1 year ago

M-Theory from the Superpoint



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

Vector Optimization: Theory, Applications, and Extensions (Second ...
Quantum Field Theory: Competitive Models
Obtaining the Witten-Reshetikhin-Turaev invariants from quantum ...
Natural Dark Energy from Lorentz Breaking Diego Blas
Preface to the Revised Second Edition
Definite descriptions in Frege's theory of cardinal numbers
Treatise on rhetoric, literally translated from the Greek, with an ...
PDF of Presentation - ICMS
Field theory from a bundle point of view
Equivariant KK-theory and noncommutative index theory
Abstract Algebra and Algebraic Number Theory
Algebraic Groups and Number Theory - Free
I-Vague Ideal Theory of Subtraction Algebras - Yang's Scientific ...
An Invitation to Galois Theory and Galois Representations
Sylow theory for table algebras, fusion rule algebras, and ... - blue
Tools of quantum field theory over curved backgrounds - Institut für ...
Solvable and Nilpotent Lie algebras -
L∞-algebras of local observables from higher prequantum bundles
Representation Theory of Lie Algebras
Group representations in entanglement theory - Department of Physics
On the birational invariants k and (2,1) genus of algebraic plane ...