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 ${\mathbb{R}}^{D-1,1}\times M$. Using the language of $E_{d(d)}\times {\mathbb{R}}^{+}$ generalised geometry, we show that, for $D\ge 4$, 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 $\mathcal{N}=1$ 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 $SU(3)\times SU(3)$ conditions as an integrable structure.

33 pages

References in corpus (12)

• Generalized Geometry and M theory
• Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB
• Exceptional Field Theory III: E${}_{8(8)}$
• Exceptional Field Theory II: E${}_{7(7)}$
• Gauged Double Field Theory
• Double Field Theory Formulation of Heterotic Strings
• AdS_6 solutions of type II supergravity
• Supersymmetric E${}_{7(7)}$ Exceptional Field Theory
• Generalised geometry for string corrections
• The gauge structure of Exceptional Field Theories and the tensor hierarchy
• Generalized geometry, calibrations and supersymmetry in diverse dimensions
• Generalized complex geometry of pure backgrounds in ten and eleven dimensions

Cited by in corpus (24)

• Half-maximal supersymmetry from exceptional field theory
• Systematics of consistent truncations from generalised geometry
• Supersymmetric AdS${}_7$ and AdS${}_6$ vacua and their minimal consistent truncations from exceptional field theory
• Supersymmetric AdS${}_7$ and AdS${}_6$ vacua and their consistent truncations with vector multiplets
• Finite deformations from a heterotic superpotential: holomorphic Chern--Simons and an $L_{\infty }$ algebra
• 7-dimensional $\mathcal{N}=2$ Consistent Truncations using $\mathrm{SL}(5)$ Exceptional Field Theory
• Exceptional Calabi--Yau spaces: the geometry of $\mathcal{N}=2$ backgrounds with flux
• The exceptional generalised geometry of supersymmetric AdS flux backgrounds
• Supersymmetric Backgrounds, the Killing Superalgebra, and Generalised Special Holonomy
• $\mathcal{N}=2$ consistent truncations from wrapped M5-branes
• Heterotic backgrounds via generalised geometry: moment maps and moduli
• From Exceptional Field Theory to Heterotic Double Field Theory via K3
• Generalised Structures for $\mathcal{N}=1$ AdS Backgrounds
• Generalising G${}_{\text{2}}$ geometry: involutivity, moment maps and moduli
• The higher-dimensional origin of five-dimensional $\mathcal{N}=2$ gauged supergravities
• Generalized geometric vacua with eight supercharges
• Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory
• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes
• Moduli identification methods in Type II compactifications
• $N=(2,0)$ AdS${}_3$ Solutions of M-theory
• Generalised holonomies and K(E${}_9$)
• Higher curvature Bianchi identities, generalised geometry and $L_{\infty }$ algebras
• Supergravity Fluxes and Generalised Geometry
• Exactly Marginal Deformations and their Supergravity Duals