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Emergence170208

**the** Lorentz group O(d − 1, 1) is equivalently a choice of super-vielbein field, hence a super-pseudo- Riemannian structure, hence a field configuration of supergravity. Observe also that **the** ma**the**matically most natural condition to demand **from** such a super- Cartan geometry is that it be (super-)torsion free, see [38]. In view of this it is worthwhile to recall **the** remarkable **the**orem of [32], based on [14]: For d = 11 **the**n **the** super-Einstein equations of motion of supergravity are implied already by (super-)torsion freedom of **the** super-vielbein. In summary, **the**orem 12 shows that **the** brane bouquet, and with it at least a fair chunk of **the** structure associated with **the** word “M-**the**ory”, has its ma**the**matical root in **the** superpoint, and proposition 3 adds that as **the** super-spacetimes grow out of **the** superpoint, **the**y consecutively discover **the**ir relevant super-Lorentzian metric structure, and finally **the**ir supergravity equations of motion. m5brane m2brane d5brane d3brane d1brane d0brane d2brane d4brane d7brane R 10,1|32 d6brane d9brane string IIB string het (pb) string IIA d8brane R 9,1|16+16 R 9,1|16 R 9,1|16+16 R 5,1|8 R 5,1|8+8 R 3,1|4+4 R 3,1|4 R 2,1|2+2 R 2,1|2 R 0|1+1 R 0|1 8

2 Automorphisms of super-Minkowski spacetimes For **the** main **the**orem 12 below we need to know **the** automorphisms (definition 16) of **the** super Minkowski super Lie algebras R d−1,1|N (definition 20). This is possibly essentially folklore (see [22, p. 95]) but since we did not find a full account in **the** literature, we provide a proof here. After some simple lemmas, **the** result is proposition 3 below. Lemma 1. The underlying vector space of **the** bosonic automorphism Lie algebras aut(R d−1,1|N ) (proposition 17) of **the** super-Minkowski Lie algebras R d−1,1|N (definition 20) in any dimension d and for any real Spin(d − 1, 1)-representation N is **the** graph of a surjective and Spin(d − 1, 1)- equivariant (with respect to **the** adjoint action) Lie algebra homomorphism K : g s −→ g v , where g s ⊕ g v ( ) := im aut(R d−1,1|N ) −→ gl(N) ⊕ gl(R d ) is **the** image of **the** canonical inclusion induced by forgetting **the** super Lie bracket on R d−1,1|N . In particular **the** kernel of K is **the** internal symmetries (definition 4) hence **the** R-symmetries (example 5): ker(K) ≃ int(R d−1,1|N ) . Proof. Consider **the** corresponding inclusion at **the** level of groups Aut(R d−1,1|N ) −→ GL(N) × GL(R d ) with image G s × G v . Observe that **the** spinor bilinear pairing [−, −] : N ⊗ N −→ R d is surjective, because it is a homomorphism of Spin(d − 1, 1)-representations, and because R d is irreducible as a Spin(d − 1, 1)-representation. Hence for every vector v ∈ R d **the**re exist ψ, φ ∈ N such that v = [ψ, φ] and so for (f, g) ∈ Aut(R d−1,1|N ) ↩→ G v × G s **the**n f is uniquely fixed by g via f(v) = [g(ψ), g(φ)] . The function K : g ↦→ f thus determined is surjective by construction of G v , is a group homomorphism because Aut(R d−1,1|N ) is a group, and is Spin(d − 1, 1|N)-equivariant (with respect to **the** adjoint action) by **the** Spin(d − 1, 1)-equivariance of **the** spinor bilinear pairing. Lemma 2. Let N be a real Spin(d − 1, 1)-representation in some dimension d. Then **the** Lie algebra g v **from** lemma 1 decomposes as a Spin(d − 1, 1)-representations into **the** direct sum of **the** adjoint representation with **the** trivial representation g v ≃ so(d − 1, 1) ⊕ R while **the** Lie algebra g s **from** lemma 1 decomposes as a direct sum of some exterior powers of **the** vector representation R d : g s ≃ ⊕ ∧ ni R d . i 9

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