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M-Theory from the Superpoint

Emergence170208

the

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 mathematically 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 theorem of [32], based on [14]: For d = 11 then the super-Einstein equations of motion of supergravity are implied already by (super-)torsion freedom of the super-vielbein. In summary, theorem 12 shows that the brane bouquet, and with it at least a fair chunk of the structure associated with the word “M-theory”, has its mathematical root in the superpoint, and proposition 3 adds that as the super-spacetimes grow out of the superpoint, they consecutively discover their relevant super-Lorentzian metric structure, and finally their 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 theorem 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 there exist ψ, φ ∈ N such that v = [ψ, φ] and so for (f, g) ∈ Aut(R d−1,1|N ) ↩→ G v × G s then 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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