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 φ in a particular orbit of the 912 representation of E 7(7) , which defines an SU(7) ⊂ 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 φ.

By resorting to recent results on the relativistic currents for mixed (flavor) fields, we calculate a space-time dependent neutrino oscillation formula in Quantum Field Theory. Our formulation provides an alternative to existing approaches for the derivation of space dependent oscillation formulas and it also accounts for the corrections due to the non-trivial nature of the flavor vacuum. By exploring different limits of our formula, we recover already known results. We study in detail the case of one-dimensional propagation with gaussian wave-packets both in the relativistic and in the non-relativistic regions: in the last case, numerical evaluations of our result show significant deviations from the standard formula.

We show that a proper field theoretical treatment of mixed (Dirac) neutrinos leads to non-trivial dispersion relations for the flavor states. We analyze such a situation in the framework of the non-linear relativity schemes recently proposed by Magueijo and Smolin. We finally examine the experimental implications of our theoretical proposals by considering the spectrum and the end-point of beta decay in tritium.