Generalising G${}_{\text{2}}$ geometry: involutivity, moment maps and moduli

Abstract

We analyse the geometry of generic Minkowski $\mathcal{N}=1$, $D=4$ flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of ${\mathrm{G}}_2$ holonomy manifolds. In type II theories, they extend the notion of Calabi--Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using ${\mathrm{E}}_{7(7)}\times {\mathbb{R}}^{+}$ generalised geometry we show that these compactifications are characterised by an $\mathrm{SU}(7)\subset {\mathrm{E}}_{7(7)}$ structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin's ${\mathrm{G}}_2$ functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of $G_2$ manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

References in corpus (16)

• Generalised Geometry for M-Theory
• Generalized N=1 Orientifold Compactifications and the Hitchin functionals
• Generalized complex geometry
• E(11) and the Embedding Tensor
• AdS_6 solutions of type II supergravity
• A Moduli Fixing Mechanism in M theory
• Deformations of calibrated D-branes in flux generalized complex manifolds
• Reformulating Supersymmetry with a Generalized Dolbeault Operator
• New aspects of the ddc-lemma
• Gravity duals to deformed SYM theories and Generalized Complex Geometry
• The ${\alpha }^{\prime }$ Expansion On A Compact Manifold Of Exceptional Holonomy
• Supersymmetric Backgrounds, the Killing Superalgebra, and Generalised Special Holonomy
• Higgs Bundles for M-theory on $G_2$-Manifolds
• Local geometry of the G2 moduli space
• Warped generalized geometry compactifications, effective theories and non-perturbative effects
• G2 Hitchin functionals at one loop

Cited by in corpus (4)

• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes
• Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory
• Y-algebroids and $E_{7(7)}\times {\mathbb{R}}^{+}$-generalised geometry
• Linearized Off-shell 4+7 Supergeometry of 11D Supergravity