Supersymmetric backgrounds and generalised special holonomy

Abstract

We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form R D−1,1 × M . Using the language of E d(d) ×R + generalised geometry, we show that, for D ≥ 4, preserving minimal supersymmetry is equivalent to the manifold M having generalised special holonomy and list the relevant holonomy groups. We conjecture that this result extends to backgrounds preserving any number of supersymmetries. As a prime example, we consider N = 1 in D = 4. The corresponding generalised special holonomy group is SU (7), giving the natural M theory extension to the notion of a G 2 manifold, and, for Type II backgrounds, reformulating the pure spinor SU (3) × SU (3) conditions as an integrable structure.

Generalised structures for N = 1 $$ \mathcal{N}=1 $$ AdS backgrounds

“…In [17], it was shown that the left-hand side of (2.16) matches exactly the intrinsic torsion T int of the generalised G-structure 3 as listed in table 1. The right-hand side, which simply vanished in the Minkowski case, now contains just the cosmological constant multiplying the (G-invariant) Killing spinor.…”

“…It was further claimed in [17] that generic minimally supersymmetric AdS compactifications could be similarly described, now by keeping certain singlet components of the generalised intrinsic torsion as a non-zero constant, corresponding to the cosmological constant Λ. We thus have that In the following we will clarify this statement and demonstrate it explicitly.…”

“…For concreteness we will focus on the M theory description, though our analysis covers the Type II cases straightforwardly [17,20,21].…”

“…Combining both approaches, it was shown in a recent work [17] that it is possible to characterise fully generic minimally supersymmetric compactifications to D ≥ 4-dimensional Minkowski space by a novel integrability condition, formulated in the language of E d(d) ×R + generalised geometry [18][19][20] (throughout d = 11 − D and E d(d) is the real split form of the rank d exceptional Lie group). Concretely, it was proven that the Killing spinor equations constrain precisely the intrinsic torsion of the generalised G-structure defined by the Killing spinor on the generalised tangent bundle.…”

“…-there exists a torsion-free generalised connection compatible with G if and only if M is the internal space of a minimally supersymmetric Minkowski flux background [17].…”