,ā„ )-indices, these generators can be found in a level decomposition of E9 with respect to its š”°š”©(9,ā„ )-subalgebra. However, it is important to note that in E 10 and E 11 the affine representations constitute merely a small subset of all representations occurring in the level decomposition, while in E9 they are actually the only ones and so they provide (together with their transposed partners) the full structure of the algebra. Moreover, in E9 the epsilon tensor is of rank 9 so all the 9-tuples of antisymmetric indices are “swallowed” by the epsilon tensor. This reflects the fact that for affine algebras the level decomposition corresponds to an infinite repetition of the low-level representations.

8.4.3 Decomposition with respect to š–˜š–”(9, 9) and š–˜š–‘(2,ā„ ) ⊕ š–˜š–‘(9,ā„ )

A level decomposition can be performed with respect to any of the regular subalgebras encoded in the Dynkin diagram. We mention here two additional cases which are specifically interesting for our purposes, since they give rise to low-level field contents that coincide with the bosonic spectrum of Type IIA and IIB supergravity. The relevant decompositions are the following:

IIA ⇐⇒ š”°š”¬(9,9) ⊂ E , 10 (8.100 ) IIB ⇐ ⇒ š”°š”©(2,ā„ ) ⊕ š”° š”©(9,ā„ ) ⊂ E10.
The corresponding levels are defined as
1∑0 š”°š”¬(9,9 ) : γ = ā„“α1 + mi αi ∈ š”„ā‹†, i=2 1∑0 (8.101 ) š”°š”©(2,ā„ ) ⊕ š”°š”©(9,ā„ ) : γ = m1α1 + ā„“α2 + mj αj ∈ š”„ā‹†. j=3
It turns out that in the š”°š”¬(9,9) decomposition the even levels correspond to vectorial representations of š”°š”¬(9,9) while the odd levels give spinorial representations. This implies that the fields in the NS-NS sector arise at even levels and the R-R fields correspond to odd level representations of š”°š”¬(9,9).

On the contrary, in the š”°š”©(2, ā„) ⊕ š”°š”©(9, ā„) decomposition the additional factor of š”°š”©(2,ā„) causes mixing between the R-R and NS-NS fields at each level. This is to be expected since we know that for example the fundamental string (F1) and the D1-brane couples to the NS-NS 2-form B2 and the R-R 2-form C2, respectively, which transform as a doublet under the SL (2,ā„)-symmetry of Type IIB supergravity.

In the š”°š”©(2,ā„ ) ⊕ š”°š”©(9,ā„) the level ā„“ = 0 content is of course just the adjoint representation in the same way as in the š”°š”©(10,ā„ ) decomposition considered above. In the other case instead we find the adjoint representation ab ba M = − M of š”°š”¬(9,9) with commutation relations

[M ab,M cd] = ηcaM bd − ηcbM ad − ηdaM bc + ηdbM ac, (8.102 )
where ηab is the split diagonal metric with (9,9)-signature.

The procedure follows a similar structure as for the previous cases so we will not give the details here. We refer the interested reader to [125126] for a detailed account. The result of the decompositions up to level 3 for the two cases discussed here is displayed in Tables 40 and 41.


Table 40: The low-level representations in a decomposition of the adjoint representation of E10 into representations of its š”°š”¬(9,9) subalgebra obtained by removing the first node in the Dynkin diagram in Figure 49View Image. Note that the lower indices at levels 1 and 3 are spinor indices of š”°š”¬(9,9).
ā„“ (ā„“) Λ = [p1,⋅⋅⋅ ,p9] E10-generator
1 [0,0,0,0,0, 0,0,1,0] Eα
2 [0,0,1,0,0, 0,0,0,0] Ea1a2a3
3 [1,0,0,0,0, 0,0,1,0] a E β


Table 41: The low-level representations in a decomposition of the adjoint representation of E10 into representations of its š”°š”©(2,ā„) ⊕ š”°š”©(9, ā„) subalgebra obtained by removing the second node in the Dynkin diagram in Figure 49View Image. The index α at levels 1 and 3 corresponds to the fundamental representation of š”°š”©(2,ā„ ).
ā„“ Λ (ā„“) = [p1,⋅⋅⋅ ,p9] ⊗ ā„› [A1] E10-generator
1 [0,0,0,0, 0,0,0,1,0] ⊗ 2 ab E α
2 [0,0,0,0, 0,1,0,0,0] ⊗ 1 Ea1 ⋅⋅⋅a4
3 [1,0,0,0, 0,0,0,1,0] ⊗ 2 a1⋅⋅⋅a6 E α


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