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5.4 Models associated with split real forms

In this section we give a complete list of all theories whose billiard description can be given in terms of a Kac–Moody algebra that is the untwisted overextension of a split real form of the associated U-duality algebra (see Table 15). These are precisely the maximally oxidized theories introduced in [22] and further examined in [37]. These theories are completely classified by their global symmetry groups 𝒰3 arising in three dimensions. For the string-related theories the group 𝒰3 is the (classical version of) the U-duality symmetry obtained by combining the S- and T-dualities in three dimensions [142]. Thereof the notation 𝒰3 for the global symmetry group in three dimensions. We extend the classification to the non-split case in Section 7.

Let us also note here that, as shown in [55], the billiard analysis sheds light on the problem of oxidation, i.e., the problem of finding the maximum spacetime dimension in which a theory with a given duality group in three dimensions can be reformulated. More on this question can be found in [118119].


Table 15: We present here the complete list of theories that exhibit extended coset symmetries of split real Lie algebras upon compactification to three spacetime dimensions. In the leftmost column we give the coset space which is relevant in each case. We also list the Kac–Moody algebras that underlie the gravitational dynamics in the BKL-limit. These appear as overextensions of the finite Lie algebras found in three dimensions. Finally we indicate which of these theories are related to string/M-theory.
𝒰 ∕𝒦 (𝒰 ) 3 3

Lagrangian in maximal dimension

Kac–Moody algebra

String/M-theory

SL-(n+1,ℝ) SO(n+1)

ℒn+3 = R ⋆ 1

++ AEn+2 ≡ An

No

---SO(n,n+1)-- SO (n)×SO (n+1)

ℒn+2 = R ⋆ 1 − ⋆dφ ∧ dφ − √- 1e2√2nφ ⋆ G(3) ∧ G (3) − 2 √2-φ 12e n ⋆ F (2) ∧ F (2),   G (3) = dB (2) + 1A (1) ∧ A(1) 2,   (2) (1) F = dA

++ BEn+2 ≡ B n

No

Sp(n) U(n)

ℒ4 = R ⋆ 1 − ⋆d⃗φ ∧ d⃗φ − ∑ 12 αe2⃗σα.⃗φ ⋆ (dχα + ⋅⋅⋅) ∧ (dχα + ⋅⋅⋅) − 1 ∑n −1 ⃗ea.⃗φ√2 a a 2 a=1 e ⋆ dA (1) ∧ dA (1)

++ CEn+2 ≡ Cn

No

---SO(n,n)-- SO (n)×SO (n)

ℒn+2 = R ⋆ 1 − ⋆dφ ∧ dφ − 4 1 e√nφ ⋆ dB (2) ∧ dB (2) 2

++ DEn+2 ≡ D n

type I (n = 8) / bosonic string (n = 24)

-G2(2) SU (8)

ℒ5 = R ⋆ 1 − 1 ⋆ F (2) ∧ F (2) + 2 -1- (2) (2) (1) 3√3F ∧ F ∧ A, F (2) = dA (1)

G++ 2

No

---F4(4)--- Sp(3)×SU (3)

ℒ6 = R ⋆ 1 − ⋆dφ ∧ dφ − 1 2φ 2e ⋆ dχ ∧ dχ − 1 −2φ (3) (3) 2e ⋆ H ∧ H − 12 ⋆ G (3)∧ G (3) − 1eφ ⋆ F + ∧ F+ − 2 (2) (2) 1 −φ − − 2e ⋆ F(2) ∧ F(2) − √1-χH (3) ∧ G (3) − 2 1 + + 2A (1) ∧ F(2)∧ H (3) − 1A+ ∧ F− ∧ G(3) 2 (1) (2),   + + F(2) = dA(1) + √12-χdA −(1),   F − = dA− (2) (1),   − − H (3) = dB (2) + 12A (1) ∧ dA(1),   G (3) = dC (2) − √1- (3) 2 χH − 12A+(1) ∧ dA−(1)

F +4+

No

E Sp(64(6)∕)ℤ2

ℒ8 = R ⋆ 1 − ⋆dφ ∧ dφ − - 1e2√2φ ⋆ d χ ∧ d χ − 2 1 −√2-φ (4) (4) 2e ⋆ G ∧ G + χ G(4) ∧ G (4),   G (4) = dC (3)

E++6

No

--E7(7)-- SU (8)∕ℤ2

ℒ9 = R ⋆ 1 − ⋆dφ ∧ dφ − 2√√2 12e 7 φ ⋆ dC (3) ∧ dC (3) − - 1 − 4√√-2φ (1) (1) 2e 7 ⋆ dA ∧ dA − 12dC (3) ∧ dC (3) ∧ A (1)

E++ 7

No

E Spin8(1(86))∕ℤ2

ℒ11 = R ⋆ 1 − 1 ⋆ dC (3) ∧ dC (3) − 2 1dC (3) ∧ dC (3) ∧ C (3) 6

E10 ≡ E+8+

M-theory, type IIA and type IIB string theory



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