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7.3 Models associated with non-split real forms

In this section we provide a list of all theories coupled to gravity which, upon compactification to three dimensions, display U-duality algebras that are not maximal split [95]. This therefore completes the classification of Section 5.

One can classify the various theories through the number 𝒩 of supersymmetries that they possess in D = 4 spacetime dimensions. All p-forms can be dualized to scalars or to 1-forms in four dimensions so the theories all take the form of pure supergravities coupled to collections of Maxwell multiplets. The analysis performed for the split forms in Section 5.3 were all concerned with the cases of 𝒩 = 0 or 𝒩 = 8 supergravity in D = 4. We consider all pure four-dimensional supergravities (𝒩 = 1,⋅⋅⋅ ,8) as well as pure 𝒩 = 4 supergravity coupled to k Maxwell multiplets.

As we have pointed out, the main new feature in the non-split cases is the possible appearance of so-called twisted overextensions. These arise when the restricted root system of 𝒰3 is of non-reduced type hence yielding a twisted affine Kac–Moody algebra in the affine extension of 𝔣 ⊂ 𝔲3. It turns out that the only cases for which the restricted root system is of non-reduced ((BC )-type) is for the pure 𝒩 = 2,3 and 𝒩 = 5 supergravities. The example of 𝒩 = 2 was already discussed in detail before, where it was found that the U-duality algebra is 𝔲3 = 𝔰𝔲(2,1) whose restricted root system is (BC )1, thus giving rise to the twisted overextension A(2)+ 2. It turns out that for the 𝒩 = 3 case the same twisted overextension appears. This is due to the fact that the U-duality algebra is 𝔲3 = 𝔰𝔲(4,1) which has the same restricted root system as 𝔰𝔲(2,1), namely (BC )1. Hence, A (12)+ controls the BKL-limit also for this theory.

The case of 𝒩 = 5 follows along similar lines. In D = 3 the non-split form E 6(−14) of E6 appears, whose maximal split subalgebra is 𝔣 = C2. However, the relevant Kac–Moody algebra is not ++ C2 but rather (2)+ A 4 because the restricted root system of E6 (− 14) is (BC )2.

In Table 37 we display the algebraic structure for all pure supergravities in four dimensions as well as for 𝒩 = 4 supergravity with k Maxwell multiplets. We give the relevant U-duality algebras 𝔲 3, the restricted root systems Σ, the maximal split subalgebras 𝔣 and, finally, the resulting overextended Kac–Moody algebras 𝔤.

Let us end this section by noting that the study of real forms of hyperbolic Kac–Moody algebras has been pursued in [17].

Table 37: Classification of theories whose U-duality symmetry in three dimensions is described by a non-split real form 𝔲 3. The leftmost column indicates the number 𝒩 of supersymmetries that the theories possess when compactified to four dimensions, and the associated number k of Maxwell multiplets. The middle column gives the restricted root system Σ of 𝔲3 and to the right of this we give the maximal split subalgebras 𝔣 ⊂ 𝔲3, constructed from a basis of Σ. Finally, the rightmost column displays the overextended Kac–Moody algebras that governs the billiard dynamics.
(𝒩,k ) 𝔲3 Σ 𝔣 𝔤
(1,0) 𝔰𝔩(2,ℝ ) A1 A1 A++1
(2,0) 𝔰𝔲(2,1) (BC )1 A1 (2)+ A 2
(3,0) 𝔰𝔲(4,1) (BC )1 A1 A (21)+
(4,0) 𝔰𝔬(8,2) C2 C2 ++ C 2
(4,k < 6) 𝔰𝔬(8,k + 2) Bk+2 Bk+2 B++k+2
(4,6) 𝔰𝔬(8,8) D 8 D 8 DE = D++ 10 8
(4,k > 6) 𝔰𝔬(8,k + 2) B8 B8 ++ BE10 = B 8
(5,0) E6(−14) (BC )2 C2 A (2)+ 4
(6,0) E7 (− 5) F4 F4 ++ F 4
(8,0) E8 (+8) E8 E8 E10 = E++8

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