Flavour symmetries have been used to constrain both quark and lepton mixing parameters. In particular, they can be used to completely fix the mixing angles. For the lepton sector, assuming that neutrinos are Majorana particles, we have derived the co
mplete list of mixing patterns achievable in this way, as well as the symmetry groups associated to each case. Partial computer scans done in the past have hinted that such list is limited, and this does indeed turn out to be the case. In addition, most mixing patterns are already 3-sigma excluded by neutrino oscillation data.
Assuming that neutrinos are Majorana particles, we perform a complete classification of all possible mixing matrices which are fully determined by residual symmetries in the charged-lepton and neutrino mass matrices. The classification is based on th
e assumption that the residual symmetries originate from a finite flavour symmetry group. The mathematical tools which allow us to accomplish this classification are theorems on sums of roots of unity. We find 17 sporadic cases plus one infinite series of mixing matrices associated with three-flavour mixing, all of which have already been discussed in the literature. Only the infinite series contains mixing matrices which are compatible with the data at the 3 sigma level.
We study the topological magnetoelectric effect on a conical topological insulator when a point charge $q$ is near the cone apex. The Hall current induced on the cone surface and the image charge configuration are determined. We also study a kind of
gravitational Aharonov-Bohm effect in this geometry and realize a phase diference betwen the components of the wavefunctions (spinors) upon closed parallel transport around the (singular) cone tip. Concretely, a net current flowing towards cone apex (or botton) shows up, yielding electric polarization of the conical topological insulator. Such an effect may be detected, for instance, by means of the net accumulated Hall charge near the apex. Once it depends only on the geometry of the material (essetially, the cone apperture angle) this may be faced as a microscopic scale realization of (2+1)-dimensional Einstein gravity.
