Generalised G 2–Manifolds

Abstract

We define new Riemannian structures on 7-manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy G2, while the constrained ones give rise to a new geometry without a classical counterpart. We characterise these structures by means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in type II supergravity theory with bosonic background fields. In particular, this geometry can be described by two linear metric connections with skew-symmetric torsion. Finally, we construct explicit examples by introducing the device of T -duality.Remark: There is a canonical embedding GL + (n) ֒→ Spin(n, n) of the identity component of GL(n) into the spin group of T ⊕ T * . As a GL + (n)-module we have S ± = Λ ev,od T * ⊗ (Λ n T ) 1/2 ,

Singular foliations for M-theory compactification

“…cit. that one can repackage the information contained in the Spin(7) ± structures into a generalized Spin(7) structure onM in the sense of [41,42]. In particular, it is easy to check that relations (4.8) of [1] are equivalent with some of the exterior differential constraints which can be obtained by expanding equation (3.5) of [8] into its rank components -exterior differential constraints which were discussed at length in [32] and in the appendix of [8].…”

Enhanced supersymmetry from vanishing Euler number

“…We hence need to confront the question of how these corrections arise upon breaking to N = 2. 13 We expect to be able to incorporate them in the couplings of the very modes we excluded from our ansatz by not considering N = 4 massive gravitino multiplets. The couplings of these fields to N = 4 supergravity have not been studied.…”

“…28) 13 An initial guess that Calabi-Yau manifolds with vanishing Euler number experience no worldsheet instanton corrections is correct in the case of the Enriques Calabi-Yau, but not true in general.…”

Non-Abelian T-duality, G 2-structure rotation and holographic duals of $ \mathcal{N} $ = 1 Chern-Simons theories

“…The original geometry supports a G 2 -structure, which is characterised by parallel ǫ 1 and ǫ 2 in ǫ = (ǫ 1 , ǫ 2 ) T . The SUSY conditions of the G 2 may be written in terms 2 real 7-d bispinors [19] …”