Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory

Abstract

We present a detailed study of a new mathematical object in ${\mathrm{E}}_{6(6)}\times {\mathbb{R}}^{+}$ generalised geometry called an `exceptional complex structure' (ECS). It is the extension of a conventional complex structure to one that includes all the degrees of freedom of M-theory or type IIB supergravity in six or five dimensions, and as such characterises, in part, the geometry of generic supersymmetric compactifications to five-dimensional Minkowkski space. We define an ECS as an integrable ${\mathrm{U}}^{\ast }(6)\times {\mathbb{R}}^{+}$ structure and show it is equivalent to a particular form of involutive subbundle of the complexified generalised tangent bundle $L_1\subset E_{\mathbb{C}}$. We also define a refinement, an ${\mathrm{SU}}^{\ast }(6)$ structure, and show that its integrability requires in addition a vanishing moment map on the space of structures. We are able to classify all possible ECSs, showing that they are characterised by two numbers denoted `type' and `class'. We then use the deformation theory of ECS to find the moduli of any ${\mathrm{SU}}^{\ast }(6)$ structure. We relate these structures to the geometry of generic minimally supersymmetric flux backgrounds of M-theory of the form ${\mathbb{R}}^{4,1}\times M$, where the ${\mathrm{SU}}^{\ast }(6)$ moduli correspond to the hypermultiplet moduli in the lower-dimensional theory. Such geometries are of class zero or one. The former are equivalent to a choice of (non-metric-compatible) conventional $\mathrm{SL}(3,\mathbb{C})$ structure and strikingly have the same space of hypermultiplet moduli as the fluxless Calabi--Yau case.

arXiv commentOrientation error fixed in equation (4.19)

References in corpus (35)

• Hierarchies from Fluxes in String Compactifications
• Supergravity description of field theories on curved manifolds and a no go theorem
• All supersymmetric solutions of minimal supergravity in five dimensions
• The Universe as a Domain Wall
• Extremal Transitions and Five-Dimensional Supersymmetric Field Theories
• Heterotic M-theory in Five Dimensions
• Five-Dimensional Supersymmetric Gauge Theories and Degenerations of Calabi-Yau Spaces
• Generalised Geometry for M-Theory
• Superstrings with Intrinsic Torsion
• The Geometry of D=11 Killing Spinors
• G-Structures and Wrapped NS5-Branes
• Supersymmetric Backgrounds from Generalized Calabi-Yau Manifolds
• Supersymmetric AdS_5 solutions of M-theory
• M-theory, exceptional generalised geometry and superpotentials
• The gauge structure of generalised diffeomorphisms
• Small Instantons, del Pezzo Surfaces and Type I' theory
• The geometry of three-forms in six and seven dimensions
• String loop corrections to the universal hypermultiplet
• Domain Walls and Superpotentials from M Theory on Calabi-Yau Three-Folds
• Loop Corrections to the Universal Hypermultiplet
• The Hitchin functionals and the topological B-model at one loop
• Quantum Corrections to the Universal Hypermultiplet and Superspace
• E7(7) formulation of N=2 backgrounds
• Supersymmetric Backgrounds and Generalised Special Holonomy
• Exactly marginal deformations from exceptional generalised geometry
• N=1 vacua in Exceptional Generalized Geometry
• 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
• Modified differentials and basic cohomology for Riemannian foliations
• Heterotic backgrounds via generalised geometry: moment maps and moduli
• Marginal deformations of 3d $\mathcal{N}=2$ CFTs from AdS${}_4$ backgrounds in generalised geometry
• Generalising G${}_{\text{2}}$ geometry: involutivity, moment maps and moduli
• Generalized geometric vacua with eight supercharges
• Supersymmetric AdS backgrounds and weak generalised holonomy

Cited by in corpus (1)

• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes