We discuss the structure of ``exceptional generalised geometry'' (EGG), an extension of Hitchin's generalised geometry that provides a unified geometrical description of backgrounds in eleven-dimensional supergravity. On a d-dimensional background, as first described by Hull, the action of the generalised geometrical O(d, d) symmetry group is replaced in EGG by the exceptional U-duality group Ed(d). The metric and form-field degrees of freedom combine into a single geometrical object, so that EGG naturally describes generic backgrounds with flux, and there is an EGG analogue of the Courant bracket which encodes the differential geometry. Our focus is on the case of seven-dimensional backgrounds with N = 1 four-dimensional supersymmetry. The corresponding EGG is the generalisation of a G2-structure manifold. We show it is characterised by an element in a particular orbit of the 912 representation of E7(7), which defines an SU(7) ⊂ E7(7) structure. As an application, we derive the generic form of the four-dimensional effective superpotential, and show that it can be written in a universal form, as a homogeneous E7(7)-invariant functional of .

https://iopscience.iop.org/article/10.1088/1126-6708/2008/09/123/pdf

M-theory, exceptional generalised geometry and superpotentials

We discuss the structure of "exceptional generalised geometry" (EGG), an extension of Hitchin's generalised geometry that provides a unified geometrical description of backgrounds in eleven-dimensional supergravity. On a d-dimensional background, as first described by Hull, the action of the generalised geometrical O(d,d) symmetry group is replaced in EGG by the exceptional U-duality group E_d(d). The metric and form-field degrees of freedom combine into a single geometrical object, so that EGG naturally describes generic backgrounds with flux, and there is an EGG analogue of the Courant bracket which encodes the differential geometry. Our focus is on the case of seven-dimensional backgrounds with N=1 four-dimensional supersymmetry. The corresponding EGG is the generalisation of a G_2-structure manifold. We show it is characterised by an element \phi in a particular orbit of the 912 representation of E_7(7), which defines an SU(7) (subset of E_7(7)) structure. As an application, we derive the generic form of the four-dimensional effective superpotential, and show that it can be written in a universal form, as a homogeneous E_7(7)-invariant functional of \phi.

We study the moduli space of torsion-free G2-structures on a fixed compact manifold, and define its associated universal intermediate Jacobian J. We define the Yukawa coupling and relate it to a natural pseudo-Kahler structure on J. We consider natural Chern-Simons type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang-Mills connections), and also deformed Donaldson-Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in J by means of G2-analogues of Abel-Jacobi maps.

Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps

/pdf/aksz-constructions-for-topological-membranes-on-g2-manifolds-3giwzhbbp7.pdf

AKSZ Constructions for Topological Membranes on G2‐Manifolds

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/pdf/generalising-g2-geometry-involutivity-moment-maps-and-moduli-1pwwnsih2o.pdf

Generalising G2 geometry: involutivity, moment maps and moduli

We consider the quantization of the effective target space description of topological M-theory in terms of the Hitchin functional whose critical points describe seven manifolds with a G2 structure. The one-loop partition function for this theory is calculated and an extended version of it, that is related to the generalized G2 geometry, is compared with the topological G2 string. We relate the reduction of the effective action for the extended G2 theory to the Hitchin functional description of the topological string in six dimensions. The dependence of the partition functions on the choice of background G2 metric is also determined.

P. de Medeiros

Papers

G(2) Hitchin functionals at one loop