The homogeneity theorem for supergravity backgrounds II: the six-dimensional theories

José Figueroa-O’Farrill,Noel Hustler

doi:10.1007/jhep04(2014)131

Abstract

We prove that supersymmetry backgrounds of (1,0) and (2,0) six-dimensional supergravity theories preserving more than one half of the supersymmetry are locally homogeneous. As a byproduct we also establish that the Killing spinors of such a background generate a Lie superalgebra.

Highlights

We prove that supersymmetric backgrounds of (1,0) and (2,0) six-dimensional supergravity theories preserving more than one half of the supersymmetry are locally homogeneous
Preserved by a background has proved to be a very useful organising principle in our efforts to tame the zoo of supergravity backgrounds, despite being a rather coarse invariant
One attractive classification problem is that of backgrounds which preserve a large fraction of the supersymmetry

Summary

The Killing superalgebra

A Killing spinor of (2, 0) supergravity is a section ε of S+ which is parallel relative to a connection D defined by. The Dirac current Vε of a spinor ε ∈ C∞(M ; S+) is the vector field defined by g(Vε, X) = ε, X · ε ,. The Dirac current of a Killing spinor is a Killing vector which preserves H. Three of the four components of the Jacobi identity vanish by construction, whence only the (g1, g1, g1) component needs to be checked This is a symmetric trilinear map g1 × g1 × g1 → g1, whence it vanishes if and only if it vanishes when restricted to the diagonal, which is the map sending a Killing spinor ε to its Lie derivative along its Dirac current: LVεε. Comparing with equation (3.11), we see that it must vanish This shows that the brackets defined on g = g0 ⊕g1 turn it into a Lie superalgebra.

Homogeneity

The Clifford module and its inner products

The spinor representations

The R-symmetry representations

The underlying real spinorial representations

Explicit matrix realisation